Repository: leanprover/theorem_proving_in_lean4
Branch: master
Commit: 63fad08fcd5f
Files: 86
Total size: 2.6 MB
Directory structure:
gitextract_265n7ear/
├── .github/
│ └── workflows/
│ └── ci.yml
├── .gitignore
├── .gitmodules
├── LICENSE
├── README.md
├── book/
│ ├── .gitignore
│ ├── .verso/
│ │ ├── verso-xref-manifest.json
│ │ └── verso-xref.json
│ ├── Main.lean
│ ├── TPiL/
│ │ ├── AxiomsComputation.lean
│ │ ├── Conv.lean
│ │ ├── DependentTypeTheory.lean
│ │ ├── Examples.lean
│ │ ├── InductionAndRecursion.lean
│ │ ├── InductiveTypes.lean
│ │ ├── InteractingWithLean.lean
│ │ ├── Intro.lean
│ │ ├── PropositionsAndProofs.lean
│ │ ├── QuantifiersEquality.lean
│ │ ├── StructuresAndRecords.lean
│ │ ├── Tactics.lean
│ │ └── TypeClasses.lean
│ ├── TPiL.lean
│ ├── lake-manifest.json
│ ├── lakefile.toml
│ ├── lean-toolchain
│ ├── static/
│ │ ├── fonts/
│ │ │ ├── noto-sans-mono/
│ │ │ │ ├── OFL.txt
│ │ │ │ └── noto-sans-mono.css
│ │ │ ├── source-code-pro/
│ │ │ │ ├── LICENSE.md
│ │ │ │ ├── OTF/
│ │ │ │ │ ├── SourceCodePro-Black.otf
│ │ │ │ │ ├── SourceCodePro-BlackIt.otf
│ │ │ │ │ ├── SourceCodePro-Bold.otf
│ │ │ │ │ ├── SourceCodePro-BoldIt.otf
│ │ │ │ │ ├── SourceCodePro-ExtraLight.otf
│ │ │ │ │ ├── SourceCodePro-ExtraLightIt.otf
│ │ │ │ │ ├── SourceCodePro-It.otf
│ │ │ │ │ ├── SourceCodePro-Light.otf
│ │ │ │ │ ├── SourceCodePro-LightIt.otf
│ │ │ │ │ ├── SourceCodePro-Medium.otf
│ │ │ │ │ ├── SourceCodePro-MediumIt.otf
│ │ │ │ │ ├── SourceCodePro-Regular.otf
│ │ │ │ │ ├── SourceCodePro-Semibold.otf
│ │ │ │ │ └── SourceCodePro-SemiboldIt.otf
│ │ │ │ └── source-code-pro.css
│ │ │ ├── source-sans/
│ │ │ │ ├── LICENSE.md
│ │ │ │ ├── OTF/
│ │ │ │ │ ├── SourceSans3-Black.otf
│ │ │ │ │ ├── SourceSans3-BlackIt.otf
│ │ │ │ │ ├── SourceSans3-Bold.otf
│ │ │ │ │ ├── SourceSans3-BoldIt.otf
│ │ │ │ │ ├── SourceSans3-ExtraLight.otf
│ │ │ │ │ ├── SourceSans3-ExtraLightIt.otf
│ │ │ │ │ ├── SourceSans3-It.otf
│ │ │ │ │ ├── SourceSans3-Light.otf
│ │ │ │ │ ├── SourceSans3-LightIt.otf
│ │ │ │ │ ├── SourceSans3-Medium.otf
│ │ │ │ │ ├── SourceSans3-MediumIt.otf
│ │ │ │ │ ├── SourceSans3-Regular.otf
│ │ │ │ │ ├── SourceSans3-Semibold.otf
│ │ │ │ │ └── SourceSans3-SemiboldIt.otf
│ │ │ │ ├── VF/
│ │ │ │ │ ├── SourceSans3VF-Italic.otf
│ │ │ │ │ └── SourceSans3VF-Upright.otf
│ │ │ │ ├── source-sans-3.css
│ │ │ │ └── source-sans-3VF.css
│ │ │ └── source-serif/
│ │ │ ├── LICENSE.md
│ │ │ ├── OTF/
│ │ │ │ ├── SourceSerif4-Black.otf
│ │ │ │ ├── SourceSerif4-BlackIt.otf
│ │ │ │ ├── SourceSerif4-Bold.otf
│ │ │ │ ├── SourceSerif4-BoldIt.otf
│ │ │ │ ├── SourceSerif4-ExtraLight.otf
│ │ │ │ ├── SourceSerif4-ExtraLightIt.otf
│ │ │ │ ├── SourceSerif4-It.otf
│ │ │ │ ├── SourceSerif4-Light.otf
│ │ │ │ ├── SourceSerif4-LightIt.otf
│ │ │ │ ├── SourceSerif4-Regular.otf
│ │ │ │ ├── SourceSerif4-Semibold.otf
│ │ │ │ └── SourceSerif4-SemiboldIt.otf
│ │ │ └── source-serif-text.css
│ │ └── theme.css
│ └── verso-sources.json
├── examples/
│ ├── .gitignore
│ ├── Examples/
│ │ └── Basic.lean
│ ├── Examples.lean
│ ├── lake-manifest.json
│ ├── lakefile.toml
│ └── lean-toolchain
└── unixode.sty
================================================
FILE CONTENTS
================================================
================================================
FILE: .github/workflows/ci.yml
================================================
name: build and test book content
# Controls when the action will run.
on:
push:
branches:
- master
pull_request:
branches:
- master
# A workflow run is made up of one or more jobs that can run sequentially or in parallel
jobs:
Build:
runs-on: ubuntu-latest
steps:
- name: Checkout
uses: actions/checkout@v2
with:
# interferes with lean4-nightly authentication
persist-credentials: false
submodules: true
- name: Setup elan toolchain on this build
run: |
curl -O --location https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh
chmod u+x elan-init.sh
./elan-init.sh -y --default-toolchain none
- name: Set elan paths
run: |
echo "$HOME/.elan/bin" >> $GITHUB_PATH
- name: Install lean toolchain for examples
run: |
cd examples
lean --version
- name: Install lean toolchain for text
run: |
cd book
lean --version
- name: Cache book/.lake
uses: actions/cache@v4
with:
path: book/.lake
# The SHA is in the key to get the most recent cache possible, rather than just saving a single one for each Lean/deps version and not touching it.
key: ${{ runner.os }}-${{ hashFiles('book/lake-manifest.json') }}-${{ hashFiles('book/lean-toolchain') }}-${{ steps.shortSHA.outputs.short_sha }}
# Try to restore cache for same OS/Lean/deps, but don't get less specific, because Lake isn't always happy to get build product version mismatches
restore-keys: |
${{ runner.os }}-${{ hashFiles('book/lake-manifest.json') }}-${{ hashFiles('book/lean-toolchain') }}-
- name: Build book
run: |
pushd book
lake exe tpil --verbose
popd
- name: Save book/.lake
uses: actions/cache/save@v4
with:
path: book/.lake
# The SHA is in the key to get the most recent cache possible, rather than just saving a single one for each Lean/deps version and not touching it.
key: ${{ runner.os }}-${{ hashFiles('book/lake-manifest.json') }}-${{ hashFiles('book/lean-toolchain') }}-${{ steps.shortSHA.outputs.short_sha }}
- name: Deploy to Netlify hosting
uses: nwtgck/actions-netlify@v2.0
if: github.event_name == 'push' && github.ref_name == 'master'
with:
publish-dir: book/_out/html-multi
production-branch: master
github-token: ${{ secrets.GITHUB_TOKEN }}
deploy-message: |
${{ github.event_name == 'pull_request' && format('pr#{0}: {1}', github.event.number, github.event.pull_request.title) || format('ref/{0}: {1}', github.ref_name, steps.deploy-info.outputs.message) }}
alias: ${{ steps.deploy-info.outputs.alias }}
enable-commit-comment: false
enable-pull-request-comment: false
github-deployment-environment: "lean-lang.org/theorem_proving_in_lean4"
fails-without-credentials: true
env:
NETLIFY_AUTH_TOKEN: ${{ secrets.NETLIFY_AUTH_TOKEN }}
NETLIFY_SITE_ID: "51300c89-2f6c-4bba-a575-8cf12e434502"
- name: GH pages deploy
uses: JamesIves/github-pages-deploy-action@4.1.5
if: github.event_name == 'push' && github.ref_name == 'master'
with:
branch: gh-pages
folder: book/_out/html-multi
================================================
FILE: .gitignore
================================================
*.olean
/_target
/leanpkg.path
out/
.vs/
.vscode/
*/_out/*
.DS_Store
================================================
FILE: .gitmodules
================================================
================================================
FILE: LICENSE
================================================
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================================================
FILE: README.md
================================================
Theorem Proving in Lean 4
-----------------------
This repository contains the source code of the book _Theorem Proving
in Lean 4_ by Jeremy Avigad, Leonardo de Moura, Soonho Kong, and
Sebastian Ullrich, with contributions from the Lean Community.
To build the book, change to the `book` directory and run `lake exe tpil`.
After this, `book/_out/html-multi` contains a multi-page Web
version of the book.
================================================
FILE: book/.gitignore
================================================
/.lake
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FILE: book/.verso/verso-xref-manifest.json
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FILE: book/.verso/verso-xref.json
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{"display": "trig_identity",
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["The Lean Language Reference",
"The mvcgen tactic",
"Enabling mvcgen For Monads",
"A Logging Monad"]}},
"address": "/The--mvcgen--tactic/Enabling--mvcgen--For-Monads/"}],
"log (in A Logging Monad)":
[{"id": "log-_LPAR_in-A-Logging-Monad_RPAR_",
"data":
{"/The--mvcgen--tactic/Enabling--mvcgen--For-Monads/#log-_LPAR_in-A-Logging-Monad_RPAR_":
{"display": "log",
"context":
["The Lean Language Reference",
"The mvcgen tactic",
"Enabling mvcgen For Monads",
"A Logging Monad"]}},
"address": "/The--mvcgen--tactic/Enabling--mvcgen--For-Monads/"}],
"location (in where for structures)":
[{"id": "location-_LPAR_in-where--for-structures_RPAR_",
"data":
{"/The-Type-System/Inductive-Types/#location-_LPAR_in-where--for-structures_RPAR_":
{"display": "location",
"context":
["The Lean Language Reference",
"The Type System",
"Inductive Types",
"Structure Declarations",
"Structure Constructors",
"where for structures"]}},
"address": "/The-Type-System/Inductive-Types/"}],
"listLen (in Matching on Complex Expressions Can Prevent Elaboration)":
[{"id":
"listLen-_LPAR_in-Matching-on-Complex-Expressions-Can-Prevent-Elaboration_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#listLen-_LPAR_in-Matching-on-Complex-Expressions-Can-Prevent-Elaboration_RPAR_":
{"display": "listLen",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Structural Recursion",
"Explicit Structural Recursion",
"Matching on Complex Expressions Can Prevent Elaboration"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"lazy":
[{"id": "lazy",
"data":
{"/Axioms/#lazy":
{"display": "lazy",
"context":
["The Lean Language Reference",
"Axioms",
"Displaying Axiom Dependencies"]}},
"address": "/Axioms/"}],
"last?' (in Pattern Equality Proofs)":
[{"id": "last______-_LPAR_in-Pattern-Equality-Proofs_RPAR_",
"data":
{"/Terms/Pattern-Matching/#last______-_LPAR_in-Pattern-Equality-Proofs_RPAR_":
{"display": "last?'",
"context":
["The Lean Language Reference",
"Terms",
"Pattern Matching",
"Types",
"Pattern Equality Proofs",
"Pattern Equality Proofs"]}},
"address": "/Terms/Pattern-Matching/"}],
"last? (in Pattern Equality Proofs)":
[{"id": "last___-_LPAR_in-Pattern-Equality-Proofs_RPAR_",
"data":
{"/Terms/Pattern-Matching/#last___-_LPAR_in-Pattern-Equality-Proofs_RPAR_":
{"display": "last?",
"context":
["The Lean Language Reference",
"Terms",
"Pattern Matching",
"Types",
"Pattern Equality Proofs",
"Pattern Equality Proofs"]}},
"address": "/Terms/Pattern-Matching/"}],
"largeNumber (in Creating and Tracking Proofs That Trust the Compiler)":
[{"id":
"largeNumber-_LPAR_in-Creating-and-Tracking-Proofs-That-Trust-the-Compiler_RPAR_",
"data":
{"/Axioms/#largeNumber-_LPAR_in-Creating-and-Tracking-Proofs-That-Trust-the-Compiler_RPAR_":
{"display": "largeNumber",
"context":
["The Lean Language Reference",
"Axioms",
"Standard Axioms",
"Creating and Tracking Proofs That Trust the Compiler"]}},
"address": "/Axioms/"}],
"iterFile (in Iterators and Effects)":
[{"id": "iterFile-_LPAR_in-Iterators-and-Effects_RPAR_",
"data":
{"/Iterators/Iterator-Definitions/#iterFile-_LPAR_in-Iterators-and-Effects_RPAR_":
{"display": "iterFile",
"context":
["The Lean Language Reference",
"Iterators",
"Iterator Definitions",
"Finite and Productive Iterators",
"Iterators and Effects"]}},
"address": "/Iterators/Iterator-Definitions/"}],
"isZero' (in Pattern-Matching Functions)":
[{"id": "isZero___-_LPAR_in-Pattern-Matching-Functions_RPAR_",
"data":
{"/Terms/Pattern-Matching/#isZero___-_LPAR_in-Pattern-Matching-Functions_RPAR_":
{"display": "isZero'",
"context":
["The Lean Language Reference",
"Terms",
"Pattern Matching",
"Pattern Matching Functions",
"Pattern-Matching Functions"]}},
"address": "/Terms/Pattern-Matching/"}],
"isZero (in Pattern-Matching Functions)":
[{"id": "isZero-_LPAR_in-Pattern-Matching-Functions_RPAR_",
"data":
{"/Terms/Pattern-Matching/#isZero-_LPAR_in-Pattern-Matching-Functions_RPAR_":
{"display": "isZero",
"context":
["The Lean Language Reference",
"Terms",
"Pattern Matching",
"Pattern Matching Functions",
"Pattern-Matching Functions"]}},
"address": "/Terms/Pattern-Matching/"}],
"isPrime (in Partial Functions in Proofs)":
[{"id": "isPrime-_LPAR_in-Partial-Functions-in-Proofs_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#isPrime-_LPAR_in-Partial-Functions-in-Proofs_RPAR_":
{"display": "isPrime",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Partial and Unsafe Definitions",
"Partial Functions",
"Partial Functions in Proofs"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"iota_succ (in Increasing E-matching Limits)":
[{"id": "iota_succ-_LPAR_in-Increasing-E-matching-Limits_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#iota_succ-_LPAR_in-Increasing-E-matching-Limits_RPAR_":
{"display": "iota_succ",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Resource Limits",
"Increasing E-matching Limits"]}},
"address": "/The--grind--tactic/E___matching/"}],
"iota (in Increasing E-matching Limits)":
[{"id": "iota-_LPAR_in-Increasing-E-matching-Limits_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#iota-_LPAR_in-Increasing-E-matching-Limits_RPAR_":
{"display": "iota",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Resource Limits",
"Increasing E-matching Limits"]}},
"address": "/The--grind--tactic/E___matching/"}],
"inv_eq (in The @[grind ←=] Attribute)":
[{"id": "inv_eq-_LPAR_in-The--____LSQ_grind-_______RSQ_--Attribute_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#inv_eq-_LPAR_in-The--____LSQ_grind-_______RSQ_--Attribute_RPAR_":
{"display": "inv_eq",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Patterns",
"The @[grind ←=] Attribute"]}},
"address": "/The--grind--tactic/E___matching/"}],
"intersperse (in Printing Equations)":
[{"id": "intersperse-_LPAR_in-Printing-Equations_RPAR_",
"data":
{"/Interacting-with-Lean/#intersperse-_LPAR_in-Printing-Equations_RPAR_":
{"display": "intersperse",
"context":
["The Lean Language Reference",
"Interacting with Lean",
"Querying the Context",
"Printing Equations"]}},
"address": "/Interacting-with-Lean/"}],
"interestingName (in Querying the Environment)":
[{"id": "interestingName-_LPAR_in-Querying-the-Environment_RPAR_",
"data":
{"/Notations-and-Macros/Elaborators/#interestingName-_LPAR_in-Querying-the-Environment_RPAR_":
{"display": "interestingName",
"context":
["The Lean Language Reference",
"Notations and Macros",
"Elaborators",
"Command Elaborators",
"Querying the Environment"]}},
"address": "/Notations-and-Macros/Elaborators/"}],
"instDecidableEqStringList (in A recursive instance for a sum class)":
[{"id":
"instDecidableEqStringList-_LPAR_in-A-recursive-instance-for-a-sum-class_RPAR_",
"data":
{"/Type-Classes/Instance-Declarations/#instDecidableEqStringList-_LPAR_in-A-recursive-instance-for-a-sum-class_RPAR_":
{"display": "instDecidableEqStringList",
"context":
["The Lean Language Reference",
"Type Classes",
"Instance Declarations",
"Instances of class inductive s",
"A recursive instance for a sum class"]}},
"address": "/Type-Classes/Instance-Declarations/"}],
"instCommRingR":
[{"id": "instCommRingR",
"data":
{"/The--grind--tactic/Bigger-Examples/#instCommRingR":
{"display": "instCommRingR",
"context":
["The Lean Language Reference",
"The grind tactic",
"Bigger Examples",
"Integrating grind 's Features"]}},
"address": "/The--grind--tactic/Bigger-Examples/"}],
"instBeqNatWrapper (in Variations in Instance Definitions)":
[{"id":
"instBeqNatWrapper-_LPAR_in-Variations-in-Instance-Definitions_RPAR_",
"data":
{"/Type-Classes/Instance-Declarations/#instBeqNatWrapper-_LPAR_in-Variations-in-Instance-Definitions_RPAR_":
{"display": "instBeqNatWrapper",
"context":
["The Lean Language Reference",
"Type Classes",
"Instance Declarations",
"Variations in Instance Definitions"]}},
"address": "/Type-Classes/Instance-Declarations/"}],
"incr_eq_plus1 (in Proofs are Private)":
[{"id": "incr_eq_plus1-_LPAR_in-Proofs-are-Private_RPAR_",
"data":
{"/Source-Files-and-Modules/#incr_eq_plus1-_LPAR_in-Proofs-are-Private_RPAR_":
{"display": "incr_eq_plus1",
"context":
["The Lean Language Reference",
"Source Files and Modules",
"Modules and Visibility",
"Proofs are Private"]}},
"address": "/Source-Files-and-Modules/"}],
"incrOrFail (in Lifting Transformed Monads)":
[{"id": "incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next",
"data":
{"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next":
{"display": "incrOrFail",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]},
"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_":
{"display": "incrOrFail",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]}},
"address": "/Functors___-Monads-and--do--Notation/Lifting-Monads/"},
{"id": "incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_",
"data":
{"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next":
{"display": "incrOrFail",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]},
"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrOrFail-_LPAR_in-Lifting-Transformed-Monads_RPAR_":
{"display": "incrOrFail",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]}},
"address": "/Functors___-Monads-and--do--Notation/Lifting-Monads/"}],
"incrBy (in Lifting Transformed Monads)":
[{"id": "incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_",
"data":
{"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next":
{"display": "incrBy",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]},
"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_":
{"display": "incrBy",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]}},
"address": "/Functors___-Monads-and--do--Notation/Lifting-Monads/"},
{"id": "incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next",
"data":
{"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_-next":
{"display": "incrBy",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]},
"/Functors___-Monads-and--do--Notation/Lifting-Monads/#incrBy-_LPAR_in-Lifting-Transformed-Monads_RPAR_":
{"display": "incrBy",
"context":
["The Lean Language Reference",
"Functors, Monads and do -Notation",
"Lifting Monads",
"Lifting Transformed Monads"]}},
"address": "/Functors___-Monads-and--do--Notation/Lifting-Monads/"}],
"incr (in Proofs are Private)":
[{"id": "incr-_LPAR_in-Proofs-are-Private_RPAR_",
"data":
{"/Source-Files-and-Modules/#incr-_LPAR_in-Proofs-are-Private_RPAR_":
{"display": "incr",
"context":
["The Lean Language Reference",
"Source Files and Modules",
"Modules and Visibility",
"Proofs are Private"]}},
"address": "/Source-Files-and-Modules/"}],
"ignore (in Ignored Terms in Notation Expansion)":
[{"id": "ignore-_LPAR_in-Ignored-Terms-in-Notation-Expansion_RPAR_",
"data":
{"/Notations-and-Macros/Notations/#ignore-_LPAR_in-Ignored-Terms-in-Notation-Expansion_RPAR_":
{"display": "ignore",
"context":
["The Lean Language Reference",
"Notations and Macros",
"Notations",
"Ignored Terms in Notation Expansion"]}},
"address": "/Notations-and-Macros/Notations/"}],
"id₃ (in The universe command when autoImplicit is false)":
[{"id":
"id___-_LPAR_in-The--universe--command-when--autoImplicit--is--false_RPAR_",
"data":
{"/The-Type-System/Universes/#id___-_LPAR_in-The--universe--command-when--autoImplicit--is--false_RPAR_":
{"display": "id₃",
"context":
["The Lean Language Reference",
"The Type System",
"Universes",
"Polymorphism",
"Universe Variable Bindings",
"The universe command when autoImplicit is false"]}},
"address": "/The-Type-System/Universes/"}],
"idiom (in Idiom Brackets)":
[{"id": "idiom-_LPAR_in-Idiom-Brackets_RPAR_",
"data":
{"/Notations-and-Macros/Macros/#idiom-_LPAR_in-Idiom-Brackets_RPAR_":
{"display": "idiom",
"context":
["The Lean Language Reference",
"Notations and Macros",
"Macros",
"Defining Macros",
"The macro_rules Command",
"Idiom Brackets"]}},
"address": "/Notations-and-Macros/Macros/"}],
"identity (in Mutual Block Scope and Automatic Implicit Parameters)":
[{"id":
"identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_-next":
{"display": "identity",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Mutual Recursion",
"Mutual Block Scope and Automatic Implicit Parameters"]},
"/Definitions/Recursive-Definitions/#identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_":
{"display": "identity",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Mutual Recursion",
"Mutual Block Scope and Automatic Implicit Parameters"]}},
"address": "/Definitions/Recursive-Definitions/"},
{"id":
"identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_-next",
"data":
{"/Definitions/Recursive-Definitions/#identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_-next":
{"display": "identity",
"context":
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"Definitions",
"Recursive Definitions",
"Mutual Recursion",
"Mutual Block Scope and Automatic Implicit Parameters"]},
"/Definitions/Recursive-Definitions/#identity-_LPAR_in-Mutual-Block-Scope-and-Automatic-Implicit-Parameters_RPAR_":
{"display": "identity",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Mutual Recursion",
"Mutual Block Scope and Automatic Implicit Parameters"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"id'":
[{"id": "id___",
"data":
{"/The-Type-System/Universes/#id___":
{"display": "id'",
"context":
["The Lean Language Reference",
"The Type System",
"Universes",
"Polymorphism"]}},
"address": "/The-Type-System/Universes/"}],
"h₆ (in Left-to-Right Generation)":
[{"id": "h___-_LPAR_in-Left-to-Right-Generation_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Left-to-Right-Generation_RPAR_":
{"display": "h₆",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Left-to-Right Generation"]}},
"address": "/The--grind--tactic/E___matching/"}],
"h₅ (in Failing Backward Pattern Generation)":
[{"id": "h___-_LPAR_in-Failing-Backward-Pattern-Generation_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Failing-Backward-Pattern-Generation_RPAR_":
{"display": "h₅",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Failing Backward Pattern Generation"]}},
"address": "/The--grind--tactic/E___matching/"}],
"h₄ (in Patterns from Conclusion and Hypotheses)":
[{"id": "h___-_LPAR_in-Patterns-from-Conclusion-and-Hypotheses_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Patterns-from-Conclusion-and-Hypotheses_RPAR_":
{"display": "h₄",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Patterns from Conclusion and Hypotheses"]}},
"address": "/The--grind--tactic/E___matching/"}],
"h₃ (in Bidirectional Equality Pattern Generation)":
[{"id": "h___-_LPAR_in-Bidirectional-Equality-Pattern-Generation_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Bidirectional-Equality-Pattern-Generation_RPAR_":
{"display": "h₃",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Bidirectional Equality Pattern Generation"]}},
"address": "/The--grind--tactic/E___matching/"}],
"h₂ (in Backward Pattern Generation)":
[{"id": "h___-_LPAR_in-Backward-Pattern-Generation_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Backward-Pattern-Generation_RPAR_":
{"display": "h₂",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Backward Pattern Generation"]}},
"address": "/The--grind--tactic/E___matching/"}],
"h₁ (in Forward Pattern Generation)":
[{"id": "h___-_LPAR_in-Forward-Pattern-Generation_RPAR_",
"data":
{"/The--grind--tactic/E___matching/#h___-_LPAR_in-Forward-Pattern-Generation_RPAR_":
{"display": "h₁",
"context":
["The Lean Language Reference",
"The grind tactic",
"E‑matching",
"Inspecting Patterns",
"Forward Pattern Generation"]}},
"address": "/The--grind--tactic/E___matching/"}],
"helper (in Recursion vs Recursors)":
[{"id": "helper-_LPAR_in-Recursion-vs-Recursors_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#helper-_LPAR_in-Recursion-vs-Recursors_RPAR_":
{"display": "helper",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Recursion vs Recursors"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"hello (in Reducibility and Instance Synthesis)":
[{"id": "hello-_LPAR_in-Reducibility-and-Instance-Synthesis_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#hello-_LPAR_in-Reducibility-and-Instance-Synthesis_RPAR_":
{"display": "hello",
"context":
["The Lean Language Reference",
"Definitions",
"Recursive Definitions",
"Controlling Reduction",
"Reducibility and Instance Synthesis"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"hello":
[{"id": "hello",
"data":
{"/Introduction/#hello":
{"display": "hello",
"context":
["The Lean Language Reference",
"Introduction",
"Typographical Conventions",
"Lean Code"]}},
"address": "/Introduction/"}],
"half.match_1' (in Elaboration Walkthrough)":
[{"id": "half___match_1___-_LPAR_in-Elaboration-Walkthrough_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#half___match_1___-_LPAR_in-Elaboration-Walkthrough_RPAR_":
{"display": "half.match_1'",
"context":
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"Definitions",
"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Elaboration Walkthrough"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"half''' (in Recursion vs Recursors)":
[{"id": "half_________-_LPAR_in-Recursion-vs-Recursors_RPAR_",
"data":
{"/Definitions/Recursive-Definitions/#half_________-_LPAR_in-Recursion-vs-Recursors_RPAR_":
{"display": "half'''",
"context":
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"Definitions",
"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Recursion vs Recursors"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"half'' (in Recursion vs Recursors)":
[{"id": "half______-_LPAR_in-Recursion-vs-Recursors_RPAR_",
"data":
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{"display": "half''",
"context":
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"Definitions",
"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Recursion vs Recursors"]}},
"address": "/Definitions/Recursive-Definitions/"}],
"half'' (in Elaboration Walkthrough)":
[{"id": "half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next-next",
"data":
{"/Definitions/Recursive-Definitions/#half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next-next-next":
{"display": "half''",
"context":
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"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Elaboration Walkthrough"]},
"/Definitions/Recursive-Definitions/#half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next-next":
{"display": "half''",
"context":
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"Recursive Definitions",
"Structural Recursion",
"Elaboration Using Course-of-Values Recursion",
"Elaboration Walkthrough"]},
"/Definitions/Recursive-Definitions/#half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next":
{"display": "half''",
"context":
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"Structural Recursion",
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"/Definitions/Recursive-Definitions/#half______-_LPAR_in-Elaboration-Walkthrough_RPAR_":
{"display": "half''",
"context":
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"address": "/Definitions/Recursive-Definitions/"},
{"id": "half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next",
"data":
{"/Definitions/Recursive-Definitions/#half______-_LPAR_in-Elaboration-Walkthrough_RPAR_-next-next-next":
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{"display": "PBool",
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"NonZero (in Universe mismatch)":
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"data": {"term": "setoid"},
"address": "/The-Type-System/Quotients/"}],
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"data": {"term": "Semireducible"},
"address": "/Definitions/Recursive-Definitions/"}],
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"data": {"term": "Semi-output parameters"},
"address": "/Type-Classes/Instance-Synthesis/"}],
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"address": "/Namespaces-and-Sections/"}],
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"section":
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"address": "/Terms/Numeric-Literals/"}],
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"data": {"term": "schematic variables"},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
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"data": {"term": "round"},
"address": "/The--grind--tactic/E___matching/"}],
"root package":
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"root directory":
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"data": {"term": "root directory"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"right hand side":
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"data": {"term": "right-hand side"},
"address": "/Terms/Pattern-Matching/"}],
"right fold":
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"data": {"term": "Right folds"},
"address": "/Basic-Types/Linked-Lists/"}],
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"data": {"term": "Right-associative"},
"address": "/Notations-and-Macros/Custom-Operators/"}],
"resolve":
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"data": {"term": "resolving"},
"address": "/Terms/Identifiers/"},
{"id": "--tech-term-resolving-next",
"data": {"term": "resolving"},
"address": "/IO/Tasks-and-Threads/"},
{"id": "--tech-term-resolve",
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"repl":
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"data": {"term": "REPL"},
"address": "/Interacting-with-Lean/"}],
"relevant":
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"data": {"term": "relevant"},
"address": "/The-Type-System/Inductive-Types/"}],
"reference count":
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"address": "/The-Type-System/"}],
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"address": "/Definitions/Recursive-Definitions/"}],
"reducibility":
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"data": {"term": "reducibility"},
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"realizing":
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"data": {"term": "realizing"},
"address": "/Terms/Identifiers/"}],
"reader monad":
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"data": {"term": "Reader monads"},
"address":
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"raw string literal":
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"address": "/Basic-Types/Strings/"}],
"raw identifier":
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"raw":
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"data": {"term": "raw"},
"address": "/Basic-Types/Maps-and-Sets/"}],
"range":
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"data": {"term": "range"},
"address": "/Basic-Types/Ranges/"}],
"quotient type":
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"address": "/The-Type-System/Quotients/"}],
"quotation":
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"address": "/Notations-and-Macros/Macros/"}],
"quasiquotation":
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"data": {"term": "Quasiquotation"},
