162 lines
5.7 KiB
Text
162 lines
5.7 KiB
Text
import Std
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inductive Expr (maxVarId : Nat) : Type where
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| var (i : Fin maxVarId)
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| op (lhs rhs : Expr maxVarId)
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deriving Repr
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def List.getIdx : (l : List α) → Fin l.length → α
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| [], ⟨_, i⟩ => by
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simp [length] at i
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apply absurd i
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cases i
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| a::as, ⟨0, _⟩ => a
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| a::as, ⟨i+1, h⟩ => by
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apply getIdx as ⟨i, _⟩
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simp [List.length_cons] at h
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apply Nat.lt_of_succ_lt_succ
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assumption
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structure Context (α : Type u) where
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op : α → α → α
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assoc : (a b c : α) → op (op a b) c = op a (op b c)
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comm : (a b : α) → op a b = op b a
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vars : List α
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theorem Context.left_comm (ctx : Context α) (a b c : α) : ctx.op a (ctx.op b c) = ctx.op b (ctx.op a c) := by
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rw [← ctx.assoc, ctx.comm a b, ctx.assoc]
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def Expr.denote (ctx : Context α) : Expr ctx.vars.length → α
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| Expr.op a b => ctx.op (denote ctx a) (denote ctx b)
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| Expr.var i => ctx.vars.getIdx i
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theorem Expr.denote_op (ctx : Context α) (a b : Expr ctx.vars.length) : denote ctx (Expr.op a b) = ctx.op (denote ctx a) (denote ctx b) :=
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rfl
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def Expr.concat : Expr n → Expr n → Expr n
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| Expr.op a b, c => Expr.op a (concat b c)
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| Expr.var i, c => Expr.op (Expr.var i) c
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theorem Expr.denote_concat (ctx : Context α) (a b : Expr ctx.vars.length) : denote ctx (concat a b) = denote ctx (Expr.op a b) := by
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induction a with
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| var i => rfl
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| op _ _ _ ih => simp [denote, ih, ctx.assoc]
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def Expr.flat : Expr n → Expr n
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| Expr.op a b => concat (flat a) (flat b)
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| Expr.var i => Expr.var i
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theorem Expr.denote_flat (ctx : Context α) (e : Expr ctx.vars.length) : denote ctx (flat e) = denote ctx e := by
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induction e with
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| var i => rfl
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| op a b ih₁ ih₂ => simp [flat, denote, denote_concat, ih₁, ih₂]
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theorem Expr.eq_of_flat (ctx : Context α) (a b : Expr ctx.vars.length) (h : flat a = flat b) : denote ctx a = denote ctx b := by
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have h := congrArg (denote ctx) h
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simp [denote_flat] at h
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assumption
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def Expr.length : Expr n → Nat
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| op a b => 1 + b.length
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| _ => 1
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def Expr.sort (e : Expr n) : Expr n :=
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loop e.length e
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where
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loop : Nat → Expr n → Expr n
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| fuel+1, Expr.op a e =>
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let (e₁, e₂) := swap a e
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Expr.op e₁ (loop fuel e₂)
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| _, e => e
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swap : Expr n → Expr n → Expr n × Expr n
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| Expr.var i, Expr.op (Expr.var j) e =>
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if i > j then
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let (e₁, e₂) := swap (Expr.var j) e
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(e₁, Expr.op (Expr.var i) e₂)
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else
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let (e₁, e₂) := swap (Expr.var i) e
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(e₁, Expr.op (Expr.var j) e₂)
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| Expr.var i, Expr.var j =>
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if i > j then
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(Expr.var j, Expr.var i)
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else
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(Expr.var i, Expr.var j)
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| e₁, e₂ => (e₁, e₂)
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theorem Expr.denote_sort (ctx : Context α) (e : Expr ctx.vars.length) : denote ctx (sort e) = denote ctx e := by
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apply denote_loop
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where
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denote_loop (n : Nat) (e : Expr ctx.vars.length) : denote ctx (sort.loop n e) = denote ctx e := by
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induction n generalizing e with
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| zero => rfl
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| succ n ih =>
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match e with
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| var _ => rfl
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| op a b =>
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simp [denote, sort.loop]
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match h:sort.swap a b with
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| (r₁, r₂) =>
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have hs := denote_swap a b
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rw [h] at hs
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simp [denote] at hs
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simp [denote, ih]
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assumption
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denote_swap (e₁ e₂ : Expr ctx.vars.length) : denote ctx (Expr.op (sort.swap e₁ e₂).1 (sort.swap e₁ e₂).2) = denote ctx (Expr.op e₁ e₂) := by
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induction e₂ generalizing e₁ with
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| op a b ih' ih =>
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clear ih'
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cases e₁ with
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| var i =>
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cases a with
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| var j =>
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byCases h : i > j
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focus
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simp [sort.swap, h]
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match h:sort.swap (var j) b with
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| (r₁, r₂) => simp; rw [denote_op (a := var i), ← ih]; simp [h, denote]; rw [Context.left_comm]
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focus
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simp [sort.swap, h]
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match h:sort.swap (var i) b with
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| (r₁, r₂) =>
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simp
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rw [denote_op (a := var i), denote_op (a := var j), Context.left_comm, ← denote_op (a := var i), ← ih]
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simp [h, denote]
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rw [Context.left_comm]
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| _ => rfl
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| _ => rfl
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| var j =>
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cases e₁ with
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| var i =>
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byCases h : i > j
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focus simp [sort.swap, h, denote, Context.comm]
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focus simp [sort.swap, h]
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| _ => rfl
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theorem Expr.eq_of_sort_flat (ctx : Context α) (a b : Expr ctx.vars.length) (h : sort (flat a) = sort (flat b)) : denote ctx a = denote ctx b := by
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have h := congrArg (denote ctx) h
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simp [denote_flat, denote_sort] at h
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assumption
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theorem ex₁ (x₁ x₂ x₃ x₄ : Nat) : (x₁ + x₂) + (x₃ + x₄) = x₁ + x₂ + x₃ + x₄ :=
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Expr.eq_of_flat
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{ op := Nat.add
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assoc := Nat.add_assoc
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comm := Nat.add_comm
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vars := [x₁, x₂, x₃, x₄] }
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(Expr.op (Expr.op (Expr.var ⟨0, by simp⟩) (Expr.var ⟨1, by simp⟩)) (Expr.op (Expr.var ⟨2, by simp⟩) (Expr.var ⟨3, by simp⟩)))
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(Expr.op (Expr.op (Expr.op (Expr.var ⟨0, by simp⟩) (Expr.var ⟨1, by simp⟩)) (Expr.var ⟨2, by simp⟩)) (Expr.var ⟨3, by simp⟩))
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(by rfl)
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theorem ex₂ (x₁ x₂ x₃ x₄ : Nat) : (x₁ + x₂) + (x₃ + x₄) = x₃ + x₁ + x₂ + x₄ :=
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Expr.eq_of_sort_flat
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{ op := Nat.add
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assoc := Nat.add_assoc
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comm := Nat.add_comm
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vars := [x₁, x₂, x₃, x₄] }
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(Expr.op (Expr.op (Expr.var ⟨0, by simp⟩) (Expr.var ⟨1, by simp⟩)) (Expr.op (Expr.var ⟨2, by simp⟩) (Expr.var ⟨3, by simp⟩)))
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(Expr.op (Expr.op (Expr.op (Expr.var ⟨2, by simp⟩) (Expr.var ⟨0, by simp⟩)) (Expr.var ⟨1, by simp⟩)) (Expr.var ⟨3, by simp⟩))
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(by rfl)
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#print ex₂
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