Move stuff around + add lean and idris experiments
This commit is contained in:
Matei Adriel
parent
a45a4e94b3
commit
ca3f83d186
|
|
@ -1,3 +1,3 @@
|
||||||
# Purescript experiments
|
# The _solar sandbox_
|
||||||
|
|
||||||
Random stuff I try in purescript
|
This is a repository containing random things I am messing around with which don't deserve their own repository.
|
||||||
|
|
|
||||||
13
idris/learning/README.md
Normal file
13
idris/learning/README.md
Normal file
|
|
@ -0,0 +1,13 @@
|
||||||
|
# Idris learning
|
||||||
|
|
||||||
|
This directory contains my experiments when first learning idris.
|
||||||
|
|
||||||
|
## File structure
|
||||||
|
|
||||||
|
| File | Description |
|
||||||
|
| ---------------------------------------------------------- | --------------------------------------------- |
|
||||||
|
| [./src/My/Nats.idr](./src/My/Nats.idr) | Natural numbers |
|
||||||
|
| [./src/My/Signs.idr](./src/My/Signs.idr) | Signs (essentially $\mathbb Z / 2 \mathbb Z$) |
|
||||||
|
| [./src/My/Integers.idr](./src/My/Integers.idr) | Integers as differences of naturals |
|
||||||
|
| [./src/My/Structures.idr](./src/My/Structures.idr) | Setoids, semigroups, monoids and groups |
|
||||||
|
| [./src/My/Syntax/Rewrite.idr](./src/My/Syntax/Rewrite.idr) | Coping with the lack of tactics |
|
||||||
575
idris/learning/flake.lock
Normal file
575
idris/learning/flake.lock
Normal file
|
|
@ -0,0 +1,575 @@
|
||||||
|
{
|
||||||
|
"nodes": {
|
||||||
|
"Prettier": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639310097,
|
||||||
|
"narHash": "sha256-+eSLEJDuy2ZRkh1h0Y5IF6RUeHEcWhAHpWhwdwW65f0=",
|
||||||
|
"owner": "Z-snails",
|
||||||
|
"repo": "prettier",
|
||||||
|
"rev": "4a90663b1d586f6d6fce25873aa0f0d7bc633b89",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "Z-snails",
|
||||||
|
"repo": "prettier",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"collie": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1631011321,
|
||||||
|
"narHash": "sha256-goYctB+WBoLgsbjA0DlqGjD8i9wr1K0lv0agqpuwflU=",
|
||||||
|
"owner": "ohad",
|
||||||
|
"repo": "collie",
|
||||||
|
"rev": "ed2eda5e04fbd02a7728e915d396e14cc7ec298e",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "ohad",
|
||||||
|
"repo": "collie",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"comonad": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1638093386,
|
||||||
|
"narHash": "sha256-kxmN6XuszFLK2i76C6LSGHe5XxAURFu9NpzJbi3nodk=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-comonad",
|
||||||
|
"rev": "06d6b551db20f1f940eb24c1dae051c957de97ad",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-comonad",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"dom": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041519,
|
||||||
|
"narHash": "sha256-4ZYc0qaUEVARxhWuH3JgejIeT+GEDNxdS6zIGhBCk34=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-dom",
|
||||||
|
"rev": "01ab52d0ffdb3b47481413a949b8f0c0688c97e4",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-dom",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"dot-parse": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1638264571,
|
||||||
|
"narHash": "sha256-VJQITz+vuQgl5HwR5QdUGwN8SRtGcb2/lJaAVfFbiSk=",
|
||||||
|
"owner": "CodingCellist",
|
||||||
|
"repo": "idris2-dot-parse",
|
||||||
|
"rev": "48fbda8bf8adbaf9e8ebd6ea740228e4394154d9",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "CodingCellist",
|
||||||
|
"repo": "idris2-dot-parse",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"effect": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1637477153,
|
||||||
|
"narHash": "sha256-Ta2Vogg/IiSBkfhhD57jjPTEf3S4DOiVRmof38hmwlM=",
|
||||||
|
"owner": "russoul",
|
||||||
|
"repo": "idris2-effect",
|
||||||
|
"rev": "ea1daf53b2d7e52f9917409f5653adc557f0ee1a",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "russoul",
|
||||||
|
"repo": "idris2-effect",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"elab-util": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041013,
|
||||||
|
"narHash": "sha256-K61s/xifFiTDXJTak5NZmZL6757CTYCY+TGywRZMD7M=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-elab-util",
|
||||||
|
"rev": "7a381c7c5dc3adb7b97c8b8be17e4fb4cc63027d",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-elab-util",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"flake-utils": {
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1644229661,
|
||||||
|
"narHash": "sha256-1YdnJAsNy69bpcjuoKdOYQX0YxZBiCYZo4Twxerqv7k=",
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
|
||||||
|
"rev": "3cecb5b042f7f209c56ffd8371b2711a290ec797",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"flake-utils_2": {
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1638122382,
|
||||||
|
"narHash": "sha256-sQzZzAbvKEqN9s0bzWuYmRaA03v40gaJ4+iL1LXjaeI=",
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
|
||||||
|
"rev": "74f7e4319258e287b0f9cb95426c9853b282730b",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"frex": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1637410704,
|
||||||
|
"narHash": "sha256-BthU1t++n0ZvS76p0fCHsE33QSoXYxf0hMUSKajDY8w=",
|
||||||
|
"owner": "frex-project",
|
||||||
|
"repo": "idris-frex",
|
||||||
|
"rev": "22c480e879c757a5cebca7bb555ec3d21ae3ac28",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "frex-project",
|
||||||
|
"repo": "idris-frex",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"fvect": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1633247988,
|
||||||
|
"narHash": "sha256-zElIze03XpcrYL4H5Aj0ZGNplJGbtOx+iWnivJMzHm0=",
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-fvect",
|
||||||
|
"rev": "1c5e3761e0cd83e711a3535ef9051bea45e6db3f",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-fvect",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"hashable": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1633965157,
|
||||||
|
"narHash": "sha256-Dggf5K//RCZ7uvtCyeiLNJS6mm+8/n0RFW3zAc7XqPg=",
|
||||||
|
"owner": "z-snails",
|
||||||
|
"repo": "idris2-hashable",
|
||||||
|
"rev": "d6fec8c878057909b67f3d4da334155de4f37907",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "z-snails",
|
||||||
|
"repo": "idris2-hashable",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"hedgehog": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041435,
|
||||||
|
"narHash": "sha256-893cPy7gGSQpVmm9co3QCpWsgjukafZHy8YFk9xts30=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-hedgehog",
|
||||||
|
"rev": "a66b1eb0bf84c4a7b743cfb217be69866bc49ad8",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-hedgehog",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"idrall": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1636495701,
|
||||||
|
"narHash": "sha256-aOdCRd4XsSxwqVGta1adlZBy8TVTxTwFDnJ1dyMZK8M=",
|
||||||
|
"owner": "alexhumphreys",
|
||||||
|
"repo": "idrall",
|
||||||
|
"rev": "13ef174290169d05c9e9abcd77c53412e3e0c944",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "alexhumphreys",
|
||||||
|
"ref": "13ef174",
|
||||||
|
"repo": "idrall",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"idris-server": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1634507315,
|
||||||
|
"narHash": "sha256-ulo23yLJXsvImoMB/1C6yRRTqmn/Odo+aUaVi+tUhJo=",
|
||||||
|
"owner": "avidela",
|
||||||
|
"repo": "idris-server",
|
||||||
|
"rev": "661a4ecf0fadaa2bd79c8e922c2d4f79b0b7a445",
|
||||||
|
"type": "gitlab"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "avidela",
|
||||||
|
"repo": "idris-server",
|
||||||
|
"type": "gitlab"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"idris2": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639427352,
|
||||||
|
"narHash": "sha256-C1K2FM1Kio8vi9FTrivdacYCX4cywIsLBeNCsZ6ft4g=",
|
||||||
|
"owner": "idris-lang",
|
||||||
|
"repo": "idris2",
|
||||||
|
"rev": "36918e618646177b1e0c2fd01f21cc8d04d9da30",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "idris-lang",
|
||||||
|
"repo": "idris2",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"idris2-pkgs": {
|
||||||
|
"inputs": {
|
||||||
|
"Prettier": "Prettier",
|
||||||
|
"collie": "collie",
|
||||||
|
"comonad": "comonad",
|
||||||
|
"dom": "dom",
|
||||||
|
"dot-parse": "dot-parse",
|
||||||
|
"effect": "effect",
|
||||||
|
"elab-util": "elab-util",
|
||||||
|
"flake-utils": "flake-utils_2",
|
||||||
|
"frex": "frex",
|
||||||
|
"fvect": "fvect",
|
||||||
|
"hashable": "hashable",
|
||||||
|
"hedgehog": "hedgehog",
|
||||||
|
"idrall": "idrall",
|
||||||
|
"idris-server": "idris-server",