"address": "/Notations-and-Macros/Macros/"}],
"pure iterator":
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"data": {"term": "pure"},
"address": "/Iterators/Iterator-Definitions/"}],
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"address": "/The-Type-System/Propositions/"}],
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"address": "/Basic-Propositions/Propositional-Equality/"}],
"proposition":
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"data": {"term": "Propositions"},
"address": "/The-Type-System/Propositions/"}],
"proof term":
[{"id": "--tech-term-proof-terms",
"data": {"term": "proof terms"},
"address": "/Tactic-Proofs/"}],
"proof state":
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"address": "/Tactic-Proofs/"}],
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"data": {"term": "proof irrelevance"},
"address": "/The-Type-System/"}],
"projection function":
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"address": "/The-Type-System/Inductive-Types/"}],
"productive iterator":
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"data": {"term": "Productive"},
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"producer":
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"data": {"term": "Producers"},
"address": "/Iterators/"}],
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"data": {"term": "private"},
"address": "/Definitions/Modifiers/"}],
"priority":
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"address": "/Type-Classes/Instance-Declarations/"},
{"id": "--tech-term-priority",
"data": {"term": "priority"},
"address": "/IO/Tasks-and-Threads/"}],
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"data": {"term": "prelude"},
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"data": {"term": "predicative"},
"address": "/The-Type-System/Universes/"}],
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"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
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"address": "/Notations-and-Macros/Custom-Operators/"}],
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"address": "/Elaboration-and-Compilation/"}],
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"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"postcondition":
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"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
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"address": "/Iterators/Iterator-Definitions/"}],
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"data": {"term": "placeholder term"},
"address": "/Terms/Holes/"}],
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[{"id": "--tech-term-patterns",
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"path dependency":
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"path":
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"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
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"package directory":
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{"id": "--tech-term-packages",
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"owned":
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"address": "/Run-Time-Code/Foreign-Function-Interface/"}],
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"address": "/Type-Classes/Instance-Synthesis/"}],
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"address": "/Notations-and-Macros/Defining-New-Syntax/"}],
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"address":
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"notation item":
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{"id": "--tech-term-notation-items",
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"namespace":
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"named pattern":
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"address":
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"monad law":
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"monad":
[{"id": "--tech-term-Monad",
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"address": "/Functors___-Monads-and--do--Notation/"}],
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"address": "/Basic-Propositions/Logical-Connectives/"}],
"module root":
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"module":
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"modifier":
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"address": "/The--grind--tactic/Linear-Integer-Arithmetic/"}],
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"method":
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"address": "/Type-Classes/"}],
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"address": "/Tactic-Proofs/Reading-Proof-States/"}],
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{"id": "--tech-term-measure-next",
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"address": "/Terms/Pattern-Matching/"}],
"match alternative":
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"marble diagram":
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"address": "/Iterators/Iterator-Combinators/"}],
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"address": "/Tactic-Proofs/"}],
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{"id": "--tech-term-Macros",
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"loop invariant":
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"address": "/The--mvcgen--tactic/Overview/"}],
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"local instance":
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"address": "/Type-Classes/Instance-Synthesis/"}],
"local cache":
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"lint driver":
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"linearly":
[{"id": "--tech-term-linearly",
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"address": "/Basic-Types/Arrays/"}],
"lift":
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"address": "/Functors___-Monads-and--do--Notation/Lifting-Monads/"}],
"library directory":
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"library":
[{"id": "--tech-term-libraries",
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"address": "/Source-Files-and-Modules/"},
{"id": "--tech-term-Libraries",
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"left fold":
[{"id": "--tech-term-Left-folds",
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"address": "/Basic-Types/Linked-Lists/"}],
"left associative":
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"address": "/Basic-Types/Lazy-Computations/"}],
"law":
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"lake script":
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"lake home":
[{"id": "--tech-term-Lake-home",
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"kleisli composition":
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"address": "/Functors___-Monads-and--do--Notation/API-Reference/"}],
"kind":
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"kernel":
[{"id": "--tech-term-kernel",
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"iterator combinator":
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"address": "/Iterators/"}],
"iterator":
[{"id": "--tech-term-iterator",
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"irrelevant":
[{"id": "--tech-term-irrelevant",
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"address": "/The-Type-System/Inductive-Types/"}],
"irreducible":
[{"id": "--tech-term-Irreducible",
"data": {"term": "Irreducible"},
"address": "/Definitions/Recursive-Definitions/"}],
"interpolated string":
[{"id": "--tech-term-interpolated-string",
"data": {"term": "interpolated string"},
"address": "/Basic-Types/Strings/"}],
"intensional":
[{"id": "--tech-term-intensional",
"data": {"term": "intensional"},
"address": "/The-Type-System/Functions/"}],
"instance implicit":
[{"id": "--tech-term-instance-implicit-next",
"data": {"term": "instance-implicit"},
"address": "/Type-Classes/"},
{"id": "--tech-term-instance-implicit",
"data": {"term": "instance-implicit"},
"address": "/Terms/Functions/"}],
"instance":
[{"id": "--tech-term-instances",
"data": {"term": "instances"},
"address": "/Type-Classes/"}],
"injection":
[{"id": "--tech-term-injections",
"data": {"term": "injections"},
"address": "/Basic-Types/Sum-Types/"}],
"initialize":
[{"id": "--tech-term-initialized",
"data": {"term": "initialized"},
"address": "/Run-Time-Code/Foreign-Function-Interface/"}],
"initialization":
[{"id": "--tech-term-initialization",
"data": {"term": "initialization"},
"address": "/Elaboration-and-Compilation/"}],
"initial target":
[{"id": "--tech-term-initial-targets",
"data": {"term": "initial targets"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"info tree":
[{"id": "--tech-term-info-trees",
"data": {"term": "info trees"},
"address": "/Elaboration-and-Compilation/"}],
"inductive type":
[{"id": "--tech-term-Inductive-types",
"data": {"term": "Inductive types"},
"address": "/The-Type-System/Inductive-Types/"}],
"indexed family":
[{"id": "--tech-term-indexed-families",
"data": {"term": "indexed families"},
"address": "/The-Type-System/Inductive-Types/"}],
"indexable":
[{"id": "--tech-term-indexable",
"data": {"term": "indexable"},
"address": "/The--grind--tactic/E___matching/"}],
"index":
[{"id": "--tech-term-indices",
"data": {"term": "indices"},
"address": "/The-Type-System/Inductive-Types/"}],
"incremental build":
[{"id": "--tech-term-incremental-build",
"data": {"term": "incremental build"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"inaccessible pattern":
[{"id": "--tech-term-Inaccessible-patterns",
"data": {"term": "Inaccessible patterns"},
"address": "/Terms/Pattern-Matching/"}],
"inaccessible":
[{"id": "--tech-term-inaccessible",
"data": {"term": "inaccessible"},
"address": "/Tactic-Proofs/Reading-Proof-States/"}],
"impredicative":
[{"id": "--tech-term-impredicative",
"data": {"term": "impredicative"},
"address": "/The-Type-System/Universes/"}],
"import statement":
[{"id": "--tech-term-import-statements",
"data": {"term": "import statements"},
"address": "/Source-Files-and-Modules/"}],
"import name":
[{"id": "--tech-term-import-name",
"data": {"term": "import name"},
"address": "/Source-Files-and-Modules/"}],
"implicit":
[{"id": "--tech-term-implicit",
"data": {"term": "implicit"},
"address": "/Terms/Functions/"}],
"identifier component":
[{"id": "--tech-term-Identifier-components",
"data": {"term": "Identifier components"},
"address": "/Source-Files-and-Modules/"}],
"identifier capture":
[{"id": "--tech-term-Identifier-capture",
"data": {"term": "Identifier capture"},
"address": "/Notations-and-Macros/Macros/"}],
"identifier":
[{"id": "--tech-term-Identifiers",
"data": {"term": "Identifiers"},
"address": "/Notations-and-Macros/Defining-New-Syntax/"}],
"hygiene":
[{"id": "--tech-term-hygienic",
"data": {"term": "hygienic"},
"address": "/Notations-and-Macros/Macros/"}],
"hole":
[{"id": "--tech-term-hole",
"data": {"term": "hole"},
"address": "/Terms/Holes/"}],
"hoare triple":
[{"id": "--tech-term-Hoare-triple",
"data": {"term": "Hoare triple"},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"hierarchical identifier":
[{"id": "--tech-term-hierarchical-identifiers",
"data": {"term": "hierarchical identifiers"},
"address": "/Source-Files-and-Modules/"}],
"heterogeneous equality":
[{"id": "--tech-term-Heterogeneous-equality",
"data": {"term": "Heterogeneous equality"},
"address": "/Basic-Propositions/Propositional-Equality/"}],
"header":
[{"id": "--tech-term-header",
"data": {"term": "header"},
"address": "/Definitions/Headers-and-Signatures/"}],
"guillemet":
[{"id": "--tech-term-guillemets",
"data": {"term": "guillemets"},
"address": "/Source-Files-and-Modules/"}],
"goal":
[{"id": "--tech-term-goals",
"data": {"term": "goals"},
"address": "/Tactic-Proofs/"}],
"global instance":
[{"id": "--tech-term-Global-instances",
"data": {"term": "Global instances"},
"address": "/Type-Classes/Instance-Synthesis/"}],
"glob":
[{"id": "--tech-term-globs",
"data": {"term": "globs"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"git dependency":
[{"id": "--tech-term-Git-dependencies",
"data": {"term": "Git dependencies"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"generation":
[{"id": "--tech-term-generation",
"data": {"term": "generation"},
"address": "/The--grind--tactic/E___matching/"}],
"generalized field notation":
[{"id": "--tech-term-generalized-field-notation",
"data": {"term": "generalized field notation"},
"address": "/Terms/Function-Application/"}],
"gap":
[{"id": "--tech-term-gaps",
"data": {"term": "gaps"},
"address": "/Basic-Types/Strings/"}],
"gadget":
[{"id": "--tech-term-gadgets",
"data": {"term": "gadgets"},
"address": "/Type-Classes/Class-Declarations/"}],
"functor":
[{"id": "--tech-term-Functor",
"data": {"term": "Functor"},
"address": "/Functors___-Monads-and--do--Notation/"}],
"function type":
[{"id": "--tech-term-function-types",
"data": {"term": "function types"},
"address": "/The-Type-System/Functions/"}],
"function abstraction":
[{"id": "--tech-term-function-abstractions",
"data": {"term": "function abstractions"},
"address": "/The-Type-System/Functions/"}],
"function":
[{"id": "--tech-term-Functions",
"data": {"term": "Functions"},
"address": "/The-Type-System/Functions/"}],
"framing":
[{"id": "--tech-term-framing",
"data": {"term": "framing"},
"address":
"/The--mvcgen--tactic/Tutorial___-Verifying-Imperative-Programs-Using--mvcgen/"}],
"frame":
[{"id": "--tech-term-frame",
"data": {"term": "frame"},
"address":
"/The--mvcgen--tactic/Tutorial___-Verifying-Imperative-Programs-Using--mvcgen/"}],
"forward rule":
[{"id": "--tech-term-forward-rules",
"data": {"term": "forward rules"},
"address": "/The--grind--tactic/Constraint-Propagation/"}],
"force":
[{"id": "--tech-term-forcing",
"data": {"term": "forcing"},
"address": "/Basic-Types/Lazy-Computations/"}],
"fixed prefix":
[{"id": "--tech-term-fixed-prefix",
"data": {"term": "fixed prefix"},
"address": "/Definitions/Recursive-Definitions/"}],
"fixed parameter":
[{"id": "--tech-term-fixed-parameter",
"data": {"term": "fixed parameter"},
"address": "/Definitions/Recursive-Definitions/"}],
"finite iterator":
[{"id": "--tech-term-Finite",
"data": {"term": "Finite"},
"address": "/Iterators/Iterator-Definitions/"}],
"file header":
[{"id": "--tech-term-file-header",
"data": {"term": "file header"},
"address": "/Source-Files-and-Modules/"}],
"file handle":
[{"id": "--tech-term-File-Handles",
"data": {"term": "File Handles"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"file":
[{"id": "--tech-term-Files",
"data": {"term": "Files"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"field resolution order":
[{"id": "--tech-term-field-resolution-order",
"data": {"term": "field resolution order"},
"address": "/The-Type-System/Inductive-Types/"}],
"field notation":
[{"id": "--tech-term-field-notation",
"data": {"term": "field notation"},
"address": "/Terms/Function-Application/"}],
"facet":
[{"id": "--tech-term-facet",
"data": {"term": "facet"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"external library":
[{"id": "--tech-term-External-libraries",
"data": {"term": "External libraries"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"extensional map":
[{"id": "--tech-term-Extensional-maps",
"data": {"term": "Extensional maps"},
"address": "/Basic-Types/Maps-and-Sets/"}],
"extension":
[{"id": "--tech-term-extensions",
"data": {"term": "extensions"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"extended splice":
[{"id": "--tech-term-extended-splices",
"data": {"term": "extended splices"},
"address": "/Notations-and-Macros/Macros/"}],
"expression":
[{"id": "--tech-term-expressions",
"data": {"term": "expressions"},
"address": "/The-Type-System/"}],
"exposed":
[{"id": "--tech-term-exposed",
"data": {"term": "exposed"},
"address": "/Source-Files-and-Modules/"}],
"exporting":
[{"id": "--tech-term-Exporting",
"data": {"term": "Exporting"},
"address": "/Namespaces-and-Sections/"}],
"expected type's namespace":
[{"id": "--tech-term-expected-type___s-namespace",
"data": {"term": "expected type's namespace"},
"address": "/Terms/Identifiers/"}],
"expansion":
[{"id": "--tech-term-expansion",
"data": {"term": "expansion"},
"address": "/Notations-and-Macros/Custom-Operators/"}],
"executable":
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"data": {"term": "Executables"},
"address": "/Source-Files-and-Modules/"},
{"id": "--tech-term-Executables",
"data": {"term": "Executables"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"exception monad":
[{"id": "--tech-term-Exception-monads",
"data": {"term": "Exception monads"},
"address":
"/Functors___-Monads-and--do--Notation/Varieties-of-Monads/"}],
"example":
[{"id": "--tech-term-example",
"data": {"term": "example"},
"address": "/Definitions/Example-Declarations/"}],
"erasing":
[{"id": "--tech-term-erasing",
"data": {"term": "erasing"},
"address": "/Attributes/"}],
"equivalence relation":
[{"id": "--tech-term-equivalence-relation",
"data": {"term": "equivalence relation"},
"address": "/The-Type-System/Quotients/"}],
"equational lemma":
[{"id": "--tech-term-equational-lemmas",
"data": {"term": "equational lemmas"},
"address": "/Elaboration-and-Compilation/"}],
"environment extension":
[{"id": "--tech-term-environment-extensions",
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"address": "/Elaboration-and-Compilation/"}],
"environment":
[{"id": "--tech-term-environments",
"data": {"term": "environments"},
"address": "/Elaboration-and-Compilation/"}],
"enum inductive":
[{"id": "--tech-term-enum-inductive",
"data": {"term": "enum inductive"},
"address": "/The-Type-System/Inductive-Types/"}],
"ellipsi":
[{"id": "--tech-term-ellipsis",
"data": {"term": "ellipsis"},
"address": "/Terms/Function-Application/"}],
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"data": {"term": "elaborators"},
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{"id": "--tech-term-elaborators",
"data": {"term": "elaborators"},
"address": "/Notations-and-Macros/Elaborators/"}],
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"data": {"term": "elaboration"},
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"data": {"term": "instance"},
"address": "/The--grind--tactic/E___matching/"}],
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[{"id": "--tech-term-E-matching",
"data": {"term": "E-matching"},
"address": "/The--grind--tactic/E___matching/"}],
"dynamic array":
[{"id": "--tech-term-dynamic-arrays",
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"address": "/Basic-Types/Arrays/"}],
"downward propagation":
[{"id": "--tech-term-downward-propagation",
"data": {"term": "downward propagation"},
"address": "/The--grind--tactic/Constraint-Propagation/"}],
"domain":
[{"id": "--tech-term-domain",
"data": {"term": "domain"},
"address": "/The-Type-System/Functions/"}],
"documentation comment":
[{"id": "--tech-term-Documentation-comments",
"data": {"term": "Documentation comments"},
"address": "/Definitions/Modifiers/"}],
"documentation":
[{"id": "--tech-term-documentation",
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"address": "/Source-Files-and-Modules/"}],
"do notation":
[{"id": "--tech-term-do-notation",
"data": {"term": "do-notation"},
"address": "/Functors___-Monads-and--do--Notation/Syntax/"}],
"do item":
[{"id": "--tech-term-do-items",
"data": {"term": "do items"},
"address": "/Functors___-Monads-and--do--Notation/Syntax/"}],
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[{"id": "--tech-term-discriminant-refinement",
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"address": "/Terms/Pattern-Matching/"}],
"directory":
[{"id": "--tech-term-Directories",
"data": {"term": "Directories"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"direct dependency":
[{"id": "--tech-term-direct-dependencies",
"data": {"term": "direct dependencies"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"dictionary":
[{"id": "--tech-term-dictionaries",
"data": {"term": "dictionaries"},
"address": "/Basic-Types/Maps-and-Sets/"}],
"diamond":
[{"id": "--tech-term-Diamonds",
"data": {"term": "Diamonds"},
"address": "/Type-Classes/Instance-Synthesis/"}],
"deriving handler":
[{"id": "--tech-term-deriving-handlers",
"data": {"term": "deriving handlers"},
"address": "/Type-Classes/Deriving-Instances/"}],
"deriving":
[{"id": "--tech-term-deriving",
"data": {"term": "deriving"},
"address": "/Type-Classes/Deriving-Instances/"}],
"derivation":
[{"id": "--tech-term-derivation",
"data": {"term": "derivation"},
"address": "/The-Type-System/"}],
"dependent sum":
[{"id": "--tech-term-dependent-sums",
"data": {"term": "dependent sums"},
"address": "/Basic-Types/Tuples/"}],
"dependent product":
[{"id": "--tech-term-dependent-products",
"data": {"term": "dependent products"},
"address": "/The-Type-System/Functions/"}],
"dependent pair":
[{"id": "--tech-term-Dependent-pairs",
"data": {"term": "Dependent pairs"},
"address": "/Basic-Types/Tuples/"}],
"dependent map":
[{"id": "--tech-term-dependent-map",
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"address": "/Basic-Types/Maps-and-Sets/"}],
"dependent coercion":
[{"id": "--tech-term-Dependent-coercions",
"data": {"term": "Dependent coercions"},
"address": "/Coercions/Coercing-Between-Types/"}],
"dependent":
[{"id": "--tech-term-Dependent",
"data": {"term": "Dependent"},
"address": "/The-Type-System/Functions/"}],
"delaborator":
[{"id": "--tech-term-delaborators",
"data": {"term": "delaborators"},
"address": "/Notations-and-Macros/Extending-Lean___s-Output/"}],
"definitional equality":
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"data": {"term": "definitional equality"},
"address": "/The-Type-System/"}],
"defined constant":
[{"id": "--tech-term-defined-constants",
"data": {"term": "defined constants"},
"address": "/The-Type-System/"}],
"default toolchain":
[{"id": "--tech-term-default-toolchain",
"data": {"term": "default toolchain"},
"address":
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"default target":
[{"id": "--tech-term-default-targets",
"data": {"term": "default targets"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"default simp set":
[{"id": "--tech-term-default-simp-set",
"data": {"term": "default simp set"},
"address": "/The-Simplifier/Simp-sets/"}],
"default instance":
[{"id": "--tech-term-default-instances",
"data": {"term": "default instances"},
"address": "/Type-Classes/Instance-Synthesis/"}],
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[{"id": "--tech-term-default-facet",
"data": {"term": "default facet"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
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[{"id": "--tech-term-decreasing-parameter",
"data": {"term": "decreasing parameter"},
"address": "/Definitions/Recursive-Definitions/"}],
"declaration name":
[{"id": "--tech-term-declaration-name",
"data": {"term": "declaration name"},
"address": "/Definitions/Headers-and-Signatures/"}],
"decidable":
[{"id": "--tech-term-decidable",
"data": {"term": "decidable"},
"address": "/Type-Classes/Basic-Classes/"}],
"de bruijn indice":
[{"id": "--tech-term-de-Bruijn-indices",
"data": {"term": "de Bruijn indices"},
"address": "/The--grind--tactic/E___matching/"}],
"cycle":
[{"id": "--tech-term-cycles",
"data": {"term": "cycles"},
"address": "/Type-Classes/Instance-Synthesis/"}],
"custom toolchain name":
[{"id": "--tech-term-custom-toolchain-name",
"data": {"term": "custom toolchain name"},
"address":
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"custom target":
[{"id": "--tech-term-Custom-targets",
"data": {"term": "Custom targets"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
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[{"id": "--tech-term-Custom-simp-sets",
"data": {"term": "Custom simp sets"},
"address": "/The-Simplifier/Simp-sets/"}],
"custom eliminator":
[{"id": "--tech-term-Custom-eliminators",
"data": {"term": "Custom eliminators"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"currying":
[{"id": "--tech-term-currying",
"data": {"term": "currying"},
"address": "/The-Type-System/Functions/"}],
"current namespace":
[{"id": "--tech-term-current-namespace",
"data": {"term": "current namespace"},
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{"id": "--tech-term-current-namespace-next",
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"address": "/Namespaces-and-Sections/"}],
"current do block":
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"data": {"term": "current do block"},
"address": "/Functors___-Monads-and--do--Notation/Syntax/"}],
"cumulative":
[{"id": "--tech-term-cumulative",
"data": {"term": "cumulative"},
"address": "/The-Type-System/Universes/"}],
"coverage":
[{"id": "--tech-term-coverage",
"data": {"term": "coverage"},
"address": "/The--grind--tactic/E___matching/"}],
"cover":
[{"id": "--tech-term-cover",
"data": {"term": "cover"},
"address": "/The--grind--tactic/E___matching/"}],
"course of values recursion":
[{"id": "--tech-term-course-of-values-recursion",
"data": {"term": "course-of-values recursion"},
"address": "/Definitions/Recursive-Definitions/"}],
"consumer":
[{"id": "--tech-term-Consumers",
"data": {"term": "Consumers"},
"address": "/Iterators/"}],
"constructor":
[{"id": "--tech-term-constructors",
"data": {"term": "constructors"},
"address": "/The-Type-System/Inductive-Types/"}],
"constraint propagation":
[{"id": "--tech-term-Constraint-propagation",
"data": {"term": "Constraint propagation"},
"address": "/The--grind--tactic/Constraint-Propagation/"}],
"conjunctive":
[{"id": "--tech-term-conjunctive",
"data": {"term": "conjunctive"},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"congruence closure":
[{"id": "--tech-term-Congruence-closure",
"data": {"term": "Congruence closure"},
"address": "/The--grind--tactic/Congruence-Closure/"}],
"configure package":
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"data": {"term": "Configuring"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"configuration item":
[{"id": "--tech-term-configuration-items",
"data": {"term": "configuration items"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"command elaborator":
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"address": "/Notations-and-Macros/Elaborators/"}],
"command":
[{"id": "--tech-term-commands",
"data": {"term": "commands"},
"address": "/Source-Files-and-Modules/"}],
"coercion insertion":
[{"id": "--tech-term-coercion-insertion",
"data": {"term": "coercion insertion"},
"address": "/Coercions/Coercion-Insertion/"}],
"coercion":
[{"id": "--tech-term-coercion",
"data": {"term": "coercion"},
"address": "/Coercions/"}],
"codomain":
[{"id": "--tech-term-codomain",
"data": {"term": "codomain"},
"address": "/The-Type-System/Functions/"}],
"class abbreviation":
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"address": "/Type-Classes/Class-Declarations/"}],
"choice node":
[{"id": "--tech-term-choice-node",
"data": {"term": "choice node"},
"address": "/Elaboration-and-Compilation/"}],
"channel":
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"data": {"term": "channel"},
"address":
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"cast":
[{"id": "--tech-term-casts",
"data": {"term": "casts"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"case label":
[{"id": "--tech-term-case-label",
"data": {"term": "case label"},
"address": "/Tactic-Proofs/Reading-Proof-States/"}],
"canonical":
[{"id": "--tech-term-canonical",
"data": {"term": "canonical"},
"address": "/Notations-and-Macros/Defining-New-Syntax/"}],
"candidate measure":
[{"id": "--tech-term-Candidate-measures",
"data": {"term": "Candidate measures"},
"address": "/Definitions/Recursive-Definitions/"}],
"cancel":
[{"id": "--tech-term-cancelled",
"data": {"term": "cancelled"},
"address": "/IO/Tasks-and-Threads/"}],
"cache scope":
[{"id": "--tech-term-cache-scope",
"data": {"term": "cache scope"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"build directory":
[{"id": "--tech-term-build-directory",
"data": {"term": "build directory"},
"address": "/Build-Tools-and-Distribution/Lake/"}],
"build":
[{"id": "--tech-term-build",
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"buffered":
[{"id": "--tech-term-buffered",
"data": {"term": "buffered"},
"address": "/IO/Files___-File-Handles___-and-Streams/"}],
"bracketed binder":
[{"id": "--tech-term-bracketed-binders",
"data": {"term": "bracketed binders"},
"address": "/Definitions/Headers-and-Signatures/"}],
"boxed":
[{"id": "--tech-term-Boxed",
"data": {"term": "Boxed"},
"address": "/Run-Time-Code/Boxing/"}],
"borrowed":
[{"id": "--tech-term-borrowed",
"data": {"term": "borrowed"},
"address": "/Run-Time-Code/Foreign-Function-Interface/"}],
"binary directory":
[{"id": "--tech-term-binary-directory",
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"basic termination measure":
[{"id": "--tech-term-basic-termination-measures",
"data": {"term": "basic termination measures"},
"address": "/Definitions/Recursive-Definitions/"}],
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"data": {"userName": "Lean.Parser.Tactic.tacIfThenElse"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.tacDepIfThenElse":
[{"id": "if-h-___-_________-then-_________-else-_________",
"data": {"userName": "Lean.Parser.Tactic.tacDepIfThenElse"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.symmSaturate":
[{"id": "symm_saturate",
"data": {"userName": "symm_saturate"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.symm":
[{"id": "symm",
"data": {"userName": "symm"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.substVars":
[{"id": "subst_vars",
"data": {"userName": "subst_vars"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.substEqs":
[{"id": "subst_eqs",
"data": {"userName": "subst_eqs"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.subst":
[{"id": "subst",
"data": {"userName": "subst"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.split":
[{"id": "split",
"data": {"userName": "split"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.specialize":
[{"id": "specialize",
"data": {"userName": "specialize"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.solveByElim":
[{"id": "solve_by_elim",
"data": {"userName": "solve_by_elim"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.sleep":
[{"id": "sleep",
"data": {"userName": "sleep"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.skip":
