53 lines
1.5 KiB
Text
53 lines
1.5 KiB
Text
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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
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import logic.eq
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inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
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refl : heq a a
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infixl `==`:50 := heq
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namespace heq
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universe variable u
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variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
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theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
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rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
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theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
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rec_on H₁ H₂
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theorem symm (H : a == b) : b == a :=
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subst H (refl a)
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theorem type_eq (H : a == b) : A = B :=
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subst H (eq.refl A)
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theorem from_eq (H : a = a') : a == a' :=
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eq.subst H (refl a)
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theorem to_eq (H : a == a') : a = a' :=
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have H₁ : ∀ (Ht : A = A), eq.rec_on Ht a = a, from
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take Ht, eq.refl (eq.rec_on Ht a),
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have H₂ : ∀ (Ht : A = A), eq.rec_on Ht a = a', from
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heq.rec_on H H₁,
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H₂ (type_eq H)
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theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
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subst H₂ H₁
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theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
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trans H₁ (from_eq H₂)
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theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
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trans (from_eq H₁) H₂
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theorem true_elim {a : Prop} (H : a == true) : a :=
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eq_true_elim (heq.to_eq H)
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end heq
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calc_trans heq.trans
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calc_trans heq.trans_left
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calc_trans heq.trans_right
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calc_symm heq.symm
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