chore(tests/lean): delete old tests
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Leonardo de Moura
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3 changed files with 0 additions and 366 deletions
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section
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variables p : nat → Prop
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variables q : nat → nat → Prop
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variables f : Π (x y : nat), p x → q x y → nat
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example : (0:nat) = 0 := sorry
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#congr @add
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#congr p
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#congr iff
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end
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exit
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section
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variables p : nat → Prop
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variables q : Π (n m : nat), p n → p m → Prop
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variables r : Π (n m : nat) (H₁ : p n) (H₂ : p m), q n m H₁ H₂ → Prop
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variables h : Π (n m : nat)
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(H₁ : p n) (H₂ : p m) (H₃ : q n n H₁ H₁) (H₄ : q n m H₁ H₂)
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(H₅ : r n m H₁ H₂ H₄) (H₆ : r n n H₁ H₁ H₃), nat
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definition h_congr (n₁ n₂ : nat) (e₁ : n₁ = n₂) (m₁ m₂ : nat) (e₂ : m₁ = m₂)
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(H₁ : p n₁) (H₂ : p m₁)
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(H₃ : q n₁ n₁ H₁ H₁)
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(H₄ : q n₁ m₁ H₁ H₂)
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(H₅ : r n₁ m₁ H₁ H₂ H₄)
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(H₆ : r n₁ n₁ H₁ H₁ H₃) :
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h n₁ m₁ H₁ H₂ H₃ H₄ H₅ H₆ =
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h n₂ m₂ (eq.drec_on e₁ H₁)
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(eq.drec_on e₂ H₂)
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(eq.drec_on e₁ H₃)
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(eq.drec_on e₁ (eq.drec_on e₂ H₄))
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(eq.drec_on e₁ (eq.drec_on e₂ H₅))
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(eq.drec_on e₁ H₆) :=
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begin
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apply eq.drec_on e₁,
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apply eq.drec_on e₂,
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apply rfl
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end
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-- set_option pp.implicit true
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-- print h_congr
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#congr h
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exit
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eq.drec_on e₁ (eq.drec_on e₂ (eq.refl (h n₂ m₂ (eq.rec_on e₁ H₁) (eq.rec_on e₂ H₂) (eq.drec_on e₁ H₃)
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(eq.drec_on e₁ (eq.drec_on e₂ H₄))
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(eq.drec_on e₁ (eq.drec_on e₂ H₅))
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(eq.drec_on e₁ H₆))))
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sorry
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exit
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q x₁ H₁) :
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h x₁ H₁ H₂ = h x₂ (eq.rec_on e H₁) (eq.drec_on e H₂) :=
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eq.drec_on e (eq.refl (h x₁ (eq.rec_on (eq.refl x₁) H₁) (eq.drec_on (eq.refl x₁) H₂)))
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exit
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variables h₂ : Π (n : nat) (H₁ : p n) (H₂ : q n H₁) (H₃ : r n H₁ H₂), nat
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definition h_congr (x₁ x₂ : nat) (e : x₁ = x₂) (H₁ : p x₁) (H₂ : q x₁ H₁) :
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h x₁ H₁ H₂ = h x₂ (eq.rec_on e H₁) (eq.drec_on e H₂) :=
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eq.drec_on e (eq.refl (h x₁ (eq.rec_on (eq.refl x₁) H₁) (eq.drec_on (eq.refl x₁) H₂)))
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definition h_congr₂ (x₁ x₂ : nat) (e : x₁ = x₂) (H₁ : p x₁) (H₂ : q x₁ H₁) (H₃ : r x₁ H₁ H₂) :
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h₂ x₁ H₁ H₂ H₃ = h₂ x₂ (eq.rec_on e H₁) (eq.drec_on e H₂) (eq.drec_on e H₃) :=
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eq.drec_on e (eq.refl (h₂ x₁ (eq.rec_on (eq.refl x₁) H₁) (eq.drec_on (eq.refl x₁) H₂) (eq.drec_on (eq.refl x₁) H₃)))
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definition h_congr₃ (x₁ x₂ : nat) (e : x₁ = x₂) (H₁ : p x₁) (H₂ : q x₁ H₁) (H₃ : r x₁ H₁ H₂) :
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h₂ x₁ H₁ H₂ H₃ = h₂ x₂ (eq.rec_on e H₁) (eq.drec_on e H₂) (eq.drec_on e H₃) :=
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begin
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congruence,
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apply e
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end
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-- print h_congr₃
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-- exit
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set_option pp.all true
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print h_congr₂
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#congr h
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exit
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set_option pp.all true
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print h_congr
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#congr h
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end