[{"id": "skip",
"data": {"userName": "skip"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpa":
[{"id": "simpa",
"data": {"userName": "simpa"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpTrace":
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"data": {"userName": "simp?"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAutoUnfold":
[{"id": "simp___",
"data": {"userName": "simp!"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpArithBang":
[{"id": "simp_arith___",
"data": {"userName": "simp_arith!"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpArith":
[{"id": "simp_arith",
"data": {"userName": "simp_arith"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAllTrace":
[{"id": "simp_all___-next",
"data": {"userName": "simp_all?"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAllAutoUnfold":
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"data": {"userName": "simp_all!"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAllArithBang":
[{"id": "simp_all_arith___",
"data": {"userName": "simp_all_arith!"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAllArith":
[{"id": "simp_all_arith",
"data": {"userName": "simp_all_arith"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simpAll":
[{"id": "simp_all",
"data": {"userName": "simp_all"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.simp":
[{"id": "simp",
"data": {"userName": "simp"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.showTerm":
[{"id": "show_term",
"data": {"userName": "show_term"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.show":
[{"id": "show",
"data": {"userName": "show"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.set_option":
[{"id": "set_option",
"data": {"userName": "Lean.Parser.Tactic.set_option"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.rwSeq":
[{"id": "rw",
"data": {"userName": "rw"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.runTac":
[{"id": "run_tac",
"data": {"userName": "run_tac"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.rotateRight":
[{"id": "rotate_right",
"data": {"userName": "rotate_right"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.rotateLeft":
[{"id": "rotate_left",
"data": {"userName": "rotate_left"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.rintro":
[{"id": "rintro",
"data": {"userName": "rintro"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.right":
[{"id": "right",
"data": {"userName": "right"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.rewrites?":
[{"id": "rw___",
"data": {"userName": "rw?"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.rewriteSeq":
[{"id": "rewrite",
"data": {"userName": "rewrite"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.revert":
[{"id": "revert",
"data": {"userName": "revert"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.replace":
[{"id": "replace",
"data": {"userName": "replace"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.repeat1'":
[{"id": "repeat1___",
"data": {"userName": "repeat1'"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.repeat'":
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"data": {"userName": "repeat'"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.renameI":
[{"id": "rename_i",
"data": {"userName": "rename_i"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.rename":
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"data": {"userName": "rename"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.refine'":
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"data": {"userName": "refine'"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.refine":
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"data": {"userName": "refine"},
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"data": {"userName": "rcases"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.pushCast":
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"data": {"userName": "push_cast"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.open":
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"data": {"userName": "Lean.Parser.Tactic.open"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.omega":
[{"id": "omega",
"data": {"userName": "omega"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.obtain":
[{"id": "obtain",
"data": {"userName": "obtain"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.nativeDecide":
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"data": {"userName": "native_decide"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mvcgenMacro":
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"data": {"userName": "mvcgen"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mstopMacro":
[{"id": "mstop",
"data": {"userName": "mstop"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mstartMacro":
[{"id": "mstart",
"data": {"userName": "mstart"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mspecializePureMacro":
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"data": {"userName": "mspecialize_pure"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mspecializeMacro":
[{"id": "mspecialize",
"data": {"userName": "mspecialize"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mspecMacro":
[{"id": "mspec",
"data": {"userName": "mspec"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mrightMacro":
[{"id": "mright",
"data": {"userName": "mright"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mreplaceMacro":
[{"id": "mreplace",
"data": {"userName": "mreplace"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mrenameIMacro":
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"data": {"userName": "mrename_i"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mrefineMacro":
[{"id": "mrefine",
"data": {"userName": "mrefine"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mpureMacro":
[{"id": "mpure",
"data": {"userName": "mpure"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mpureIntroMacro":
[{"id": "mpure_intro",
"data": {"userName": "mpure_intro"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mleftMacro":
[{"id": "mleft",
"data": {"userName": "mleft"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mleaveMacro":
[{"id": "mleave",
"data": {"userName": "mleave"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mintroMacro":
[{"id": "mintro",
"data": {"userName": "mintro"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mhaveMacro":
[{"id": "mhave",
"data": {"userName": "mhave"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mframeMacro":
[{"id": "mframe",
"data": {"userName": "mframe"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mexistsMacro":
[{"id": "mexists",
"data": {"userName": "mexists"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mexfalsoMacro":
[{"id": "mexfalso",
"data": {"userName": "mexfalso"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mexactMacro":
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"data": {"userName": "mexact"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mdup":
[{"id": "mdup",
"data": {"userName": "mdup"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mconstructorMacro":
[{"id": "mconstructor",
"data": {"userName": "mconstructor"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mclearMacro":
[{"id": "mclear",
"data": {"userName": "mclear"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.mcasesMacro":
[{"id": "mcases",
"data": {"userName": "mcases"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.match":
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"data": {"userName": "Lean.Parser.Tactic.match"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.massumptionMacro":
[{"id": "massumption",
"data": {"userName": "massumption"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.letrec":
[{"id": "let-rec",
"data": {"userName": "let"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.left":
[{"id": "left",
"data": {"userName": "left"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.intros":
[{"id": "intros",
"data": {"userName": "intros"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.introMatch":
[{"id":
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"data": {"userName": "Lean.Parser.Tactic.introMatch"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.intro":
[{"id": "intro",
"data": {"userName": "intro"},
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"Lean.Parser.Tactic.injections":
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"data": {"userName": "injections"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.injection":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.induction":
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"data": {"userName": "induction"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
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"data": {"userName": "guard_target"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.guardHyp":
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"data": {"userName": "guard_hyp"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.guardExpr":
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"data": {"userName": "guard_expr"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.grind":
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"data": {"userName": "grind"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.generalize":
[{"id": "generalize",
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"Lean.Parser.Tactic.funInduction":
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"Lean.Parser.Tactic.funCases":
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"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.first":
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"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.falseOrByContra":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.failIfSuccess":
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"data": {"userName": "fail_if_success"},
"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.fail":
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"Lean.Parser.Tactic.exposeNames":
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"data": {"userName": "expose_names"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.exact":
[{"id": "exact",
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"Lean.Parser.Tactic.eqRefl":
[{"id": "eq_refl",
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"Lean.Parser.Tactic.dsimpTrace":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.dsimpAutoUnfold":
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"Lean.Parser.Tactic.dsimp":
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"Lean.Parser.Tactic.done":
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"data": {"userName": "done"},
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"Lean.Parser.Tactic.delta":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.decide":
[{"id": "decide",
"data": {"userName": "decide"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
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"data": {"userName": "dbg_trace"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.contradiction":
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"data": {"userName": "contradiction"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.constructor":
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"Lean.Parser.Tactic.congr":
[{"id": "congr",
"data": {"userName": "congr"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.clear":
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"data": {"userName": "clear"},
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"Lean.Parser.Tactic.classical":
[{"id": "classical",
"data": {"userName": "classical"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.changeWith":
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"data": {"userName": "change"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.change":
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"data": {"userName": "change"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.cases":
[{"id": "cases",
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.case'":
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"Lean.Parser.Tactic.case":
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"Lean.Parser.Tactic.bvTraceMacro":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.bvNormalizeMacro":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.bvDecideMacro":
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"Lean.Parser.Tactic.bvCheck":
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"data": {"userName": "bv_check"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.assumption":
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"data": {"userName": "assumption"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.as_aux_lemma":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.applyRules":
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"Lean.Parser.Tactic.applyRfl":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.applyAssumption":
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"data": {"userName": "apply_assumption"},
"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.apply?":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.apply":
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"Lean.Parser.Tactic.anyGoals":
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"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.allGoals":
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"address": "/Tactic-Proofs/The-Tactic-Language/"}],
"Lean.Parser.Tactic.acRfl":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.acNf0":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Parser.Tactic.Conv.convTactic":
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"data": {"userName": "conv'"},
"address": "/Tactic-Proofs/Targeted-Rewriting-with--conv/"},
{"id": "conv___-next",
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"Lean.Parser.Tactic.Conv.conv":
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"Lean.Elab.Tactic.Ext.tacticExt1___":
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"address": "/Tactic-Proofs/Tactic-Reference/"}],
"Lean.Elab.Tactic.Ext.ext":
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"Verso.Genre.Manual.doc.syntaxKind":
{"title": "Verso.Genre.Manual.doc.syntaxKind",
"description": "Verso.Genre.Manual.doc.syntaxKind",
"contents":
{"«term∃_,_»":
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"data": null,
"address": "/Basic-Propositions/Quantifiers/"},
{"id": "_FLQQ_term_________FLQQ_-next",
"data": null,
"address": "/Basic-Propositions/Quantifiers/"}],
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"address": "/Build-Tools-and-Distribution/Lake/"}],
"--repo":
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"--rehash":
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"--platform":
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"--packages":
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"--old":
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"test":
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"script list":
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[{"id": "lean",
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"init":
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"exec":
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"exe":
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"env":
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"clean":
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"Manual.examples":
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"Zero-Overhead Subtypes": [],
"Widths and Newlines": [],
"Vector length index as a parameter": [],
"Variations in Instance Definitions": [],
"Variable not in scope": [],
"Using Any Local Variable": [],
"Using simp?": [],
"Using in for Local Scopes": [],
"Using Ord Instances for LT and LE Instances": [],
"Using NoNatZeroDivisors": [],
"Using #print axioms with #guard_msgs": [],
"Updating arrays": [],
"Unresolvable name inferred by dotted-identifier notation": [],
"Unnecessary fallback clause": [],
"Universes, constructors, and parameters": [],
"Universe-polymorphic identity function": [],
"Universe-polymorphic map": [],
"Universe polymorphism and definitional equality": [],
"Universe monomorphism in auto-bound implicit parameters": [],
"Universe mismatch": [],
"Universe levels of function types": [],
"Universe level expressions": [],
"Universe Level Metavariables": [],
"Unit in the ABI": [],
"Uninferred binder type due to resulting type annotation": [],
"Unexpansion and Arguments": [],
"Typo in resulting type": [],
"Types are irrelevant": [],
"Type Refinement": [],
"Type Metavariables": [],
"Turning Off Field Notation": [],
"Truth-Functional Implication": [],
"True is a subsingleton": [],
"Translating Nested Inductive Types": [],
"Tracing Instance Search": [],
"Too Many Instances": [],
"Throwing and Catching Errors": [],
"The recursor for List": [],
"The recursor for Bool": [],
"The Macro Attribute": [],
"The Integers as a Quotient Type": [],
"The Identity Monad Transformer ": [],
"The universe command when autoImplicit is false": [],
"The @[grind ←=] Attribute": [],
"The @[grind ext] Attribute": [],
"Testing Return Values": [],
"Termination failure": [],
"Termination Proof Tactics During Inference": [],
"Termination Proof Obligations": [],
"Terminating Loops in the Option Monad": [],
"Term vs Command Quotation Syntax": [],
"Target and Facet Specifications": [],
"Taking Advantage of Run-Time Representations": [],
"Tail Recursion with Branching": [],
"Tactic hygiene: inaccessible assumptions": [],
"Tactic hygiene: accessible assumptions": [],
"Synthesizing a Type Class Instance": [],
"Syntax as Strings": [],
"Suffixed Splices": [],
"Subtype Coercions": [],
"Subgoal Sequencing": [],
"Sub-Array Syntax": [],
"Structure type inheritance with overlapping fields": [],
"Structural Recursion vs Subtraction": [],
"Structural Recursion and Definitional Equality": [],
"Structural Recursion Under Pairs": [],
"Stateful Proofs": [],
"Stateful Predicates": [],
"State Types": [],
"Standard Library Types": [],
"Spuriously mutual types": [],
"Splitting Predicates": [],
"Splitting Pattern Matching": [],
"Splitting Conditional Expressions": [],
"Specifying Decreasing Parameters": [],
"Sources as Tables": [],
"Sort Coercions as Ordinary Coercions": [],
"Sort Coercions": [],
"Slicing Lists": [],
"Simultaneous Matching vs Matching Pairs for Structural Recursion": [],
"Signed Integers": [],
"Semi-Output Parameters with Pre-Existing Values": [],
"Selecting Patterns": [],
"Section Variables": [],
"Scoped Macros": [],
"Scope Information": [],
"Schematic Variables": [],
"Schematic Postconditions": [],
"Running a Program on a File": [],
"Running a Program": [],
"Ring Characteristics": [],
"Right pipeline notation": [],
"Rewriting Under Binders with conv": [],
"Rewriting Propositions": [],
"Requiring Reservoir Packages from a Git tag": [],
"Requiring Packages from a Git tag": [],
"Requiring Packages from Reservoir": [],
"Requiring Packages from Paths": [],
"Requiring Packages from Git": [],
"Representing Syntax as Constructors": [],
"Relaxed vs Non-Relaxed Automatic Implicit Parameters": [],
"Refined Parameters": [],
"Reference Counts in IR": [],
"Reference Cells as Locks": [],
"Reducing Functions": [],
"Reducibility and Tactics": [],
"Reducibility and Instance Synthesis": [],
"Reducibility and Generalized Field Notation": [],
"Redirecting Standard I/O to Strings": [],
"Recursor with parameters and indices": [],
"Recursive tactic macro": [],
"Recursive Equational Lemmas": [],
"Recursion vs Recursors": [],
"Reasoning via Lists": [],
"Reasoning by cases with match": [],
"Reasoning by cases with if": [],
"Reading from Standard Input": [],
"Rational Solutions": [],
"Ranges of Natural Numbers": [],
"Querying the Environment": [],
"Quasiquotation": [],
"Proving Existential Statements": [],
"Provided Operator Names": [],
"Protected constant name without namespace prefix": [],
"Propagating Equalities": [],
"Proofs in Public": [],
"Proofs by Induction": [],
"Proofs are irrelevant": [],
"Proofs are Private": [],
"Proofs About Quotients": [],
"Private and Public Definitions": [],
"Private Methods": [],
"Private Field Values": [],
"Printing Equations": [],
"Printing Axioms of Simple Definitions": [],
"Printing Axioms": [],
"Printing": [],
"Pretty-Printed Syntax": [],
"Preprocessing for a custom data type": [],
"Predicativity of Type": [],
"Precedence for Prefix and Infix Operators": [],
"Postfix Ascription vs show": [],
"Popcount": [],
"Pipeline Fields": [],
"Patterns from Conclusion and Hypotheses": [],
"Patterns for Constructors": [],
"Patterns and default values": [],
"Pattern-Matching Functions": [],
"Pattern treated as variable, not constructor": [],
"Pattern Matching and Lists": [],
"Pattern Equality Proofs": [],
"Partial Functions in Proofs": [],
"Partial Correctness Theorem": [],
"Parsing Matched Parentheses and Brackets": [],
"Parsing Comma-Separated Repetitions": [],
"Parameters before and after the colon": [],
"Parameter Scope": [],
"Panic": [],
"Overly-universe-polymorphic Bool": [],
"Overflowing Fixed-Width Literals": [],
"Output Parameters with Pre-Existing Values": [],
"Output Parameters and Stuck Search": [],
"Ordinary vs Strict Implicit Parameters": [],
"Or is not a subsingleton": [],
"Options as Safe Nullability": [],
"Options as Nullability": [],
"Optional Splices": [],
"Opening Scoped Declarations": [],
"Opening Nested Namespaces": [],
"One vs. Two Sets of Macro Rules": [],
"One File, Multiple Handles": [],
"Omitting parameter in argument to higher-order predicate": [],
"Observing Uniqueness": [],
"Numeric Literals for Bitvectors": [],
"Numeric Literals for Fin": [],
"Nonlinear Terms": [],
"Noncomputable instance in namespace": [],
"Noncomputable dependency can be made computable": [],
"Non-strictly-positive inductive types": [],
"Non-default constructor name": [],
"Non-Dependent Functions Don't Bind Variables": [],
"Non-Comma Separators": [],
"No structure subtyping": [],
"No cumulativity": [],
"No Kernel Reasoning About Floating-Point Numbers": [],
"No Instances of Non-Classes": [],
"No Backtracking of Lexicographic Order": [],
"No Automatic Implicit Parameters": [],
"Newline Characters in Strings": [],
"Nesting Namespaces and Sections": [],
"Nested positions": [],
"Nested Patterns and Sub-Terms": [],
"Nested Inductive Types with Std.HashMap": [],
"Nested Inductive Types and Quotients": [],
"Nested Inductive Types": [],
"Nested Dependent Pair Types": [],
"Nested do and Branches": [],
"Necessarily noncomputable function not appropriately marked": [],
"Navigation and Rewriting with conv": [],
"Natural Numbers as Linear Integer Arithmetic": [],
"Named goals": [],
"Named Section": [],
"Named Arguments": [],
"Mutually Defined Inductive Types": [],
"Mutual strict positivity": [],
"Mutual recursion without parameter decrease": [],
"Mutual inductive type constructors may not mention each other": [],
"Mutual Structural Recursion Over Non-Mutual Types": [],
"Mutual Structural Recursion Over Mutual Types": [],
"Mutual Block Scope and Automatic Implicit Parameters": [],
"Mutual Block Scope": [],
"Multiple Parameters, Same Type": [],
"Multi-Patterns": [],
"More Commutative Ring Proofs": [],
"Monadic functions": [],
"Monadic Collection Operations": [],
"Monad Lifts in Function Signatures": [],
"Modifying Values in Maps": [],
"Modifiers on structure constructor": [],
"Mixing Collections": [],
"Missing resulting type after constructor parameter": [],
"Missing namespace": [],
"Missing import": [],
"Missing WPMonad Instance": [],
"Minimal TOML Package Configuration": [],
"Minimal Library Target": [],
"Minimal Executable Target": [],
"Metavariables in Proofs": [],
"Meta Definitions": [],
"Memory Re-Use in IR": [],
"Maximal Parentheses": [],
"Matching, With and Without Generalization": [],
"Matching with an Explicit Motive": [],
"Matching on Complex Expressions Can Prevent Elaboration": [],
"Match Patterns Follow Reduction": [],
"Manually Quotiented Integers": [],
"Macro Error Reporting": [],
"Loops are Tail Recursive Functions": [],
"Longer Prefixes of Current Namespace Take Precedence": [],
"Longer Identifier Prefixes Take Precedence": [],
"Long List Literals": [],
"Location specifiers for simp": [],
"Local Names Take Precedence": [],
"Local Instances Have Priority": [],
"Local Instances": [],
"Local Effects with the Identity Monad": [],
"Lists with Lengths as Applicative Functors": [],
"Linear Integer Arithmetic with Case Analysis": [],
"Linear Division and Modulo": [],
"Limiting ring Steps": [],
"Lifting Transformed Monads": [],
"Lifting IO Monads": [],
"Library TOML Package Configuration": [],
"Left-to-Right Generation": [],
"Left pipeline notation": [],
"Leading . and Unfolding Definitions": [],
"Leading .": [],
"Lazy Lists": [],
"Iterators and Effects": [],
"Iterator State Types": [],
"Iteration over Array Indices with for": [],
"Iteration Over Multiple Collections": [],
"Iterating Over Triples": [],
"Iterating Over Nat": [],
"Iterated Automatic Implicit Parameters": [],
"Invalid Nested Inductive Types": [],
"Integer Negation and Addition": [],
"Insufficient type information": [],
"Instances for nested types": [],
"Instances are not recursive": [],
"Instances are Not Unique": [],
"Instance Name Generation": [],
"Inspecting Patterns": [],
"Injectivity Patterns": [],
"Inheriting Documentation": [],
"Inhabited Sum Types": [],
"Infix Syntax": [],
"Infix Functor and Applicative Operators": [],
"Inferred Termination Annotations": [],
"Inferred Characteristic": [],
"Inference too powerful": [],
"Ineligible decreasing parameters": [],
"Inductive Types with Constructors": [],
"Increasing E-matching Limits": [],
"Incorrect ordering of pattern matches": [],
"Inconsistent Indices": [],
"Inconsistencies From Axioms": [],
"Included and Omitted Section Variables": [],
"Inaccessible Patterns": [],
"Inaccessible Assumption Names": [],
"Impredicativity": [],
"Impossible Functions": [],
"Impossible Code Paths": [],
"Importing Private Information": [],
"Implicit argument cannot be inferred": [],
"Implicit and Explicit Parameters to Instances": [],
"Implicit Parameters from Types": [],
"Implementing Ranges": [],
"Implementing Coercions": [],
"Ignored Terms in Notation Expansion": [],
"Idiom Brackets": [],
"Identifying Missing Facts": [],
"Hierarchical Goal Names": [],
"Hiding Proof Terms": [],
"Heterogeneous Equality": [],
"Grouping and Filling": [],
"Goals Decided by linarith": [],
"Generalized Field Notation": [],
"From Characters to Strings": [],
"Forward Pattern Generation": [],
"Floating-Point Literals": [],
"Floating-Point Equality Is Not a Congruence": [],
"Floating-Point Equality Is Not Reflexive": [],
"Fixed-Width Literals": [],
"Fixed-Width Bitvector Literals": [],
"Finitely Many Skips": [],
"Finite-Field Reasoning": [],
"Finite and Infinite Ranges": [],
"Filling Holes with Unification": [],
"Fields and grind": [],
"Field Indices and Structure Inheritance": [],
"False is a subsingleton": [],
"Failing Backward Pattern Generation": [],
"Extracting the witness from an existential proof": [],
"Extensional Equality of Subtypes": [],
"Extending trivial": [],
"Exposed and Unexposed Definitions": [],
"Exporting Private Definitions": [],
"Exported Names": [],
"Explicitly-Created Metavariables": [],
"Explicit Root Namespace": [],
"Expansion Backtracking": [],
"Expanding Quasiquotation": [],
"Excluded Middle and Decidable": [],
"Exceptions and Lifting": [],
"Examples of Linear Integer Arithmetic": [],
"Even and odd": [],
"Even Numbers": [],
"Equational Lemmas": [],
"Eq is a subsingleton": [],
"Enriched Proof Obligation Contexts": [],
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"E-matching and Pattern Matching": [],
"E-matching Patterns": [],
"Duplicated Terms in Notation Expansion": [],
"Duplicate Evaluation in Coercions": [],
"Dotted identifier where type universe expected": [],
"Division by Iterated Subtraction: Termination Proof": [],
"Division by Iterated Subtraction": [],
"Division by 0": [],
"Displaying Output": [],
"Displaying Differences": [],
"Discriminant Refinement": [],
"Disambiguation via Typing": [],
"Differing parameter types": [],
"Differing numbers of parameters": [],
"Deriving Multiple Classes": [],
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"Dependent projection types": [],
"Dependent Pairs with Data": [],
"Dependent Pairs as Sums": [],
"Dependent Function Types": [],
"Dependent Coercion to Functions": [],
"Dependent Coercion": [],
"Definitional Equality of Unit-Like Types": [],
"Definitional Equality of Subtypes": [],
"Definitional Equality of Implicit and Explicit Function Types": [],
"Definitional Equality of Dependent and Non-Dependent Functions": [],
"Definitional Equality of Unit": [],
"Definition type uninferrable due to unknown parameter type": [],
"Definition by Partial Fixpoint": [],
"Defining an intermediate data value within a proof": [],
"Defining Dependent Coercions": [],
"Defining Coercions: Decimal Numbers": [],
"Defining Coercions": [],
"Default values": [],
"Default Size Instance": [],
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"Dead Code and Subsingleton Elimination": [],
"Data races with get and set": [],
"Custom Unit Type": [],
"Custom Slices": [],
"Custom Operators in Lean's Output": [],
"Custom Natural Number Literals": [],
"Current Namespace Contents Take Precedence Over Opened Namespaces": [],
"Current Namespace": [],
"Cross-Phase Code Re-Use": [],
"Creating and Tracking Proofs That Trust the Compiler": [],
"Course-of-Values Tables": [],
"Converting Iterators to Lists": [],
"Converting Infinite Iterators to Lists": [],
"Controlling Evaluation with Coercions": [],
"Controlling Coercion Insertion": [],
"Consuming Iterators in Loops": [],
"Consuming Iterators Directly": [],
"Constructing Lists": [],
"Congruence Closure": [],
"Configured Library Target": [],
"Configured Executable Target": [],
"Computing types": [],
"Commutative Rings": [],
"Commutative Ring Goals Decided by linarith": [],
"Commutative (Semi)ring Normalization": [],
"Coercions and Generalized Field Notation": [],
"Coercions and Option": [],
"Coercions and OfNat": [],
"Coercions": [],
"Coercing to Function Types": [],
"Coercing from Fin to Nat": [],
"Coercing Decorated Functions to Function Types": [],