|
||||||
|
"idris2": "idris2",
|
||||||
|
"indexed": "indexed",
|
||||||
|
"inigo": "inigo",
|
||||||
|
"ipkg-to-json": "ipkg-to-json",
|
||||||
|
"json": "json",
|
||||||
|
"katla": "katla",
|
||||||
|
"lsp": "lsp",
|
||||||
|
"nixpkgs": "nixpkgs",
|
||||||
|
"odf": "odf",
|
||||||
|
"pretty-show": "pretty-show",
|
||||||
|
"python": "python",
|
||||||
|
"rhone": "rhone",
|
||||||
|
"rhone-js": "rhone-js",
|
||||||
|
"snocvect": "snocvect",
|
||||||
|
"sop": "sop",
|
||||||
|
"tailrec": "tailrec",
|
||||||
|
"xml": "xml"
|
||||||
|
},
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1642030375,
|
||||||
|
"narHash": "sha256-J1uXnpPR72mjFjLBuYcvDHStBxVya6/MjBNNwqxGeD0=",
|
||||||
|
"owner": "claymager",
|
||||||
|
"repo": "idris2-pkgs",
|
||||||
|
"rev": "ac33a49d4d4bd2b50fddb040cd889733a02c8f09",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "claymager",
|
||||||
|
"repo": "idris2-pkgs",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"indexed": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1638685238,
|
||||||
|
"narHash": "sha256-FceB7o88yKYzjTfRC6yfhOL6oDPMmCQAsJZu/pjE2uA=",
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-indexed",
|
||||||
|
"rev": "ff3ba99b0063da6a74c96178e7f3c58a4ac1693e",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-indexed",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"inigo": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1637596767,
|
||||||
|
"narHash": "sha256-LNx30LO0YWDVSPTxRLWGTFL4f3d5ANG6c60WPdmiYdY=",
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "Inigo",
|
||||||
|
"rev": "57f5b5c051222d8c630010a0a3cf7d7138910127",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "Inigo",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"ipkg-to-json": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1634937414,
|
||||||
|
"narHash": "sha256-LhSmWRpI7vyIQE7QTo38ZTjlqYPVSvV/DIpIxzPmqS0=",
|
||||||
|
"owner": "claymager",
|
||||||
|
"repo": "ipkg-to-json",
|
||||||
|
"rev": "2969b6b83714eeddc31e41577a565778ee5922e6",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "claymager",
|
||||||
|
"repo": "ipkg-to-json",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"json": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041459,
|
||||||
|
"narHash": "sha256-TP/V1jBBP1hFPm/cJ5O2EJiaNoZ19KvBOAI0S9lvAR4=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-json",
|
||||||
|
"rev": "7c0c028acad0ba0b63b37b92199f37e6ec73864a",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-json",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"katla": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1636542431,
|
||||||
|
"narHash": "sha256-X83NA/P3k1iPcBa8g5z8JldEmFEz/jxVeViJX0/FikY=",
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "katla",
|
||||||
|
"rev": "d841ec243f96b4762074211ee81033e28884c858",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "katla",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"lsp": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639486283,
|
||||||
|
"narHash": "sha256-po396FnUu8iqiipwPxqpFZEU4rtpX3jnt3cySwjLsH8=",
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "idris2-lsp",
|
||||||
|
"rev": "7ebb6caf6bb4b57c5107579aba2b871408e6f183",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "idris-community",
|
||||||
|
"repo": "idris2-lsp",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nixpkgs": {
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639213685,
|
||||||
|
"narHash": "sha256-Evuobw7o9uVjAZuwz06Al0fOWZ5JMKOktgXR0XgWBtg=",
|
||||||
|
"owner": "nixos",
|
||||||
|
"repo": "nixpkgs",
|
||||||
|
"rev": "453bcb8380fd1777348245b3c44ce2a2b93b2e2d",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "nixos",
|
||||||
|
"ref": "nixpkgs-21.11-darwin",
|
||||||
|
"repo": "nixpkgs",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"odf": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1638184051,
|
||||||
|
"narHash": "sha256-usSdPx+UqOGImHHdHcrytdzi2LXtIRZuUW0fkD/Wwnk=",
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-odf",
|
||||||
|
"rev": "d2f532437321c8336f1ca786b44b6ebef4117126",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-odf",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"pretty-show": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041411,
|
||||||
|
"narHash": "sha256-BzEe1fpX+lqGEk8b1JZoQT1db5I7s7SZnLCttRVGXdY=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-pretty-show",
|
||||||
|
"rev": "a4bc6156b9dac43699f87504cbdb8dada5627863",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-pretty-show",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"python": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1635936936,
|
||||||
|
"narHash": "sha256-c9mcMApN0qgu0AXQVu0V+NXt2poP258wCPkyvtQvv4I=",
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-python",
|
||||||
|
"rev": "0eab028933c65bebe744e879881416f5136d6943",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-python",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"rhone": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041532,
|
||||||
|
"narHash": "sha256-2g43shlWQIT/1ogesUBUBV9N8YiD3RwaCbbhdKLVp1s=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-rhone",
|
||||||
|
"rev": "c4d828b0b8efea495d9a5f1e842a9c67cad57724",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-rhone",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"rhone-js": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041546,
|
||||||
|
"narHash": "sha256-ddWVsSRbfA6ghmwiRMzDpHBPM+esGdutuqm1qQZgs88=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-rhone-js",
|
||||||
|
"rev": "520dd59549f5b14075045314b6805c7492ed636e",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-rhone-js",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"root": {
|
||||||
|
"inputs": {
|
||||||
|
"flake-utils": "flake-utils",
|
||||||
|
"idris2-pkgs": "idris2-pkgs",
|
||||||
|
"nixpkgs": [
|
||||||
|
"idris2-pkgs",
|
||||||
|
"nixpkgs"
|
||||||
|
]
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"snocvect": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1641633224,
|
||||||
|
"narHash": "sha256-6zTU4sDzd/R/dFCTNZaX41H4L3/USGLFghMS0Oc9liY=",
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-snocvect",
|
||||||
|
"rev": "ff1e7afba360a62f7e522e9bbb856096a79702c4",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "mattpolzin",
|
||||||
|
"repo": "idris-snocvect",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"sop": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1639041379,
|
||||||
|
"narHash": "sha256-PDTf1Wx6EygiWszguvoVPiqIISYFLabI4e0lXHlrjcA=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-sop",
|
||||||
|
"rev": "e4354d1883cd73616019457cb9ebf864d99df6a0",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-sop",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"tailrec": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1637146655,
|
||||||
|
"narHash": "sha256-0yi7MQIrISPvAwkgDC1M5PHDEeVyIaISF0HjKDaT0Rw=",
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-tailrec",
|
||||||
|
"rev": "dd0bc6381b3a2e69aa37f9a8c1b165d4b1516ad7",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "stefan-hoeck",
|
||||||
|
"repo": "idris2-tailrec",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"xml": {
|
||||||
|
"flake": false,
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1637939752,
|
||||||
|
"narHash": "sha256-yYJBhPfwYoi7amlHmeNGrVCOAc3BjZpKTCd9wDs3XEM=",
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-xml",
|
||||||
|
"rev": "1292ccfcd58c551089ef699e4560343d5c473d64",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "madman-bob",
|
||||||
|
"repo": "idris2-xml",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"root": "root",
|
||||||
|
"version": 7
|
||||||
|
}
|
||||||
28
idris/learning/flake.nix
Normal file
28
idris/learning/flake.nix
Normal file
|
|
@ -0,0 +1,28 @@
|
||||||
|
{
|
||||||
|
description = "My Idris 2 package";
|
||||||
|
|
||||||
|
inputs = {
|
||||||
|
flake-utils.url = "github:numtide/flake-utils";
|
||||||
|
idris2-pkgs.url = "github:claymager/idris2-pkgs";
|
||||||
|
nixpkgs.follows = "idris2-pkgs/nixpkgs";
|
||||||
|
};
|
||||||
|
|
||||||
|
outputs = { self, nixpkgs, idris2-pkgs, flake-utils }:
|
||||||
|
flake-utils.lib.eachSystem [ "x86_64-darwin" "x86_64-linux" "i686-linux" ] (system:
|
||||||
|
let
|
||||||
|
pkgs = import nixpkgs { inherit system; overlays = [ idris2-pkgs.overlay ]; };
|
||||||
|
inherit (pkgs.idris2-pkgs._builders) idrisPackage devEnv;
|
||||||
|