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exit
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variables g : Π (A : Type) (x y : A) (B : Type) (z : B), x = y → y == z → nat
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#congr g
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exit
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lemma f_congr
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(x₁ x₂ : nat) (e₁ : x₁ = x₂)
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(y₁ y₂ : nat) (e₂ : y₁ = y₂)
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(H₁ : p x₁)
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(H₂ : q x₁ y₁) :
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f x₁ y₁ H₁ H₂ =
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f x₂ y₂ (@eq.rec_on nat x₁ (λ (a : ℕ), p a) x₂ e₁ H₁)
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(@eq.rec_on nat x₁ (λ (a : ℕ), q a y₂) x₂ e₁ (@eq.rec_on nat y₁ (λ (a : ℕ), q x₁ a) y₂ e₂ H₂)) :=
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let R := (eq.refl (f x₁ y₁ (@eq.rec_on nat x₁ (λ (a : ℕ), p a) x₁ (eq.refl x₁) H₁) (@eq.rec_on nat x₁ (λ (a : ℕ), q a y₁) x₁ (eq.refl x₁) (@eq.rec_on nat y₁ (λ (a : ℕ), q x₁ a) y₁ (eq.refl y₁) H₂)))) in
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@eq.drec_on nat x₁
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(λ (z : ℕ) (H : x₁ = z),
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f x₁ y₁ H₁ H₂ =
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f z y₂ (@eq.rec_on nat x₁ (λ a, p a) z H H₁)
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(@eq.rec_on nat x₁ (λ a, q a y₂) z H (@eq.rec_on nat y₁ (λ a, q x₁ a) y₂ e₂ H₂)))
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x₂ e₁
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(@eq.drec_on nat y₁
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(λ (z : ℕ) (H : y₁ = z),
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f x₁ y₁ H₁ H₂ =
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f x₁ z (@eq.rec_on nat x₁ (λ a, p a) x₁ (eq.refl x₁) H₁)
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(@eq.rec_on nat x₁ (λ a, q a z) x₁ (eq.refl x₁) (@eq.rec_on nat y₁ (λ a, q x₁ a) z H H₂)))
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y₂ e₂ R)
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/-
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f x₁ y₁ H₁ H₂ =
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f x₁ y₂ (@eq.rec_on nat x₁ (λ a, p a) x₁ (eq.refl x₁) H₁)
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(@eq.rec_on nat x₁ (λ a, q a y₂) x₁ (eq.refl x₁) (@eq.rec_on nat y₁ (λ a, q x₁ a) y₂ e₂ H₂)))
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-/
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@ -1,64 +0,0 @@
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura, Jeremy Avigad
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open inhabited decidable
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namespace play
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-- TODO: take this outside the namespace when the inductive package handles it better
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inductive sum (A B : Type) : Type
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| inl : A → sum
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| inr : B → sum
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namespace sum
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reserve infixr `+`:25
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infixr `+` := sum
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open eq
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theorem inl_inj {A B : Type} {a1 a2 : A} (H : inl B a1 = inl B a2) : a1 = a2 :=
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let f := λs, sum.rec_on s (λa, a1 = a) (λb, false) in
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have H1 : f (inl B a1), from rfl,
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have H2 : f (inl B a2), from subst H H1,
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H2
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theorem inl_neq_inr {A B : Type} {a : A} {b : B} (H : inl B a = inr A b) : false :=
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let f := λs, sum.rec_on s (λa', a = a') (λb, false) in
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have H1 : f (inl B a), from rfl,
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have H2 : f (inr A b), from subst H H1,
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H2
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theorem inr_inj {A B : Type} {b1 b2 : B} (H : inr A b1 = inr A b2) : b1 = b2 :=
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let f := λs, sum.rec_on s (λa, false) (λb, b1 = b) in
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have H1 : f (inr A b1), from rfl,
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have H2 : f (inr A b2), from subst H H1,
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H2
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attribute [instance]
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theorem sum_inhabited_left {A B : Type} (H : inhabited A) : inhabited (A + B) :=
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inhabited.mk (inl B (default A))
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attribute [instance]
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theorem sum_inhabited_right {A B : Type} (H : inhabited B) : inhabited (A + B) :=
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inhabited.mk (inr A (default B))
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attribute [instance]
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theorem sum_eq_decidable {A B : Type} (s1 s2 : A + B)
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(H1 : ∀a1 a2 : A, decidable (inl B a1 = inl B a2))
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(H2 : ∀b1 b2 : B, decidable (inr A b1 = inr A b2)) : decidable (s1 = s2) :=
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sum.rec_on s1
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(take a1, show decidable (inl B a1 = s2), from
