66bc9ae177
This PR deprecates `levelZero` in favor of `Level.zero` and `levelOne`
in favor of the new `Level.one`, and updates all usages throughout the
codebase. The `levelZero` alias was previously required for computed
field `data` to work, but this is no longer needed.
🤖 Prepared with Claude Code
112 lines
3.2 KiB
Text
112 lines
3.2 KiB
Text
import Lean
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open Lean
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open Meta
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def execShow (x : MetaM Expr) : MetaM Unit := do
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let e ← x
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IO.println (← ppExpr e)
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/-
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The `MetaM` has an ambient local context.
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The local context contains the types of the free variables.
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-/
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def mkLam1 : MetaM Expr :=
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/- The following `withLocalDecl` extends the local context with `(x : Nat)`, and executes the lambda
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with an expression `x`. -/
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withLocalDecl `x BinderInfo.default (mkConst `Nat) fun x =>
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/- Similar to the method above, but sets `BinderInfo.default`. -/
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withLocalDeclD `y (mkConst `Nat) fun y => do
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-- Double backticks instruct Lean to resolve the names at compilation time
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let b ← mkAppM ``HAdd.hAdd #[x, y] -- `x + y`
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let b ← mkAppM ``HAdd.hAdd #[b, x] -- `x + y + x`
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/- `mkLambdaFVars` converts the free variables into de-Bruijn bound variables, and construct the lambda for us. -/
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mkLambdaFVars #[x, y] b
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/-- info: fun x y => x + y + x -/
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#guard_msgs in
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#eval execShow mkLam1
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/-- info: Nat → Nat → Nat -/
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#guard_msgs in
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#eval execShow do let e ← mkLam1; inferType e
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def mkForall1 : MetaM Expr :=
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withLocalDeclD `x (mkConst `Nat) fun x =>
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withLocalDeclD `y (mkConst `Nat) fun y => do
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let b ← mkEq x y
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mkForallFVars #[x, y] b
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/-- info: ∀ (x y : Nat), x = y -/
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#guard_msgs in
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#eval execShow mkForall1
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def mkForall2Lambda : MetaM Expr := do
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let e ← mkForall1
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/- The following method `opens` a `forall` term, and expands the local context with
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new free variables corresponding to the forall binders.
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We also have a telescope function for `lambda` terms. -/
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forallTelescopeReducing e fun xs b => do
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IO.println s!">> {← xs.mapM Meta.ppExpr}"
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IO.println s!">> {← ppExpr b}"
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-- xs is an array of `Expr`
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let b ← mkAppM ``HMul.hMul xs
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mkLambdaFVars xs b
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/--
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info: >> #[x, y]
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>> x = y
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fun x y => x * y
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-/
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#guard_msgs in
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#eval execShow mkForall2Lambda
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def mkGoal : MetaM Expr :=
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withLocalDeclD `x (mkConst `Nat) fun x =>
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withLocalDeclD `y (mkConst `Nat) fun y => do
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withLocalDeclD `h (← mkEq x y) fun h => do
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let b ← mkEq y x
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mkForallFVars #[x, y, h] b
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/-- info: ∀ (x y : Nat), x = y → y = x -/
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#guard_msgs in
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#eval execShow mkGoal
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def proveGoal : MetaM Unit := do
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let g ← mkGoal
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-- `main` is a metavariable representing the term we want to construct
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let main ← mkFreshExprSyntheticOpaqueMVar g
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let m := main.mvarId!
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IO.println (← ppGoal m)
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IO.println "-----"
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let (xs, m) ← m.introNP 3 -- `intro` 3 variables using the binder names to name the new locals
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IO.println (← ppGoal m)
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IO.println "-----"
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-- The `apply` tactic generates 0 or more subgoals
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let [m] ← m.apply (mkConst ``Eq.symm [Level.one]) | throwError "unexpected number of subgoals"
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IO.println (← ppGoal m)
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IO.println "-----"
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m.assumption
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-- `main` contains the whole proof now
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IO.println (← ppExpr main)
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check main -- type check the proof. This is not the kernel type checker.
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IO.println (← ppExpr (← inferType main))
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/--
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info: ⊢ ∀ (x y : Nat), x = y → y = x
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-----
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x y : Nat
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h : x = y
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⊢ y = x
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-----
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case h
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x y : Nat
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h : x = y
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⊢ x = y
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-----
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fun x y h => Eq.symm h
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∀ (x y : Nat), x = y → y = x
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-/
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#guard_msgs in
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#eval proveGoal
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