mypkg = idrisPackage ./. { };
|
||||||
|
runTests = idrisPackage ./test { extraPkgs.mypkg = mypkg; };
|
||||||
|
in
|
||||||
|
{
|
||||||
|
defaultPackage = mypkg;
|
||||||
|
|
||||||
|
packages = { inherit mypkg runTests; };
|
||||||
|
|
||||||
|
devShell = pkgs.mkShell {
|
||||||
|
buildInputs = [ (devEnv mypkg) ];
|
||||||
|
};
|
||||||
|
}
|
||||||
|
);
|
||||||
|
}
|
||||||
10
idris/learning/sandbox.ipkg
Normal file
10
idris/learning/sandbox.ipkg
Normal file
|
|
@ -0,0 +1,10 @@
|
||||||
|
package mypkg
|
||||||
|
|
||||||
|
-- modules = Main
|
||||||
|
-- main = Main
|
||||||
|
|
||||||
|
depends = contrib
|
||||||
|
|
||||||
|
executable = mypkg
|
||||||
|
|
||||||
|
sourcedir = "src"
|
||||||
207
idris/learning/src/My/Integers.idr
Normal file
207
idris/learning/src/My/Integers.idr
Normal file
|
|
@ -0,0 +1,207 @@
|
||||||
|
module My.Integers
|
||||||
|
|
||||||
|
import My.Nats
|
||||||
|
import My.Structures
|
||||||
|
import Syntax.PreorderReasoning
|
||||||
|
import My.Syntax.Rewrite
|
||||||
|
import My.Signs
|
||||||
|
|
||||||
|
%default total
|
||||||
|
|
||||||
|
public export
|
||||||
|
ℤ : Type
|
||||||
|
ℤ = (ℕ, ℕ)
|
||||||
|
|
||||||
|
public export
|
||||||
|
addIntegers : ℤ -> ℤ -> ℤ
|
||||||
|
addIntegers (x, y) (z, w) = (x + z, y + w)
|
||||||
|
|
||||||
|
public export
|
||||||
|
toNat : ℤ -> (Sign, ℕ)
|
||||||
|
toNat (Z, S x) = (Negative, S x)
|
||||||
|
toNat (x, Z) = (Positive, x)
|
||||||
|
toNat ((S x), (S y)) = toNat $ assert_smaller (S x, S y) (x, y)
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplyIntegers : ℤ -> ℤ -> ℤ
|
||||||
|
multiplyIntegers (x, y) (z, w) = (x * z + y * w, x * w + y * z)
|
||||||
|
|
||||||
|
public export
|
||||||
|
negateInteger : ℤ -> ℤ
|
||||||
|
negateInteger (x, y) = (y, x)
|
||||||
|
|
||||||
|
public export
|
||||||
|
substractIntegers : ℤ -> ℤ -> ℤ
|
||||||
|
substractIntegers a b = (?l, ?r)
|
||||||
|
|
||||||
|
public export
|
||||||
|
normalForm : ℤ -> ℤ
|
||||||
|
normalForm ((S x), (S y)) = normalForm (assert_smaller (S x, S y) (x, y))
|
||||||
|
normalForm a = a
|
||||||
|
|
||||||
|
-- absoluteValue : ℤ -> ℕ
|
||||||
|
-- absoluteValue (Z, b) = b
|
||||||
|
-- absoluteValue ((S x), y) = absoluteValue (x, y)
|
||||||
|
|
||||||
|
public export
|
||||||
|
fromNat : ℕ -> ℤ
|
||||||
|
fromNat n = (n, Z)
|
||||||
|
|
||||||
|
public export
|
||||||
|
fromActualInteger : Integer -> ℤ
|
||||||
|
fromActualInteger 0 = (Z, Z)
|
||||||
|
fromActualInteger n =
|
||||||
|
if n > 0 then
|
||||||
|
fromNat (fromInteger n)
|
||||||
|
else
|
||||||
|
negateInteger (fromNat (fromInteger (-n)))
|
||||||
|
|
||||||
|
public export
|
||||||
|
Num ℤ where
|
||||||
|
(+) = addIntegers
|
||||||
|
(*) = multiplyIntegers
|
||||||
|
fromInteger = fromActualInteger
|
||||||
|
|
||||||
|
public export
|
||||||
|
Neg ℤ where
|
||||||
|
negate = negateInteger
|
||||||
|
(-) = substractIntegers
|
||||||
|
|
||||||
|
---------- Equivalence
|
||||||
|
public export
|
||||||
|
integersAreEquivalent : ℤ -> ℤ -> Type
|
||||||
|
integersAreEquivalent (x, y) (z, w) = x + w = z + y
|
||||||
|
|
||||||
|
public export
|
||||||
|
equivalenceIsReflexive : integersAreEquivalent a a
|
||||||
|
equivalenceIsReflexive {a = (x, y)} = Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
equivalenceIsTransitive : {a, b, c: ℤ} -> integersAreEquivalent a b -> integersAreEquivalent b c -> integersAreEquivalent a c
|
||||||
|
equivalenceIsTransitive {a = (z, w)} {b = (v, s)} {c = (t, u)} zsISvw vuISts = id
|
||||||
|
$ … (z + u = t + w)
|
||||||
|
$ My.Nats.substractionPreservesEquality s
|
||||||
|
$ … ((z + u) + s = (t + w) + s)
|
||||||
|
$ rewrite My.Nats.additionIsCommutative z u
|
||||||
|
in …l ((u + z) + s)
|
||||||
|
$ rewrite My.Nats.additionIsAssociative u z s
|
||||||
|
in …l (u + (z + s))
|
||||||
|
$ rewrite zsISvw
|
||||||
|
in …l (u + (v + w))
|
||||||
|
$ rewrite sym $ My.Nats.additionIsAssociative u v w
|
||||||
|
in …l ((u + v) + w)
|
||||||
|
$ rewrite My.Nats.additionIsCommutative t w
|
||||||
|
in …r ((w + t) + s)
|
||||||
|
$ rewrite My.Nats.additionIsAssociative w t s
|
||||||
|
in …r (w + (t + s))
|
||||||
|
$ rewrite My.Nats.additionIsCommutative w (t + s)
|
||||||
|
in …r ((t + s) + w)
|
||||||
|
$ My.Nats.additionPreservesEquality w
|
||||||
|
$ … (u + v = t + s)
|
||||||
|
$ rewrite My.Nats.additionIsCommutative u v
|
||||||
|
in …l (v + u)
|
||||||
|
$ vuISts
|
||||||
|
|
||||||
|
public export
|
||||||
|
equivalenceIsSymmetric : {a, b: ℤ} -> integersAreEquivalent a b -> integersAreEquivalent b a
|
||||||
|
equivalenceIsSymmetric {a = (y, z)} {b = (w, v)} x = sym x
|
||||||
|
|
||||||
|
public export
|
||||||
|
My.Structures.Setoid ℤ where
|
||||||
|
(<->) = integersAreEquivalent
|
||||||
|
reflexivity = equivalenceIsReflexive
|
||||||
|
transitivity = equivalenceIsTransitive
|
||||||
|
symmetry = equivalenceIsSymmetric
|
||||||
|
|
||||||
|
---------- Addition proofs
|
||||||
|
public export
|
||||||
|
additionIsCommutative : CommutativityProof My.Integers.addIntegers
|
||||||
|
additionIsCommutative (Z, y) (x, z) = (x + (z + y) = (x + 0) + (y + z))
|
||||||
|
.... rewrite My.Nats.additionRightIdentity x in (x + (z + y) = x + (y + z))
|
||||||
|
.... rewrite My.Nats.additionIsCommutative y z in (x + (z + y) = x + (z + y))
|
||||||
|
.... Refl
|
||||||
|
additionIsCommutative ((S x), y) (z, w) = id
|
||||||
|
$ … (1 + ((x + z) + (w + y)) = (z + (1 + x)) + (y + w))
|
||||||
|
$ rewrite My.Nats.additionIsCommutative y w
|
||||||
|
in …r ((z + (1 + x)) + (w + y))
|
||||||
|
$ rewrite My.Nats.additionIsCommutative z (1 + x)
|
||||||
|
in …r (1 + (x + z) + (w + y))
|
||||||
|
$ Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionIsAssociative : AssociativityProof My.Integers.addIntegers
|
||||||
|
additionIsAssociative (x, y) (z, w) (v, s) = id
|
||||||
|
$ … ((((x + z) + v) + (y + (w + s))) = ((x + (z + v) + ((y + w) + s))))
|
||||||
|
$ rewrite My.Nats.additionIsAssociative x z v
|
||||||
|
in …l ((x + (z + v) + (y + (w + s))))
|
||||||
|
$ rewrite My.Nats.additionIsAssociative y w s
|
||||||
|
in …r ((x + (z + v) + (y + (w + s))))
|
||||||
|
$ Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionRightIdentity : RightIdentityProof My.Integers.addIntegers 0
|
||||||
|
additionRightIdentity (x, y) = id
|
||||||
|
$ … ((x + 0) + y = x + (y + 0))
|
||||||
|
$ rewrite My.Nats.additionRightIdentity x
|
||||||
|
in …l (x + y)
|
||||||
|
$ rewrite My.Nats.additionRightIdentity y
|
||||||
|
in …r (x + y)
|
||||||
|
$ Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionLeftIdentity : LeftIdentityProof My.Integers.addIntegers 0
|
||||||
|
additionLeftIdentity (x, y) = Refl
|
||||||
|
|
||||||
|
---------- Multiplication proofs
|
||||||
|
multiplyToNatResults : (Sign, ℕ) -> (Sign, ℕ) -> (Sign, ℕ)
|
||||||
|
multiplyToNatResults (x, z) (y, w) = (multiplySigns x y, z * w)
|
||||||
|
|
||||||
|
ToNatDistributesMultiplication : ℤ -> ℤ -> Type
|
||||||
|
ToNatDistributesMultiplication a b = toNat (a * b) = multiplyToNatResults (toNat a) (toNat b)
|
||||||
|
|
||||||
|
toNatDistributesMultiplication : (a, b: ℤ) -> ToNatDistributesMultiplication a b
|
||||||
|
toNatDistributesMultiplication (Z, Z) (Z, Z) = Refl
|
||||||
|
toNatDistributesMultiplication (Z, Z) (Z, (S x)) = ?toNatDistributesMultiplication_rhs_9
|
||||||
|
|
||||||
|
toNatDistributesMultiplication (Z, Z) ((S x), w) = ?toNatDistributesMultiplication_rhs_7
|
||||||
|
|
||||||
|
toNatDistributesMultiplication (Z, (S x)) (z, w) = ?toNatDistributesMultiplication_rhs_5
|
||||||
|
|
||||||
|
toNatDistributesMultiplication ((S x), y) (z, w) = ?toNatDistributesMultiplication_rhs_3
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
multiplicationIsAssociative : AssociativityProof My.Integers.multiplyIntegers
|
||||||
|
multiplicationIsAssociative 0 b c = ?multiplicationIsAssociative_rhs
|
||||||
|
multiplicationIsAssociative a b c = ?multiplicationIsAssociative_rhs_1
|
||||||
|
|
||||||
|
|
||||||
|
---------- Interface implementations
|
||||||
|
public export
|
||||||
|
[additionSemigroup] My.Structures.Semigroup ℤ where
|
||||||
|
∘ = addIntegers
|
||||||
|
associativityProof = additionIsAssociative
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
[additionMonoid] My.Structures.Monoid ℤ using My.Integers.additionSemigroup where
|
||||||
|
empty = 0
|
||||||
|
rightIdentityProof = My.Integers.additionRightIdentity
|
||||||
|
leftIdentityProof = My.Integers.additionLeftIdentity
|
||||||
|
|
||||||
|
[multiplicationSemigroup] My.Structures.Semigroup ℤ where
|
||||||