"CoeOut vs Coe instances": [],
"Closing a Subprocess's Standard Input": [],
"Class vs Structure Constructors": [],
"Class Abbreviations": [],
"Choosing Eliminators": [],
"Checking for Type Errors": [],
"Checking Exit Codes": [],
"Checking Equality with Pointers": [],
"Checking Array Bounds": [],
"Characteristic Zero": [],
"Built-In Integers as Quotients": [],
"Bounded Bitvector Literals": [],
"Binder type requires new type variable": [],
"Bidirectional Equality Pattern Generation": [],
"Backward Pattern Generation": [],
"Axioms and Stuck Reduction": [],
"Axioms and Compilation": [],
"Avoiding data races with modify": [],
"Avoiding a Type": [],
"Avoiding Intermediate Structures": [],
"Automatic universe parameters and the universe command": [],
"Automatic Implicit Parameters and Universe Polymorphism": [],
"Automatic Implicit Parameters": [],
"Auto-implicit parameters in structure fields": [],
"Attempting to use index projection on existential proof": [],
"Attempting to name an example declaration": [],
"Attempting to define multiple structure fields on the same line": [],
"Attempting to define multiple opaque constants at once": [],
"Atomic Types and Repr": [],
"Asynchronous Subprocesses": [],
"Assumptions by Type, Outside Proofs": [],
"Assumptions by Type": [],
"Assumption Names and Conclusion": [],
"Assigned Operator Names": [],
"Ascribing Types to do Blocks": [],
"Ascribing Statements to Proofs": [],
"Array Literals": [],
"Array Indexing": [],
"Antiquotation Annotations": [],
"Anonymous constructors": [],
"Anonymous Goals": [],
"And is a subsingleton": [],
"An instance for a sum class": [],
"Ambiguous Operators and Priorities": [],
"Ambiguous Identifiers": [],
"Aligned Columns": [],
"Algebraic Reasoning": [],
"Accessible Assumption Names": [],
"A unit type": [],
"A type with parameter and index": [],
"A true proposition": [],
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"A constructorless type": [],
"A constructorless proposition": [],
"A Logging Monad": [],
"#print and Structure Types": [],
"#check and Underdetermined Types": []}},
"Manual.errorExplanation":
{"title": "Error Explanations",
"description": "Error Explanations",
"contents":
{"lean.unknownIdentifier":
[{"id": "lean___unknownIdentifier",
"data":
{"summary": "Failed to resolve identifier to variable or constant."},
"address": "/Error-Explanations/lean___unknownIdentifier/"}],
"lean.redundantMatchAlt":
[{"id": "lean___redundantMatchAlt",
"data": {"summary": "Match alternative will never be reached."},
"address": "/Error-Explanations/lean___redundantMatchAlt/"}],
"lean.propRecLargeElim":
[{"id": "lean___propRecLargeElim",
"data":
{"summary":
"Attempted to eliminate a proof into a higher type universe."},
"address": "/Error-Explanations/lean___propRecLargeElim/"}],
"lean.projNonPropFromProp":
[{"id": "lean___projNonPropFromProp",
"data": {"summary": "Tried to project data from a proof."},
"address": "/Error-Explanations/lean___projNonPropFromProp/"}],
"lean.invalidDottedIdent":
[{"id": "lean___invalidDottedIdent",
"data":
{"summary":
"Dotted identifier notation used with invalid or uninferrable expected type."},
"address": "/Error-Explanations/lean___invalidDottedIdent/"}],
"lean.inferDefTypeFailed":
[{"id": "lean___inferDefTypeFailed",
"data": {"summary": "The type of a definition could not be inferred."},
"address": "/Error-Explanations/lean___inferDefTypeFailed/"}],
"lean.inferBinderTypeFailed":
[{"id": "lean___inferBinderTypeFailed",
"data": {"summary": "The type of a binder could not be inferred."},
"address": "/Error-Explanations/lean___inferBinderTypeFailed/"}],
"lean.inductiveParamMissing":
[{"id": "lean___inductiveParamMissing",
"data":
{"summary":
"Parameter not present in an occurrence of an inductive type in one of its constructors."},
"address": "/Error-Explanations/lean___inductiveParamMissing/"}],
"lean.inductiveParamMismatch":
[{"id": "lean___inductiveParamMismatch",
"data":
{"summary":
"Invalid parameter in an occurrence of an inductive type in one of its constructors."},
"address": "/Error-Explanations/lean___inductiveParamMismatch/"}],
"lean.dependsOnNoncomputable":
[{"id": "lean___dependsOnNoncomputable",
"data":
{"summary":
"Declaration depends on noncomputable definitions but is not marked as noncomputable"},
"address": "/Error-Explanations/lean___dependsOnNoncomputable/"}],
"lean.ctorResultingTypeMismatch":
[{"id": "lean___ctorResultingTypeMismatch",
"data":
{"summary":
"Resulting type of constructor was not the inductive type being declared."},
"address": "/Error-Explanations/lean___ctorResultingTypeMismatch/"}]}},
"Manual.envVar":
{"title": "Environment Variables",
"description": "Environment Variables",
"contents":
{"LEAN_SYSROOT":
[{"id": "LEAN_SYSROOT",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LEAN_NUM_THREADS":
[{"id": "LEAN_NUM_THREADS",
"data": null,
"address": "/IO/Tasks-and-Threads/"}],
"LEAN_CC":
[{"id": "LEAN_CC",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LEAN_AR":
[{"id": "LEAN_AR",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LEAN":
[{"id": "LEAN",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_OVERRIDE_LEAN":
[{"id": "LAKE_OVERRIDE_LEAN",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_NO_CACHE":
[{"id": "LAKE_NO_CACHE",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_HOME":
[{"id": "LAKE_HOME",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_CACHE_REVISION_ENDPOINT":
[{"id": "LAKE_CACHE_REVISION_ENDPOINT",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_CACHE_KEY":
[{"id": "LAKE_CACHE_KEY",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_CACHE_ARTIFACT_ENDPOINT":
[{"id": "LAKE_CACHE_ARTIFACT_ENDPOINT",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE_ARTIFACT_CACHE":
[{"id": "LAKE_ARTIFACT_CACHE",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"LAKE":
[{"id": "LAKE",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"ELAN_HOME":
[{"id": "ELAN_HOME",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"},
{"id": "ELAN_HOME-next",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"ELAN":
[{"id": "ELAN",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"CC":
[{"id": "CC",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}],
"AR":
[{"id": "AR",
"data": null,
"address": "/Build-Tools-and-Distribution/Lake/"}]}},
"Manual.elanOpt":
{"title": "Manual.elanOpt",
"description": "Manual.elanOpt",
"contents":
{"-v":
[{"id": "elan-flag-v",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"-h":
[{"id": "elan-flag-h",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"-V":
[{"id": "elan-flag-V",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--version":
[{"id": "elan-flag--version",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--verbose":
[{"id": "elan-flag--verbose",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--path":
[{"id": "elan-option--path",
"data": "option",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--nonexistent":
[{"id": "elan-flag--nonexistent",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--json":
[{"id": "elan-flag--json",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--install":
[{"id": "elan-flag--install",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--help":
[{"id": "elan-flag--help",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"--delete":
[{"id": "elan-flag--delete",
"data": "flag",
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}]}},
"Manual.elanCommand":
{"title": "Elan commands",
"description": "Elan commands",
"contents":
{"which":
[{"id": "which",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"toolchain uninstall":
[{"id": "toolchain-uninstall",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"toolchain list":
[{"id": "toolchain-list",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"toolchain link":
[{"id": "toolchain-link",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"toolchain install":
[{"id": "toolchain-install",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"toolchain gc":
[{"id": "toolchain-gc",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"show":
[{"id": "show-next",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"self update":
[{"id": "self-update",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"self uninstall":
[{"id": "self-uninstall",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"run":
[{"id": "run",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"override unset":
[{"id": "override-unset",
"data": null,
"address":
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"override set":
[{"id": "override-set",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"override list":
[{"id": "override-list",
"data": null,
"address":
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"default":
[{"id": "default",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}],
"completions":
[{"id": "completions",
"data": null,
"address":
"/Build-Tools-and-Distribution/Managing-Toolchains-with-Elan/"}]}},
"Manual.Syntax.production":
{"title": "Manual.Syntax.production",
"description": "Manual.Syntax.production",
"contents":
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"address": "/Tactic-Proofs/Reading-Proof-States/"}],
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[{"id": "_FLQQ_prio_LPAR___RPAR__FLQQ_",
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{"kind": "«prio(_)»",
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"address": "/Type-Classes/Instance-Declarations/"}],
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[{"id": "_FLQQ_prec_LPAR___RPAR__FLQQ_",
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"address": "/Notations-and-Macros/Precedence/"}],
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[{"id": "termIfThenElse",
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{"kind": "termIfThenElse",
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"address": "/Terms/Conditionals/"}],
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{"string": "…", "kind": "meta"},
{"string": " ", "kind": "ws"},
{"string": ":=", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"},
{"string": " ", "kind": "ws"},
{"string": "then", "kind": "keyword"},
{"string": " ", "kind": "ws"},
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{"string": " ", "kind": "ws"},
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"address": "/Terms/Conditionals/"}],
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[{"id": "termDepIfThenElse",
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{"string": " ", "kind": "ws"},
{"string": ":", "kind": "keyword"},
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{"string": " ", "kind": "ws"},
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{"string": " ", "kind": "ws"},
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[{"id": "stx____",
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"stx!_ [{\"string\": \"!\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "stx____-next",
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"str.antiquot [{\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "str___antiquot",
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{"kind": "str.antiquot",
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"address": "/Notations-and-Macros/Notations/"}],
"prioMid [{\"string\": \"mid\", \"kind\": \"keyword\"}]":
[{"id": "prioMid",
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{"kind": "prioMid",
"forms": [{"string": "mid", "kind": "keyword"}],
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"address": "/Type-Classes/Instance-Declarations/"}],
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[{"id": "prioLow",
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{"kind": "prioLow",
"forms": [{"string": "low", "kind": "keyword"}],
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"address": "/Type-Classes/Instance-Declarations/"}],
"prioHigh [{\"string\": \"high\", \"kind\": \"keyword\"}]":
[{"id": "prioHigh",
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{"kind": "prioHigh",
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"address": "/Type-Classes/Instance-Declarations/"}],
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[{"id": "prioDefault",
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{"kind": "prioDefault",
"forms": [{"string": "default", "kind": "keyword"}],
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"address": "/Type-Classes/Instance-Declarations/"}],
"precMin [{\"string\": \"min\", \"kind\": \"keyword\"}]":
[{"id": "precMin",
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{"kind": "precMin",
"forms": [{"string": "min", "kind": "keyword"}],
"category": null},
"address": "/Notations-and-Macros/Precedence/"}],
"precMax [{\"string\": \"max\", \"kind\": \"keyword\"}]":
[{"id": "precMax",
"data":
{"kind": "precMax",
"forms": [{"string": "max", "kind": "keyword"}],
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"address": "/Notations-and-Macros/Precedence/"}],
"precLead [{\"string\": \"lead\", \"kind\": \"keyword\"}]":
[{"id": "precLead",
"data":
{"kind": "precLead",
"forms": [{"string": "lead", "kind": "keyword"}],
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"address": "/Notations-and-Macros/Precedence/"}],
"precArg [{\"string\": \"arg\", \"kind\": \"keyword\"}]":
[{"id": "precArg",
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{"kind": "precArg",
"forms": [{"string": "arg", "kind": "keyword"}],
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"address": "/Notations-and-Macros/Precedence/"}],
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[{"id": "num___antiquot-next",
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{"kind": "num.antiquot",
"forms": [{"string": "…", "kind": "meta"}],
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"address": "/Notations-and-Macros/Precedence/"},
{"id": "num___antiquot",
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{"kind": "num.antiquot",
"forms": [{"string": "…", "kind": "meta"}],
"category": null},
"address": "/Type-Classes/Instance-Declarations/"}],
"ident.antiquot [{\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "ident___antiquot",
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{"kind": "ident.antiquot",
"forms": [{"string": "…", "kind": "meta"}],
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"address": "/Terms/Functions/"}],
"coeSortNotation [{\"string\": \"↥\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "coeSortNotation",
"data":
{"kind": "coeSortNotation",
"forms":
[{"string": "↥", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/Coercions/Coercing-to-Sorts/"}],
"coeNotation [{\"string\": \"↑\", \"kind\": \"keyword\"}, {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "coeNotation",
"data":
{"kind": "coeNotation",
"forms":
[{"string": "↑", "kind": "keyword"}, {"string": "…", "kind": "meta"}],
"category": null},
"address": "/Coercions/Coercing-Between-Types/"}],
"coeFunNotation [{\"string\": \"⇑\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "coeFunNotation",
"data":
{"kind": "coeFunNotation",
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[{"string": "⇑", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/Coercions/Coercing-to-Function-Types/"}],
"boolIfThenElse [{\"string\": \"bif\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"then\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"else\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "boolIfThenElse",
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{"string": " ", "kind": "ws"},
{"string": "else", "kind": "keyword"},
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[{"id": "Std____FLQQ_term__LT_______________FLQQ_",
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"Std.«term_<...<_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \"<...<\", \"kind\": \"keyword\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std____FLQQ_term__LT___________LT___FLQQ_",
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"Std.«term_<...*» [{\"string\": \"…\", \"kind\": \"meta\"}, {\"string\": \"<...*\", \"kind\": \"keyword\"}]":
[{"id": "Std____FLQQ_term__LT______________FLQQ_",
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"Std.«term_..._» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \"...\", \"kind\": \"keyword\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
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"Std.«term_...=_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \"...=\", \"kind\": \"keyword\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std____FLQQ_term_______________FLQQ_",
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"Std.«term_...<_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \"...<\", \"kind\": \"keyword\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std____FLQQ_term___________LT___FLQQ_",
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"Std.«term_...*» [{\"string\": \"…\", \"kind\": \"meta\"}, {\"string\": \"...*\", \"kind\": \"keyword\"}]":
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"Std.«term*..._» [{\"string\": \"*...\", \"kind\": \"keyword\"}, {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std____FLQQ_term______________FLQQ_-next",
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"Std.«term*...=_» [{\"string\": \"*...=\", \"kind\": \"keyword\"}, {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std____FLQQ_term_________________FLQQ_",
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{"kind": "Std.«term*...=_»",
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[{"id": "Std____FLQQ_term________________FLQQ_",
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{"kind": "Std.«term*...*»",
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"address": "/Basic-Types/Ranges/"}],
"Std.Tactic.Do.mgoalStx [{\"string\": \"(\", \"kind\": \"meta\"},\n {\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \":\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \")*\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"⊢ₛ\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___Tactic___Do___mgoalStx-next",
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{"kind": "Std.Tactic.Do.mgoalStx",
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{"string": "…", "kind": "meta"},
{"string": " ", "kind": "ws"},
{"string": ":", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"},
{"string": ")*", "kind": "meta"},
{"string": " ", "kind": "ws"},
{"string": "⊢ₛ", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/The--mvcgen--tactic/Proof-Mode/"}],
"Std.HashMap.«term_~m_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"~m\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___HashMap____FLQQ_term____m__FLQQ_",
"data":
{"kind": "Std.HashMap.«term_~m_»",
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{"string": " ", "kind": "ws"},
{"string": "~m", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/Basic-Types/Maps-and-Sets/"},
{"id": "Std___HashMap____FLQQ_term____m__FLQQ_-next",
"data":
{"kind": "Std.HashMap.«term_~m_»",
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{"string": " ", "kind": "ws"},
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"category": null},
"address": "/Basic-Types/Maps-and-Sets/"}],
"Std.Do.«term⌜_⌝» [{\"string\": \"⌜\", \"kind\": \"keyword\"},\n {\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \"⌝\", \"kind\": \"keyword\"}]":
[{"id": "Std___Do____FLQQ_term________FLQQ_",
"data":
{"kind": "Std.Do.«term⌜_⌝»",
"forms":
[{"string": "⌜", "kind": "keyword"},
{"string": "…", "kind": "meta"},
{"string": "⌝", "kind": "keyword"}],
"category": null},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"Std.Do.«term⊢ₛ_» [{\"string\": \"⊢ₛ\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___Do____FLQQ_term_VDASH______FLQQ_",
"data":
{"kind": "Std.Do.«term⊢ₛ_»",
"forms":
[{"string": "⊢ₛ", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"Std.Do.«term_⊣⊢ₛ_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"⊣⊢ₛ\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___Do____FLQQ_term_____VDASH______FLQQ_",
"data":
{"kind": "Std.Do.«term_⊣⊢ₛ_»",
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{"string": " ", "kind": "ws"},
{"string": "⊣⊢ₛ", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"Std.Do.«term_⊢ₛ_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"⊢ₛ\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___Do____FLQQ_term__VDASH______FLQQ_",
"data":
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{"string": " ", "kind": "ws"},
{"string": "⊢ₛ", "kind": "keyword"},
{"string": " ", "kind": "ws"},
{"string": "…", "kind": "meta"}],
"category": null},
"address": "/The--mvcgen--tactic/Predicate-Transformers/"}],
"Std.Do.«term_⊢ₚ_» [{\"string\": \"…\", \"kind\": \"meta\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"⊢ₚ\", \"kind\": \"keyword\"},\n {\"string\": \" \", \"kind\": \"ws\"},\n {\"string\": \"…\", \"kind\": \"meta\"}]":
[{"id": "Std___Do____FLQQ_term__VDASH______FLQQ_-next",
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{"string": "⊢ₚ", "kind": "keyword"},
{"string": " ", "kind": "ws"},
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[{"id": "Lean___Parser___Command___notation",
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[{"id": "Lean___Parser___Command___mixfix-next-next",
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{"string": "…", "kind": "meta"},
{"string": " ", "kind": "ws"},
{"string": ":", "kind": "keyword"},
{"string": "]", "kind": "keyword"}],
"category": null},
"address": "/Basic-Types/Arrays/"}]}}}}
================================================
FILE: book/Main.lean
================================================
import VersoManual
import TPiL
open Verso.Genre Manual
open Verso Code External
open Verso.Output.Html in
def plausible := {{
}}
def config : Config where
emitTeX := false
emitHtmlSingle := false
emitHtmlMulti := true
htmlDepth := 1
extraFiles := [("static", "static")]
extraCss := Std.HashSet.ofList [
"/static/theme.css",
"/static/fonts/source-serif/source-serif-text.css",
"/static/fonts/source-code-pro/source-code-pro.css",
"/static/fonts/source-sans/source-sans-3.css",
"/static/fonts/noto-sans-mono/noto-sans-mono.css"
]
extraHead := #[plausible]
logo := some "/static/lean_logo.svg"
sourceLink := some "https://github.com/leanprover/theorem_proving_in_lean4"
issueLink := some "https://github.com/leanprover/theorem_proving_in_lean4/issues"
linkTargets := fun st => st.localTargets ++ st.remoteTargets
def main := manualMain (%doc TPiL) (config := config)
================================================
FILE: book/TPiL/AxiomsComputation.lean
================================================
import VersoManual
import TPiL.Examples
open Verso.Genre Manual
open TPiL
#doc (Manual) "Axioms and Computation" =>
%%%
tag := "axioms-and-computation"
%%%
We have seen that the version of the Calculus of Constructions that
has been implemented in Lean includes dependent function types,
inductive types, and a hierarchy of universes that starts with an
{tech}[impredicative], {tech (key := "proof irrelevance")}[proof-irrelevant] {lean}`Prop` at the bottom. In this
chapter, we consider ways of extending the CIC with additional axioms
and rules. Extending a foundational system in such a way is often
convenient; it can make it possible to prove more theorems, as well as
make it easier to prove theorems that could have been proved
otherwise. But there can be negative consequences of adding additional
axioms, consequences which may go beyond concerns about their
correctness. In particular, the use of axioms bears on the
computational content of definitions and theorems, in ways we will
explore here.
Lean is designed to support both computational and classical
reasoning. Users that are so inclined can stick to a “computationally
pure” fragment, which guarantees that closed expressions in the system
evaluate to canonical normal forms. In particular, any closed
computationally pure expression of type {lean}`Nat`, for example, will
reduce to a numeral.
Lean's standard library defines an additional axiom, propositional
extensionality, and a quotient construction which in turn implies the
principle of function extensionality. These extensions are used, for
example, to develop theories of sets and finite sets. We will see
below that using these theorems can block evaluation in Lean's kernel,
so that closed terms of type {lean}`Nat` no longer evaluate to numerals. But
Lean erases types and propositional information when compiling
definitions to executable code, and since
these axioms only add new propositions, they are compatible with that
computational interpretation. Even computationally inclined users may
wish to use the classical law of the excluded middle to reason about
computation. This also blocks evaluation in the kernel, but it is
compatible with compiled code.
The standard library also defines a choice principle that is entirely
antithetical to a computational interpretation, since it magically
produces “data” from a proposition asserting its existence. Its use is
essential to some classical constructions, and users can import it
when needed. But expressions that use this construction to produce
data do not have computational content, and in Lean we are required to
mark such definitions as {kw}`noncomputable` to flag that fact.
Using a clever trick (known as Diaconescu's theorem), one can use
propositional extensionality, function extensionality, and choice to
derive the law of the excluded middle. As noted above, however, use of
the law of the excluded middle is still compatible with
compilation, as are other classical principles, as
long as they are not used to manufacture data.
To summarize, then, on top of the underlying framework of universes,
dependent function types, and inductive types, the standard library
adds three additional components:
- the axiom of propositional extensionality
- a quotient construction, which implies function extensionality
- a choice principle, which produces data from an existential proposition.
The first two of these block normalization within Lean, but are
compatible with code generation, whereas the third is not amenable
to computational interpretation. We will spell out the details more
precisely below.
# Historical and Philosophical Context
%%%
tag := "historical-and-philosophical-context"
%%%
:::setup
```
variable (x : α) (y : β)
```
For most of its history, mathematics was essentially computational:
geometry dealt with constructions of geometric objects, algebra was
concerned with algorithmic solutions to systems of equations, and
analysis provided means to compute the future behavior of systems
evolving over time. From the proof of a theorem to the effect that
“for every {lean}`x`, there is a {lean}`y` such that ...”, it was generally
straightforward to extract an algorithm to compute such a {lean}`y` given
{lean}`x`.
:::
In the nineteenth century, however, increases in the complexity of
mathematical arguments pushed mathematicians to develop new styles of
reasoning that suppress algorithmic information and invoke
descriptions of mathematical objects that abstract away the details of
how those objects are represented. The goal was to obtain a powerful
“conceptual” understanding without getting bogged down in
computational details, but this had the effect of admitting
mathematical theorems that are simply _false_ on a direct
computational reading.
There is still fairly uniform agreement today that computation is
important to mathematics. But there are different views as to how best
to address computational concerns. From a _constructive_ point of
view, it is a mistake to separate mathematics from its computational
roots; every meaningful mathematical theorem should have a direct
computational interpretation. From a _classical_ point of view, it is
more fruitful to maintain a separation of concerns: we can use one
language and body of methods to write computer programs, while
maintaining the freedom to use nonconstructive theories and methods
to reason about them. Lean is designed to support both of these
approaches. Core parts of the library are developed constructively,
but the system also provides support for carrying out classical
mathematical reasoning.
:::setup
```
open Nat
notation "… " e "…" => e
```
Computationally, the purest part of dependent type theory avoids the
use of {lean}`Prop` entirely. Inductive types and dependent function types
can be viewed as data types, and terms of these types can be
“evaluated” by applying reduction rules until no more rules can be
applied. In principle, any closed term (that is, term with no free
variables) of type {lean}`Nat` should evaluate to a numeral, {lean}`succ (… (succ zero)…)`.
:::
:::setup
```
variable (p : Prop) (s t : α) (prf : p)
notation x " = " y " : " α => @Eq α x y
```
Introducing a proof-irrelevant {lean}`Prop` and marking theorems
irreducible represents a first step towards separation of
concerns. The intention is that elements of a type {lean}`p : Prop` should
play no role in computation, and so the particular construction of a
term {lean}`prf : p` is “irrelevant” in that sense. One can still define
computational objects that incorporate elements of type {lean}`Prop`; the
point is that these elements can help us reason about the effects of
the computation, but can be ignored when we extract “code” from the
term. Elements of type {lean}`Prop` are not entirely innocuous,
however. They include equations {lean}`s = t : α` for any type {lean}`α`, and
such equations can be used as casts, to type check terms. Below, we
will see examples of how such casts can block computation in the
system. However, computation is still possible under an evaluation
scheme that erases propositional content, ignores intermediate typing
constraints, and reduces terms until they reach a normal form. This is
precisely what Lean's virtual machine does.
Having adopted a proof-irrelevant {lean}`Prop`, one might consider it
legitimate to use, for example, the law of the excluded middle,
{lean}`p ∨ ¬p`, where {lean}`p` is any proposition. Of course, this, too, can block
computation according to the rules of CIC, but it does not prevent the generation
of executable code, as described above. It is only the choice
principles discussed in {ref "choice"}[the section on choice] that completely erase the
distinction between the proof-irrelevant and data-relevant parts of
the theory.