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sum.rec_on s2
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(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
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(take b2,
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have H3 : (inl B a1 = inr A b2) ↔ false,
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from iff.intro inl_neq_inr (assume H4, false.elim H4),
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show decidable (inl B a1 = inr A b2), from decidable_of_decidable_of_iff _ (iff.symm H3)))
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(take b1, show decidable (inr A b1 = s2), from
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sum.rec_on s2
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(take a2,
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have H3 : (inr A b1 = inl B a2) ↔ false,
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from iff.intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false.elim H4),
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show decidable (inr A b1 = inl B a2), from decidable_of_decidable_of_iff _ (iff.symm H3))
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(take b2, show decidable (inr A b1 = inr A b2), from H2 b1 b2))
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end sum
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end play
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@ -1,153 +0,0 @@
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prelude
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-- Porting Vladimir's file to Lean
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notation `assume` binders `,` r:(scoped f, f) := r
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notation `take` binders `,` r:(scoped f, f) := r
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inductive empty : Type
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inductive unit : Type
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| tt : unit
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definition tt := @unit.tt
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inductive nat : Type
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| O : nat
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| S : nat → nat
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inductive paths {A : Type} (a : A) : A → Type
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| idpath : paths a
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definition idpath := @paths.idpath
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inductive sum (A : Type) (B : Type) : Type
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| inl : A -> sum
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| inr : B -> sum
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definition coprod := sum
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definition ii1fun {A : Type} (B : Type) (a : A) := sum.inl B a
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definition ii2fun (A : Type) {B : Type} (b : B) := sum.inr A b
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definition ii1 {A : Type} {B : Type} (a : A) := sum.inl B a
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definition ii2 {A : Type} {B : Type} (b : B) := sum.inl A b
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inductive total2 {T: Type} (P: T → Type) : Type
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| tpair : Π (t : T) (tp : P t), total2
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definition tpair := @total2.tpair
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definition pr1 {T : Type} {P : T → Type} (tp : total2 P) : T
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:= total2.rec (λ a b, a) tp
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definition pr2 {T : Type} {P : T → Type} (tp : total2 P) : P (pr1 tp)
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:= total2.rec (λ a b, b) tp
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inductive Phant (T : Type) : Type
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| phant : Phant
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definition fromempty {X : Type} : empty → X
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:= λe, empty.rec (λe, X) e
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definition tounit {X : Type} : X → unit
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:= λx, tt
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definition termfun {X : Type} (x : X) : unit → X
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:= λt, x
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definition idfun (T : Type) := λt : T, t
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definition funcomp {X : Type} {Y : Type} {Z : Type} (f : X → Y) (g : Y → Z)
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:= λx, g (f x)
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infixl `∘`:60 := funcomp
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definition iteration {T : Type} (f : T → T) (n : nat) : T → T
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:= nat.rec (idfun T) (λ m fm, funcomp fm f) n
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definition adjev {X : Type} {Y : Type} (x : X) (f : X → Y) := f x
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definition adjev2 {X : Type} {Y : Type} (phi : ((X → Y) → Y ) → Y ) : X → Y
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:= λx, phi (λf, f x)
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definition dirprod (X : Type) (Y : Type) := total2 (λ x : X, Y)
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definition dirprodpair {X : Type} {Y : Type} := @tpair _ (λ x : X, Y)
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definition dirprodadj {X : Type} {Y : Type} {Z : Type} (f : dirprod X Y → Z ) : X → Y → Z
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:= λx y, f (dirprodpair x y)
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definition dirprodf {X : Type} {Y : Type} {X' : Type} {Y' : Type} (f : X → Y) (f' : X' → Y') (xx' : dirprod X X') : dirprod Y Y'