|
∘ = multiplyIntegers
|
||||||
|
associativityProof = multiplicationIsAssociative
|
||||||
|
|
||||||
|
---------- Constants to play around with
|
||||||
|
seven : ℤ
|
||||||
|
seven = 7
|
||||||
|
|
||||||
|
minusFour : ℤ
|
||||||
|
minusFour = -4
|
||||||
|
|
||||||
|
three : ℤ
|
||||||
|
three = 3
|
||||||
|
|
||||||
|
three' : ℤ
|
||||||
|
three' = minusFour + seven
|
||||||
206
idris/learning/src/My/Nats.idr
Normal file
206
idris/learning/src/My/Nats.idr
Normal file
|
|
@ -0,0 +1,206 @@
|
||||||
|
module My.Nats
|
||||||
|
|
||||||
|
import My.Structures
|
||||||
|
|
||||||
|
%default total
|
||||||
|
%hide Z
|
||||||
|
%hide S
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
data ℕ : Type where
|
||||||
|
Z : ℕ
|
||||||
|
S : ℕ -> ℕ
|
||||||
|
|
||||||
|
public export
|
||||||
|
fromIntegerNat : Integer -> ℕ
|
||||||
|
fromIntegerNat 0 = Z
|
||||||
|
fromIntegerNat n =
|
||||||
|
if (n > 0) then
|
||||||
|
S (fromIntegerNat (assert_smaller n (n - 1)))
|
||||||
|
else
|
||||||
|
Z
|
||||||
|
|
||||||
|
one : ℕ
|
||||||
|
one = S Z
|
||||||
|
|
||||||
|
public export
|
||||||
|
add : ℕ -> ℕ -> ℕ
|
||||||
|
add Z a = a
|
||||||
|
add (S a) b = S (add a b)
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiply : ℕ -> ℕ -> ℕ
|
||||||
|
multiply Z a = Z
|
||||||
|
multiply (S a) b = add b (multiply a b)
|
||||||
|
|
||||||
|
public export
|
||||||
|
raiseToPower : ℕ -> ℕ -> ℕ
|
||||||
|
raiseToPower a Z = one
|
||||||
|
raiseToPower a (S b) = multiply a (raiseToPower a b)
|
||||||
|
|
||||||
|
public export
|
||||||
|
monus : ℕ -> ℕ -> ℕ
|
||||||
|
monus (S a) (S b) = monus a b
|
||||||
|
monus a Z = a
|
||||||
|
monus _ (S _) = Z
|
||||||
|
|
||||||
|
public export
|
||||||
|
naturalInduction : (P: ℕ -> Type) -> P Z -> ({x: ℕ} -> P x -> P (S x)) -> (x: ℕ) -> P x
|
||||||
|
naturalInduction p base recurse Z = base
|
||||||
|
naturalInduction p base recurse (S a) = recurse (naturalInduction p base recurse a)
|
||||||
|
|
||||||
|
public export
|
||||||
|
Num ℕ where
|
||||||
|
fromInteger = fromIntegerNat
|
||||||
|
(+) = add
|
||||||
|
(*) = multiply
|
||||||
|
|
||||||
|
public export
|
||||||
|
%hint
|
||||||
|
setoidNats : My.Structures.Setoid ℕ
|
||||||
|
setoidNats = trivialSetoid ℕ
|
||||||
|
|
||||||
|
---------- Proofs
|
||||||
|
public export
|
||||||
|
succCommutesAddition : (a, b: ℕ) -> add (S a) b = add a (S b)
|
||||||
|
succCommutesAddition Z a = Refl
|
||||||
|
succCommutesAddition (S c) b = let
|
||||||
|
rec = succCommutesAddition c b
|
||||||
|
in rewrite rec in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionIsAssociative : AssociativityProof My.Nats.add
|
||||||
|
additionIsAssociative Z b c = Refl
|
||||||
|
additionIsAssociative (S a) b c = let
|
||||||
|
rec = additionIsAssociative a b c
|
||||||
|
in rewrite rec in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionRightIdentity : RightIdentityProof My.Nats.add 0
|
||||||
|
additionRightIdentity Z = Refl
|
||||||
|
additionRightIdentity (S x) = rewrite additionRightIdentity x in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionIsCommutative : CommutativityProof My.Nats.add
|
||||||
|
additionIsCommutative Z b = sym (additionRightIdentity b)
|
||||||
|
additionIsCommutative (S x) Z = rewrite additionIsCommutative x Z in Refl
|
||||||
|
additionIsCommutative (S x) (S y) =
|
||||||
|
rewrite sym (succCommutesAddition x y) in
|
||||||
|
rewrite additionIsCommutative y (S x) in
|
||||||
|
Refl
|
||||||
|
|
||||||
|
----- Multiplication proofs
|
||||||
|
public export
|
||||||
|
multiplicationRightNullification : (a: ℕ) -> multiply a 0 = 0
|
||||||
|
multiplicationRightNullification Z = Refl
|
||||||
|
multiplicationRightNullification (S x) = rewrite multiplicationRightNullification x in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationRightIdentity : (a: ℕ) -> a * 1 = a
|
||||||
|
multiplicationRightIdentity Z = Refl
|
||||||
|
multiplicationRightIdentity (S x) = rewrite multiplicationRightIdentity x in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationLeftIdentity : (a: ℕ) -> a = 1 * a
|
||||||
|
multiplicationLeftIdentity a = rewrite additionRightIdentity a in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationDistributesAddition : (a, b, c: ℕ) -> a * (b + c) = a * b + a * c
|
||||||
|
multiplicationDistributesAddition Z b c = Refl
|
||||||
|
multiplicationDistributesAddition (S x) b c
|
||||||
|
= let rec = multiplicationDistributesAddition x b c
|
||||||
|
in rewrite rec
|
||||||
|
in rewrite additionIsAssociative b c ((x * b) + (x * c))
|
||||||
|
in rewrite additionIsAssociative b (x * b) (c + (x * c))
|
||||||
|
in rewrite additionIsCommutative (x * b) (c + (x * c))
|
||||||
|
in rewrite additionIsAssociative c (x * c) (x * b)
|
||||||
|
in rewrite additionIsCommutative (x * b) (x * c)
|
||||||
|
in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
succIsPlusOne : (a: ℕ) -> S a = a + 1
|
||||||
|
succIsPlusOne Z = Refl
|
||||||
|
succIsPlusOne (S x) = rewrite additionIsCommutative x 1 in Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationisCommutative : CommutativityProof My.Nats.multiply
|
||||||
|
multiplicationisCommutative Z b = sym (multiplicationRightNullification b)
|
||||||
|
multiplicationisCommutative (S x) b =
|
||||||
|
rewrite succIsPlusOne x in
|
||||||
|
rewrite multiplicationDistributesAddition b x 1 in
|
||||||
|
rewrite multiplicationRightIdentity b in
|
||||||
|
rewrite additionIsCommutative b (x * b) in
|
||||||
|
rewrite multiplicationisCommutative x b in
|
||||||
|
Refl
|
||||||
|
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationIsAssociative : AssociativityProof My.Nats.multiply
|
||||||
|
multiplicationIsAssociative Z b c = Refl
|
||||||
|
multiplicationIsAssociative (S x) y c =
|
||||||
|
rewrite multiplicationisCommutative (y + (x * y)) c in
|
||||||
|
rewrite multiplicationDistributesAddition c y (x * y) in
|
||||||
|
rewrite multiplicationisCommutative y c in
|
||||||
|
rewrite sym (multiplicationIsAssociative c x y) in
|
||||||
|
rewrite sym (multiplicationIsAssociative x c y) in
|
||||||
|
rewrite sym (multiplicationisCommutative x c) in
|
||||||
|
Refl
|
||||||
|
|
||||||
|
---------- Monus proofs
|
||||||
|
public export
|
||||||
|
xMinusXIsZero : (a: ℕ) -> (monus a a) = 0
|
||||||
|
xMinusXIsZero Z = Refl
|
||||||
|
xMinusXIsZero (S x) = xMinusXIsZero x
|
||||||
|
|
||||||
|
public export
|
||||||
|
additionNullifiesMonus : (a, b: ℕ) -> (monus (a + b) b) = a
|
||||||
|
additionNullifiesMonus Z b = xMinusXIsZero b
|
||||||
|
additionNullifiesMonus (S x) Z = rewrite additionRightIdentity x in Refl
|
||||||
|
additionNullifiesMonus x (S y) = rewrite sym $ succCommutesAddition x y in additionNullifiesMonus x y
|
||||||
|
|
||||||
|
---------- Equality proofs
|
||||||
|
public export
|
||||||
|
additionPreservesEquality : {a, b: ℕ} -> (c: ℕ) -> (a = b) -> (a + c = b + c)
|
||||||
|
additionPreservesEquality c prf = cong (+c) prf
|
||||||
|
|
||||||
|
public export
|
||||||
|
substractionPreservesEquality : {a, b: ℕ} -> (c: ℕ) -> (a + c = b + c) -> (a = b)
|
||||||
|
substractionPreservesEquality c prf = let
|
||||||
|
middle : (monus (a + c) c = monus (b + c) c)
|
||||||
|
middle = cong (\e => monus e c) prf
|
||||||
|
|
||||||
|
left : (a = monus (a + c) c)
|
||||||
|
left = sym $ additionNullifiesMonus a c
|
||||||
|
|
||||||
|
right : (monus (b + c) c = b)
|
||||||
|
right = additionNullifiesMonus b c
|
||||||
|
in (left `trans` middle) `trans` right
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplicationPreservesEquality : {a, b: ℕ} -> (c: ℕ) -> (a = b) -> (a * c = b * c)
|
||||||
|
multiplicationPreservesEquality c prf = cong (*c) prf
|
||||||
|
|
||||||
|
---------- Interace implementations
|
||||||
|
public export
|
||||||
|
[additionSemigroup] My.Structures.Semigroup ℕ where
|
||||||
|
∘ = add
|
||||||
|
associativityProof = additionIsAssociative
|
||||||
|
|
||||||
|
public export
|
||||||
|
[additionMonoid] My.Structures.Monoid ℕ using additionSemigroup where
|
||||||
|
empty = 0
|
||||||
|
rightIdentityProof a = additionRightIdentity a
|
||||||
|
leftIdentityProof a = Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
[multiplicationSemigroup] My.Structures.Semigroup ℕ where
|
||||||
|
∘ = multiply
|
||||||
|
associativityProof = multiplicationIsAssociative
|
||||||
|
|
||||||
|
public export
|
||||||
|
[multiplicationMonoid] My.Structures.Monoid ℕ using multiplicationSemigroup where
|
||||||
|
empty = 1
|
||||||
|
rightIdentityProof = multiplicationRightIdentity
|
||||||
|