:::
# Propositional Extensionality
%%%
tag := "propositional-extensionality"
%%%
Propositional extensionality is the following axiom:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
------
axiom propext {a b : Prop} : (a ↔ b) → a = b
------
end Hidden
```
:::setup
```
variable (a : Prop)
```
It asserts that when two propositions imply one another, they are
actually equal. This is consistent with set-theoretic interpretations
in which any element {lean}`a : Prop` is either empty or the singleton set
$`\{\ast\}`, for some distinguished element $`\ast`. The axiom has the
effect that equivalent propositions can be substituted for one another
in any context:
:::
```lean
variable (a b c d e : Prop)
theorem thm₁ (h : a ↔ b) : (c ∧ a ∧ d → e) ↔ (c ∧ b ∧ d → e) :=
propext h ▸ Iff.refl _
theorem thm₂ (p : Prop → Prop) (h : a ↔ b) (h₁ : p a) : p b :=
propext h ▸ h₁
```
:::comment
```
```
:::
# Function Extensionality
%%%
tag := "function-extensionality"
%%%
:::leanFirst
Similar to propositional extensionality, function extensionality
asserts that any two functions of type {leanRef}`(x : α) → β x` that agree on
all their inputs are equal:
```signature
funext.{u, v}
{α : Sort u} {β : α → Sort v}
{f g : (x : α) → β x}
(h : ∀ (x : α), f x = g x) :
f = g
```
:::
From a classical, set-theoretic perspective, this is exactly what it
means for two functions to be equal. This is known as an “extensional”
view of functions. From a constructive perspective, however, it is
sometimes more natural to think of functions as algorithms, or
computer programs, that are presented in some explicit way. It is
certainly the case that two computer programs can compute the same
answer for every input despite the fact that they are syntactically
quite different. In much the same way, you might want to maintain a
view of functions that does not force you to identify two functions
that have the same input / output behavior. This is known as an
“intensional” view of functions.
In fact, function extensionality follows from the existence of
quotients, which we describe in the next section. In the Lean standard
library, therefore, {leanRef}`funext` is thus
[proved from the quotient construction](https://github.com/leanprover/lean4/blob/master/src/Init/Core.lean).
:::leanFirst
Suppose that for {leanRef}`α : Type u` we define the {leanRef}`Set `{leanRef (in := "(α : Type u)")}`α`{leanRef}` := α → Prop` to
denote the type of subsets of {leanRef (in := "(α : Type u)")}`α`, essentially identifying subsets
with predicates. By combining {leanRef}`funext` and {leanRef}`propext`, we obtain an
extensional theory of such sets:
```lean
def Set (α : Type u) := α → Prop
namespace Set
def mem (x : α) (a : Set α) := a x
infix:50 (priority := high) "∈" => mem
theorem setext {a b : Set α} (h : ∀ x, x ∈ a ↔ x ∈ b) : a = b :=
funext (fun x => propext (h x))
end Set
```
:::
We can then proceed to define the empty set and set intersection, for
example, and prove set identities:
```lean
def Set (α : Type u) := α → Prop
namespace Set
def mem (x : α) (a : Set α) := a x
infix:50 (priority := high) "∈" => mem
theorem setext {a b : Set α} (h : ∀ x, x ∈ a ↔ x ∈ b) : a = b :=
funext (fun x => propext (h x))
------
def empty : Set α := fun _ => False
notation (priority := high) "∅" => empty
def inter (a b : Set α) : Set α :=
fun x => x ∈ a ∧ x ∈ b
infix:70 " ∩ " => inter
theorem inter_self (a : Set α) : a ∩ a = a :=
setext fun x => Iff.intro
(fun ⟨h, _⟩ => h)
(fun h => ⟨h, h⟩)
theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ :=
setext fun _ => Iff.intro
(fun ⟨_, h⟩ => h)
(fun h => False.elim h)
theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ :=
setext fun _ => Iff.intro
(fun ⟨h, _⟩ => h)
(fun h => False.elim h)
theorem inter.comm (a b : Set α) : a ∩ b = b ∩ a :=
setext fun _ => Iff.intro
(fun ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩)
(fun ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩)
-----
end Set
```
The following is an example of how function extensionality blocks
computation inside the Lean kernel:
```lean
def f (x : Nat) := x
def g (x : Nat) := 0 + x
theorem f_eq_g : f = g :=
funext fun x => (Nat.zero_add x).symm
def val : Nat :=
Eq.recOn (motive := fun _ _ => Nat) f_eq_g 0
-- does not reduce to 0
#reduce val
-- evaluates to 0
#eval val
```
First, we show that the two functions {leanRef}`f` and {leanRef}`g` are equal using
function extensionality, and then we cast {leanRef}`0` of type {lean}`Nat` by
replacing {leanRef}`f` by {leanRef}`g` in the type. Of course, the cast is
vacuous, because {lean}`Nat` does not depend on {leanRef}`f`. But that is enough
to do the damage: under the computational rules of the system, we now
have a closed term of {lean}`Nat` that does not reduce to a numeral. In this
case, we may be tempted to reduce the expression to {lean}`0`. But in
nontrivial examples, eliminating cast changes the type of the term,
which might make an ambient expression type incorrect. The virtual
machine, however, has no trouble evaluating the expression to
{lean}`0`. Here is a similarly contrived example that shows how
{lean}`propext` can get in the way:
```lean
theorem tteq : (True ∧ True) = True :=
propext (Iff.intro (fun ⟨h, _⟩ => h) (fun h => ⟨h, h⟩))
def val : Nat :=
Eq.recOn (motive := fun _ _ => Nat) tteq 0
-- does not reduce to 0
#reduce val
-- evaluates to 0
#eval val
```
Current research programs, including work on _observational type
theory_ and _cubical type theory_, aim to extend type theory in ways
that permit reductions for casts involving function extensionality,
quotients, and more. But the solutions are not so clear-cut, and the
rules of Lean's underlying calculus do not sanction such reductions.
In a sense, however, a cast does not change the meaning of an
expression. Rather, it is a mechanism to reason about the expression's
type. Given an appropriate semantics, it then makes sense to reduce
terms in ways that preserve their meaning, ignoring the intermediate
bookkeeping needed to make the reductions type-correct. In that case,
adding new axioms in {lean}`Prop` does not matter; by {tech}[proof irrelevance],
an expression in {lean}`Prop` carries no information, and can be safely
ignored by the reduction procedures.
# Quotients
%%%
tag := "quotients"
%%%
:::setup
```
variable (α : Sort u) (r : α → α → Prop) (f : α → β) (x y : α) (f' : Quot r → β)
notation α " / " r:max => Quot (α := α) r
notation "⟦" x "⟧" => Quot.mk _ x
```
Let {lean}`α` be any type, and let {lean}`r` be an equivalence relation on
{lean}`α`. It is mathematically common to form the “quotient” {lean}`α / r`,
that is, the type of elements of {lean}`α` “modulo” {lean}`r`. Set
theoretically, one can view {lean}`α / r` as the set of equivalence
classes of {lean}`α` modulo {lean}`r`. If {lean}`f : α → β` is any function that
respects the equivalence relation in the sense that for every
{lean}`x y : α`, {lean}`r x y` implies {lean}`f x = f y`, then {lean}`f` “lifts” to a function
{lean}`f' : α / r → β` defined on each equivalence class {lean (type := "Quot r")}`⟦x⟧` by
{lean}`f' ⟦x⟧ = f x`. Lean's standard library extends the Calculus of
Constructions with additional constants that perform exactly these
constructions, and installs this last equation as a definitional
reduction rule.
In its most basic form, the quotient construction does not even
require {lean}`r` to be an equivalence relation. The following constants
are built into Lean:
:::
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
------
universe u v
axiom Quot : {α : Sort u} → (α → α → Prop) → Sort u
axiom Quot.mk : {α : Sort u} → (r : α → α → Prop) → α → Quot r
axiom Quot.ind :
∀ {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop},
(∀ a, β (Quot.mk r a)) → (q : Quot r) → β q
axiom Quot.lift :
{α : Sort u} → {r : α → α → Prop} → {β : Sort u} → (f : α → β)
→ (∀ a b, r a b → f a = f b) → Quot r → β
------
end Hidden
```
:::setup
```
variable (α : Type u) (r : α → α → Prop) (a : α) (f : α → β) (h : ∀ a b, r a b → f a = f b)
```
The first one forms a type {lean}`Quot r` given a type {lean}`α` by any binary
relation {lean}`r` on {lean}`α`. The second maps {lean}`α` to {lit}`Quot α`, so that
if {lean}`r : α → α → Prop` and {lit}`a : α`, then {lean}`Quot.mk r a` is an
element of {lean}`Quot r`. The third principle, {lean}`Quot.ind`, says that
every element of {lean}`Quot.mk r a` is of this form. As for
{lean}`Quot.lift`, given a function {lean}`f : α → β`, if {lean}`h` is a proof
that {lean}`f` respects the relation {lean}`r`, then {lean}`Quot.lift f h` is the
corresponding function on {lean}`Quot r`. The idea is that for each
element {lean}`a` in {lean}`α`, the function {lean}`Quot.lift f h` maps
{lean}`Quot.mk r a` (the {lean}`r`-class containing {lean}`a`) to {lean}`f a`, wherein {lean}`h`
shows that this function is well defined. In fact, the computation
principle is declared as a reduction rule, as the proof below makes
clear.
```lean
def mod7Rel (x y : Nat) : Prop :=
x % 7 = y % 7
-- the quotient type
#check (Quot mod7Rel : Type)
-- the class of numbers equivalent to 4
#check (Quot.mk mod7Rel 4 : Quot mod7Rel)
def f (x : Nat) : Bool :=
x % 7 = 0
theorem f_respects (a b : Nat) (h : mod7Rel a b) : f a = f b := by
simp [mod7Rel, f] at *
rw [h]
#check (Quot.lift f f_respects : Quot mod7Rel → Bool)
-- the computation principle
example (a : Nat) : Quot.lift f f_respects (Quot.mk mod7Rel a) = f a :=
rfl
```
The four constants, {lean}`Quot`, {lean}`Quot.mk`, {lean}`Quot.ind`, and
{lean}`Quot.lift` in and of themselves are not very strong. You can check
that the {lean}`Quot.ind` is satisfied if we take {lean}`Quot r` to be simply
{lean}`α`, and take {lean}`Quot.lift` to be the identity function (ignoring
{lean}`h`). For that reason, these four constants are not viewed as
additional axioms.
:::
:::comment
```
```
:::
They are, like inductively defined types and the associated
constructors and recursors, viewed as part of the logical framework.
What makes the {lean}`Quot` construction into a bona fide quotient is the
following additional axiom:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
universe u v
------
axiom Quot.sound :
∀ {α : Type u} {r : α → α → Prop} {a b : α},
r a b → Quot.mk r a = Quot.mk r b
```
This is the axiom that asserts that any two elements of {leanRef}`α` that are
related by {leanRef}`r` become identified in the quotient. If a theorem or
definition makes use of {leanRef}`Quot.sound`, it will show up in the
{kw}`#print axioms` command.
:::setup
```
variable (α : Type u) (r : α → α → Prop) (r' r'': α → α → Prop) (a b : α)
```
Of course, the quotient construction is most commonly used in
situations when {lean}`r` is an equivalence relation. Given {lean}`r` as
above, if we define {lean}`r'` according to the rule {lean}`r' a b` iff
{lean}`Quot.mk r a = Quot.mk r b`, then it's clear that {lean}`r'` is an
equivalence relation. Indeed, {lean}`r'` is the _kernel_ of the function
{lean}`fun a => Quot.mk r a`. The axiom {lean}`Quot.sound` says that {lean}`r a b`
implies {lean}`r' a b`. Using {lean}`Quot.lift` and {lean}`Quot.ind`, we can show
that {lean}`r'` is the smallest equivalence relation containing {lean}`r`, in
the sense that if {lean}`r''` is any equivalence relation containing
{lean}`r`, then {lean}`r' a b` implies {lean}`r'' a b`. In particular, if {lean}`r`
was an equivalence relation to start with, then for all {lean}`a` and
{lean}`b` we have {lean}`r a b` iff {lean}`r' a b`.
:::
To support this common use case, the standard library defines the
notion of a _setoid_, which is simply a type with an associated
equivalence relation:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
------
class Setoid (α : Sort u) where
r : α → α → Prop
iseqv : Equivalence r
instance {α : Sort u} [Setoid α] : HasEquiv α :=
⟨Setoid.r⟩
namespace Setoid
variable {α : Sort u} [Setoid α]
theorem refl (a : α) : a ≈ a :=
iseqv.refl a
theorem symm {a b : α} (hab : a ≈ b) : b ≈ a :=
iseqv.symm hab
theorem trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c :=
iseqv.trans hab hbc
end Setoid
------
end Hidden
```
Given a type {leanRef (in := "Setoid (α")}`α`, a relation {leanRef (in := "Equivalence r")}`r`
on {leanRef (in := "Setoid (α")}`α`, and a proof {leanRef}`iseqv`
that {leanRef (in := "Equivalence r")}`r` is an equivalence relation, we can define an
instance of the {leanRef (in := "class Setoid")}`Setoid` class.
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
------
def Quotient {α : Sort u} (s : Setoid α) :=
@Quot α Setoid.r
------
end Hidden
```
:::setup
```
variable (α : Type u) [Setoid α] (a b : α)
```
The constants {lean}`Quotient.mk`, {lean}`Quotient.ind`, {lean}`Quotient.lift`,
and {lean}`Quotient.sound` are nothing more than the specializations of
the corresponding elements of {lean}`Quot`. The fact that type class
inference can find the setoid associated to a type {lean}`α` brings a
number of benefits. First, we can use the notation {lean}`a ≈ b` (entered
with {kbd}`\approx`) for {lean}`Setoid.r a b`, where the instance of
{lean}`Setoid` is implicit in the notation {lean}`Setoid.r`. We can use the
generic theorems {lean}`Setoid.refl`, {lean}`Setoid.symm`, {lean}`Setoid.trans` to
reason about the relation. Specifically with quotients we can use the
theorem {lean}`Quotient.exact`:
```signature
Quotient.exact {α : Sort u} {s : Setoid α} {a b : α} :
Quotient.mk s a = Quotient.mk s b →
a ≈ b
```
Together with {lean}`Quotient.sound`, this implies that the elements of
the quotient correspond exactly to the equivalence classes of elements
in {lean}`α`.
:::
:::setup
```
variable (α : Type u) (β : Type v)
```
Recall that in the standard library, {lean}`α × β` represents the
Cartesian product of the types {lean}`α` and {lean}`β`. To illustrate the use
of quotients, let us define the type of _unordered_ pairs of elements
of a type {lean}`α` as a quotient of the type {lean}`α × α`. First, we define
the relevant equivalence relation:
:::
```lean
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
```
The next step is to prove that {leanRef}`eqv` is in fact an equivalence
relation, which is to say, it is reflexive, symmetric and
transitive. We can prove these three facts in a convenient and
readable way by using dependent pattern matching to perform
case-analysis and break the hypotheses into pieces that are then
reassembled to produce the conclusion.
```lean
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
------
private theorem eqv.refl (p : α × α) : p ~ p :=
Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
| (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
```
:::leanFirst
Now that we have proved that {leanRef}`eqv` is an equivalence relation, we
can construct a {leanRef}`Setoid (α × α)`, and use it to define the type
{leanRef}`UProd α` of unordered pairs.
```lean
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
| (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
------
instance uprodSetoid (α : Type u) : Setoid (α × α) where
r := eqv
iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
end UProd
```
:::
:::setup
```
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
| (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
r := eqv
iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
end UProd
variable (a₁ a₂ : α)
```
Notice that we locally define the notation {lean}`{a₁, a₂}` for unordered
pairs as {lean}`Quotient.mk' (a₁, a₂)`. This is useful for illustrative
purposes, but it is not a good idea in general, since the notation
will shadow other uses of curly brackets, such as for records and
sets.
We can easily prove that {lean}`{a₁, a₂} = {a₂, a₁}` using {lean}`Quot.sound`,
since we have {lean}`(a₁, a₂) ~ (a₂, a₁)`.
:::
```lean
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
| (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
r := eqv
iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
------
theorem mk_eq_mk (a₁ a₂ : α) : {a₁, a₂} = {a₂, a₁} :=
Quot.sound (Or.inr ⟨rfl, rfl⟩)
------
end UProd
```
:::leanFirst
To complete the example, given {leanRef}`a : α` and {leanRef}`u : UProd α`, we
define the proposition {leanRef (in := "mem (a : α) (u : UProd α)")}`a`{lit}` ∈ `{leanRef (in := "mem (a : α) (u : UProd α)")}`u` which should hold if {leanRef (in := "mem (a : α) (u : UProd α)")}`a` is one of
the elements of the unordered pair {leanRef (in := "mem (a : α) (u : UProd α)")}`u`. First, we define a similar
proposition {leanRef}`mem_fn`{leanRef (in := "mem (a : α) (u : UProd α)")}` a`{leanRef (in := "mem (a : α) (u : UProd α)")}` u` on (ordered) pairs; then we show that
{leanRef}`mem_fn` respects the equivalence relation {leanRef}`eqv` with the lemma
{leanRef}`mem_respects`. This is an idiom that is used extensively in the
Lean standard library.
```lean
set_option linter.unusedVariables false
private def eqv (p₁ p₂ : α × α) : Prop :=
(p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)
infix:50 " ~ " => eqv
private theorem eqv.refl (p : α × α) : p ~ p :=
Or.inl ⟨rfl, rfl⟩
private theorem eqv.symm : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁
| (a₁, a₂), (b₁, b₂), (Or.inl ⟨a₁b₁, a₂b₂⟩) =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (Or.inr ⟨a₁b₂, a₂b₁⟩) =>
Or.inr (by simp_all)
private theorem eqv.trans : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inl (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inl ⟨a₁b₁, a₂b₂⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inl ⟨b₁c₁, b₂c₂⟩ =>
Or.inr (by simp_all)
| (a₁, a₂), (b₁, b₂), (c₁, c₂), Or.inr ⟨a₁b₂, a₂b₁⟩, Or.inr ⟨b₁c₂, b₂c₁⟩ =>
Or.inl (by simp_all)
private theorem is_equivalence : Equivalence (@eqv α) :=
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
instance uprodSetoid (α : Type u) : Setoid (α × α) where
r := eqv
iseqv := is_equivalence
def UProd (α : Type u) : Type u :=
Quotient (uprodSetoid α)
namespace UProd
def mk {α : Type} (a₁ a₂ : α) : UProd α :=
Quotient.mk' (a₁, a₂)
notation "{ " a₁ ", " a₂ " }" => mk a₁ a₂
theorem mk_eq_mk (a₁ a₂ : α) : {a₁, a₂} = {a₂, a₁} :=
Quot.sound (Or.inr ⟨rfl, rfl⟩)
------
private def mem_fn (a : α) : α × α → Prop
| (a₁, a₂) => a = a₁ ∨ a = a₂
-- auxiliary lemma for proving mem_respects
private theorem mem_swap {a : α} :
∀ {p : α × α}, mem_fn a p = mem_fn a (⟨p.2, p.1⟩)
| (a₁, a₂) => by
apply propext
apply Iff.intro
. intro
| Or.inl h => exact Or.inr h
| Or.inr h => exact Or.inl h
. intro
| Or.inl h => exact Or.inr h
| Or.inr h => exact Or.inl h
private theorem mem_respects : {p₁ p₂ : α × α} → (a : α) → p₁ ~ p₂ → mem_fn a p₁ = mem_fn a p₂
| (a₁, a₂), (b₁, b₂), a, Or.inl ⟨a₁b₁, a₂b₂⟩ => by
simp_all
| (a₁, a₂), (b₁, b₂), a, Or.inr ⟨a₁b₂, a₂b₁⟩ => by
simp_all only
apply mem_swap
def mem (a : α) (u : UProd α) : Prop :=
Quot.liftOn u (fun p => mem_fn a p) (fun p₁ p₂ e => mem_respects a e)
infix:50 (priority := high) " ∈ " => mem
theorem mem_mk_left (a b : α) : a ∈ {a, b} :=
Or.inl rfl
theorem mem_mk_right (a b : α) : b ∈ {a, b} :=
Or.inr rfl
theorem mem_or_mem_of_mem_mk {a b c : α} : c ∈ {a, b} → c = a ∨ c = b :=
fun h => h
---------
end UProd
```
:::
For convenience, the standard library also defines {lean}`Quotient.lift₂`
for lifting binary functions, and {lit}`Quotient.ind₂` for induction on
two variables.
:::setup
```
variable (α : Sort u) (β : α → Sort v) (f₁ f₂ f : (x : α) → β x) (a : α)
def extfun (α : Sort u) (β : α → Sort v) := Quot (fun (f g : (x : α) → β x) => ∀ x, f x = g x)
def extfun_app {α β} : extfun α β → (x : α) → β x := fun f x =>
Quot.lift (· x) (by intros; simp [*]) f
```
We close this section with some hints as to why the quotient
construction implies function extensionality. It is not hard to show
that extensional equality on the {lean}`(x : α) → β x` is an equivalence
relation, and so we can consider the type {lean}`extfun α β` of functions
“up to equivalence.” Of course, application respects that equivalence
in the sense that if {lean}`f₁` is equivalent to {lean}`f₂`, then {lean}`f₁ a` is
equal to {lean}`f₂ a`. Thus application gives rise to a function
{lean}`extfun_app : extfun α β → (x : α) → β x`. But for every {lean}`f`,
{lean}`extfun_app (.mk _ f)` is definitionally equal to {lean}`fun x => f x`, which is
in turn definitionally equal to {lean}`f`. So, when {lean}`f₁` and {lean}`f₂` are
extensionally equal, we have the following chain of equalities:
```lean
variable {α : Sort u} {β : α → Sort v}
def extfun (α : Sort u) (β : α → Sort v) := Quot (fun (f g : (x : α) → β x) => ∀ x, f x = g x)
def extfun_app {α β} (f : extfun α β) (x : α) : β x :=
Quot.lift (· x) (by intros; simp [*]) f
----------
example (f₁ f₂ : (x : α) → β x) (h : ∀ x, f₁ x = f₂ x) :=
calc f₁
_ = extfun_app (.mk _ f₁) := rfl
_ = extfun_app (.mk _ f₂) := by rw [Quot.sound]; trivial
_ = f₂ := rfl
```
As a result, {leanRef}`f₁` is equal to {leanRef}`f₂`.
:::
# Choice
%%%
tag := "choice"
%%%
:::leanFirst
To state the final axiom defined in the standard library, we need the
{leanRef}`Nonempty` type, which is defined as follows:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
------
class inductive Nonempty (α : Sort u) : Prop where
| intro (val : α) : Nonempty α
------
end Hidden
```
:::
:::setup
```
variable {α : Sort u}
```
Because {lean}`Nonempty α` has type {lean}`Prop` and its constructor contains data, it can only eliminate to {lean}`Prop`.
In fact, {lean}`Nonempty α` is equivalent to {lean}`∃ x : α, True`:
:::
```lean
example (α : Type u) : Nonempty α ↔ ∃ x : α, True :=
Iff.intro (fun ⟨a⟩ => ⟨a, trivial⟩) (fun ⟨a, h⟩ => ⟨a⟩)
```
Our axiom of choice is now expressed simply as follows:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
universe u
------
axiom choice {α : Sort u} : Nonempty α → α
------
end Hidden
```
:::setup
```
variable {α : Sort u} {h : Nonempty α}
open Classical
```
Given only the assertion {lean}`h` that {lean}`α` is nonempty, {lean}`choice h`
magically produces an element of {lean}`α`. Of course, this blocks any
meaningful computation: by the interpretation of {lean}`Prop`, {lean}`h`
contains no information at all as to how to find such an element.
:::
This is found in the {lit}`Classical` namespace, so the full name of the
theorem is {lean}`Classical.choice`. The choice principle is equivalent to
the principle of *indefinite description*, which can be expressed with
subtypes as follows:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
namespace Hidden
universe u
axiom choice {α : Sort u} : Nonempty α → α
------
noncomputable def indefiniteDescription {α : Sort u}
(p : α → Prop) (h : ∃ x, p x) : {x // p x} :=
choice <| let ⟨x, px⟩ := h; ⟨⟨x, px⟩⟩
------
end Hidden
```
:::setup
```
variable {α : Sort u} {h : Nonempty α}
open Classical
```
Because it depends on {lean}`choice`, Lean cannot generate executable code for
{lean}`indefiniteDescription`, and so requires us to mark the definition
as {kw}`noncomputable`. Also in the {lit}`Classical` namespace, the
function {lean}`choose` and the property {lean}`choose_spec` decompose the two
parts of the output of {lean}`indefiniteDescription`:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
open Classical
namespace Hidden
------
variable {α : Sort u} {p : α → Prop}
noncomputable def choose (h : ∃ x, p x) : α :=
(indefiniteDescription p h).val
theorem choose_spec (h : ∃ x, p x) : p (choose h) :=
(indefiniteDescription p h).property
------
end Hidden
```
The {lean}`choice` principle also erases the distinction between the
property of being {lean}`Nonempty` and the more constructive property of
being {lean}`Inhabited`:
```lean
open Classical
------
noncomputable def inhabited_of_nonempty (h : Nonempty α) : Inhabited α :=
choice (let ⟨a⟩ := h; ⟨⟨a⟩⟩)
```
:::
In the next section, we will see that {lean}`propext`, {lean}`funext`, and
{leanRef}`choice`, taken together, imply the law of the excluded middle and
the decidability of all propositions. Using those, one can strengthen
the principle of indefinite description as follows:
::::setup
```
open Classical
```
```signature
strongIndefiniteDescription {α : Sort u} (p : α → Prop)
(h : Nonempty α) :
{x // (∃ (y : α), p y) → p x}
```
Assuming the ambient type {leanRef}`α` is nonempty,
{leanRef}`strongIndefiniteDescription`{lit}` `{leanRef}`p` produces an element of {leanRef}`α`
satisfying {leanRef}`p` if there is one. The data component of this
definition is conventionally known as *Hilbert's epsilon function*:
```signature
epsilon {α : Sort u} [h : Nonempty α] (p : α → Prop) : α
```
```signature
epsilon_spec {α : Sort u} {p : α → Prop}
(hex : ∃ (y : α), p y) :
p (@epsilon _ (nonempty_of_exists hex) p)
```
::::
# The Law of the Excluded Middle
%%%
tag := "the-law-of-the-excluded-middle"
%%%
The law of the excluded middle is the following:
```signature
Classical.em : ∀ (p : Prop), p ∨ ¬p
```
[Diaconescu's theorem](https://en.wikipedia.org/wiki/Diaconescu%27s_theorem) states
that the axiom of choice is sufficient to derive the law of excluded
middle. More precisely, it shows that the law of the excluded middle
follows from {lean}`Classical.choice`, {lean}`propext`, and {lean}`funext`. We
sketch the proof that is found in the standard library.
