63 lines
2.2 KiB
Text
63 lines
2.2 KiB
Text
import data.nat
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open tactic expr
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meta_definition is_op_app (op : expr) (e : expr) : option (expr × expr) :=
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match e with
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| app (app fn a1) a2 := if op = fn then some (a1, a2) else none
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| _ := none
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end
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meta_definition flat_with (op : expr) (assoc : expr) (e : expr) (rhs : expr) : tactic (expr × expr) :=
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match is_op_app op e with
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| some (a1, a2) := do
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-- H₁ is a proof for a2 + rhs = rhs₁
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(rhs₁, H₁) ← flat_with op assoc a2 rhs,
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-- H₂ is a proof for a1 + rhs₁ = rhs₂
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(new_app, H₂) ← flat_with op assoc a1 rhs₁,
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-- We need to generate a proof that (a1 + a2) + rhs = a1 + (a2 + rhs)
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-- H₃ is a proof for (a1 + a2) + rhs = a1 + (a2 + rhs)
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H₃ : expr ← return (app (app (app assoc a1) a2) rhs),
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-- H₃ is a proof for a1 + (a2 + rhs) = a1 + rhs1
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H₄ : expr ← mk_app `congr_arg [app op a1, H₁],
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H₅ ← mk_app `eq.trans [H₃, H₄],
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H ← mk_app `eq.trans [H₅, H₂],
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return (new_app, H)
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| none := do
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new_app ← return $ app (app op e) rhs,
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H ← mk_app `eq.refl [new_app],
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return (new_app, H)
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end
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meta_definition flat (op : expr) (assoc : expr) (e : expr) : tactic (expr × expr) :=
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match is_op_app op e with
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| some (a1, a2) := do
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-- H₁ is a proof that a2 = new_a2
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(new_a2, H₁) ← flat op assoc a2,
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-- H₂ is a proof that a1 + new_a2 = new_app
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(new_app, H₂) ← flat_with op assoc a1 new_a2,
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-- We need a proof that (a1 + a2) = new_app
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-- H₃ is a proof for a1 + a2 = a1 + new_a2
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H₃ : expr ← mk_app `congr_arg [app op a1, H₁],
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H ← mk_app `eq.trans [H₃, H₂],
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return (new_app, H)
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| none :=
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do pr ← mk_app `eq.refl [e],
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return (e, pr)
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end
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local infix `+` := nat.add
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set_option trace.app_builder true
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set_option pp.all true
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example (a b c d e f g : nat) : ((a + b) + c) + ((d + e) + (f + g)) = a + (b + (c + (d + (e + (f + g))))) :=
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by do
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assoc : expr ← mk_const $ `nat.add_assoc,
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op : expr ← mk_const $ `nat.add,
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tgt ← target,
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match is_eq tgt with
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| some (lhs, rhs) := do
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r ← flat op assoc lhs,
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trace (prod.pr2 r),
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exact (prod.pr2 r)
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| none := failed
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end
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