leftIdentityProof = multiplicationLeftIdentity
|
||||||
|
|
||||||
53
idris/learning/src/My/Signs.idr
Normal file
53
idris/learning/src/My/Signs.idr
Normal file
|
|
@ -0,0 +1,53 @@
|
||||||
|
module My.Signs
|
||||||
|
|
||||||
|
import My.Structures
|
||||||
|
|
||||||
|
public export
|
||||||
|
data Sign = Positive | Negative
|
||||||
|
|
||||||
|
public export
|
||||||
|
negateSign : Sign -> Sign
|
||||||
|
negateSign Positive = Negative
|
||||||
|
negateSign Negative = Positive
|
||||||
|
|
||||||
|
public export
|
||||||
|
multiplySigns : Sign -> Sign -> Sign
|
||||||
|
multiplySigns Positive b = b
|
||||||
|
multiplySigns Negative b = negateSign b
|
||||||
|
|
||||||
|
public export
|
||||||
|
%hint
|
||||||
|
setoidSign : My.Structures.Setoid Sign
|
||||||
|
setoidSign = trivialSetoid Sign
|
||||||
|
|
||||||
|
doubleNegationIdentity : (a: Sign) -> negateSign (negateSign a) = a
|
||||||
|
doubleNegationIdentity Positive = Refl
|
||||||
|
doubleNegationIdentity Negative = Refl
|
||||||
|
|
||||||
|
multiplicationIsAssociative : AssociativityProof My.Signs.multiplySigns
|
||||||
|
multiplicationIsAssociative Positive b c = Refl
|
||||||
|
multiplicationIsAssociative Negative Positive c = Refl
|
||||||
|
multiplicationIsAssociative Negative Negative c = sym $ doubleNegationIdentity c
|
||||||
|
|
||||||
|
multiplicationRightIdentity : RightIdentityProof My.Signs.multiplySigns Positive
|
||||||
|
multiplicationRightIdentity Positive = Refl
|
||||||
|
multiplicationRightIdentity Negative = Refl
|
||||||
|
|
||||||
|
multiplicationLeftIdentity : LeftIdentityProof My.Signs.multiplySigns Positive
|
||||||
|
multiplicationLeftIdentity _ = Refl
|
||||||
|
|
||||||
|
multiplicationIsCommutative : CommutativityProof My.Signs.multiplySigns
|
||||||
|
multiplicationIsCommutative Positive b = sym $ multiplicationRightIdentity b
|
||||||
|
multiplicationIsCommutative Negative Positive = Refl
|
||||||
|
multiplicationIsCommutative Negative Negative = Refl
|
||||||
|
|
||||||
|
public export
|
||||||
|
My.Structures.Semigroup Sign where
|
||||||
|
∘ = multiplySigns
|
||||||
|
associativityProof = multiplicationIsAssociative
|
||||||
|
|
||||||
|
public export
|
||||||
|
My.Structures.Monoid Sign where
|
||||||
|
empty = Positive
|
||||||
|
rightIdentityProof = multiplicationRightIdentity
|
||||||
|
leftIdentityProof = multiplicationLeftIdentity
|
||||||
79
idris/learning/src/My/Structures.idr
Normal file
79
idris/learning/src/My/Structures.idr
Normal file
|
|
@ -0,0 +1,79 @@
|
||||||
|
module My.Structures
|
||||||
|
|
||||||
|
%hide Semigroup
|
||||||
|
%hide Monoid
|
||||||
|
%hide empty
|
||||||
|
|
||||||
|
infix 5 <->
|
||||||
|
|
||||||
|
public export
|
||||||
|
interface Setoid t where
|
||||||
|
constructor MkSetoid
|
||||||
|
(<->) : t -> t -> Type
|
||||||
|
reflexivity : {0 a: t} -> a <-> a
|
||||||
|
transitivity : {a,b,c: t} -> (a <-> b) -> (b <-> c) -> (a <-> c)
|
||||||
|
symmetry : {a, b: t} -> (a <-> b) -> (b <-> a)
|
||||||
|
|
||||||
|
public export
|
||||||
|
trivialSetoid : (t: Type) -> Setoid t
|
||||||
|
trivialSetoid _ = MkSetoid Equal Refl (\a,b => trans a b) (\a => sym a)
|
||||||
|
|
||||||
|
public export
|
||||||
|
AssociativityProof : {x: Type} -> Setoid x => (x -> x -> x) -> Type
|
||||||
|
AssociativityProof {x} t = (a, b, c: x) -> (t (t a b) c) <-> (t a (t b c))
|
||||||
|
|
||||||
|
public export
|
||||||
|
RightIdentityProof : {x: Type} -> Setoid x => (x -> x -> x) -> x -> Type
|
||||||
|
RightIdentityProof t e = (a: x) -> t a e <-> a
|
||||||
|
|
||||||
|
public export
|
||||||
|
LeftIdentityProof : {x: Type} -> Setoid x => (x -> x -> x) -> x -> Type
|
||||||
|
LeftIdentityProof t e = (a: x) -> a <-> t e a
|
||||||
|
|
||||||
|
public export
|
||||||
|
RightInverseProof : {x: Type} -> Setoid x => (x -> x -> x) -> (x -> x) -> x -> Type
|
||||||
|
RightInverseProof t inverse e = (a: x) -> t a (inverse a) <-> e
|
||||||
|
|
||||||
|
public export
|
||||||
|
LeftInverseProof : {x: Type} -> Setoid x => (x -> x -> x) -> (x -> x) -> x -> Type
|
||||||
|
LeftInverseProof t inverse e = (a: x) -> t (inverse a) a <-> e
|
||||||
|
|
||||||
|
public export
|
||||||
|
CommutativityProof : {x: Type} -> Setoid x => (x -> x -> x) -> Type
|
||||||
|
CommutativityProof t = (a, b: x) -> t a b <-> t b a
|
||||||
|
|
||||||
|
public export
|
||||||
|
rightToLeftIdentity :
|
||||||
|
{x: Type} -> Setoid x =>
|
||||||
|
(f: x -> x -> x) -> (e: x) ->
|
||||||
|
CommutativityProof f ->
|
||||||
|
RightIdentityProof f e ->
|
||||||
|
LeftIdentityProof f e
|
||||||
|
rightToLeftIdentity f e commutativity rightIdentity x = symmetry $ transitivity (commutativity e x) (rightIdentity x)
|
||||||
|
|
||||||
|
public export
|
||||||
|
leftToRightIdentity :
|
||||||
|
{x: Type} -> Setoid x =>
|
||||||
|
(f: x -> x -> x) -> (e: x) ->
|
||||||
|
CommutativityProof f ->
|
||||||
|
LeftIdentityProof f e ->
|
||||||
|
RightIdentityProof f e
|
||||||
|
leftToRightIdentity f e commutativity leftIdentity x = symmetry $ transitivity (leftIdentity x) (commutativity e x)
|
||||||
|
|
||||||
|
public export
|
||||||
|
interface Setoid t => Semigroup t where
|
||||||
|
∘ : t -> t -> t
|
||||||
|
associativityProof : AssociativityProof ∘
|
||||||
|
|
||||||
|
public export
|
||||||
|
interface Semigroup t => Monoid t where
|
||||||
|
empty : t
|
||||||
|
rightIdentityProof : RightIdentityProof ∘ empty
|
||||||
|
leftIdentityProof : LeftIdentityProof ∘ empty
|
||||||
|
|
||||||
|
public export
|
||||||
|
interface Monoid t => Group t where
|
||||||
|
⁻¹ : t -> t
|
||||||
|
rightInverseProof : RightInverseProof ∘ (⁻¹) My.Structures.empty
|
||||||
|
leftInverseProof : LeftInverseProof ∘ (⁻¹) My.Structures.empty
|
||||||
|
|
||||||
19
idris/learning/src/My/Syntax/Rewrite.idr
Normal file
19
idris/learning/src/My/Syntax/Rewrite.idr
Normal file
|
|
@ -0,0 +1,19 @@
|
||||||
|
module My.Syntax.Rewrite
|
||||||
|
|
||||||
|
infix 1 ....
|
||||||
|
|
||||||
|
public export
|
||||||
|
(....) : (0 a : Type) -> a -> a
|
||||||
|
(....) _ a = a
|
||||||
|
|
||||||
|
public export
|
||||||
|
… : (0 a : Type) -> a -> a
|
||||||
|
… _ a = a
|
||||||
|
|
||||||
|
public export
|
||||||
|
…l : (0 a : t) -> {0 b : t} -> a = b -> a = b
|
||||||
|
…l _ a = a
|
||||||
|
|
||||||
|
public export
|
||||||
|
…r : {0 a : t} -> (0 b : t) -> a = b -> a = b
|
||||||
|
…r _ a = a
|
||||||
6
idris/learning/test/Tests.idr
Normal file
6
idris/learning/test/Tests.idr
Normal file
|
|
@ -0,0 +1,6 @@
|
||||||
|
module Tests
|
||||||
|
|
||||||
|
import Main
|
||||||
|
|
||||||
|
main : IO ()
|
||||||
|
main = putStrLn "tests passed"
|
||||||
8
idris/learning/test/runTests.ipkg
Normal file
8
idris/learning/test/runTests.ipkg
Normal file
|
|
@ -0,0 +1,8 @@
|
||||||
|
package runTests
|
||||||
|
|
||||||
|
depends = mypkg
|
||||||
|
|
||||||
|
modules = Tests
|
||||||
|
main = Tests
|
||||||
|
|
||||||
|
executable = runTests
|
||||||
1
lean/learning/.gitignore
vendored
Normal file
1
lean/learning/.gitignore
vendored
Normal file
|
|
@ -0,0 +1 @@
|
||||||
|
build
|
||||||
1
lean/learning/LeanSandbox.lean
Normal file
1
lean/learning/LeanSandbox.lean
Normal file
|
|
@ -0,0 +1 @@
|
||||||
|
def hello := "world"
|
||||||
397
lean/learning/LeanSandbox/Integers.lean
Normal file
397
lean/learning/LeanSandbox/Integers.lean
Normal file
|
|
@ -0,0 +1,397 @@
|
||||||
|
import LeanSandbox.Nat
|
||||||
|
|
||||||
|
macro "nat_ring_all" : tactic => `(simp_all [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm, Nat.add_assoc, Nat.add_left_comm, Nat.add_comm, Nat.left_distrib, Nat.right_distrib])
|
||||||
|
macro "nat_ring" : tactic => `(simp [Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm, Nat.add_assoc, Nat.add_left_comm, Nat.add_comm, Nat.left_distrib, Nat.right_distrib])
|
||||||
|
macro "quotient_madness" : tactic => `(simp [Quotient.mk', Quotient.mk, Quotient.liftOn₂, Quotient.lift₂, Quotient.lift])
|
||||||
|
|
||||||
|
structure RawInt where
|
||||||
|
pos : Nat
|
||||||
|
neg : Nat
|
||||||
|
deriving Repr
|
||||||
|
|
||||||
|
private def eqv : (x y: RawInt) → Prop
|
||||||
|
| ⟨a, b⟩, ⟨c, d⟩ => a + d = c + b
|
||||||
|
|
||||||
|
infix:50 " ~ " => eqv
|
||||||
|
|
||||||
|
private theorem eqv.refl (x: RawInt) : x ~ x := rfl
|
||||||
|
private theorem eqv.symm {x y: RawInt} (xy: x ~ y): y ~ x := Eq.symm xy
|
||||||
|
/-
|
||||||
|
a - b c - d e - f
|
||||||
|
a + d = c + b
|
||||||
|
c + f = e + d
|
||||||
|
=> a + f = e + b -- the target
|
||||||
|
|
||||||
|
a + d + c + f = c + b + e + d
|
||||||
|
a + f + e + b -- done
|
||||||
|