```save emProof
-- ANCHOR: emSetup
open Classical
theorem em (p : Prop) : p ∨ ¬p := by
let U (x : Prop) : Prop := x = True ∨ p
let V (x : Prop) : Prop := x = False ∨ p
have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
-- ^ PROOF_STATE: em1
-- ANCHOR_END: emSetup
-- ANCHOR: emChoose
let u : Prop := choose exU
let v : Prop := choose exV
have u_def : U u := choose_spec exU
have v_def : V v := choose_spec exV
-- ^ PROOF_STATE: em2
-- ANCHOR_END: emChoose
-- ANCHOR: emCases
have not_uv_or_p : u ≠ v ∨ p := by
match u_def, v_def with
| Or.inr h, _ => exact Or.inr h
| _, Or.inr h => exact Or.inr h
| Or.inl hut, Or.inl hvf =>
apply Or.inl
simp [hvf, hut, true_ne_false]
-- ANCHOR_END: emCases
-- ANCHOR: emNext
have p_implies_uv : p → u = v :=
fun hp =>
have hpred : U = V :=
funext fun x =>
have hl : (x = True ∨ p) → (x = False ∨ p) :=
fun _ => Or.inr hp
have hr : (x = False ∨ p) → (x = True ∨ p) :=
fun _ => Or.inr hp
show (x = True ∨ p) = (x = False ∨ p) from
propext (Iff.intro hl hr)
have h₀ : ∀ exU exV, @choose _ U exU = @choose _ V exV := by
rw [hpred]; intros; rfl
show u = v from h₀ _ _
-- ANCHOR_END: emNext
-- ANCHOR: emDone
match not_uv_or_p with
| Or.inl hne =>
exact Or.inr (mt p_implies_uv hne)
| Or.inr h =>
exact Or.inl h
-- ANCHOR_END: emDone
```
:::leanFirst
First, we import the necessary axioms, and define two predicates {leanRef}`U` and {leanRef}`V`:
```savedAnchor emSetup
open Classical
theorem em (p : Prop) : p ∨ ¬p := by
let U (x : Prop) : Prop := x = True ∨ p
let V (x : Prop) : Prop := x = False ∨ p
have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
```
:::
If {leanRef}`p` is true, then every element of {lean}`Prop` is in both {leanRef}`U` and {leanRef}`V`.
If {leanRef}`p` is false, then {leanRef}`U` is the singleton {leanRef}`True`, and {leanRef}`V` is the singleton {leanRef}`False`.
:::leanFirst
Next, we use {leanRef}`choose` to choose an element from each of {leanRef}`U` and {leanRef}`V`:
```savedAnchor emChoose
let u : Prop := choose exU
let v : Prop := choose exV
have u_def : U u := choose_spec exU
have v_def : V v := choose_spec exV
```
:::
:::leanFirst
Each of {leanRef}`U` and {leanRef}`V` is a disjunction, so {leanRef}`u_def` and {leanRef}`v_def`
represent four cases. In one of these cases, {leanRef}`u = True` and
{leanRef}`v = False`, and in all the other cases, {leanRef}`p` is true. Thus we have:
```savedAnchor emCases
have not_uv_or_p : u ≠ v ∨ p := by
match u_def, v_def with
| Or.inr h, _ => exact Or.inr h
| _, Or.inr h => exact Or.inr h
| Or.inl hut, Or.inl hvf =>
apply Or.inl
simp [hvf, hut, true_ne_false]
```
:::
On the other hand, if {leanRef}`p` is true, then, by function extensionality
and propositional extensionality, {leanRef}`U` and {leanRef}`V` are equal. By the
definition of {leanRef}`u` and {leanRef}`v`, this implies that they are equal as well.
```savedAnchor emNext
have p_implies_uv : p → u = v :=
fun hp =>
have hpred : U = V :=
funext fun x =>
have hl : (x = True ∨ p) → (x = False ∨ p) :=
fun _ => Or.inr hp
have hr : (x = False ∨ p) → (x = True ∨ p) :=
fun _ => Or.inr hp
show (x = True ∨ p) = (x = False ∨ p) from
propext (Iff.intro hl hr)
have h₀ : ∀ exU exV, @choose _ U exU = @choose _ V exV := by
rw [hpred]; intros; rfl
show u = v from h₀ _ _
```
Putting these last two facts together yields the desired conclusion:
```savedAnchor emDone
match not_uv_or_p with
| Or.inl hne =>
exact Or.inr (mt p_implies_uv hne)
| Or.inr h =>
exact Or.inl h
```
Consequences of excluded middle include double-negation elimination,
proof by cases, and proof by contradiction, all of which are described
in the section on {ref "classical-logic"}[classical logic].
The law of the excluded middle and propositional extensionality imply propositional completeness:
```lean (suppressNamespaces := "Hidden") (allowVisible := false)
open Classical
theorem propComplete (a : Prop) : a = True ∨ a = False :=
match em a with
| Or.inl ha =>
Or.inl (propext (Iff.intro (fun _ => True.intro) (fun _ => ha)))
| Or.inr hn =>
Or.inr (propext (Iff.intro (fun h => hn h) (fun h => False.elim h)))
```
Together with choice, we also get the stronger principle that every
proposition is decidable. Recall that the class of {lean}`Decidable`
propositions is defined as follows:
```lean
namespace Hidden
------
class inductive Decidable (p : Prop) where
| isFalse (h : ¬p) : Decidable p
| isTrue (h : p) : Decidable p
------
end Hidden
```
::::setup
```
variable {p : Prop} {f : α → β} {c : Prop} [Decidable c] {t e : α}
open Classical (choose propDecidable)
```
:::leanFirst
In contrast to {lean}`p ∨ ¬ p`, which can only eliminate to {lean}`Prop`, the
type {lean}`Decidable p` is equivalent to the sum type {lit}`Sum p (¬ p)`, which
can eliminate to any type. It is this data that is needed to write an
if-then-else expression.
As an example of classical reasoning, we use {lean}`choose` to show that if
{lean}`f : α → β` is injective and {lean}`α` is inhabited, then {lean}`f` has a
left inverse. To define the left inverse {leanRef}`linv`, we use a dependent
if-then-else expression. Recall that {lean}`if h : c then t else e` is
notation for {lean}`dite c (fun h : c => t) (fun h : ¬ c => e)`. In the definition
of {leanRef}`linv`, choice is used twice: first, to show that
{leanRef}`(∃ a : α, f a = b)` is “decidable,” and then to choose an {leanRef}`a` such that
{leanRef}`f a = b`. Notice that {lean}`propDecidable` is a scoped instance and is activated
by the {leanRef}`open Classical` command. We use this instance to justify
the {kw}`if`-{kw}`then`-{kw}`else` expression. (See also the discussion in
{ref "decidable-propositions"}[Decidable Propositions]).
```lean
open Classical
noncomputable def linv [Inhabited α] (f : α → β) : β → α :=
fun b : β => if ex : (∃ a : α, f a = b) then choose ex else default
theorem linv_comp_self {f : α → β} [Inhabited α]
(inj : ∀ {a b}, f a = f b → a = b)
: linv f ∘ f = id :=
funext fun a =>
have ex : ∃ a₁ : α, f a₁ = f a := ⟨a, rfl⟩
have feq : f (choose ex) = f a := choose_spec ex
calc linv f (f a)
_ = choose ex := rfl
_ = a := inj feq
```
From a classical point of view, {leanRef}`linv` is a function. From a
constructive point of view, it is unacceptable; because there is no
way to implement such a function in general, the construction is not
informative.
:::
::::
================================================
FILE: book/TPiL/Conv.lean
================================================
import VersoManual
import TPiL.Examples
open Verso.Genre
open Verso.Genre.Manual hiding tactic
open TPiL
#doc (Manual) "The Conversion Tactic Mode" =>
%%%
tag := "conv"
%%%
Inside a tactic block, one can use the keyword {tactic}`conv` to enter
_conversion mode_. This mode allows to travel inside assumptions and
goals, even inside function abstractions and dependent arrows, to apply rewriting or
simplifying steps.
# Basic navigation and rewriting
%%%
tag := "basic-navigation-and-rewriting"
%%%
:::leanFirst
As a first example, let us prove example
{leanRef}`(a b c : Nat) : a * (b * c) = a * (c * b)`
(examples in this file are somewhat artificial since
other tactics could finish them immediately). The naive
first attempt is to enter tactic mode and try {leanRef}`rw [Nat.mul_comm]`. But this
transforms the goal into {leanRef}`b * c * a = a * (c * b)`, after commuting the
very first multiplication appearing in the term. There are several
ways to fix this issue, and one way is to use a more precise tool:
the conversion mode. The following code block shows the current target
after each line.
```lean (showProofStates := "oops conv1 conv2 conv3 conv4")
#guard_msgs (drop all) in
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
rw [Nat.mul_comm]
-- ^ PROOF_STATE: oops
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
conv =>
-- ^ PROOF_STATE: conv1
lhs
-- ^ PROOF_STATE: conv2
congr
-- ^ PROOF_STATE: conv3
rfl
-- ^ PROOF_STATE: conv4
rw [Nat.mul_comm]
```
:::
The above snippet shows three navigation commands:
- {leanRef}`lhs` navigates to the left-hand side of a relation (equality, in this case).
There is also a {tactic}`rhs` to navigate to the right-hand side.
- {leanRef}`congr` creates as many targets as there are (nondependent and explicit) arguments to the current head function
(here the head function is multiplication).
- {leanRef}`rfl` closes target using reflexivity.
Once arrived at the relevant target, we can use {leanRef}`rw` as in normal
tactic mode.
:::leanFirst
The second main reason to use conversion mode is to rewrite under
binders. Suppose we want to prove example
{leanRef}`(fun x : Nat => 0 + x) = (fun x => x)`.
The naive first attempt is to enter tactic mode and try
{leanRef}`rw [Nat.zero_add]`. But this fails with a frustrating
```
error: tactic 'rewrite' failed, did not find instance of the pattern
in the target expression
0 + ?n
⊢ (fun x => 0 + x) = fun x => x
```
The solution is:
```lean
example : (fun x : Nat => 0 + x) = (fun x => x) := by
conv =>
lhs
intro x
rw [Nat.zero_add]
```
:::
where {leanRef}`intro x` is the navigation command entering inside the {kw}`fun` binder.
Note that this example is somewhat artificial, one could also do:
```lean
example : (fun x : Nat => 0 + x) = (fun x => x) := by
funext x; rw [Nat.zero_add]
```
or just
```lean
example : (fun x : Nat => 0 + x) = (fun x => x) := by
simp
```
{leanRef}`conv` can also rewrite a hypothesis {lit}`h` from the local context, using {kw}`conv at`{lit}` h`.
# Pattern matching
%%%
tag := "pattern-matching-conv"
%%%
Navigation using the above commands can be tedious. One can shortcut it using pattern matching as follows:
```lean
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
conv in b * c =>
rw [Nat.mul_comm]
```
which is just syntax sugar for
```lean
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
conv =>
pattern b * c
rw [Nat.mul_comm]
```
Of course, wildcards are allowed:
```lean
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
conv in _ * c => rw [Nat.mul_comm]
```
# Structuring conversion tactics
%%%
tag := "structuring-conversion-tactics"
%%%
Curly brackets and {lit}`.` can also be used in {leanRef}`conv` mode to structure tactics:
```lean
example (a b c : Nat) : (0 + a) * (b * c) = a * (c * b) := by
conv =>
lhs
congr
. rw [Nat.zero_add]
. rw [Nat.mul_comm]
```
# Other tactics inside conversion mode
%%%
tag := "other-tactics-inside-conversion-mode"
%%%
- :::leanFirst
{leanRef}`arg`{lit}` i` enter the {lit}`i`-th nondependent explicit argument of an application.
```lean (showProofStates := "arg2 arg3")
example (a b c : Nat) : a * (b * c) = a * (c * b) := by
conv =>
-- ^ PROOF_STATE: arg1
lhs
-- ^ PROOF_STATE: arg2
arg 2
-- ^ PROOF_STATE: arg3
rw [Nat.mul_comm]
```
:::
- {tactic}`args` is an alternative name for {leanRef}`congr`.
- {leanRef}`simp` applies the simplifier to the current goal. It supports the same options available in regular tactic mode.
```lean
def f (x : Nat) :=
if x > 0 then x + 1 else x + 2
example (g : Nat → Nat)
(h₁ : g x = x + 1) (h₂ : x > 0) :
g x = f x := by
conv =>
rhs
simp [f, h₂]
exact h₁
```
- {kw}`enter`{lit}` [1, x, 2, y]` iterate {leanRef}`arg` and {leanRef}`intro` with the given arguments.
- {tactic}`done` fail if there are unsolved goals.
- {tactic}`trace_state` display the current tactic state.
- {tactic}`whnf` put term in weak head normal form.
- {kw}`tactic`{lit}` => ` go back to regular tactic mode. This
is useful for discharging goals not supported by {leanRef}`conv` mode, and
applying custom congruence and extensionality lemmas.
```lean (showProofStates := "convTac1 convTac2 convTac4")
example (g : Nat → Nat → Nat)
(h₁ : ∀ x, x ≠ 0 → g x x = 1)
(h₂ : x ≠ 0)
: g x x + x = 1 + x := by
conv =>
lhs
-- ^ PROOF_STATE: convTac1
arg 1
-- ^ PROOF_STATE: convTac2
rw [h₁]
. skip
. tactic =>
-- ^ PROOF_STATE: convTac4
exact h₂
```
- {kw}`apply`{lit}` ` is syntax sugar for {kw}`tactic`{lit}` => apply `.
```lean
example (g : Nat → Nat → Nat)
(h₁ : ∀ x, x ≠ 0 → g x x = 1)
(h₂ : x ≠ 0)
: g x x + x = 1 + x := by
conv =>
lhs
arg 1
rw [h₁]
. skip
. apply h₂
```
================================================
FILE: book/TPiL/DependentTypeTheory.lean
================================================
import VersoManual
import TPiL.Examples
open TPiL
open Verso.Genre Manual
#doc (Manual) "Dependent Type Theory" =>
%%%
tag := "dependent-type-theory"
htmlSplit := .never
%%%
Dependent type theory is a powerful and expressive language, allowing
you to express complex mathematical assertions, write complex hardware
and software specifications, and reason about both of these in a
natural and uniform way. Lean is based on a version of dependent type
theory known as the _Calculus of Constructions_, with a countable
hierarchy of non-cumulative universes and inductive types. By the end
of this chapter, you will understand much of what this means.
# Simple Type Theory
%%%
tag := "simple-type-theory"
%%%
“Type theory” gets its name from the fact that every expression has an
associated _type_. For example, in a given context, {lit}`x + 0` may
denote a natural number and {lit}`f` may denote a function on the natural
numbers. For those who like precise definitions, a Lean natural number
is an arbitrary-precision unsigned integer.
Here are some examples of how you can declare objects in Lean and
check their types.
```lean
/- Define some constants. -/
def m : Nat := 1 -- m is a natural number
def n : Nat := 0
def b1 : Bool := true -- b1 is a Boolean
def b2 : Bool := false
/- Check their types. -/
#check m -- m : Nat
#check n
#check n + 0 -- n + 0 : Nat
#check m * (n + 0) -- m * (n + 0) : Nat
#check b1 -- b1 : Bool
-- "&&" is the Boolean and
#check b1 && b2 -- b1 && b2 : Bool
-- Boolean or
#check b1 || b2 -- b1 || b2 : Bool
-- Boolean "true"
#check true -- Bool.true : Bool
/- Evaluate -/
#eval 5 * 4 -- 20
#eval m + 2 -- 3
#eval b1 && b2 -- false
```
Any text between {lit}`/-` and {lit}`-/` constitutes a comment block that is
ignored by Lean. Similarly, two dashes {lean}`--` indicate that the rest of
the line contains a comment that is also ignored. Comment blocks can
be nested, making it possible to “comment out” chunks of code, just as
in many programming languages.
The {kw}`def` keyword declares new constant symbols into the
working environment. In the example above, {leanRef}`def m : Nat := 1`
defines a new constant {leanRef}`m` of type {lean}`Nat` whose value is {leanRef}`1`.
The {kw}`#check` command asks Lean to report their
types; in Lean, auxiliary commands that query the system for
information typically begin with the hash (#) symbol.
The {kw}`#eval` command asks Lean to evaluate the given expression.
You should try
declaring some constants and type checking some expressions on your
own. Declaring new objects in this manner is a good way to experiment
with the system.
:::setup
```
variable (a b : Type)
```
What makes simple type theory powerful is that you can build new types
out of others. For example, if {lean}`a` and {lean}`b` are types, {lean}`a -> b`
denotes the type of functions from {lean}`a` to {lean}`b`, and {lean}`a × b`
denotes the type of pairs consisting of an element of {lean}`a` paired
with an element of {lean}`b`, also known as the _Cartesian product_. Note
that {lit}`×` is a Unicode symbol. The judicious use of Unicode improves
legibility, and all modern editors have great support for it. In the
Lean standard library, you often see Greek letters to denote types,
and the Unicode symbol {lit}`→` as a more compact version of {lit}`->`.
:::
```lean (check := false)
#check Nat → Nat -- type the arrow as “\to” or "\r"
#check Nat -> Nat -- alternative ASCII notation
#check Nat × Nat -- type the product as "\times"
#check Prod Nat Nat -- alternative notation
#check Nat → Nat → Nat
#check Nat → (Nat → Nat) -- same type as above
#check Nat × Nat → Nat
#check (Nat → Nat) → Nat -- a "functional"
```
```lean
#check Nat.succ -- Nat.succ (n : Nat) : Nat
#check (0, 1) -- (0, 1) : Nat × Nat
#check Nat.add -- Nat.add : Nat → Nat → Nat
#check Nat.succ 2 -- Nat.succ 2 : Nat
#check Nat.add 3 -- Nat.add 3 : Nat → Nat
#check Nat.add 5 2 -- Nat.add 5 2 : Nat
#check (5, 9).1 -- (5, 9).fst : Nat
#check (5, 9).2 -- (5, 9).snd : Nat
#eval Nat.succ 2 -- 3
#eval Nat.add 5 2 -- 7
#eval (5, 9).1 -- 5
#eval (5, 9).2 -- 9
```
Once again, you should try some examples on your own.
Let's take a look at some basic syntax. You can enter the Unicode
arrow {lit}`→` by typing {kbd}`\to` or {kbd}`\r` or {kbd}`\->`. You can also use the
ASCII alternative {lit}`->`, so the expressions {lean}`Nat -> Nat` and {lean}`Nat → Nat`
mean the same thing. Both expressions denote the type of
functions that take a natural number as input and return a natural
number as output. The Unicode symbol {lit}`×` for the Cartesian product
is entered as {kbd}`\times`. You will generally use lower-case Greek
letters like {lit}`α`, {lit}`β`, and {lit}`γ` to range over types. You can
enter these particular ones with {kbd}`\a`, {kbd}`\b`, and {kbd}`\g`.
::::setup
```
variable (α β : Type) (f : α → β) (x : α) (m n : Nat) (p : Nat × Nat)
```
There are a few more things to notice here. First, the application of
a function {lean}`f` to a value {lean}`x` is denoted {lean}`f x` (e.g., {lean}`Nat.succ 2`).
Second, when writing type expressions, arrows associate to the _right_; for
example, the type of {lean}`Nat.add` is {lean}`Nat → Nat → Nat` which is equivalent
to {lean}`Nat → (Nat → Nat)`. Thus you can
view {lean}`Nat.add` as a function that takes a natural number and returns
another function that takes a natural number and returns a natural
number. In type theory, this is generally more convenient than
writing {lean}`Nat.add` as a function that takes a pair of natural numbers as
input and returns a natural number as output. For example, it allows
you to “partially apply” the function {lean}`Nat.add`. The example above shows
that {lean}`Nat.add 3` has type {lean}`Nat → Nat`, that is, {lean}`Nat.add 3` returns a
function that “waits” for a second argument, {lean}`n`, which is then
equivalent to writing {lean}`Nat.add 3 n`.
:::comment
```
```
:::
You have seen that if you have {lean}`m : Nat` and {lean}`n : Nat`, then
{lean}`(m, n)` denotes the ordered pair of {lean}`m` and {lean}`n` which is of
type {lean}`Nat × Nat`. This gives you a way of creating pairs of natural
numbers. Conversely, if you have {lean}`p : Nat × Nat`, then you can write
{lean}`p.1 : Nat` and {lean}`p.2 : Nat`. This gives you a way of extracting
its two components.
::::
# Types as objects
%%%
tag := "types-as-objects"
%%%
One way in which Lean's dependent type theory extends simple type
theory is that types themselves—entities like {lean}`Nat` and {lean}`Bool`—are first-class citizens, which is to say that they themselves are
objects. For that to be the case, each of them also has to have a
type.
```lean
#check Nat
#check Bool
#check Nat → Bool
#check Nat × Bool
#check Nat → Nat
#check Nat × Nat → Nat
#check Nat → Nat → Nat
#check Nat → (Nat → Nat)
#check Nat → Nat → Bool
#check (Nat → Nat) → Nat
```
You can see that each one of the expressions above is an object of
type {lean}`Type`. You can also declare new constants for types:
```lean
def α : Type := Nat
def β : Type := Bool
def F : Type → Type := List
def G : Type → Type → Type := Prod
#check α -- α : Type
#check F α -- F α : Type
#check F Nat -- F Nat : Type
#check G α -- G α : Type → Type
#check G α β -- G α β : Type
#check G α Nat -- G α Nat : Type
```
As the example above suggests, you have already seen an example of a function of type
{lean}`Type → Type → Type`, namely, the Cartesian product {lean}`Prod`:
```lean
def α : Type := Nat
def β : Type := Bool
#check Prod α β -- α × β : Type
#check α × β -- α × β : Type
#check Prod Nat Nat -- Nat × Nat : Type
#check Nat × Nat -- Nat × Nat : Type
```
:::leanFirst
Here is another example: given any type {leanRef}`α`, the type {leanRef}`List α`
denotes the type of lists of elements of type {leanRef}`α`.
```lean
def α : Type := Nat
#check List α -- List α : Type
#check List Nat -- List Nat : Type
```
:::
Given that every expression in Lean has a type, it is natural to ask:
what type does {lean}`Type` itself have?
```lean
#check Type -- Type : Type 1
```
You have actually come up against one of the most subtle aspects of
Lean's typing system. Lean's underlying foundation has an infinite
hierarchy of types:
```lean
#check Type -- Type : Type 1
#check Type 1 -- Type 1 : Type 2
#check Type 2 -- Type 2 : Type 3
#check Type 3 -- Type 3 : Type 4
#check Type 4 -- Type 4 : Type 5
```
:::setup
```
universe n
variable (n : Nat)
```
Think of {lean}`Type 0` as a universe of “small” or “ordinary” types.
{lean}`Type 1` is then a larger universe of types, which contains {lean}`Type 0`
as an element, and {lean}`Type 2` is an even larger universe of types,
which contains {lean}`Type 1` as an element. The list is infinite:
there is a {lean}`Type n` for every natural number {lean}`n`. {lean}`Type` is
an abbreviation for {lean}`Type 0`:
:::
```lean
#check Type
#check Type 0
```
The following table may help concretize the relationships being discussed.
Movement along the x-axis represents a change in the universe, while movement
along the y-axis represents a change in what is sometimes referred to as
“degree”.
:::table
*
* sort
* {lean}`Prop` ({lean}`Sort 0`)
* {lean}`Type` ({lean}`Sort 1`)
* {lean}`Type 1` ({lean}`Sort 2`)
* {lean}`Type 2` ({lean}`Sort 3`)
* ...
*
* type
* {lean}`True`
* {lean}`Bool`
* {lean}`Nat -> Type`
* {lean}`Type -> Type 1`
* ...
*
* term
* {lean}`True.intro`
* {lean}`true`
* {lean}`fun n => Fin n`
* {lean}`fun (_ : Type) => Type`
* ...
:::
:::setup
```
universe u
variable (α : Type u)
```
Some operations, however, need to be _polymorphic_ over type
universes. For example, {lean}`List α` should make sense for any type
{lean}`α`, no matter which type universe {lean}`α` lives in. This explains the
type signature of the function {lean}`List`:
```lean
#check List -- List.{u} (α : Type u) : Type u
```
Here {lit}`u` is a variable ranging over type levels. The output of the
{kw}`#check` command means that whenever {lean}`α` has type {lean}`Type u`,
{lean}`List α` also has type {lean}`Type u`. The function {lean}`Prod` is
similarly polymorphic:
:::
```lean
#check Prod -- Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)
```
To define polymorphic constants, Lean allows you to
declare universe variables explicitly using the {kw}`universe` command:
```lean
universe u
def F (α : Type u) : Type u := Prod α α
#check F -- F.{u} (α : Type u) : Type u
```
:::leanFirst
You can avoid the {kw}`universe` command by providing the universe parameters when defining {leanRef}`F`:
```lean
def F.{u} (α : Type u) : Type u := Prod α α
#check F -- F.{u} (α : Type u) : Type u
```
:::
# Function Abstraction and Evaluation
%%%
tag := "function-abstraction-and-evaluation"
%%%
Lean provides a {kw}`fun` (or {kw}`λ`) keyword to create a function
from an expression as follows:
```lean
#check fun (x : Nat) => x + 5 -- fun x => x + 5 : Nat → Nat
-- λ and fun mean the same thing
#check λ (x : Nat) => x + 5 -- fun x => x + 5 : Nat → Nat
```
The type {lean}`Nat` can be inferred in this example:
```lean
#check fun x => x + 5 -- fun x => x + 5 : Nat → Nat
#check λ x => x + 5 -- fun x => x + 5 : Nat → Nat
```
You can evaluate a lambda function by passing the required parameters:
```lean
#eval (λ x : Nat => x + 5) 10 -- 15
```
:::setup
```
variable {x : α} {t : β}
```
Creating a function from another expression is a process known as
_lambda abstraction_. Suppose you have the variable {lean}`x : α` and you can
construct an expression {lean}`t : β`, then the expression {lean}`fun (x : α) => t`,
or, equivalently, {lean}`λ (x : α) => t`, is an object of type {lean}`α → β`. Think of
this as the function from {lean}`α` to {lean}`β` which maps
any value {leanRef}`x` to the value {leanRef}`t`.
:::
Here are some more examples
```lean
#check fun x : Nat => fun y : Bool => if not y then x + 1 else x + 2
#check fun (x : Nat) (y : Bool) => if not y then x + 1 else x + 2
#check fun x y => if not y then x + 1 else x + 2 -- fun x y => if (!y) = true then x + 1 else x + 2 : Nat → Bool → Nat
```
Lean interprets the final three examples as the same expression; in
the last expression, Lean infers the type of {leanRef}`x` and {leanRef}`y` from the
expression {leanRef}`if not y then x + 1 else x + 2`.
