47 lines
2.7 KiB
Text
47 lines
2.7 KiB
Text
import logic data.nat.basic data.sigma
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open nat eq.ops sigma
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inductive vector (A : Type) : nat → Type :=
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| nil : vector A zero
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| cons : A → (Π{n}, vector A n → vector A (succ n))
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namespace vector
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definition vec (A : Type) : Type := Σ n : nat, vector A n
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definition to_vec {A : Type} {n : nat} (v : vector A n) : vec A := ⟨n, v⟩
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inductive direct_subterm (A : Type) : vec A → vec A → Prop :=
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cons : Π (n : nat) (a : A) (v : vector A n), direct_subterm A (to_vec v) (to_vec (cons a v))
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definition direct_subterm.wf (A : Type) : well_founded (direct_subterm A) :=
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well_founded.intro (λ (bv : vec A),
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sigma.rec_on bv (λ (n : nat) (v : vector A n),
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vector.rec_on v
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(show acc (direct_subterm A) (to_vec (nil A)), from
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acc.intro (to_vec (nil A)) (λ (v₂ : vec A) (H : direct_subterm A v₂ (to_vec (nil A))),
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have gen : ∀ (bv : vec A) (H : direct_subterm A v₂ bv) (Heq : bv = (to_vec (nil A))), acc (direct_subterm A) v₂, from
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λ bv H, direct_subterm.induction_on H (λ n₁ a₁ v₁ e,
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have e₁ : succ n₁ = zero, from sigma.no_confusion e (λ e₁ e₂, e₁),
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nat.no_confusion e₁),
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gen (to_vec (nil A)) H rfl))
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(λ (a₁ : A) (n₁ : nat) (v₁ : vector A n₁) (ih : acc (direct_subterm A) (to_vec v₁)),
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acc.intro (to_vec (cons a₁ v₁)) (λ (w₁ : vec A) (lt₁ : direct_subterm A w₁ (to_vec (cons a₁ v₁))),
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have gen : ∀ (bv : vec A) (H : direct_subterm A w₁ bv) (Heq : bv = (to_vec (cons a₁ v₁))), acc (direct_subterm A) w₁, from
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λ bv H, direct_subterm.induction_on H (λ n₂ a₂ v₂ e,
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sigma.no_confusion e (λ (e₁ : succ n₂ = succ n₁) (e₂ : @cons A a₂ n₂ v₂ == @cons A a₁ n₁ v₁),
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nat.no_confusion e₁ (λ (e₃ : n₂ = n₁),
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have gen₂ : ∀ (m : nat) (Heq₁ : n₂ = m) (v : vector A m) (ih : acc (direct_subterm A) (to_vec v))
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(Heq₂ : @cons A a₂ n₂ v₂ == @cons A a₁ m v), acc (direct_subterm A) (to_vec v₂), from
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λ m Heq₁, eq.rec_on Heq₁ (λ (v : vector A n₂) (ih : acc (direct_subterm A) (to_vec v)) (Heq₂ : @cons A a₂ n₂ v₂ == @cons A a₁ n₂ v),
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vector.no_confusion (heq.to_eq Heq₂) (λ (e₄ : a₂ = a₁) (e₅ : n₂ = n₂) (e₆ : v₂ == v),
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eq.rec_on (heq.to_eq (heq.symm e₆)) ih)),
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gen₂ n₁ e₃ v₁ ih e₂))),
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gen (to_vec (cons a₁ v₁)) lt₁ rfl))))
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definition subterm (A : Type) := tc (direct_subterm A)
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definition subterm.wf (A : Type) : well_founded (subterm A) :=
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tc.wf (direct_subterm.wf A)
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end vector
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