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:= dirprodpair (f (pr1 xx')) (f' (pr2 xx'))
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definition ddualand {X : Type} {Y : Type} {P : Type} (xp : (X → P) → P) (yp : (Y → P) → P) : (dirprod X Y → P) → P
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:= λ X0,
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let int1 := λ (ypp : (Y → P) → P) (x : X), yp (λ y : Y, X0 (dirprodpair x y)) in
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xp (int1 yp)
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definition neg (X : Type) : Type := X → empty
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definition negf {X : Type} {Y : Type} (f : X → Y) : neg Y → neg X
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:= λ (phi : Y → empty) (x : X), phi (f x)
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definition dneg (X : Type) : Type := (X → empty) → empty
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definition dnegf {X : Type} {Y : Type} (f : X → Y) : dneg X → dneg Y
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:= negf (negf f)
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definition todneg (X : Type) : X → dneg X
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:= adjev
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definition dnegnegtoneg {X : Type} : dneg (neg X) → neg X
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:= adjev2
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lemma dneganddnegl1 {X : Type} {Y : Type} (dnx : dneg X) (dny : dneg Y) : neg (X → neg Y)
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:= take X2 : X → neg Y,
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have X3 : dneg X → neg Y, from
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take xx : dneg X, dnegnegtoneg (dnegf X2 xx),
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dny (X3 dnx)
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definition logeq (X : Type) (Y : Type) := dirprod (X → Y) (Y → X)
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infix `<->`:25 := logeq
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infix `↔`:25 := logeq
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definition logeqnegs {X : Type} {Y : Type} (l : X ↔ Y) : (neg X) ↔ (neg Y)
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:= dirprodpair (negf (pr2 l)) (negf (pr1 l))
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infix `=`:50 := paths
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definition pathscomp0 {X : Type} {a b c : X} (e1 : a = b) (e2 : b = c) : a = c
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:= paths.rec e1 e2
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definition pathscomp0rid {X : Type} {a b : X} (e1 : a = b) : pathscomp0 e1 (idpath b) = e1
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:= idpath _
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definition pathsinv0 {X : Type} {a b : X} (e : a = b) : b = a
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:= paths.rec (idpath _) e
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definition transport {A : Type} {a b : A} {P : A → Type} (H1 : a = b) (H2 : P a) : P b
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:= paths.rec H2 H1
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infixr `▸`:75 := transport
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infixr `⬝`:75 := pathscomp0
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postfix `⁻¹`:100 := pathsinv0
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definition idinv {X : Type} (x : X) : (idpath x)⁻¹ = idpath x
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:= idpath (idpath x)
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definition idtrans {A : Type} (x : A) : (idpath x) ⬝ (idpath x) = (idpath x)
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:= idpath (idpath x)
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definition pathsinv0l {X : Type} {a b : X} (e : a = b) : e⁻¹ ⬝ e = idpath b
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:= paths.rec (idinv a⁻¹ ▸ idtrans a) e
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definition pathsinv0r {A : Type} {x y : A} (p : x = y) : p⁻¹ ⬝ p = idpath y
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:= paths.rec (idinv x⁻¹ ▸ idtrans x) p
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definition pathsinv0inv0 {A : Type} {x y : A} (p : x = y) : (p⁻¹)⁻¹ = p
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:= paths.rec (idpath (idpath x)) p
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definition pathsdirprod {X : Type} {Y : Type} {x1 x2 : X} {y1 y2 : Y} (ex : x1 = x2) (ey : y1 = y2 ) : dirprodpair x1 y1 = dirprodpair x2 y2
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:= ex ▸ ey ▸ idpath (dirprodpair x1 y1)
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definition maponpaths {T1 : Type} {T2 : Type} (f : T1 → T2) {t1 t2 : T1} (e : t1 = t2) : f t1 = f t2
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:= e ▸ idpath (f t1)
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definition ap {T1 : Type} {T2 : Type} := @maponpaths T1 T2
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definition maponpathscomp0 {X : Type} {Y : Type} {x y z : X} (f : X → Y) (p : x = y) (q : y = z) : ap f (p ⬝ q) = (ap f p) ⬝ (ap f q)
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:= paths.rec (idpath _) q
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definition maponpathsinv0 {X : Type} {Y : Type} (f : X → Y) {x1 x2 : X} (e : x1 = x2 ) : ap f (e⁻¹) = (ap f e)⁻¹
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:= paths.rec (idpath _) e
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lemma maponpathsidfun {X : Type} {x x' : X} (e : x = x') : ap (idfun X) e = e
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:= paths.rec (idpath _) e
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lemma maponpathscomp {X : Type} {Y : Type} {Z : Type} {x x' : X} (f : X → Y) (g : Y → Z) (e : x = x') : ap g (ap f e) = ap (f ∘ g) e
|
||||
:= paths.rec (idpath _) e
|
||||
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Add table
Reference in a new issue