Some mathematically common examples of operations of functions can be
described in terms of lambda abstraction:
```lean
def f (n : Nat) : String := toString n
def g (s : String) : Bool := s.length > 0
#check fun x : Nat => x -- fun x => x : Nat → Nat
#check fun x : Nat => true -- fun x => true : Nat → Bool
#check fun x : Nat => g (f x) -- fun x => g (f x) : Nat → Bool
#check fun x => g (f x) -- fun x => g (f x) : Nat → Bool
```
Think about what these expressions mean. The expression
{lean}`fun x : Nat => x` denotes the identity function on {lean}`Nat`, the
expression {lean}`fun x : Nat => true` denotes the constant function that
always returns {lean}`true`, and {leanRef}`fun x : Nat => g (f x)` denotes the
composition of {leanRef}`f` and {leanRef}`g`. You can, in general, leave off the
type annotation and let Lean infer it for you. So, for example, you
can write {leanRef}`fun x => g (f x)` instead of {leanRef}`fun x : Nat => g (f x)`.
:::leanFirst
You can pass functions as parameters and by giving them names {leanRef}`f`
and {leanRef}`g` you can then use those functions in the implementation:
```lean
#check fun (g : String → Bool) (f : Nat → String) (x : Nat) => g (f x)
```
:::
You can also pass types as parameters:
```lean
#check fun (α β γ : Type) (g : β → γ) (f : α → β) (x : α) => g (f x)
```
The last expression, for example, denotes the function that takes
three types, {leanRef}`α`, {leanRef}`β`, and {leanRef}`γ`, and two functions, {leanRef}`g : β → γ`
and {leanRef}`f : α → β`, and returns the composition of {leanRef}`g` and {leanRef}`f`.
(Making sense of the type of this function requires an understanding
of _dependent products_, which will be explained below.)
:::setup
```
variable (α : Type) (t : β)
-- Avoid warnings
axiom whatever : α
def b : γ := whatever
```
The general form of a lambda expression is {lean}`fun (x : α) => t`, where
the variable {leanRef}`x` is a “bound variable”: it is really a placeholder,
whose “scope” does not extend beyond the expression {leanRef}`t`. For
example, the variable {lit}`b` in the expression {lean}`fun (b : β) (x : α) => b`
has nothing to do with the constant {lean}`b` declared earlier. In fact,
the expression denotes the same function as {lean}`fun (u : β) (z : α) => u`.
Formally, expressions that are the same up to a renaming of bound
variables are called _alpha equivalent_, and are considered “the
same.” Lean recognizes this equivalence.
:::
:::setup
```
variable (t : α → β) (s : α)
```
Notice that applying a term {lean}`t : α → β` to a term {lean}`s : α` yields
an expression {lean}`t s : β`. Returning to the previous example and
renaming bound variables for clarity, notice the types of the
following expressions:
:::
```lean
#check (fun x : Nat => x) 1 -- (fun x => x) 1 : Nat
#check (fun x : Nat => true) 1 -- (fun x => true) 1 : Bool
def f (n : Nat) : String := toString n
def g (s : String) : Bool := s.length > 0
#check
(fun (α β γ : Type) (u : β → γ) (v : α → β) (x : α) => u (v x)) Nat String Bool g f 0
```
As expected, the expression {lean}`(fun x : Nat => x) 1` has type {lean}`Nat`.
In fact, more should be true: applying the expression {lean}`(fun x : Nat => x)` to
{lean}`1` should “return” the value {lean}`1`. And, indeed, it does:
```lean
#eval (fun x : Nat => x) 1 -- 1
#eval (fun x : Nat => true) 1 -- true
```
You will see later how these terms are evaluated. For now, notice that
this is an important feature of dependent type theory: every term has
a computational behavior, and supports a notion of _normalization_. In
principle, two terms that reduce to the same value are called
_definitionally equal_. They are considered “the same” by Lean's type
checker, and Lean does its best to recognize and support these
identifications.
Lean is a complete programming language. It has a compiler that
generates a binary executable and an interactive interpreter. You can
use the command {kw}`#eval` to execute expressions, and it is the
preferred way of testing your functions.
:::comment
```
```
:::
# Definitions
%%%
tag := "definitions"
%%%
Recall that the {kw}`def` keyword provides one important way of declaring new named
objects.
```lean
def double (x : Nat) : Nat :=
x + x
```
This might look more familiar to you if you know how functions work in
other programming languages. The name {leanRef}`double` is defined as a
function that takes an input parameter {leanRef}`x` of type {lean}`Nat`, where the
result of the call is {leanRef}`x + x`, so it is returning type {lean}`Nat`. You
can then invoke this function using:
```lean
def double (x : Nat) : Nat :=
x + x
-----
#eval double 3 -- 6
```
In this case you can think of {kw}`def` as a kind of named {kw}`fun`.
The following yields the same result:
```lean
def double : Nat → Nat :=
fun x => x + x
#eval double 3 -- 6
```
You can omit the type declarations when Lean has enough information to
infer it. Type inference is an important part of Lean:
```lean
def double :=
fun (x : Nat) => x + x
```
The general form of a definition is {lit}`def foo : α := bar` where
{lit}`α` is the type returned from the expression {lit}`bar`. Lean can
usually infer the type {lit}`α`, but it is often a good idea to write it
explicitly. This clarifies your intention, and Lean will flag an
error if the right-hand side of the definition does not have a matching
type.
The right hand side {lit}`bar` can be any expression, not just a lambda.
So {kw}`def` can also be used to simply name a value like this:
```lean
def pi := 3.141592654
```
{kw}`def` can take multiple input parameters. Let's create one
that adds two natural numbers:
```lean
def add (x y : Nat) :=
x + y
#eval add 3 2 -- 5
```
The parameter list can be separated like this:
```lean
def double (x : Nat) : Nat :=
x + x
-----
def add (x : Nat) (y : Nat) :=
x + y
#eval add (double 3) (7 + 9) -- 22
```
Notice here we called the {leanRef}`double` function to create the first
parameter to {leanRef}`add`.
You can use other more interesting expressions inside a {kw}`def`:
```lean
def greater (x y : Nat) :=
if x > y then x
else y
```
You can probably guess what this one will do.
You can also define a function that takes another function as input.
The following calls a given function twice passing the output of the
first invocation to the second:
```lean
def double (x : Nat) : Nat :=
x + x
-----
def doTwice (f : Nat → Nat) (x : Nat) : Nat :=
f (f x)
#eval doTwice double 2 -- 8
```
Now to get a bit more abstract, you can also specify arguments that
are like type parameters:
```lean
def compose (α β γ : Type) (g : β → γ) (f : α → β) (x : α) : γ :=
g (f x)
```
This means {leanRef}`compose` is a function that takes any two functions as input
arguments, so long as those functions each take only one input.
The type algebra {leanRef}`β → γ` and {leanRef}`α → β` means it is a requirement
that the type of the output of the second function must match the
type of the input to the first function—which makes sense, otherwise
the two functions would not be composable.
{leanRef}`compose` also takes a 3rd argument of type {leanRef}`α` which
it uses to invoke the second function (locally named {leanRef}`f`) and it
passes the result of that function (which is type {leanRef}`β`) as input to the
first function (locally named {leanRef}`g`). The first function returns a type
{leanRef}`γ` so that is also the return type of the {leanRef}`compose` function.
{leanRef}`compose` is also very general in that it works over any type
{leanRef}`α β γ`. This means {leanRef}`compose` can compose just about any 2 functions
so long as they each take one parameter, and so long as the type of
output of the second matches the input of the first. For example:
```lean
def compose (α β γ : Type) (g : β → γ) (f : α → β) (x : α) : γ :=
g (f x)
def double (x : Nat) : Nat :=
x + x
-----
def square (x : Nat) : Nat :=
x * x
#eval compose Nat Nat Nat double square 3 -- 18
```
# Local Definitions
%%%
tag := "local-definitions"
%%%
:::setup
```
variable (t1 : α) (t2 : β)
```
Lean also allows you to introduce “local” definitions using the
{kw}`let` keyword. The expression {lean}`let a := t1; t2` is
definitionally equal to the result of replacing every occurrence of
{leanRef}`a` in {leanRef}`t2` by {leanRef}`t1`.
:::
```lean
#check let y := 2 + 2; y * y -- let y := 2 + 2; y * y : Nat
#eval let y := 2 + 2; y * y -- 16
def twice_double (x : Nat) : Nat :=
let y := x + x; y * y
#eval twice_double 2 -- 16
```
:::setup
```
def twice_double (x : Nat) : Nat :=
let y := x + x; y * y
variable (x : Nat)
```
Here, {lean}`twice_double x` is definitionally equal to the term {lean}`(x + x) * (x + x)`.
:::
You can combine multiple assignments by chaining {kw}`let` statements:
```lean
#check let y := 2 + 2; let z := y + y; z * z
#eval let y := 2 + 2; let z := y + y; z * z -- 64
```
The {lit}`;` can be omitted when a line break is used.
```lean
def t (x : Nat) : Nat :=
let y := x + x
y * y
```
::::leanFirst
:::setup
```
variable (t1 : α) (t2 : β)
```
Notice that the meaning of the expression {lean}`let a := t1; t2` is very
similar to the meaning of {lean}`(fun a => t2) t1`, but the two are not
the same. In the first expression, you should think of every instance
of {leanRef (in:="let a := t1; t2")}`a` in {leanRef (in:="let a := t1; t2")}`t2` as a syntactic abbreviation for {leanRef (in:="let a := t1; t2")}`t1`. In the
second expression, {leanRef (in:="(fun a => t2) t1")}`a` is a variable, and the expression
{leanRef (in:="(fun a => t2) t1")}`fun a => t2` has to make sense independently of the value of {leanRef (in:="(fun a => t2) t1")}`a`.
The {kw}`let` construct is a stronger means of abbreviation, and there
are expressions of the form {lean}`let a := t1; t2` that cannot be
expressed as {lean}`(fun a => t2) t1`. As an exercise, try to understand
why the definition of {leanRef}`foo` below type checks, but the definition of
{lit}`bar` does not.
:::
```lean
def foo := let a := Nat; fun x : a => x + 2
/-
def bar := (fun a => fun x : a => x + 2) Nat
-/
```
::::
# Variables and Sections
%%%
tag := "variables-and-sections"
%%%
Consider the following three function definitions:
```lean
def compose (α β γ : Type) (g : β → γ) (f : α → β) (x : α) : γ :=
g (f x)
def doTwice (α : Type) (h : α → α) (x : α) : α :=
h (h x)
def doThrice (α : Type) (h : α → α) (x : α) : α :=
h (h (h x))
```
Lean provides you with the {kw}`variable` command to make such
declarations look more compact:
```lean
variable (α β γ : Type)
def compose (g : β → γ) (f : α → β) (x : α) : γ :=
g (f x)
def doTwice (h : α → α) (x : α) : α :=
h (h x)
def doThrice (h : α → α) (x : α) : α :=
h (h (h x))
```
You can declare variables of any type, not just {lean}`Type` itself:
```lean
variable (α β γ : Type)
variable (g : β → γ) (f : α → β) (h : α → α)
variable (x : α)
def compose := g (f x)
def doTwice := h (h x)
def doThrice := h (h (h x))
#print compose
#print doTwice
#print doThrice
```
Printing them out shows that all three groups of definitions have
exactly the same effect.
The {kw}`variable` command instructs Lean to insert the declared
variables as bound variables in definitions that refer to them by
name. Lean is smart enough to figure out which variables are used
explicitly or implicitly in a definition. You can therefore proceed as
though {leanRef}`α`, {leanRef}`β`, {leanRef}`γ`, {leanRef}`g`, {leanRef}`f`, {leanRef}`h`, and {leanRef}`x` are fixed
objects when you write your definitions, and let Lean abstract the
definitions for you automatically.
When declared in this way, a variable stays in scope until the end of
the file you are working on. Sometimes, however, it is useful to limit
the scope of a variable. For that purpose, Lean provides the notion of
a {kw}`section`:
```lean
section useful
variable (α β γ : Type)
variable (g : β → γ) (f : α → β) (h : α → α)
variable (x : α)
def compose := g (f x)
def doTwice := h (h x)
def doThrice := h (h (h x))
end useful
```
When the section is closed, the variables go out of scope, and cannot
be referenced any more.
You do not have to indent the lines within a section. Nor do you have
to name a section, which is to say, you can use an anonymous
{kw}`section` / {kw}`end` pair. If you do name a section, however, you
have to close it using the same name. Sections can also be nested,
which allows you to declare new variables incrementally.
# Namespaces
%%%
tag := "namespaces"
%%%
Lean provides you with the ability to group definitions into nested,
hierarchical _namespaces_:
```lean
namespace Foo
def a : Nat := 5
def f (x : Nat) : Nat := x + 7
def fa : Nat := f a
def ffa : Nat := f (f a)
#check a
#check f
#check fa
#check ffa
#check Foo.fa
end Foo
-- #check a -- error
-- #check f -- error
#check Foo.a
#check Foo.f
#check Foo.fa
#check Foo.ffa
open Foo
#check a
#check f
#check fa
#check Foo.fa
```
When you declare that you are working in the namespace {leanRef}`Foo`, every
identifier you declare has a full name with prefix “{lit}`Foo.`”. Within
the namespace, you can refer to identifiers by their shorter names,
but once you end the namespace, you have to use the longer names.
Unlike {kw}`section`, namespaces require a name. There is only one
anonymous namespace at the root level.
The {leanRef}`open` command brings the shorter names into the current
context. Often, when you import a module, you will want to open one or
more of the namespaces it contains, to have access to the short
identifiers. But sometimes you will want to leave this information
protected by a fully qualified name, for example, when they conflict
with identifiers in another namespace you want to use. Thus namespaces
give you a way to manage names in your working environment.
For example, Lean groups definitions and theorems involving lists into
a namespace {lit}`List`.
```lean
#check List.nil
#check List.cons
#check List.map
```
:::leanFirst
The command {leanRef}`open List` allows you to use the shorter names:
```lean
open List
#check nil
#check cons
#check map
```
:::
Like sections, namespaces can be nested:
```lean
namespace Foo
def a : Nat := 5
def f (x : Nat) : Nat := x + 7
def fa : Nat := f a
namespace Bar
def ffa : Nat := f (f a)
#check fa
#check ffa
end Bar
#check fa
#check Bar.ffa
end Foo
#check Foo.fa
#check Foo.Bar.ffa
open Foo
#check fa
#check Bar.ffa
```
Namespaces that have been closed can later be reopened, even in another file:
```lean
namespace Foo
def a : Nat := 5
def f (x : Nat) : Nat := x + 7
def fa : Nat := f a
end Foo
#check Foo.a
#check Foo.f
namespace Foo
def ffa : Nat := f (f a)
end Foo
```
Like sections, nested namespaces have to be closed in the order they
are opened. Namespaces and sections serve different purposes:
namespaces organize data and sections declare variables for insertion
in definitions. Sections are also useful for delimiting the scope of
commands such as {kw}`set_option` and {kw}`open`.
In many respects, however, a {kw}`namespace`{lit}` ... `{kw}`end` block behaves the
same as a {kw}`section`{lit}` ... `{kw}`end` block. In particular, if you use the
{kw}`variable` command within a namespace, its scope is limited to the
namespace. Similarly, if you use an {kw}`open` command within a
namespace, its effects disappear when the namespace is closed.
# What makes dependent type theory dependent?
%%%
tag := "what-makes-dependent-type-theory-dependent"
%%%
:::setup
```
variable (α : Type) (n : Nat)
```
The short explanation is that types can depend on parameters. You
have already seen a nice example of this: the type {lean}`List α` depends
on the argument {lean}`α`, and this dependence is what distinguishes
{lean}`List Nat` and {lean}`List Bool`. For another example, consider the
type {lean}`Vector α n`, the type of vectors of elements of {lean}`α` of
length {lean}`n`. This type depends on _two_ parameters: the type of the
elements in the vector ({lean}`α : Type`) and the length of the vector
{lean}`n : Nat`.
:::
::::setup
```
def cons (α : Type) (a : α) (as : List α) : List α :=
List.cons a as
variable (α : Type) (a : α) (as : List α)
```
:::leanFirst
Suppose you wish to write a function {leanRef}`cons` which inserts a new
element at the head of a list. What type should {leanRef}`cons` have? Such a
function is _polymorphic_: you expect the {leanRef}`cons` function for
{lean}`Nat`, {lean}`Bool`, or an arbitrary type {leanRef}`α` to behave the same way.
So it makes sense to take the type to be the first argument to
{leanRef}`cons`, so that for any type, {lean}`α`, {lean}`cons α` is the insertion
function for lists of type {lean}`α`. In other words, for every {lean}`α`,
{lean}`cons α` is the function that takes an element {lean}`a : α` and a list
{lean}`as : List α`, and returns a new list, so you have {lean}`cons α a as : List α`.
It is clear that {lean}`cons α` should have type {lean}`α → List α → List α`.
But what type should {leanRef}`cons` have? A first guess might be
{lean}`Type → α → List α → List α`, but, on reflection, this does not make
sense: the {leanRef}`α` in this expression does not refer to anything,
whereas it should refer to the argument of type {lean}`Type`. In other
words, _assuming_ {lean}`α : Type` is the first argument to the function,
the type of the next two elements are {lean}`α` and {lean}`List α`. These
types vary depending on the first argument, {leanRef}`α`.
```lean
def cons (α : Type) (a : α) (as : List α) : List α :=
List.cons a as
#check cons Nat -- cons Nat : Nat → List Nat → List Nat
#check cons Bool -- cons Bool : Bool → List Bool → List Bool
#check cons -- cons (α : Type) (a : α) (as : List α) : List α
```
:::
::::
:::setup
```
variable (α : Type) (β : α → Type) (a : α) (f : (a : α) → β a)
```
This is an instance of a _dependent function type_, or *dependent
arrow type*. Given {lean}`α : Type` and {lean}`β : α → Type`, think of {lean}`β`
as a family of types over {lean}`α`, that is, a type {lean}`β a` for each
{lean}`a : α`. In that case, the type {lean}`(a : α) → β a` denotes the type
of functions {lean}`f` with the property that, for each {lean}`a : α`, {lean}`f a`
is an element of {lean}`β a`. In other words, the type of the value
returned by {lean}`f` depends on its input.
:::
:::setup
```
variable (α : Type) (β : Type) (a : α) (f : (a : α) → β a)
```
Notice that {lean}`(a : α) → β` makes sense for any expression {lean}`β : Type`.
When the value of {lean}`β` depends on {leanRef}`a` (as does, for
example, the expression {leanRef}`β a` in the previous paragraph),
{leanRef}`(a : α) → β` denotes a dependent function type. When {lean}`β` doesn't
depend on {leanRef}`a`, {leanRef}`(a : α) → β` is no different from the type
{lean}`α → β`. Indeed, in dependent type theory (and in Lean), {lean}`α → β`
is just notation for {lean}`(a : α) → β` when {lean}`β` does not depend on {leanRef (in := "a : α")}`a`.
:::
Returning to the example of lists, you can use the command {kw}`#check` to
inspect the type of the following {lean}`List` functions. The {lit}`@` symbol
and the difference between the round and curly braces will be
explained momentarily.
```lean
#check @List.cons -- @List.cons : {α : Type u_1} → α → List α → List α
#check @List.nil -- @List.nil : {α : Type u_1} → List α
#check @List.length -- @List.length : {α : Type u_1} → List α → Nat
#check @List.append -- @List.append : {α : Type u_1} → List α → List α → List α
```
:::setup
```
variable (α : Type) (β : α → Type) (a : α) (b : β a)
```
Just as dependent function types {lean}`(a : α) → β a` generalize the
notion of a function type {leanRef}`α → β` by allowing {leanRef (in := "α → β")}`β` to depend on
{lean}`a`, dependent Cartesian product types {lean}`(a : α) × β a` generalize
the Cartesian product {lit}`α × β` in the same way. Dependent products
are also called _sigma_ types, and you can also write them as
{lean}`Σ a : α, β a`. You can use {lean (type := "(a : α) × β a")}`⟨a, b⟩` or {lean}`Sigma.mk a b` to create a
dependent pair. The {lit}`⟨` and {lit}`⟩` characters may be typed with
{kbd}`\langle` and {kbd}`\rangle` or {kbd}`\<` and {kbd}`\>`, respectively.
:::
```lean
universe u v
def f (α : Type u) (β : α → Type v) (a : α) (b : β a) : (a : α) × β a :=
⟨a, b⟩
def g (α : Type u) (β : α → Type v) (a : α) (b : β a) : Σ a : α, β a :=
Sigma.mk a b
def h1 (x : Nat) : Nat :=
(f Type (fun α => α) Nat x).2
#eval h1 5 -- 5
def h2 (x : Nat) : Nat :=
(g Type (fun α => α) Nat x).2
#eval h2 5 -- 5
```
The functions {leanRef}`f` and {leanRef}`g` above denote the same function.
# Implicit Arguments
%%%
tag := "implicit-arguments"
%%%
Suppose we have an implementation of lists as:
```lean
universe u
def Lst (α : Type u) : Type u := List α
def Lst.cons (α : Type u) (a : α) (as : Lst α) : Lst α := List.cons a as
def Lst.nil (α : Type u) : Lst α := List.nil
def Lst.append (α : Type u) (as bs : Lst α) : Lst α := List.append as bs
-----
#check Lst -- Lst.{u} (α : Type u) : Type u
#check Lst.cons -- Lst.cons.{u} (α : Type u) (a : α) (as : Lst α) : Lst α
#check Lst.nil -- Lst.nil.{u} (α : Type u) : Lst α
#check Lst.append -- Lst.append.{u} (α : Type u) (as bs : Lst α) : Lst α
```
Then, you can construct lists of {lean}`Nat` as follows:
```lean
universe u
def Lst (α : Type u) : Type u := List α
def Lst.cons (α : Type u) (a : α) (as : Lst α) : Lst α := List.cons a as
def Lst.nil (α : Type u) : Lst α := List.nil
def Lst.append (α : Type u) (as bs : Lst α) : Lst α := List.append as bs
-----
#check Lst.cons Nat 0 (Lst.nil Nat)
def as : Lst Nat := Lst.nil Nat
def bs : Lst Nat := Lst.cons Nat 5 (Lst.nil Nat)
#check Lst.append Nat as bs
```
:::setup
```
def Lst (α : Type u) : Type u := List α
variable (α : Type)
```
Because the constructors are polymorphic over types, we have to insert
the type {lean}`Nat` as an argument repeatedly. But this information is
redundant: one can infer the argument {leanRef}`α` in
{leanRef}`Lst.cons Nat 5 (Lst.nil Nat)` from the fact that the second argument, {leanRef}`5`, has
type {lean}`Nat`. One can similarly infer the argument in {leanRef}`Lst.nil Nat`, not
from anything else in that expression, but from the fact that it is
sent as an argument to the function {leanRef}`Lst.cons`, which expects an element
of type {lean}`Lst α` in that position.
:::
This is a central feature of dependent type theory: terms carry a lot
of information, and often some of that information can be inferred
from the context. In Lean, one uses an underscore, {lit}`_`, to specify
that the system should fill in the information automatically. This is
known as an “implicit argument.”
```lean
universe u
def Lst (α : Type u) : Type u := List α
def Lst.cons (α : Type u) (a : α) (as : Lst α) : Lst α := List.cons a as
def Lst.nil (α : Type u) : Lst α := List.nil
def Lst.append (α : Type u) (as bs : Lst α) : Lst α := List.append as bs
#check Lst
#check Lst.cons
#check Lst.nil
#check Lst.append
-----
#check Lst.cons _ 0 (Lst.nil _)
def as : Lst Nat := Lst.nil _
def bs : Lst Nat := Lst.cons _ 5 (Lst.nil _)
#check Lst.append _ as bs -- Lst.append Nat as bs : Lst Nat
```
It is still tedious, however, to type all these underscores. When a
function takes an argument that can generally be inferred from
context, Lean allows you to specify that this argument should, by
default, be left implicit. This is done by putting the arguments in
curly braces, as follows:
```lean
universe u
def Lst (α : Type u) : Type u := List α
def Lst.cons {α : Type u} (a : α) (as : Lst α) : Lst α := List.cons a as
def Lst.nil {α : Type u} : Lst α := List.nil
def Lst.append {α : Type u} (as bs : Lst α) : Lst α := List.append as bs
#check Lst.cons 0 Lst.nil
def as : Lst Nat := Lst.nil
def bs : Lst Nat := Lst.cons 5 Lst.nil
#check Lst.append as bs
```
All that has changed are the braces around {leanRef}`α : Type u` in the
declaration of the variables. We can also use this device in function
definitions:
```lean
universe u
def ident {α : Type u} (x : α) := x
```
Checking the type of {leanRef}`ident` requires wrapping it in parentheses to avoid having its signature shown:
```lean
universe u
def ident {α : Type u} (x : α) := x
---------
#check (ident) -- ident : ?m.22 → ?m.22
#check ident 1 -- ident 1 : Nat
#check ident "hello" -- ident "hello" : String
#check @ident -- @ident : {α : Type u_1} → α → α
```
The makes the first argument to {leanRef}`ident` implicit. Notationally,
this hides the specification of the type, making it look as though
{leanRef}`ident` simply takes an argument of any type. In fact, the function
{lean}`id` is defined in the standard library in exactly this way. We have
chosen a nontraditional name here only to avoid a clash of names.
Variables can also be specified as implicit when they are declared with
the {kw}`variable` command:
```lean
universe u
section
variable {α : Type u}
variable (x : α)
def ident := x
end
#check ident
#check ident 4
#check ident "hello"
```
This definition of {leanRef}`ident` here has the same effect as the one
above.
Lean has very complex mechanisms for instantiating implicit arguments,
and we will see that they can be used to infer function types,
predicates, and even proofs. The process of instantiating these
“holes,” or “placeholders,” in a term is often known as
_elaboration_. The presence of implicit arguments means that at times
there may be insufficient information to fix the meaning of an
expression precisely. An expression like {lean}`id` or {lean}`List.nil` is
said to be _polymorphic_, because it can take on different meanings in
different contexts.
:::setup
```
variable (T : Type) (e : T)
```
One can always specify the type {lean}`T` of an expression {lean}`e` by
writing {lean}`(e : T)`. This instructs Lean's elaborator to use the value
{lean}`T` as the type of {lean}`e` when trying to resolve implicit
arguments. In the second pair of examples below, this mechanism is
used to specify the desired types of the expressions {lean}`id` and
{lean}`List.nil`:
:::
```lean
#check (List.nil) -- [] : List ?m.2
#check (id) -- id : ?m.1 → ?m.1
#check (List.nil : List Nat) -- [] : List Nat
#check (id : Nat → Nat) -- id : Nat → Nat
```
Numerals are overloaded in Lean, but when the type of a numeral cannot
be inferred, Lean assumes, by default, that it is a natural number. So
the expressions in the first two {kw}`#check` commands below are
elaborated in the same way, whereas the third {kw}`#check` command
interprets {lean (type := "Int")}`2` as an integer.
```lean
#check 2 -- 2 : Nat
#check (2 : Nat) -- 2 : Nat
#check (2 : Int) -- 2 : Int
```
:::setup
```
variable (foo : {α : Type} → α → β)
```
Sometimes, however, we may find ourselves in a situation where we have
declared an argument to a function to be implicit, but now want to
provide the argument explicitly. If {lean}`foo` is such a function, the
notation {lean}`@foo` denotes the same function with all the arguments
made explicit.