-/
|
||||||
|
private theorem eqv.trans {x y z: RawInt} (xy: x ~ y) (yz: y ~ z): x ~ z := by
|
||||||
|
have summed: _ := Nat.add_equations xy yz
|
||||||
|
apply @Nat.add_right_cancel _ (y.pos + y.neg) _
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
private theorem is_equivalence: Equivalence eqv :=
|
||||||
|
{ refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }
|
||||||
|
|
||||||
|
instance rawIntSetoid: Setoid RawInt where
|
||||||
|
r := eqv
|
||||||
|
iseqv := is_equivalence
|
||||||
|
|
||||||
|
def MyInt: Type :=
|
||||||
|
Quotient rawIntSetoid
|
||||||
|
|
||||||
|
private theorem eqv.sound: x ~ y → Quotient.mk' x = Quotient.mk' y := Quot.sound
|
||||||
|
|
||||||
|
@[simp]
|
||||||
|
def MyInt.mk (pos neg: Nat): MyInt := Quotient.mk' ⟨pos, neg⟩
|
||||||
|
|
||||||
|
notation "{ " a₁ ", " a₂ " }" => MyInt.mk a₁ a₂
|
||||||
|
|
||||||
|
@[simp, inline]
|
||||||
|
private def MyInt.ofRawInt(raw: RawInt) := MyInt.mk raw.pos raw.neg
|
||||||
|
|
||||||
|
@[simp, inline]
|
||||||
|
private def RawInt.ofNat(nat: Nat): RawInt := ⟨nat, 0⟩
|
||||||
|
|
||||||
|
@[simp, inline]
|
||||||
|
private def MyInt.ofNat(nat: Nat): MyInt := {nat, 0}
|
||||||
|
|
||||||
|
private instance rawIntOfNat: OfNat RawInt n where
|
||||||
|
ofNat := RawInt.ofNat n
|
||||||
|
|
||||||
|
instance myIntOfNat: OfNat MyInt n where
|
||||||
|
ofNat := MyInt.ofNat n
|
||||||
|
|
||||||
|
namespace MyInt
|
||||||
|
private def negateRawInt: RawInt → MyInt
|
||||||
|
| ⟨pos, neg⟩ => {neg, pos}
|
||||||
|
|
||||||
|
/-
|
||||||
|
a - b = c - d
|
||||||
|
a + d = c + b
|
||||||
|
|
||||||
|
b + c = d + a
|
||||||
|
b - a = d - c
|
||||||
|
-/
|
||||||
|
private theorem negateRawInt.respects {x y: RawInt} (xy: x ~ y): negateRawInt x = negateRawInt y := by
|
||||||
|
apply eqv.sound
|
||||||
|
simp_all [eqv, Nat.add_comm]
|
||||||
|
|
||||||
|
def negate (τ: MyInt): MyInt :=
|
||||||
|
Quotient.liftOn τ negateRawInt @negateRawInt.respects
|
||||||
|
|
||||||
|
instance negMyInt: Neg MyInt where
|
||||||
|
neg := negate
|
||||||
|
|
||||||
|
private theorem double_neg_elim: ∀x, x = negate (negate x) := by
|
||||||
|
intro x
|
||||||
|
induction x using Quotient.ind
|
||||||
|
rfl
|
||||||
|
|
||||||
|
private def addRawInts: RawInt → RawInt → MyInt
|
||||||
|
| ⟨a, b⟩, ⟨c, d⟩ => {a + c, b + d}
|
||||||
|
|
||||||
|
private theorem addRawInts.respects
|
||||||
|
{a b c d: RawInt}
|
||||||
|
(ac: a ~ c)
|
||||||
|
(bd: b ~ d): addRawInts a b = addRawInts c d := by
|
||||||
|
have summed: _ := Nat.add_equations ac bd
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv] at summed ⊢
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
private theorem addRawInts.comm (a b: RawInt): addRawInts a b = addRawInts b a := by
|
||||||
|
simp_all [addRawInts, Nat.add_comm]
|
||||||
|
|
||||||
|
def add (τ β: MyInt): MyInt :=
|
||||||
|
Quotient.liftOn₂ τ β addRawInts @addRawInts.respects
|
||||||
|
|
||||||
|
private instance hAddRawInts: HAdd RawInt RawInt MyInt where
|
||||||
|
hAdd := addRawInts
|
||||||
|
instance addMyInts: Add MyInt where
|
||||||
|
add := add
|
||||||
|
|
||||||
|
def sub (a b: MyInt): MyInt := a + (-b)
|
||||||
|
|
||||||
|
instance subMyInt: Sub MyInt where
|
||||||
|
sub := sub
|
||||||
|
|
||||||
|
@[simp]
|
||||||
|
theorem sub.x_minus_x_is_zero (a: MyInt): a - a = 0 := by
|
||||||
|
simp_all [HSub.hSub, sub, HAdd.hAdd, add, negate, Neg.neg, MyInt.ofNat]
|
||||||
|
induction a using Quotient.ind
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
apply Nat.add_comm
|
||||||
|
|
||||||
|
theorem add.comm: ∀x y: MyInt, x + y = y + x := by
|
||||||
|
intro x y
|
||||||
|
simp_all [HAdd.hAdd, add]
|
||||||
|
induction x, y using Quotient.ind₂
|
||||||
|
quotient_madness
|
||||||
|
apply addRawInts.comm
|
||||||
|
|
||||||
|
theorem add.assoc(x y z: MyInt): x + (y + z) = (x + y) + z := by
|
||||||
|
simp_all [HAdd.hAdd, add]
|
||||||
|
|
||||||
|
induction x, y using Quotient.ind₂
|
||||||
|
induction z using Quotient.ind
|
||||||
|
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
@[simp]
|
||||||
|
theorem add.zero(x: MyInt): x + 0 = x := by
|
||||||
|
simp_all [HAdd.hAdd, add]
|
||||||
|
induction x using Quotient.ind
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
|
||||||
|
/-
|
||||||
|
(a - b) * (c - d)
|
||||||
|
ac - bc - ad + bd
|
||||||
|
-/
|
||||||
|
private def multiplyRawInts: RawInt → RawInt → MyInt
|
||||||
|
| ⟨a, b⟩, ⟨c, d⟩ => {a * c + b * d, b * c + a * d}
|
||||||
|
|
||||||
|
/-
|
||||||
|
ac : c.neg + a.pos = a.neg + c.pos
|
||||||
|
bd : d.neg + b.pos = b.neg + d.pos
|
||||||
|
⊢ a.neg * b.neg + (a.pos * b.pos + (c.pos * d.neg + c.neg * d.pos)) =
|
||||||
|
c.neg * d.neg + (a.pos * b.neg + (a.neg * b.pos + c.pos * d.pos))
|
||||||
|
|
||||||
|
a - b c - d e - f g - h
|
||||||
|
|
||||||
|
f + a = b + e
|
||||||
|
h + c = d + g
|
||||||
|
|
||||||
|
bd + ac + eh + fg = fh + ad + bc + eg
|
||||||
|
|
||||||
|
bd + ac + fc + eh + fg + ec = fh + ad + bc + ec + eg + fc
|
||||||
|
+ cf + ce
|
||||||
|
bd + c(a + f) + eh + fg + ec = fh + ad + c(b + e) + eg + fc
|
||||||
|
bd + eh + fg + ec = fh + ad + eg + fc
|
||||||
|
bd + e(h + c) + fg = f(h + c) + ad + eg
|
||||||
|
+ bg + ag
|
||||||
|
b(d + g) + e(h + c) + fg + ag = f(h + c) + a(d + g) + bg + eg
|
||||||
|
(h + c)(b + e) + g(a + f) = (h + c)(f + a) + g(b + e)
|
||||||
|
-/
|
||||||
|
private theorem multiplyRawInts.respects: ∀
|
||||||
|
{x y z w: RawInt}
|
||||||
|
(xz: x ~ z)
|
||||||
|
(yw: y ~ w), (multiplyRawInts x y = multiplyRawInts z w)
|
||||||
|
| ⟨a, b⟩, ⟨c, d⟩, ⟨e, f⟩, ⟨g, h⟩ => by
|
||||||
|
intro xz yw
|
||||||
|
apply eqv.sound
|
||||||
|
simp_all [eqv]
|
||||||
|
|
||||||
|
have first: (c + h) * (b + e) + g * (a + f) + c * (a + f)
|
||||||
|
= (c + h) * (f + a) + g * (b + e) + c * (b + e) := by
|
||||||
|
simp [Nat.add_comm, xz, yw]
|
||||||
|
|
||||||
|
have second: b * (d + g) + e * (c + h) + c * (a + f) + f * g + a * g
|
||||||
|
= f * (c + h) + a * (d + g) + c * (b + e) + b * g + e * g := by
|
||||||
|
simp [yw, xz] at first ⊢
|
||||||
|
conv at first in g * (e + b) => rw [<-xz]
|
||||||
|
conv at first => tactic => nat_ring
|
||||||
|
nat_ring
|
||||||
|
exact first
|
||||||
|
|
||||||
|
conv at second => tactic => nat_ring
|
||||||
|
apply @Nat.subtract_to_equation_left _ _
|
||||||
|
(a * g + b * g + c * f + c * e)
|
||||||
|
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
def multiply (τ β: MyInt): MyInt :=
|
||||||
|
Quotient.liftOn₂ τ β multiplyRawInts @multiplyRawInts.respects
|
||||||
|
|
||||||
|
private instance hMulRawInt: HMul RawInt RawInt MyInt where
|
||||||
|
hMul := multiplyRawInts
|
||||||
|
instance mulMyInt: Mul MyInt where
|
||||||
|
mul := multiply
|
||||||
|
|
||||||
|
private theorem multiplyRawInts.comm (a b: RawInt): a * b = b * a := by
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
simp_all [multiplyRawInts, Nat.mul_comm]
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
theorem multiply.comm (a b: MyInt): a * b = b * a := by
|
||||||
|
simp_all [Mul.mul, multiply]
|
||||||
|
induction a, b using Quotient.ind₂
|
||||||
|
quotient_madness
|
||||||
|
apply multiplyRawInts.comm
|
||||||
|
|
||||||
|
theorem multiply.assoc(x y z: MyInt): x * (y * z) = (x * y) * z := by
|
||||||
|
simp_all [Mul.mul, multiply]
|
||||||
|
|
||||||
|
induction x, y using Quotient.ind₂
|
||||||
|
induction z using Quotient.ind
|
||||||
|
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
@[simp]
|
||||||
|
theorem multiply.one(x: MyInt): x * 1 = 1 * x := by
|
||||||
|
simp_all [Mul.mul, multiply]
|
||||||
|
induction x using Quotient.ind
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
|
||||||
|
@[simp]
|
||||||
|
theorem multiply.zero(x: MyInt): x * 0 = 0 := by
|
||||||
|
simp_all [Mul.mul, multiply]
|
||||||
|
induction x using Quotient.ind
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
|
||||||
|
theorem left_distrib(x y z: MyInt): x * (y + z) = x * y + x * z := by
|
||||||
|
simp_all [Mul.mul, Add.add, add, multiply]
|
||||||
|
|
||||||
|
induction x, y using Quotient.ind₂
|
||||||
|
induction z using Quotient.ind
|
||||||
|
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
theorem right_distrib(x y z: MyInt): (x + y) * z = x * z + y * z := by
|
||||||
|
simp_all [Mul.mul, Add.add, add, multiply]
|
||||||
|
|
||||||
|
induction x, y using Quotient.ind₂
|
||||||
|
induction z using Quotient.ind
|
||||||
|
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
nat_ring_all
|
||||||
|
|
||||||
|
/-
|
||||||
|
notes on division?
