From dc5f7a2a883302bd9f51dcbd693bec0a4e9a2063 Mon Sep 17 00:00:00 2001 From: ice1000 Date: Tue, 18 Jun 2024 23:24:01 -0400 Subject: [PATCH] blog: write more about HIT --- aya/guide/prover-tutorial.aya.md | 32 +- package.json | 2 +- pnpm-lock.yaml | 524 ++++++++++++++++--------------- 3 files changed, 306 insertions(+), 252 deletions(-) diff --git a/aya/guide/prover-tutorial.aya.md b/aya/guide/prover-tutorial.aya.md index 8da916d..9e9497e 100644 --- a/aya/guide/prover-tutorial.aya.md +++ b/aya/guide/prover-tutorial.aya.md @@ -271,10 +271,34 @@ open inductive Interval ``` This is an uninteresting quotient type, that is basically `Bool`{} but saying its two values are equal, -so it's really just `Unit`. The type of `line` will be translated into `I -> Interval`{} -together with the judgmental equality that `line 0`{} is `left`{} and `line 1`{} is `right`{}, -basically a desugaring of the equality with additional features. -Every time we do pattern matching, we need to make sure it preserves these judgmental equalities. +so it's really just a unit type, with its unique element being the equivalence class of `left`{} and `right`{}. + +If you're familiar with a proof assistant with an intensional equality like +Coq/Agda/Lean/etc., you might find this surprising because a unit type shall not have two distinct elements, +and an equality shall not be stated between two distinct constructors. How does this work in Aya? + +Actually, in these systems, the equality is defined _inductively_, and it only has one constructor -- `refl`. +This is not how equality is defined in Aya, so we can cook some interesting equality proofs into it, +which includes these equality-looking constructors. + +1. The type of `line` will be translated into `I -> Interval`{} + together with the judgmental equality that `line 0`{} is `left`{} and `line 1`{} is `right`{}, + basically a desugaring of the equality with additional features. + This makes `line` a valid constructor in normal type theory: it takes some parameters and returns `Interval`{}. +2. These judgmental equalities need to be preserved by the elimination rule of `Interval`{}. + Here is an example elimination: + +```aya +example def Interval-elim {a b : A} {p : a = b} (i : Interval) : A elim i +| left => a +| right => b +| line j => p j +``` + +Note that the term `pmap Interval-elim line`, which reduces to `p`, +has type `Interval-elim left = Interval-elim right`, +so we need to check if `p 0` equals `Interval-elim left`, and `p 1` equals `Interval-elim right`. +This is a _confluence check_ that ensures the elimination is well-defined. What's interesting about this type, is that its elimination implies function extensionality: diff --git a/package.json b/package.json index f9755aa..d43da98 100644 --- a/package.json +++ b/package.json @@ -28,7 +28,7 @@ "@types/markdown-it-footnote": "^3.0.4", "@vitejs/plugin-vue-jsx": "^4.0.0", "vitepress": "1.2.3", - "vue": "^3.4.27" + "vue": "^3.4.29" }, "pnpm": { "overrides": { diff --git a/pnpm-lock.yaml b/pnpm-lock.yaml index 1e9f4f7..abf884c 100644 --- a/pnpm-lock.yaml +++ b/pnpm-lock.yaml @@ -23,13 +23,13 @@ importers: version: 3.0.4 '@vitejs/plugin-vue-jsx': specifier: ^4.0.0 - version: 4.0.0(vite@5.2.12)(vue@3.4.27) + version: 4.0.0(vite@5.3.1)(vue@3.4.29) vitepress: specifier: 