66 lines
2.3 KiB
Text
66 lines
2.3 KiB
Text
import Std
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inductive Expr where
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| var (i : Nat)
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| op (lhs rhs : Expr)
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def List.getIdx : List α → Nat → α → α
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| [], i, u => u
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| a::as, 0, u => a
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| a::as, i+1, u => getIdx as i u
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structure Context (α : Type u) where
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op : α → α → α
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unit : α
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assoc : (a b c : α) → op (op a b) c = op a (op b c)
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vars : List α
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def Expr.denote (ctx : Context α) : Expr → 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 ctx.unit
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theorem Expr.denote_op (ctx : Context α) (a b : Expr) : denote ctx (Expr.op a b) = ctx.op (denote ctx a) (denote ctx b) :=
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rfl
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theorem Expr.denote_var (ctx : Context α) (i : Nat) : denote ctx (Expr.var i) = ctx.vars.getIdx i ctx.unit :=
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rfl
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def Expr.concat : Expr → Expr → Expr
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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.concat_op (a b c : Expr) : concat (Expr.op a b) c = Expr.op a (concat b c) :=
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rfl
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theorem Expr.concat_var (i : Nat) (c : Expr) : concat (Expr.var i) c = Expr.op (Expr.var i) c :=
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rfl
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theorem Expr.denote_concat (ctx : Context α) (a b : Expr) : 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 => rw [concat_op, denote_op, ih, denote_op, denote_op, denote_op, ctx.assoc]
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def Expr.flat : Expr → Expr
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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.flat_op (a b : Expr) : flat (Expr.op a b) = concat (flat a) (flat b) :=
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rfl
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theorem Expr.denote_flat (ctx : Context α) (a : Expr) : denote ctx (flat a) = denote ctx a := by
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induction a with
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| var i => rfl
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| op a b ih₁ ih₂ => rw [flat_op, denote_concat, denote_op, denote_op, ih₁, ih₂]
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theorem Expr.eq_of_flat (ctx : Context α) (a b : Expr) (h : flat a = flat b) : denote ctx a = denote ctx b := by
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rw [← Expr.denote_flat _ a, ← Expr.denote_flat _ b, h]
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theorem test (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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unit := Nat.zero
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vars := [x₁, x₂, x₃, x₄] }
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(Expr.op (Expr.op (Expr.var 0) (Expr.var 1)) (Expr.op (Expr.var 2) (Expr.var 3)))
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(Expr.op (Expr.op (Expr.op (Expr.var 0) (Expr.var 1)) (Expr.var 2)) (Expr.var 3))
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rfl
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