|
||||||
|
t * (c - d) + r = a - b
|
||||||
|
t * c + b + r = a + t * d
|
||||||
|
-/
|
||||||
|
@[simp]
|
||||||
|
def is_even(x: MyInt) := ∃h, h + h = x
|
||||||
|
@[simp]
|
||||||
|
def is_odd(x: MyInt) := ∃h, h + h + 1 = x
|
||||||
|
|
||||||
|
theorem double_is_even(x: MyInt): is_even (2 * x) := by
|
||||||
|
simp
|
||||||
|
exists x
|
||||||
|
induction x using Quotient.ind
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv, Nat.double.addition_is_multiplication]
|
||||||
|
|
||||||
|
theorem raw_int_induction
|
||||||
|
(P: MyInt → Prop)
|
||||||
|
(pz: P 0)
|
||||||
|
(pn: ∀k, P k ↔ P (k + 1)):
|
||||||
|
(x: RawInt) → ∃k, k ~ x ∧ P (MyInt.ofRawInt k)
|
||||||
|
| ⟨0, 0⟩ => ⟨0, ⟨rfl, pz⟩⟩
|
||||||
|
| ⟨Nat.succ a, 0⟩ => by
|
||||||
|
have ⟨⟨kp, kn⟩, pk⟩ := raw_int_induction P pz pn ⟨a, 0⟩
|
||||||
|
exists (⟨kp + 1, kn⟩ : RawInt)
|
||||||
|
apply And.intro
|
||||||
|
. simp [eqv, Nat.succ_add]
|
||||||
|
rw [<-pk.left]
|
||||||
|
simp [Nat.add_zero]
|
||||||
|
. apply (@pn {kp, kn}).mp
|
||||||
|
exact pk.right
|
||||||
|
| ⟨Nat.succ a, Nat.succ b⟩ => by
|
||||||
|
have ⟨k, pk⟩ := raw_int_induction P pz pn ⟨a, b⟩
|
||||||
|
exists k
|
||||||
|
apply And.intro
|
||||||
|
. simp [eqv, Nat.succ_add]
|
||||||
|
rw [<-pk.left]
|
||||||
|
simp_arith
|
||||||
|
. exact pk.right
|
||||||
|
| ⟨0, Nat.succ a⟩ => by
|
||||||
|
have ⟨⟨kp, kn⟩, pk⟩ := raw_int_induction P pz pn ⟨0, a⟩
|
||||||
|
exists (⟨kp, kn + 1⟩ : RawInt)
|
||||||
|
apply And.intro
|
||||||
|
. have pkl := pk.left
|
||||||
|
simp [eqv, Nat.succ_add, Nat.add_zero] at pkl ⊢
|
||||||
|
rw [<-pkl]
|
||||||
|
simp_arith
|
||||||
|
. have recurse := (@pn {kp, kn + 1}).mpr
|
||||||
|
have rewriter: {kp, kn + 1} + 1 = {kp, kn} := by
|
||||||
|
apply eqv.sound
|
||||||
|
simp [eqv]
|
||||||
|
simp_arith
|
||||||
|
rw [rewriter] at recurse
|
||||||
|
exact (recurse pk.right)
|
||||||
|
|
||||||
|
theorem int_induction
|
||||||
|
(P: MyInt → Prop)
|
||||||
|
(zero: P 0)
|
||||||
|
(succ: ∀k, P k ↔ P (k + 1)):
|
||||||
|
∀k, P k := by
|
||||||
|
intro k
|
||||||
|
induction k using Quotient.ind
|
||||||
|
rename RawInt => kRaw
|
||||||
|
have ⟨e, ⟨eIsK, proof⟩⟩ := raw_int_induction P zero succ kRaw
|
||||||
|
have eIsKQuot : MyInt.ofRawInt e = MyInt.ofRawInt kRaw := by
|
||||||
|
exact (eqv.sound eIsK)
|
||||||
|
simp [Quotient.mk'] at eIsKQuot
|
||||||
|
rw [<-eIsKQuot]
|
||||||
|
exact proof
|
||||||
|
|
||||||
|
theorem add_left_cancel {a b c: MyInt}: a + b = a + c → b = c := by
|
||||||
|
intro hip
|
||||||
|
induction b, c using Quotient.ind₂
|
||||||
|
induction a using Quotient.ind
|
||||||
|
rename RawInt => c
|
||||||
|
simp_all [HAdd.hAdd, Add.add, add]
|
||||||
|
conv at hip => tactic => quotient_madness
|
||||||
|
/- apply eqv.sound -/
|
||||||
|
/- simp [eqv] -/
|
||||||
|
/- nat_ring -/
|
||||||
|
simp_all [MyInt.addRawInts, Quotient.mk', Quotient.mk]
|
||||||
|
sorry
|
||||||
|
|
||||||
|
/- induction a using int_induction with -/
|
||||||
|
/- | zero => -/
|
||||||
|
/- rw [add.comm 0 c, add.comm 0 b] -/
|
||||||
|
/- simp_all -/
|
||||||
|
/- | succ k => -/
|
||||||
|
/- apply Iff.intro -/
|
||||||
|
/- . intro win -/
|
||||||
|
/- intro previous -/
|
||||||
|
/- have p: k + b = k + c := by -/
|
||||||
|
/- rw [add.comm k 1] at previous -/
|
||||||
|
/- induction b, c using Quotient.ind₂ -/
|
||||||
|
/- induction k using Quotient.ind -/
|
||||||
|
/- rename RawInt => c -/
|
||||||
|
/- simp [HAdd.hAdd, Add.add, add] -/
|
||||||
|
/- apply eqv.sound -/
|
||||||
|
/- simp [eqv] -/
|
||||||
|
/- nat_ring -/
|
||||||
|
/- -/
|
||||||
|
/- sorry -/
|
||||||
|
/- exact (win p) -/
|
||||||
|
/- . sorry -/
|
||||||
|
/- -/
|
||||||
|
|
||||||
|
theorem odd_and_even_contradict(x: MyInt): ¬(is_odd x ∧ is_even x)
|
||||||
|
| ⟨⟨h₁, oddProof⟩, ⟨h₂, evenProof⟩⟩ => by
|
||||||
|
have wrong: (1: MyInt) = 0 := by
|
||||||
|
apply @add_left_cancel (h₁ + h₂)
|
||||||
|
exact oddProof.trans evenProof.symm
|
||||||
|
sorry
|
||||||
|
contradiction
|
||||||
|
|
||||||
|
theorem odds_not_even(x: MyInt): is_odd x ↔ ¬(is_even x) := by
|
||||||
|
apply Iff.intro
|
||||||
|
case mp =>
|
||||||
|
intro oddProof
|
||||||
|
intro evenProof
|
||||||
|
apply odd_and_even_contradict x
|
||||||
|
exact ⟨oddProof, evenProof⟩
|
||||||
|
case mpr =>
|
||||||
|
simp [is_even, is_odd]
|
||||||
|
sorry
|
||||||
|
|
||||||
|
end MyInt
|
||||||
55
lean/learning/LeanSandbox/Nat.lean
Normal file
55
lean/learning/LeanSandbox/Nat.lean
Normal file
|
|
@ -0,0 +1,55 @@
|
||||||
|
namespace Nat
|
||||||
|
theorem add_equations: ∀{a b c d: Nat}, a = b → c = d → a + c = b + d := by
|
||||||
|
intro a b c d ab cd
|
||||||
|
rw [ab, cd]
|
||||||
|
|
||||||
|
theorem add_to_equation_right: ∀{a b c: Nat}, a = b → a + c = b + c := by
|
||||||
|
intro a b c ab
|
||||||
|
exact add_equations ab rfl
|
||||||
|
|
||||||
|
theorem succ_equation : ∀{a b: Nat}, Nat.succ a = Nat.succ b → a = b := by
|
||||||
|
intro a b
|
||||||
|
apply Eq.mp -- (a + 1 = b + 1) = a + b
|
||||||
|
apply Nat.succ.injEq -- this was already here?
|
||||||
|
|
||||||
|
inductive A : Nat -> Type
|
||||||
|
| ooo: (n: Nat) → A n
|
||||||
|
|
||||||
|
theorem subtract_to_equation_right: ∀{a b c: Nat}, a + c = b + c → a = b := by
|
||||||
|
intro a b c acbc
|
||||||
|
induction c with
|
||||||
|
| zero =>
|
||||||
|
repeat rw [Nat.add_zero] at acbc
|
||||||
|
exact acbc
|
||||||
|
| succ pc inner =>
|
||||||
|
/-
|
||||||
|
a + S pc = b + S pc -- comm
|
||||||
|
S pc + a = S pc + b -- addition definition
|
||||||
|
S (pc + a) = S (pc + b) -- injective constructor
|
||||||
|
pc + a = pc + b
|
||||||
|
-/
|
||||||
|
rw [Nat.add_comm a (Nat.succ pc), Nat.add_comm b (Nat.succ pc)] at acbc
|
||||||
|
simp [Nat.succ_add, Nat.add_comm] at acbc
|
||||||
|
exact (inner acbc)
|
||||||
|
|
||||||
|
theorem subtract_to_equation_left: ∀{a b c: Nat}, c + a = c + b → a = b := by
|
||||||
|
intro a b c cacb
|
||||||
|
rw [Nat.add_comm c a, Nat.add_comm c b] at cacb
|
||||||
|
exact (subtract_to_equation_right cacb)
|
||||||
|
|
||||||
|
theorem add_equation_both_sides_right: ∀{a b c: Nat}, (a = b) ↔ (a + c = b + c) := by
|
||||||
|
intro a b c
|
||||||
|
apply Iff.intro
|
||||||
|
. exact add_to_equation_right
|
||||||
|
. exact subtract_to_equation_right
|
||||||
|
|
||||||
|
theorem multiply_equation_left: ∀{a b c: Nat}, (a = b) → (c * a = c * b) := by
|
||||||
|
intro a b c ab
|
||||||
|
rw [ab]
|
||||||
|
|
||||||
|
theorem double.addition_is_multiplication (x: Nat): 2 * x = x + x := by
|
||||||
|
induction x with
|
||||||
|
| zero => simp
|
||||||
|
| succ px ic =>
|
||||||
|
simp [<-Nat.add_one, Nat.left_distrib, ic, Nat.add_left_comm, Nat.add_assoc]
|
||||||
|
end Nat
|
||||||
289
lean/learning/LeanSandbox/Noob.lean
Normal file
289
lean/learning/LeanSandbox/Noob.lean
Normal file
|
|
@ -0,0 +1,289 @@
|
||||||
|
-- single line comment
|
||||||
|
universe u
|
||||||
|
|
||||||
|
def m: Nat := 1
|
||||||
|
|
||||||
|
#check m
|
||||||
|
#check m
|
||||||
|
|
||||||
|
#check (m + m)
|
||||||
|
#eval (m + m)
|
||||||
|
|
||||||
|
#check Nat × Nat
|
||||||
|
|
||||||
|
def Γ := 2 + m
|
||||||
|
|
||||||
|
|
||||||
|
def double (a: Nat) := a + a
|
||||||
|
|
||||||
|
#eval double (double 3)
|
||||||
|
|
||||||
|
section composition
|
||||||
|
variable (α β γ: Type)
|
||||||
|
|
||||||
|
def compose (f: α → β) (g: β → γ): (α -> γ) := fun x: α => g (f x)
|
||||||
|
end composition
|
||||||
|
|
||||||
|
def quadruble := compose _ _ _ double double
|
||||||
|
|
||||||
|
#print compose
|
||||||
|
|
||||||
|
section composition'
|
||||||
|
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 composition'
|
||||||
|
|
||||||
|
def a: Nat := 3
|
||||||
|
|
||||||
|
#eval double a
|
||||||
|
#print compose'
|
||||||
|
#check (Nat : Sort 1)
|
||||||
|
#check (Prop : Sort 1)
|
||||||
|
#check (Prop : Type)
|
||||||
|
|
||||||
|
|
||||||
|
section proofs
|
||||||
|
theorem hmm (α: Prop) (β: Prop): Prop := α
|
||||||
|
theorem hmm2 : Prop -> Prop -> Prop :=
|
||||||
|
fun α β => show Prop from α
|
||||||
|
|
||||||
|
theorem hmm3 (hm: α): α := hm
|
||||||
|
|
||||||
|
axiom myProof: Prop
|
||||||
|
|
||||||
|
#check hmm
|
||||||
|
#check hmm2
|
||||||
|
#check hmm3 myProof
|
||||||
|
#check @hmm3
|
||||||
|