1.2.3 version: 1.2.3(@algolia/client-search@4.23.3)(postcss@8.4.38)(search-insights@2.14.0) vue: - specifier: ^3.4.27 - version: 3.4.27 + specifier: ^3.4.29 + version: 3.4.29 packages: @@ -266,140 +266,140 @@ packages: search-insights: optional: true - '@esbuild/aix-ppc64@0.20.2': - resolution: {integrity: sha512-D+EBOJHXdNZcLJRBkhENNG8Wji2kgc9AZ9KiPr1JuZjsNtyHzrsfLRrY0tk2H2aoFu6RANO1y1iPPUCDYWkb5g==} + '@esbuild/aix-ppc64@0.21.5': + resolution: {integrity: sha512-1SDgH6ZSPTlggy1yI6+Dbkiz8xzpHJEVAlF/AM1tHPLsf5STom9rwtjE4hKAF20FfXXNTFqEYXyJNWh1GiZedQ==} engines: {node: '>=12'} cpu: [ppc64] os: [aix] - 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shiki: 1.6.2 + shiki: 1.7.0 '@types/estree@1.0.5': {} @@ -1411,20 +1426,20 @@ snapshots: '@types/web-bluetooth@0.0.20': {} - '@vitejs/plugin-vue-jsx@4.0.0(vite@5.2.12)(vue@3.4.27)': + '@vitejs/plugin-vue-jsx@4.0.0(vite@5.3.1)(vue@3.4.29)': dependencies: '@babel/core': 7.24.7 '@babel/plugin-transform-typescript': 7.24.7(@babel/core@7.24.7) '@vue/babel-plugin-jsx': 1.2.2(@babel/core@7.24.7) - vite: 5.2.12 - vue: 3.4.27 + vite: 5.3.1 + vue: 3.4.29 transitivePeerDependencies: - supports-color - '@vitejs/plugin-vue@5.0.5(vite@5.2.12)(vue@3.4.27)': + '@vitejs/plugin-vue@5.0.5(vite@5.3.1)(vue@3.4.29)': dependencies: - vite: 5.2.12 - vue: 3.4.27 + vite: 5.3.1 + vue: 3.4.29 '@vscode/markdown-it-katex@1.0.3': dependencies: @@ -1457,106 +1472,109 @@ snapshots: '@babel/helper-module-imports': 7.22.15 '@babel/helper-plugin-utils': 7.24.7 '@babel/parser': 7.24.7 - '@vue/compiler-sfc': 3.4.27 + '@vue/compiler-sfc': 3.4.29 - '@vue/compiler-core@3.4.27': + '@vue/compiler-core@3.4.29': dependencies: '@babel/parser': 7.24.7 - '@vue/shared': 3.4.27 + '@vue/shared': 3.4.29 entities: 4.5.0 estree-walker: 2.0.2 source-map-js: 1.2.0 - '@vue/compiler-dom@3.4.27': + '@vue/compiler-dom@3.4.29': dependencies: - '@vue/compiler-core': 3.4.27 - '@vue/shared': 3.4.27 + '@vue/compiler-core': 3.4.29 + '@vue/shared': 3.4.29 - '@vue/compiler-sfc@3.4.27': + '@vue/compiler-sfc@3.4.29': dependencies: '@babel/parser': 7.24.7 - '@vue/compiler-core': 3.4.27 - '@vue/compiler-dom': 3.4.27 - '@vue/compiler-ssr': 3.4.27 - '@vue/shared': 3.4.27 + '@vue/compiler-core': 3.4.29 + '@vue/compiler-dom': 3.4.29 + '@vue/compiler-ssr': 3.4.29 + '@vue/shared': 3.4.29 estree-walker: 2.0.2 magic-string: 0.30.10 postcss: 8.4.38 source-map-js: 1.2.0 - '@vue/compiler-ssr@3.4.27': + '@vue/compiler-ssr@3.4.29': dependencies: - '@vue/compiler-dom': 3.4.27 - '@vue/shared': 3.4.27 + '@vue/compiler-dom': 3.4.29 + '@vue/shared': 3.4.29 - '@vue/devtools-api@7.2.1(vue@3.4.27)': + '@vue/devtools-api@7.3.1(vue@3.4.29)': dependencies: - '@vue/devtools-kit': 7.2.1(vue@3.4.27) + '@vue/devtools-kit': 7.3.1(vue@3.4.29) transitivePeerDependencies: - vue - '@vue/devtools-kit@7.2.1(vue@3.4.27)': + '@vue/devtools-kit@7.3.1(vue@3.4.29)': dependencies: - '@vue/devtools-shared': 7.2.1 + '@vue/devtools-shared': 7.3.1 + birpc: 0.2.17 hookable: 5.5.3 mitt: 3.0.1 perfect-debounce: 1.0.0 speakingurl: 14.0.1 - vue: 3.4.27 + superjson: 2.2.1 + vue: 3.4.29 - '@vue/devtools-shared@7.2.1': + '@vue/devtools-shared@7.3.1': dependencies: - rfdc: 1.3.1 + rfdc: 1.4.1 - '@vue/reactivity@3.4.27': + '@vue/reactivity@3.4.29': dependencies: - '@vue/shared': 3.4.27 + '@vue/shared': 3.4.29 - '@vue/runtime-core@3.4.27': + '@vue/runtime-core@3.4.29': dependencies: - '@vue/reactivity': 3.4.27 - '@vue/shared': 3.4.27 + '@vue/reactivity': 3.4.29 + '@vue/shared': 3.4.29 - '@vue/runtime-dom@3.4.27': + '@vue/runtime-dom@3.4.29': dependencies: - '@vue/runtime-core': 3.4.27 - '@vue/shared': 3.4.27 + '@vue/reactivity': 3.4.29 + '@vue/runtime-core': 3.4.29 + '@vue/shared': 3.4.29 csstype: 3.1.3 - '@vue/server-renderer@3.4.27(vue@3.4.27)': + '@vue/server-renderer@3.4.29(vue@3.4.29)': dependencies: - '@vue/compiler-ssr': 3.4.27 - '@vue/shared': 3.4.27 - vue: 3.4.27 + '@vue/compiler-ssr': 3.4.29 + '@vue/shared': 3.4.29 + vue: 3.4.29 - '@vue/shared@3.4.27': {} + '@vue/shared@3.4.29': {} - '@vueuse/core@10.10.0(vue@3.4.27)': + '@vueuse/core@10.11.0(vue@3.4.29)': dependencies: '@types/web-bluetooth': 0.0.20 - '@vueuse/metadata': 10.10.0 - '@vueuse/shared': 10.10.0(vue@3.4.27) - vue-demi: 0.14.8(vue@3.4.27) + '@vueuse/metadata': 10.11.0 + '@vueuse/shared': 10.11.0(vue@3.4.29) + vue-demi: 0.14.8(vue@3.4.29) transitivePeerDependencies: - '@vue/composition-api' - vue - '@vueuse/integrations@10.10.0(focus-trap@7.5.4)(vue@3.4.27)': + '@vueuse/integrations@10.11.0(focus-trap@7.5.4)(vue@3.4.29)': dependencies: - '@vueuse/core': 10.10.0(vue@3.4.27) - '@vueuse/shared': 10.10.0(vue@3.4.27) - vue-demi: 0.14.8(vue@3.4.27) + '@vueuse/core': 10.11.0(vue@3.4.29) + '@vueuse/shared': 10.11.0(vue@3.4.29) + vue-demi: 0.14.8(vue@3.4.29) optionalDependencies: focus-trap: 7.5.4 transitivePeerDependencies: - '@vue/composition-api' - vue - '@vueuse/metadata@10.10.0': {} + '@vueuse/metadata@10.11.0': {} - '@vueuse/shared@10.10.0(vue@3.4.27)': + '@vueuse/shared@10.11.0(vue@3.4.29)': dependencies: - vue-demi: 0.14.8(vue@3.4.27) + vue-demi: 0.14.8(vue@3.4.29) transitivePeerDependencies: - '@vue/composition-api' - vue @@ -1583,16 +1601,18 @@ snapshots: dependencies: color-convert: 1.9.3 - browserslist@4.23.0: + birpc@0.2.17: {} + + browserslist@4.23.1: dependencies: - caniuse-lite: 1.0.30001629 - electron-to-chromium: 1.4.792 + caniuse-lite: 1.0.30001636 + electron-to-chromium: 1.4.806 node-releases: 2.0.14 - update-browserslist-db: 1.0.16(browserslist@4.23.0) + update-browserslist-db: 1.0.16(browserslist@4.23.1) camelcase@6.3.0: {} - caniuse-lite@1.0.30001629: {} + caniuse-lite@1.0.30001636: {} chalk@2.4.2: dependencies: @@ -1610,41 +1630,45 @@ snapshots: convert-source-map@2.0.0: {} + copy-anything@3.0.5: + dependencies: + is-what: 4.1.16 + csstype@3.1.3: {} debug@4.3.5: dependencies: ms: 2.1.2 - electron-to-chromium@1.4.792: {} + electron-to-chromium@1.4.806: {} entities@4.5.0: {} - esbuild@0.20.2: + esbuild@0.21.5: optionalDependencies: - '@esbuild/aix-ppc64': 0.20.2 - '@esbuild/android-arm': 0.20.2 - '@esbuild/android-arm64': 0.20.2 - '@esbuild/android-x64': 0.20.2 - '@esbuild/darwin-arm64': 0.20.2 - '@esbuild/darwin-x64': 0.20.2 - '@esbuild/freebsd-arm64': 0.20.2 - '@esbuild/freebsd-x64': 0.20.2 - '@esbuild/linux-arm': 0.20.2 - '@esbuild/linux-arm64': 0.20.2 - '@esbuild/linux-ia32': 0.20.2 - '@esbuild/linux-loong64': 0.20.2 - '@esbuild/linux-mips64el': 0.20.2 - '@esbuild/linux-ppc64': 0.20.2 - '@esbuild/linux-riscv64': 0.20.2 - '@esbuild/linux-s390x': 0.20.2 - '@esbuild/linux-x64': 0.20.2 - '@esbuild/netbsd-x64': 0.20.2 - '@esbuild/openbsd-x64': 0.20.2 - '@esbuild/sunos-x64': 