:::
```lean
#check @id -- @id : {α : Sort u_1} → α → α
#check @id Nat -- id : Nat → Nat
#check @id Bool -- id : Bool → Bool
#check @id Nat 1 -- id 1 : Nat
#check @id Bool true -- id true : Bool
```
Notice that now the first {kw}`#check` command gives the type of the
identifier, {leanRef}`id`, without inserting any placeholders. Moreover, the
output indicates that the first argument is implicit.
================================================
FILE: book/TPiL/Examples.lean
================================================
import SubVerso.Examples
import Lean.Data.NameMap
import Lean.DocString.Syntax
import VersoManual
import Verso.Code.HighlightedToTex
open Lean (NameMap MessageSeverity)
open Lean.Doc.Syntax
namespace TPiL
open Verso Doc Elab Genre.Manual ArgParse Code Highlighted WebAssets Output Html Log Code External
open SubVerso.Highlighting
open SubVerso.Examples.Messages
open Lean
open Std
export Verso.Code.External (lit)
private def projectDir : System.FilePath := "../examples/"
def alphabet := "abcdefghijklmnopqrstuvwxyz0123456789"
def hashString (n : UInt64) : String := Id.run do
let mut n : Nat := n.toNat
let mut out : String := "Example" -- always start with a letter
while n > 0 do
out := out.push ({ byteIdx := n % 36 : String.Pos.Raw} |>.get! alphabet )
n := n / 36
return out
section
open System
open SubVerso.Module
variable [Monad m] [MonadError m] [MonadLiftT BaseIO m] [MonadLiftT IO m] [MonadFinally m]
variable [MonadTrace m] [AddMessageContext m] [MonadOptions m] [MonadAlwaysExcept ε m]
def extractFile (contents : String) (suppressNamespaces : Option String) : m (Array ModuleItem) := do
let codeHash := hash contents
let modBase := hashString codeHash
let filename := modBase ++ ".lean"
let mod := "Examples." ++ modBase
let jsonFile := s!"{modBase}.json"
let jsonPath := (projectDir / "Examples" / jsonFile)
let jsonExists : Bool ←
if (← jsonPath.pathExists) then (IO.FS.readFile jsonPath) <&> (!·.isEmpty)
else pure false
unless jsonExists do
IO.FS.writeFile (projectDir / "Examples" / filename) contents
-- Validate that the path is really a Lean project
let lakefile := projectDir / "lakefile.lean"
let lakefile' := projectDir / "lakefile.toml"
if !(← lakefile.pathExists) && !(← lakefile'.pathExists) then
throwError m!"Neither {lakefile} nor {lakefile'} exist, couldn't load project"
let toolchainfile := projectDir / "lean-toolchain"
let toolchain ← do
if !(← toolchainfile.pathExists) then
throwError m!"File {toolchainfile} doesn't exist, couldn't load project"
pure (← IO.FS.readFile toolchainfile).trimAscii.copy
-- Kludge: remove variables introduced by Lake. Clearing out DYLD_LIBRARY_PATH and
-- LD_LIBRARY_PATH is useful so the version selected by Elan doesn't get the wrong shared
-- libraries.
let lakeVars :=
#["LAKE", "LAKE_HOME", "LAKE_PKG_URL_MAP",
"LEAN_SYSROOT", "LEAN_AR", "LEAN_PATH", "LEAN_SRC_PATH",
"LEAN", "ELAN", "ELAN_HOME", "LEAN_GITHASH",
"ELAN_TOOLCHAIN", "DYLD_LIBRARY_PATH", "LD_LIBRARY_PATH"]
let toolchainFile ← IO.FS.Handle.mk toolchainfile .read
toolchainFile.lock (exclusive := true)
try
let cmd := "elan"
let runCmd' (args : Array String) : m Unit := do
let res ← IO.Process.output {
cmd, args, cwd := projectDir
-- Unset Lake's environment variables
env := lakeVars.map (·, none)
}
if res.exitCode != 0 then reportFail projectDir cmd args res
let runCmd (trace : MessageData) (args : Array String) : m Unit :=
withTraceNode `Elab.Verso.Code.External.loadModule (fun _ => pure trace) (runCmd' args)
runCmd m!"loadModuleContent': building subverso" #["run", "--install", toolchain, "lake", "build", "subverso-extract-mod"]
runCmd m!"loadModuleContent': building example project's module" #["run", "--install", toolchain, "lake", "build", "+" ++ mod]
let suppressArgs :=
if let some nss := suppressNamespaces then
nss |>.splitToList (· == ' ') |>.filter (!String.isEmpty ·) |>.map (#["--suppress-namespace", ·]) |>.toArray |>.flatten
else #[]
withTraceNode `Elab.Verso.Code.External.loadModule (fun _ => pure m!"loadModuleContent': extracting '{mod}'") do
let args :=
#["run", "--install", toolchain, "lake", "exe", "subverso-extract-mod"] ++
suppressArgs ++
#[mod, "Examples/" ++ jsonFile]
runCmd' args
finally
toolchainFile.unlock
let jsonString ← IO.FS.readFile (projectDir / "Examples" / jsonFile)
let .ok json := Json.parse jsonString
| if jsonString.isEmpty then
throwError s!"Expected JSON in {projectDir / "Examples" / jsonFile}, got empty output"
else
throwError s!"Expected JSON in {projectDir / "Examples" / jsonFile}, got {jsonString}"
match Module.fromJson? json with
| .error err =>
throwError s!"Couldn't parse JSON from output file: {err}\nIn:\n{json}"
| .ok m => pure m.items
where
decorateOut (name : String) (out : String) : String :=
if out.isEmpty then "" else s!"\n{name}:\n{out}\n"
reportFail {α} (projectDir : FilePath) (cmd : String) (args : Array String) (res : IO.Process.Output) : m α := do
throwError ("Build process failed." ++
"\nCWD: " ++ projectDir.toString ++
"\nCommand: " ++ cmd ++
"\nArgs: " ++ repr args ++
"\nExit code: " ++ toString res.exitCode ++
"\nstdout: " ++ res.stdout ++
"\nstderr: " ++ res.stderr)
end
open Std.Iterators in
private def hasAtLeastM [Monad m] [Iterator α m β] [Productive α m] (it : IterM (α := α) m β) : Nat → m Bool
| 0 => pure true
| n + 1 => do
match (← it.step).inflate with
| .done .. => pure false
| .skip it' .. => hasAtLeastM it' (n + 1)
| .yield it' .. => hasAtLeastM it' n
termination_by n => (n, it.finitelyManySkips)
open Std.Iterators in
private def hasAtLeast [Iterator α Id β] [Productive α Id] (it : Iter (α := α) β) (n : Nat) : Bool :=
hasAtLeastM it.toIterM n
def splitExample (code : Highlighted) : Option Highlighted × Highlighted := Id.run do
let lines := code.lines
let mut out := .empty
for h : i in [0:lines.size] do
let line := lines[i]
if isSplit line then
return (some out, lines.extract (i+1) lines.size |>.foldl (init := .empty) (· ++ ·))
out := out ++ line
return (none, out)
where
isSplit (line : Highlighted) : Bool :=
let trimmed := line.toString.trimAscii
hasAtLeast trimmed.positions 4 && trimmed.all (· == '-')
partial def hlIsWs (hl : Highlighted) : Bool :=
match hl with
| .text s | .unparsed s => s.all Char.isWhitespace
| .seq xs => xs.all hlIsWs
| .span _ x | .tactics _ _ _ x => hlIsWs x
| .point .. => true
| .token .. => false
section
open SubVerso.Module
private inductive LineType where
| whitespace
| split
| other
private def lineType (line : Highlighted) : LineType :=
let trimmed := line.toString.trimAscii
if trimmed.isEmpty then .whitespace
else if hasAtLeast trimmed.positions 4 && trimmed.all (· == '-') then .split
else .other
def detachPrefix (code : Array ModuleItem) : Option (Array ModuleItem) × Array ModuleItem := Id.run do
let mut out : Array ModuleItem := #[]
for h : i in [0:code.size] do
let lines := code[i].code.lines
-- Check for pre-split
for h : j in [0:lines.size] do
match lineType lines[j] with
| .whitespace => continue
| .other => break -- not a pre-split
| .split =>
let leadingWs := lines.extract 0 j
let pre := out.modify (out.size - 1) (fun c => {c with code := c.code ++ .seq leadingWs})
return (pre, #[{code[i] with code := .seq <| lines.extract (j+1)}] ++ code.extract (i + 1) code.size)
-- Check for post-split
for h : j in [0:lines.size] do
let j' := lines.size - (j + 1)
have : j' < lines.size := by
have : j < lines.size := by get_elem_tactic
omega
match lineType lines[j'] with
| .whitespace => continue
| .other => break -- not a post-split
| .split =>
let trailingWs := lines.extract (j' + 1) lines.size
let thisItem := { code[i] with code := lines.extract 0 j' |>.foldl (init := .empty) (· ++ ·) }
let nextItem? := code[i + 1]? |>.map (fun c => {c with code := Highlighted.seq trailingWs ++ c.code})
return (out.push thisItem, nextItem?.toArray ++ code.extract (i + 2) code.size)
-- No split here
out := out.push code[i]
-- No split anywhere
return (none, out)
def detachSuffix (code : Array ModuleItem) : Array ModuleItem × Option (Array ModuleItem) := Id.run do
let mut out : Array ModuleItem := #[]
for h : i in [0:code.size] do
let i := code.size - (i + 1)
have : i < code.size := by
rename_i i' _ _
have : i' < code.size := by get_elem_tactic
omega
let lines := code[i].code.lines
-- Check for post-split
for h : j in [0:lines.size] do
let j' := lines.size - (j + 1)
have : j' < lines.size := by
have : j < lines.size := by get_elem_tactic
omega
match lineType lines[j'] with
| .whitespace => continue
| .other => break -- not a post-split
| .split =>
let trailingWs := lines.extract (j' + 1) lines.size
let thisItem := { code[i] with code := lines.extract 0 j' |>.foldl (init := .empty) (· ++ ·) }
let nextItem? := if i > 0 then some {code[i - 1] with code := Highlighted.seq trailingWs ++ code[i-1].code} else none
return (nextItem?.toArray ++ code.extract (i + 2) code.size, out.push thisItem |>.reverse)
-- Check for pre-split
for h : j in [0:lines.size] do
match lineType lines[j] with
| .whitespace => continue
| .other => break -- not a pre-split
| .split =>
let leadingWs := lines.extract 0 j
return (code.extract 0 i |>.modify (i-1) (fun c => {c with code := c.code ++ .seq leadingWs}), some (out.push {code[i] with code := .seq (lines.extract (j+1))} |>.reverse))
-- No split here
out := out.push code[i]
-- No split anywhere
return (out.reverse, none)
def splitExample' (code : Array ModuleItem) : Option (Array ModuleItem) × Array ModuleItem × Option (Array ModuleItem) :=
let (pre, code) := detachPrefix code
let (code, suffix) := detachSuffix code
(pre, code, suffix)
end
def copyButtonCss : String :=
r#"
.tpil-code-container {
position: relative;
margin: 20px 0;
}
.copy-btn,
.toggle-btn {
position: absolute;
top: 0px;
background: inherit;
color: black;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 12px;
display: flex;
align-items: center;
opacity: 0.7;
transition: opacity 0.2s;
padding: 8px;
}
.copy-btn:hover,
.toggle-btn:hover {
opacity: 1;
background: #555;
}
.copy-btn {
right: 10px;
gap: 5px;
}
.toggle-btn {
right: 45px; /* Position to the left of copy button */
}
.copy-btn.copied {
background: #28a745;
}
.copy-icon,
.toggle-icon {
width: 14px;
height: 14px;
}
.hidden {
display: grid;
grid-template-rows: 1fr;
transition: grid-template-rows 0.3s ease-out, opacity 0.2s ease-out, margin 0.3s ease-out;
opacity: 0.8;
}
.tpil-hide-prefix .hidden {
grid-template-rows: 0fr;
opacity: 0;
margin: 0;
}
.hidden > .hl.lean {
overflow: hidden;
margin-bottom: 0;
}
"#
def copyButtonJs : String :=
r#"
function addToggleButtonToElement(elementId, className = 'tpil-hide-prefix') {
const element = document.getElementById(elementId);
if (!element) {
console.error(`Element with ID '${elementId}' not found`);
return false;
}
// Create container wrapper if it doesn't exist
let container = element.parentElement;
if (!container.classList.contains('code-container')) {
container = document.createElement('div');
container.className = 'code-container';
container.classList.toggle(className);
// Insert container before element
element.parentNode.insertBefore(container, element);
// Move element into container
container.appendChild(element);
}
// Remove existing toggle button if present
const existingBtn = container.querySelector('.toggle-btn');
if (existingBtn) {
existingBtn.remove();
}
// Create toggle button
const toggleBtn = document.createElement('button');
toggleBtn.className = 'toggle-btn';
toggleBtn.title = 'Show hidden lines';
toggleBtn.innerHTML = `
`;
// Add click event listener
toggleBtn.addEventListener('click', () => {
container.classList.toggle(className);
// Update icon based on state
const isHidden = container.classList.contains(className);
toggleBtn.innerHTML = !isHidden ? `
` : `
`;
toggleBtn.title = isHidden ? 'Show hidden lines' : 'Hide lines';
});
// Position toggle button to the left of copy button if it exists
const copyBtn = container.querySelector('.copy-btn');
if (copyBtn) {
container.insertBefore(toggleBtn, copyBtn);
} else {
container.appendChild(toggleBtn);
}
return true;
}
function addCopyButtonToElement(elementId, codeText) {
const element = document.getElementById(elementId);
if (!element) {
console.error(`Element with ID '${elementId}' not found`);
return false;
}
// Create container wrapper if it doesn't exist
let container = element.parentElement;
if (!container.classList.contains('code-container')) {
container = document.createElement('div');
container.className = 'tpil-code-container';
// Insert container before element
element.parentNode.insertBefore(container, element);
// Move element into container
container.appendChild(element);
}
// Remove existing copy button if present
const existingBtn = container.querySelector('.copy-btn');
if (existingBtn) {
existingBtn.remove();
}
// Create copy button
const copyBtn = document.createElement('button');
copyBtn.className = 'copy-btn';
copyBtn.title = 'Copy to clipboard';
copyBtn.innerHTML = `
`;
// Add click event listener
copyBtn.addEventListener('click', async () => {
try {
// Copy the provided code text to clipboard
await navigator.clipboard.writeText(codeText);
// Show success feedback
const originalText = copyBtn.innerHTML;
copyBtn.innerHTML = `
Copied!
`;
copyBtn.classList.add('copied');
// Reset after 2 seconds
setTimeout(() => {
copyBtn.innerHTML = originalText;
copyBtn.classList.remove('copied');
}, 2000);
} catch (err) {
// Fallback for older browsers
fallbackCopyTextToClipboard(codeText);
// Show feedback
copyBtn.textContent = 'Copied!';
setTimeout(() => {
copyBtn.innerHTML = `
`;
}, 2000);
}
});
// Add button to container
container.appendChild(copyBtn);
return true;
}
// Fallback function for older browsers
function fallbackCopyTextToClipboard(text) {
const textArea = document.createElement('textarea');
textArea.value = text;
textArea.style.position = 'fixed';
textArea.style.left = '-999999px';
textArea.style.top = '-999999px';
document.body.appendChild(textArea);
textArea.focus();
textArea.select();
try {
document.execCommand('copy');
} catch (err) {
console.error('Fallback: Oops, unable to copy', err);
}
document.body.removeChild(textArea);
}
// Expose API function globally
window.addCopyButtonToElement = addCopyButtonToElement;
window.addToggleButtonToElement = addToggleButtonToElement;
"#
def examplesCss := r#"
.example {
margin-left: 0.75em;
}
.example .hl.lean.block {
margin: 0;
}
.example .information pre {
margin: 0 0 0.25em 0;
}
"#
def tpilBlock (block : BlockDescr) : BlockDescr :=
{ block with
extraJsFiles :=
block.extraJsFiles
|>.insert { filename := "copybutton.js", contents := copyButtonJs, sourceMap? := none }
extraCssFiles :=
block.extraCssFiles
|>.insert { filename := "copybutton.css", contents := copyButtonCss }
|>.insert { filename := "examples.css", contents := examplesCss }
}
def tpilInline (inline : InlineDescr) : InlineDescr :=
{ inline with
extraJsFiles :=
inline.extraJsFiles
|>.insert {filename := "copybutton.js", contents := copyButtonJs, sourceMap? := none}
extraCssFiles :=
inline.extraCssFiles
|>.insert { filename := "copybutton.css", contents := copyButtonCss }
|>.insert { filename := "examples.css", contents := examplesCss }
}
structure ExampleItem where
code : Highlighted
output : Option Highlighted.Message
trailing : String
deriving ToJson, FromJson, Repr, Quote
def verbatimBlock (cmd : Highlighted) : TeX :=
let contents := cmd.trimOneTrailingNl.trimOneLeadingNl.toVerbatimTeX
.seq #[.raw s!"\\begin\{LeanVerbatim}[vspace=0pt]\n", contents, .raw "\n\\end{LeanVerbatim}\n"]
block_extension Block.lean
(allowToggle : Bool)
(pre : Option Highlighted)
(code : Array ExampleItem)
(post : Option Highlighted)
(goalVisibility : HighlightHtmlM.VisibleProofStates := .none)
via withHighlighting, tpilBlock where
data :=
let defined : Array (Name × String) := code.flatMap (definedNames ·.code)
.arr #[.bool allowToggle, toJson pre, toJson code, toJson post, toJson goalVisibility, toJson defined]
traverse id data _ := do
let .arr #[_allowToggle, _pre, _code, _post, _visibility, definesJson] := data
| logError s!"Expected array for Lean block, got {data.compress}"; return none
match FromJson.fromJson? definesJson with
| .error err =>
logError <| "Failed to deserialize code config during traversal:" ++ err
return none
| .ok (defines : Array (Name × String)) =>
for (d, s) in defines do
if d.isAnonymous then continue
let d := d.toString
let path ← (·.path) <$> read
let _ ← externalTag id path d
let context := (← read).headers.map (·.titleString)
modify (·.saveDomainObject exampleDomain d id)
if let some link := (← get).externalTags[id]? then
modify (·.modifyDomainObjectData exampleDomain d fun v =>
let v := if let .obj _ := v then v else .obj {}
v.setObjVal! link.link (json%{"context": $context, "display": $s}))
pure none
toTeX :=
open Verso.Output.TeX in
open Verso.Doc.TeX in
some <| fun _ _ _ data _ => do
let .arr #[.bool _allowToggle, hlPreJson, hlJson, hlPostJson, goalVisibilityJson, _defs] := data
| logError "Expected five-element JSON for Lean code"
pure .empty
let pre ←
match FromJson.fromJson? (α := Option Highlighted) hlPreJson with
| .error err =>
logError <| "Couldn't deserialize Lean code intro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let code ←
match FromJson.fromJson? (α := Array ExampleItem) hlJson with
| .error err =>
logError <| "Couldn't deserialize Lean code block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let _post ←
match FromJson.fromJson? (α := Option Highlighted) hlPostJson with
| .error err =>
logError <| "Couldn't deserialize Lean code outro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let _visibility ←
match FromJson.fromJson? (α := HighlightHtmlM.VisibleProofStates) goalVisibilityJson with
| .error err =>
logError <| "Couldn't deserialize Lean code outro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let codeIndent := code.foldl (init := pre.map (·.indentation)) (fun i? y => i?.map (min · y.1.indentation)) |>.getD 0
let mut codeTeX : TeX := .empty
for ⟨cmd, out?, ws⟩ in code do
let cmd := cmd.deIndent codeIndent
codeTeX := codeTeX ++ verbatimBlock cmd
if let some msg := out? then
codeTeX := codeTeX ++ (← msg.toTeX)
unless ws.isEmpty do
codeTeX := codeTeX ++ (← (Highlighted.text ws).toTeX)
pure codeTeX
extraJsFiles := .ofList [{ filename := "copybutton.js", contents := copyButtonJs, sourceMap? := none }]
extraCssFiles := .ofList [
{ filename := "copybutton.css", contents := copyButtonCss },
{ filename := "examples.css", contents := examplesCss }
]
toHtml :=
open Verso.Output.Html in
some <| fun _ _ _ data _ => do
let .arr #[.bool allowToggle, hlPreJson, hlJson, hlPostJson, goalVisibilityJson, _defs] := data
| HtmlT.logError "Expected five-element JSON for Lean code"
pure .empty
let pre ←
match FromJson.fromJson? (α := Option Highlighted) hlPreJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code intro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let code ←
match FromJson.fromJson? (α := Array ExampleItem) hlJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let post ←
match FromJson.fromJson? (α := Option Highlighted) hlPostJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code outro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let visibility ←
match FromJson.fromJson? (α := HighlightHtmlM.VisibleProofStates) goalVisibilityJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code outro block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let codeIndent := code.foldl (init := pre.map (·.indentation)) (fun i? y => i?.map (min · y.1.indentation)) |>.getD 0
let mut codeHtml : Html := .empty
let mut codeString := ""
if allowToggle then
if let some p := pre then
let p := p.deIndent codeIndent
let inner ←
withDefinitionsAsTargets false <|
withVisibleProofStates visibility <|
p.trimOneLeadingNl |>.blockHtml "examples" (trim := false) (g := Verso.Genre.Manual)
codeHtml := codeHtml ++ {{
{{ inner }}
}}
codeString := codeString ++ p.toString
for ⟨cmd, out?, ws⟩ in code do
let cmd := cmd.deIndent codeIndent
let moreCode ←
withDefinitionsAsTargets true <|
withVisibleProofStates visibility <|
cmd.trimOneLeadingNl |>.blockHtml "examples" (trim := false) (g := Verso.Genre.Manual)
codeHtml := codeHtml ++ moreCode
codeString := codeString ++ cmd.toString
if let some msg := out? then
let msgHtml ← msg.toHtml (g := Verso.Genre.Manual) []
codeHtml := codeHtml ++ {{
{{msgHtml}}
}}
unless ws.isEmpty do
codeHtml := codeHtml ++ (← (Highlighted.text ws).blockHtml "examples" (trim := false) (g := Verso.Genre.Manual))
if allowToggle then
if let some p := post then
let p := p.deIndent codeIndent
let inner ←
withDefinitionsAsTargets false <|
withVisibleProofStates visibility <|
p.trimOneLeadingNl |>.blockHtml "examples" (trim := false) (g := Verso.Genre.Manual)
codeHtml := codeHtml ++ {{
{{ inner }}
}}
codeString := codeString ++ p.toString
let i ← uniqueId (g := Verso.Genre.Manual)
let toCopy := (pre.map (·.toString)).getD "" ++ codeString
let mut script := s!"addCopyButtonToElement({i.quote}, {toCopy.quote});"
if allowToggle && (pre.isSome || post.isSome) then
script := script ++ s!"\naddToggleButtonToElement({i.quote});"
return {{
{{codeHtml}}
}}
block_extension Block.leanAnchor (code : Highlighted) (completeCode : String)
via withHighlighting, tpilBlock where
data := .arr #[toJson code, toJson completeCode]
traverse _ _ _ := pure none
toTeX :=
open Verso.Output.TeX in
open Verso.Doc.TeX in
some <| fun _ _ _ data _ => do
let .arr #[hlJson, completeCodeJson] := data
| logError "Expected two-element JSON for Lean code"
pure .empty
let code ←
match FromJson.fromJson? (α := Highlighted) hlJson with
| .error err =>
logError <| "Couldn't deserialize Lean code block while rendering TeX: " ++ err
return .empty
| .ok hl => pure hl
let _completeCode ←
match FromJson.fromJson? (α := String) completeCodeJson with
| .error err =>
logError <| "Couldn't deserialize Lean code string while rendering TeX: " ++ err
return .empty
| .ok hl => pure hl
let code := code.deIndent code.indentation
pure (verbatimBlock code)
toHtml :=
open Verso.Output.Html in
some <| fun _ _ _ data _ => do
let .arr #[hlJson, completeCodeJson] := data
| HtmlT.logError "Expected two-element JSON for Lean code"
pure .empty
let code ←
match FromJson.fromJson? (α := Highlighted) hlJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code block while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let completeCode ←
match FromJson.fromJson? (α := String) completeCodeJson with
| .error err =>
HtmlT.logError <| "Couldn't deserialize Lean code string while rendering HTML: " ++ err
return .empty
| .ok hl => pure hl
let code := code.deIndent code.indentation
let codeHtml : Html := (← code.blockHtml "examples" (trim := false) (g := Verso.Genre.Manual))
let i ← uniqueId (g := Verso.Genre.Manual)
let mut script := s!"addCopyButtonToElement({i.quote}, {completeCode.quote});"
return {{
{{codeHtml}}
}}
def proofStateStyle := r#"
.hl.lean.proof-state-view {
white-space: collapse;
margin-left: 0.75em;
}
.hl.lean.proof-state-view .hypothesis {
display: table !important;
border-spacing: 0 0.2rem;
border-collapse: separate;
}
.hl.lean.proof-state-view .tactic-state {
display: block;
left: 0;
padding: 0;
border: none;
}
.hl.lean.proof-state-view .tactic-state:has(.goal + .goal) {
display: flex;
flex-wrap: wrap;
gap: 2rem;
justify-content: space-evenly;
width: 100%;
}
.hl.lean.proof-state-view .tactic-state .goal {
margin: 0 !important;
align-self: flex-start;
width: fit-content;
}
"#
block_extension Block.goals (goals : Array (Highlighted.Goal Highlighted))
via withHighlighting, tpilBlock where
data := toJson goals
traverse _ _ _ := pure none
toTeX :=
open Verso.Output.TeX in
open Verso.Doc.TeX in
some <| fun _ _ _ data _ => do
let goals ←
match fromJson? (α := Array (Highlighted.Goal Highlighted)) data with
| .ok v => pure v
| .error e =>
logError <| "Failed to deserialize proof state: " ++ e
return .empty
-- TODO: lay these out side-by-side
pure <| .seq (← goals.mapM (·.toTeX))
extraCssFiles := .ofList [{ filename := "proof-state.css", contents := proofStateStyle }]
toHtml :=
open Verso.Output.Html in
some <| fun _ _ _ data _ => do
let goals ←
match fromJson? (α := Array (Highlighted.Goal Highlighted)) data with
| .ok v => pure v
| .error e =>
HtmlT.logError <| "Failed to deserialize proof state: " ++ e
return .empty
pure {{
{{← if goals.isEmpty then
pure {{"All goals completed! 🐙"}}
else
withCollapsedSubgoals (g := Verso.Genre.Manual) .never <|
.seq <$> (goals.mapIdxM (fun i x => x.toHtml (·.toHtml) i))}}