#check @hmm3 myProof
|
||||||
|
end proofs
|
||||||
|
|
||||||
|
section logic
|
||||||
|
variable {α β π: Prop}
|
||||||
|
variable (ha: α) (hb: β)
|
||||||
|
|
||||||
|
theorem pair: α -> β -> α ∧ β := fun a b => And.intro a b
|
||||||
|
theorem unpair: (α → β → π) → α ∧ β → π := fun f a =>
|
||||||
|
show π from f (And.left a) (And.right a)
|
||||||
|
|
||||||
|
theorem pairsCommute: (α ∧ β) → (β ∧ α) := fun p =>
|
||||||
|
show (β ∧ α) from And.intro (And.right p) (And.left p)
|
||||||
|
|
||||||
|
example (h: α ∧ β): β ∧ α := ⟨h.right, h.left⟩
|
||||||
|
|
||||||
|
theorem thrice: (f: α) -> α ∧ α ∧ α := fun f => ⟨ f, f, f ⟩
|
||||||
|
|
||||||
|
theorem negateFunc: (β → α) → ¬α → ¬β :=
|
||||||
|
fun f notₐ b =>
|
||||||
|
show False from notₐ (f b)
|
||||||
|
|
||||||
|
#print pairsCommute
|
||||||
|
#check (⟨ha, hb⟩: α ∧ β)
|
||||||
|
|
||||||
|
theorem exFalso: (α ∧ ¬ α) → β :=
|
||||||
|
fun contradiction =>
|
||||||
|
show β from (contradiction.right contradiction.left).elim
|
||||||
|
|
||||||
|
theorem pairIso: α ∧ β ↔ β ∧ α := ⟨ pairsCommute, pairsCommute ⟩
|
||||||
|
end logic
|
||||||
|
|
||||||
|
section classical
|
||||||
|
open Classical
|
||||||
|
|
||||||
|
variable (α : Prop)
|
||||||
|
|
||||||
|
def dneT: Prop := ¬¬α → α
|
||||||
|
|
||||||
|
theorem doubleNegation: α ↔ ¬¬α :=
|
||||||
|
suffices l: α → ¬¬α from
|
||||||
|
suffices r: ¬¬α → α from ⟨ l, r ⟩
|
||||||
|
show dneT α from fun doubleNegA => (em α).elim
|
||||||
|
(fun p: α => p)
|
||||||
|
(fun p: ¬α => absurd p doubleNegA)
|
||||||
|
show α → ¬¬α from (fun a f => f a)
|
||||||
|
|
||||||
|
#print byCases
|
||||||
|
|
||||||
|
theorem dne: dneT α := fun nnₚ =>
|
||||||
|
byCases id
|
||||||
|
(fun nₚ => (nnₚ nₚ).elim)
|
||||||
|
|
||||||
|
#print byContradiction
|
||||||
|
|
||||||
|
theorem dne': dneT α := fun nnₚ =>
|
||||||
|
byContradiction fun nₚ => nnₚ nₚ
|
||||||
|
end classical
|
||||||
|
|
||||||
|
section exercises
|
||||||
|
variable (p q r : Prop)
|
||||||
|
|
||||||
|
theorem noContradictions : ¬(p ↔ ¬p) :=
|
||||||
|
fun contradiction =>
|
||||||
|
suffices someP: p from contradiction.mp someP someP
|
||||||
|
suffices someNotP: ¬p from show p from contradiction.mpr someNotP
|
||||||
|
show ¬p from fun p => contradiction.mp p p
|
||||||
|
end exercises
|
||||||
|
|
||||||
|
section quantifiers
|
||||||
|
variable (α: Type) (r: α → α → Prop)
|
||||||
|
variable (transitivity: ∀x y z, r x y → r y z → r x z)
|
||||||
|
variable (n: Nat)
|
||||||
|
|
||||||
|
variable (trans2: ∀x y z, r x y → r y z → r x z)
|
||||||
|
variable (trans3: ∀τ β ω, r τ β → r β ω → r τ ω)
|
||||||
|
|
||||||
|
axiom someA: α
|
||||||
|
|
||||||
|
#print someA
|
||||||
|
#check transitivity
|
||||||
|
|
||||||
|
#print Eq.subst
|
||||||
|
|
||||||
|
theorem substTest: (a b c: Nat) → (myProp: Nat -> Prop) →
|
||||||
|
(a + b = c) → myProp c → myProp (a + b) :=
|
||||||
|
fun a b c myProp p₁ l => @Eq.subst Nat (fun a => myProp a) c (a + b) p₁.symm l
|
||||||
|
theorem substTest': {a b c: Nat} → {myProp: Nat -> Prop} →
|
||||||
|
(a + b = c) → myProp c → myProp (a + b) :=
|
||||||
|
fun p₁ l => p₁ ▸ l
|
||||||
|
|
||||||
|
#check @congr
|
||||||
|
#print congrArg
|
||||||
|
#print congrFun
|
||||||
|
|
||||||
|
theorem congrArgTest: {a b: Nat} → (c: Nat) → (a = b) → (a + c = b + c) :=
|
||||||
|
fun c p₁ => congrArg (fun a => a + c) p₁
|
||||||
|
|
||||||
|
end quantifiers
|
||||||
|
|
||||||
|
section calc_
|
||||||
|
variable (a b c d e : Nat)
|
||||||
|
variable (h1 : a = b)
|
||||||
|
variable (h2 : b = c + 1)
|
||||||
|
variable (h3 : c = d)
|
||||||
|
variable (h4 : e = 1 + d)
|
||||||
|
|
||||||
|
theorem T : a = e :=
|
||||||
|
calc
|
||||||
|
a = b := h1
|
||||||
|
_ = c + 1 := h2
|
||||||
|
_ = d + 1 := congrArg Nat.succ h3
|
||||||
|
_ = 1 + d := Nat.add_comm d 1
|
||||||
|
_ = e := h4.symm
|
||||||
|
|
||||||
|
theorem T₂ : a = e :=
|
||||||
|
calc
|
||||||
|
a = b := h1
|
||||||
|
_ = c + 1 := h2
|
||||||
|
_ = d + 1 := by rw [h3]
|
||||||
|
_ = 1 + d := by rw [Nat.add_comm]
|
||||||
|
_ = e := by rw [h4]
|
||||||
|
|
||||||
|
theorem T₃ : a = e :=
|
||||||
|
by rw [h1, h2, h3, Nat.add_comm, h4]
|
||||||
|
|
||||||
|
theorem T₄ : a = e :=
|
||||||
|
by simp [h1, h2, h3, Nat.add_comm, h4]
|
||||||
|
|
||||||
|
example (a b c d : Nat) (h1 : a = b) (h2 : b ≤ c) (h3 : c + 1 < d) : a < d :=
|
||||||
|
calc
|
||||||
|
a = b := h1
|
||||||
|
_ < b + 1 := Nat.lt_succ_self b
|
||||||
|
_ ≤ c + 1 := Nat.succ_le_succ h2
|
||||||
|
_ < d := h3
|
||||||
|
|
||||||
|
variable (x y: Nat)
|
||||||
|
|
||||||
|
theorem A₁: (x + y) * (x + y) = x * x + y * x + x * y + y * y :=
|
||||||
|
calc
|
||||||
|
(x + y) * (x + y) = (x + y) * x + (x + y) * y := by rw [Nat.mul_add]
|
||||||
|
_ = x * x + y * x + (x * y + y * y) := by simp [Nat.add_mul]
|
||||||
|
_ = x * x + y * x + x * y + y * y := by rw [←Nat.add_assoc]
|
||||||
|
|
||||||
|
theorem A₂: (x + y) * (x + y) = x * x + y * x + x * y + y * y :=
|
||||||
|
by simp [Nat.add_mul, Nat.mul_add, Nat.add_assoc]
|
||||||
|
|
||||||
|
axiom forallTest: ¬(∀a: Nat, a + a = a + 1)
|
||||||
|
|
||||||
|
theorem existsOne: ∃a: Nat, a + a = a + 1 := ⟨ 1, rfl ⟩
|
||||||
|
theorem existsBiggerThanOne: ∃a: Nat, a > 1 :=
|
||||||
|
have r: 1 < 2 := Nat.succ_lt_succ (Nat.zero_lt_succ 0)
|
||||||
|
⟨ 2, r ⟩
|
||||||
|
|
||||||
|
example (x : Nat) (h : x > 0) : ∃ y, y < x :=
|
||||||
|
⟨ 0, h ⟩
|
||||||
|
end calc_
|
||||||
|
|
||||||
|
section exists_print
|
||||||
|
variable (g : Nat → Nat → Nat)
|
||||||
|
variable (hg : g 0 0 = 0)
|
||||||
|
|
||||||
|
theorem gex1 : ∃ x, g x x = x := ⟨0, hg⟩
|
||||||
|
theorem gex2 : ∃ x, g x 0 = x := ⟨0, hg⟩
|
||||||
|
theorem gex3 : ∃ x, g 0 0 = x := ⟨0, hg⟩
|
||||||
|
theorem gex4 : ∃ x, g x x = 0 := ⟨0, hg⟩
|
||||||
|
|
||||||
|
theorem gex5 : ∃ x, g x x = 0 := ⟨0, hg⟩
|
||||||
|
theorem gex6 : ∃ x, g x x = 0 := ⟨0, hg⟩
|
||||||
|
theorem gex7 : ∃ x, g x x = 0 := ⟨0, hg⟩
|
||||||
|
|
||||||
|
set_option pp.explicit true -- display implicit arguments
|
||||||
|
#print gex1
|
||||||
|
#print gex2
|
||||||
|
#print gex3
|
||||||
|
#print gex4
|
||||||
|
end exists_print
|
||||||
|
|
||||||
|
section existentials
|
||||||
|
variable (α : Type) (p q : α → Prop)
|
||||||
|
|
||||||
|
example (h : ∃ x, p x ∧ q x) : ∃ x, q x ∧ p x :=
|
||||||
|
match h with
|
||||||
|
| ⟨w, hw⟩ => ⟨w, hw.right, hw.left⟩
|
||||||
|
|
||||||
|
theorem _matches : (n: Nat) → n + 1 = 1 + n := fun n =>
|
||||||
|
match n with
|
||||||
|
| Nat.succ w => calc
|
||||||
|
Nat.succ w + 1 = Nat.succ (w + 1) := rfl
|
||||||
|
_ = Nat.succ (1 + w) := by rw [_matches w]
|
||||||
|
_ = 1 + Nat.succ w := rfl
|
||||||
|
| Nat.zero => rfl
|
||||||
|
end existentials
|
||||||
|
|
||||||
|
section tactics
|
||||||
|
theorem test (p q : Prop) (hp: p) (hq: q): p ∧ q ∧ p := by
|
||||||
|
apply And.intro
|
||||||
|
case left =>
|
||||||
|
exact hp
|
||||||
|
case right =>
|
||||||
|
apply And.intro hq hp
|
||||||
|
|
||||||
|
#print test
|
||||||
|
|
||||||
|
theorem test₂ (p q r : Prop) : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) := by
|
||||||
|
apply Iff.intro
|
||||||
|
. intro h
|
||||||
|
apply Or.elim h.right
|
||||||
|
. intro hq
|
||||||
|
apply Or.inl
|
||||||
|
exact ⟨h.left, hq⟩
|
||||||
|
. intro hr
|
||||||
|
apply Or.inr
|
||||||
|
exact ⟨h.left, hr⟩
|
||||||
|
. intro h
|
||||||
|
apply Or.elim h
|
||||||
|
. intro pq
|
||||||
|
apply And.intro
|
||||||
|
. exact pq.left
|
||||||
|
. exact Or.inl pq.right
|
||||||
|
. intro pr
|
||||||
|
apply And.intro
|
||||||
|
. exact pr.left
|
||||||
|
. exact Or.inr pr.right
|
||||||
|
|
||||||
|
#print test₂
|
||||||
|
end tactics
|
||||||
4
lean/learning/Main.lean
Normal file
4
lean/learning/Main.lean
Normal file
|
|
@ -0,0 +1,4 @@
|
||||||
|
import LeanSandbox
|
||||||
|
|
||||||
|
def main : IO Unit :=
|
||||||
|
IO.println s!"Hello, {hello}!"
|
||||||
6
lean/learning/lakefile.lean
Normal file
6
lean/learning/lakefile.lean
Normal file
|
|
@ -0,0 +1,6 @@
|
||||||
|
import Lake
|
||||||
|
open Lake DSL
|
||||||
|
|
||||||
|
package «lean-sandbox» {
|
||||||
|
-- add configuration options here
|
||||||
|
}
|
||||||
Some files were not shown because too many files have changed in this diff Show more
Loading…
Reference in a new issue