0.20.2 - '@esbuild/win32-arm64': 0.20.2 - '@esbuild/win32-ia32': 0.20.2 - '@esbuild/win32-x64': 0.20.2 + '@esbuild/aix-ppc64': 0.21.5 + '@esbuild/android-arm': 0.21.5 + '@esbuild/android-arm64': 0.21.5 + '@esbuild/android-x64': 0.21.5 + '@esbuild/darwin-arm64': 0.21.5 + '@esbuild/darwin-x64': 0.21.5 + '@esbuild/freebsd-arm64': 0.21.5 + '@esbuild/freebsd-x64': 0.21.5 + '@esbuild/linux-arm': 0.21.5 + '@esbuild/linux-arm64': 0.21.5 + '@esbuild/linux-ia32': 0.21.5 + '@esbuild/linux-loong64': 0.21.5 + '@esbuild/linux-mips64el': 0.21.5 + '@esbuild/linux-ppc64': 0.21.5 + '@esbuild/linux-riscv64': 0.21.5 + '@esbuild/linux-s390x': 0.21.5 + '@esbuild/linux-x64': 0.21.5 + '@esbuild/netbsd-x64': 0.21.5 + '@esbuild/openbsd-x64': 0.21.5 + '@esbuild/sunos-x64': 0.21.5 + '@esbuild/win32-arm64': 0.21.5 + '@esbuild/win32-ia32': 0.21.5 + '@esbuild/win32-x64': 0.21.5 escalade@3.1.2: {} @@ -1669,6 +1693,8 @@ snapshots: html-tags@3.3.1: {} + is-what@4.1.16: {} + js-tokens@4.0.0: {} jsesc@2.5.2: {} @@ -1713,7 +1739,7 @@ snapshots: preact@10.22.0: {} - rfdc@1.3.1: {} + rfdc@1.4.1: {} rollup@4.18.0: dependencies: @@ -1741,14 +1767,18 @@ snapshots: semver@6.3.1: {} - shiki@1.6.2: + shiki@1.7.0: dependencies: - '@shikijs/core': 1.6.2 + '@shikijs/core': 1.7.0 source-map-js@1.2.0: {} speakingurl@14.0.1: {} + superjson@2.2.1: + dependencies: + copy-anything: 3.0.5 + supports-color@5.5.0: dependencies: has-flag: 3.0.0 @@ -1759,15 +1789,15 @@ snapshots: to-fast-properties@2.0.0: {} - update-browserslist-db@1.0.16(browserslist@4.23.0): + update-browserslist-db@1.0.16(browserslist@4.23.1): dependencies: - browserslist: 4.23.0 + browserslist: 4.23.1 escalade: 3.1.2 picocolors: 1.0.1 - vite@5.2.12: + vite@5.3.1: dependencies: - esbuild: 0.20.2 + esbuild: 0.21.5 postcss: 8.4.38 rollup: 4.18.0 optionalDependencies: @@ -1777,20 +1807,20 @@ snapshots: dependencies: '@docsearch/css': 3.6.0 '@docsearch/js': 3.6.0(@algolia/client-search@4.23.3)(search-insights@2.14.0) - '@shikijs/core': 1.6.2 - '@shikijs/transformers': 1.6.2 + '@shikijs/core': 1.7.0 + '@shikijs/transformers': 1.7.0 '@types/markdown-it': 14.1.1 - '@vitejs/plugin-vue': 5.0.5(vite@5.2.12)(vue@3.4.27) - '@vue/devtools-api': 7.2.1(vue@3.4.27) - '@vue/shared': 3.4.27 - '@vueuse/core': 10.10.0(vue@3.4.27) - '@vueuse/integrations': 10.10.0(focus-trap@7.5.4)(vue@3.4.27) + '@vitejs/plugin-vue': 5.0.5(vite@5.3.1)(vue@3.4.29) + '@vue/devtools-api': 7.3.1(vue@3.4.29) + '@vue/shared': 3.4.29 + '@vueuse/core': 10.11.0(vue@3.4.29) + '@vueuse/integrations': 10.11.0(focus-trap@7.5.4)(vue@3.4.29) focus-trap: 7.5.4 mark.js: 8.11.1 minisearch: 6.3.0 - shiki: 1.6.2 - vite: 5.2.12 - vue: 3.4.27 + shiki: 1.7.0 + vite: 5.3.1 + vue: 3.4.29 optionalDependencies: postcss: 8.4.38 transitivePeerDependencies: @@ -1820,16 +1850,16 @@ snapshots: - typescript - universal-cookie - vue-demi@0.14.8(vue@3.4.27): + vue-demi@0.14.8(vue@3.4.29): dependencies: - vue: 3.4.27 + vue: 3.4.29 - vue@3.4.27: + vue@3.4.29: dependencies: - '@vue/compiler-dom': 3.4.27 - '@vue/compiler-sfc': 3.4.27 - '@vue/runtime-dom': 3.4.27 - '@vue/server-renderer': 3.4.27(vue@3.4.27) - '@vue/shared': 3.4.27 + '@vue/compiler-dom': 3.4.29 + '@vue/compiler-sfc': 3.4.29 + '@vue/runtime-dom': 3.4.29 + '@vue/server-renderer': 3.4.29(vue@3.4.29) + '@vue/shared': 3.4.29 yallist@3.1.1: {}