83 lines
3 KiB
Text
83 lines
3 KiB
Text
-- 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
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definition cast {A B : Type} (H : A = B) (a : A) : B
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:= eq_rec a H
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theorem cast_refl {A : Type} (a : A) : cast (refl A) a = a
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:= refl (cast (refl A) a)
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theorem cast_proof_irrel {A B : Type} (H1 H2 : A = B) (a : A) : cast H1 a = cast H2 a
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:= refl (cast H1 a)
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theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a
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:= calc cast H a = cast (refl A) a : cast_proof_irrel H (refl A) a
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... = a : cast_refl a
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definition heq {A B : Type} (a : A) (b : B) := ∃ H, cast H a = b
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infixl `==`:50 := heq
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theorem heq_elim {A B : Type} {C : Bool} {a : A} {b : B} (H1 : a == b) (H2 : ∀ (Hab : A = B), cast Hab a = b → C) : C
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:= obtain w Hw, from H1, H2 w Hw
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theorem heq_type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
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:= obtain w Hw, from H, w
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theorem eq_to_heq {A : Type} {a b : A} (H : a = b) : a == b
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:= exists_intro (refl A) (trans (cast_refl a) H)
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theorem heq_to_eq {A : Type} {a b : A} (H : a == b) : a = b
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:= obtain (w : A = A) (Hw : cast w a = b), from H,
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calc a = cast w a : symm (cast_eq w a)
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... = b : Hw
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theorem hrefl {A : Type} (a : A) : a == a
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:= eq_to_heq (refl a)
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theorem heqt_elim {a : Bool} (H : a == true) : a
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:= eqt_elim (heq_to_eq H)
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opaque_hint (hiding cast)
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theorem hsubst {A B : Type} {a : A} {b : B} {P : ∀ (T : Type), T → Bool} (H1 : a == b) (H2 : P A a) : P B b
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:= have Haux1 : ∀ H : A = A, P A (cast H a), from
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assume H : A = A, subst (symm (cast_eq H a)) H2,
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obtain (Heq : A = B) (Hw : cast Heq a = b), from H1,
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have Haux2 : P B (cast Heq a), from subst Heq Haux1 Heq,
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subst Hw Haux2
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theorem hsymm {A B : Type} {a : A} {b : B} (H : a == b) : b == a
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:= hsubst H (hrefl a)
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theorem htrans {A B C : Type} {a : A} {b : B} {c : C} (H1 : a == b) (H2 : b == c) : a == c
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:= hsubst H2 H1
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theorem htrans_left {A B : Type} {a : A} {b c : B} (H1 : a == b) (H2 : b = c) : a == c
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:= htrans H1 (eq_to_heq H2)
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theorem htrans_right {A C : Type} {a b : A} {c : C} (H1 : a = b) (H2 : b == c) : a == c
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:= htrans (eq_to_heq H1) H2
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calc_trans htrans
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calc_trans htrans_left
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calc_trans htrans_right
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theorem type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
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:= hsubst H (refl A)
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theorem cast_heq {A B : Type} (H : A = B) (a : A) : cast H a == a
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:= have H1 : ∀ (H : A = A) (a : A), cast H a == a, from
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λ H a, eq_to_heq (cast_eq H a),
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subst H H1 H a
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theorem cast_eq_to_heq {A B : Type} {a : A} {b : B} {H : A = B} (H1 : cast H a = b) : a == b
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:= calc a == cast H a : hsymm (cast_heq H a)
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... = b : H1
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theorem cast_trans {A B C : Type} (Hab : A = B) (Hbc : B = C) (a : A) : cast Hbc (cast Hab a) = cast (trans Hab Hbc) a
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:= heq_to_eq (calc cast Hbc (cast Hab a) == cast Hab a : cast_heq Hbc (cast Hab a)
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... == a : cast_heq Hab a
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... == cast (trans Hab Hbc) a : hsymm (cast_heq (trans Hab Hbc) a))
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