235 lines
7 KiB
Text
235 lines
7 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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definition Bool [inline] := Type.{0}
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inductive false : Bool :=
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-- No constructors
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theorem false_elim (c : Bool) (H : false)
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:= false_rec c H
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inductive true : Bool :=
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| trivial : true
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definition not (a : Bool) := a → false
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precedence `¬`:40
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notation `¬` a:prev := not a
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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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theorem not_intro {a : Bool} (H : a → false) : ¬ a
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:= H
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theorem not_elim {a : Bool} (H1 : ¬ a) (H2 : a) : false
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:= H1 H2
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theorem absurd {a : Bool} (H1 : a) (H2 : ¬ a) : false
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:= H2 H1
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theorem mt {a b : Bool} (H1 : a → b) (H2 : ¬ b) : ¬ a
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:= assume Ha : a, absurd (H1 Ha) H2
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theorem contrapos {a b : Bool} (H : a → b) : ¬ b → ¬ a
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:= assume Hnb : ¬ b, mt H Hnb
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theorem absurd_elim {a : Bool} (b : Bool) (H1 : a) (H2 : ¬ a) : b
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:= false_elim b (absurd H1 H2)
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inductive and (a b : Bool) : Bool :=
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| and_intro : a → b → and a b
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infixr `/\`:35 := and
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infixr `∧`:35 := and
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theorem and_elim {a b c : Bool} (H1 : a → b → c) (H2 : a ∧ b) : c
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:= and_rec H1 H2
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theorem and_elim_left {a b : Bool} (H : a ∧ b) : a
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:= and_rec (λ a b, a) H
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theorem and_elim_right {a b : Bool} (H : a ∧ b) : b
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:= and_rec (λ a b, b) H
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inductive or (a b : Bool) : Bool :=
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| or_intro_left : a → or a b
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| or_intro_right : b → or a b
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infixr `\/`:30 := or
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infixr `∨`:30 := or
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theorem or_elim {a b c : Bool} (H1 : a ∨ b) (H2 : a → c) (H3 : b → c) : c
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:= or_rec H2 H3 H1
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theorem resolve_right {a b : Bool} (H1 : a ∨ b) (H2 : ¬ a) : b
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:= or_elim H1 (assume Ha, absurd_elim b Ha H2) (assume Hb, Hb)
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theorem resolve_left {a b : Bool} (H1 : a ∨ b) (H2 : ¬ b) : a
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:= or_elim H1 (assume Ha, Ha) (assume Hb, absurd_elim a Hb H2)
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theorem or_flip {a b : Bool} (H : a ∨ b) : b ∨ a
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:= or_elim H (assume Ha, or_intro_right b Ha) (assume Hb, or_intro_left a Hb)
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inductive eq {A : Type} (a : A) : A → Bool :=
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| refl : eq a a
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infix `=`:50 := eq
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theorem subst {A : Type} {a b : A} {P : A → Bool} (H1 : a = b) (H2 : P a) : P b
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:= eq_rec H2 H1
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theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c
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:= subst H2 H1
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calc_subst subst
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calc_refl refl
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calc_trans trans
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theorem symm {A : Type} {a b : A} (H : a = b) : b = a
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:= subst H (refl a)
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theorem congr1 {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a
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:= subst H (refl (f a))
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theorem congr2 {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b
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:= subst H (refl (f a))
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theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) : f a = g b
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:= subst H1 (subst H2 (refl (f a)))
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theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀ x, f x = g x
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:= take x, congr1 H x
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theorem not_congr {a b : Bool} (H : a = b) : (¬ a) = (¬ b)
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:= congr2 not H
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theorem eqmp {a b : Bool} (H1 : a = b) (H2 : a) : b
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:= subst H1 H2
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infixl `<|`:100 := eqmp
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infixl `◂`:100 := eqmp
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theorem eqmpr {a b : Bool} (H1 : a = b) (H2 : b) : a
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:= (symm H1) ◂ H2
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theorem eqt_elim {a : Bool} (H : a = true) : a
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:= (symm H) ◂ trivial
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theorem eqf_elim {a : Bool} (H : a = false) : ¬ a
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:= not_intro (assume Ha : a, H ◂ Ha)
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theorem imp_trans {a b c : Bool} (H1 : a → b) (H2 : b → c) : a → c
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:= assume Ha, H2 (H1 Ha)
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theorem imp_eq_trans {a b c : Bool} (H1 : a → b) (H2 : b = c) : a → c
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:= assume Ha, H2 ◂ (H1 Ha)
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theorem eq_imp_trans {a b c : Bool} (H1 : a = b) (H2 : b → c) : a → c
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:= assume Ha, H2 (H1 ◂ Ha)
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definition ne {A : Type} (a b : A) := ¬ (a = b)
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infix `≠`:50 := ne
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theorem ne_intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b
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:= H
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theorem ne_elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false
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:= H1 H2
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theorem ne_irrefl {A : Type} {a : A} (H : a ≠ a) : false
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:= H (refl a)
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theorem ne_symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a
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:= assume H1 : b = a, H (symm H1)
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theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c
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:= subst (symm H1) H2
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theorem ne_eq_trans {A : Type} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c
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:= subst H2 H1
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calc_trans eq_ne_trans
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calc_trans ne_eq_trans
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definition iff (a b : Bool) := (a → b) ∧ (b → a)
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infix `↔`:50 := iff
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theorem iff_intro {a b : Bool} (H1 : a → b) (H2 : b → a) : a ↔ b
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:= and_intro H1 H2
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theorem iff_elim {a b c : Bool} (H1 : (a → b) → (b → a) → c) (H2 : a ↔ b) : c
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:= and_rec H1 H2
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theorem iff_elim_left {a b : Bool} (H : a ↔ b) : a → b
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:= iff_elim (assume H1 H2, H1) H
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theorem iff_elim_right {a b : Bool} (H : a ↔ b) : b → a
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:= iff_elim (assume H1 H2, H2) H
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theorem iff_mp_left {a b : Bool} (H1 : a ↔ b) (H2 : a) : b
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:= (iff_elim_left H1) H2
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theorem iff_mp_right {a b : Bool} (H1 : a ↔ b) (H2 : b) : a
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:= (iff_elim_right H1) H2
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inductive Exists {A : Type} (P : A → Bool) : Bool :=
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| exists_intro : ∀ (a : A), P a → Exists P
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notation `∃` binders `,` r:(scoped P, Exists P) := r
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theorem exists_elim {A : Type} {P : A → Bool} {B : Bool} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B) : B
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:= Exists_rec H2 H1
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definition exists_unique {A : Type} (p : A → Bool) := ∃ x, p x ∧ ∀ y, y ≠ x → ¬ p y
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notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
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theorem exists_unique_intro {A : Type} {p : A → Bool} (w : A) (H1 : p w) (H2 : ∀ y, y ≠ w → ¬ p y) : ∃! x, p x
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:= exists_intro w (and_intro H1 H2)
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theorem exists_unique_elim {A : Type} {p : A → Bool} {b : Bool} (H2 : ∃! x, p x) (H1 : ∀ x, p x → (∀ y, y ≠ x → ¬ p y) → b) : b
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:= exists_elim H2 (λ w Hw, H1 w (and_elim_left Hw) (and_elim_right Hw))
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inductive inhabited (A : Type) : Bool :=
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| inhabited_intro : A → inhabited A
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theorem inhabited_elim {A : Type} {B : Bool} (H1 : inhabited A) (H2 : A → B) : B
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:= inhabited_rec H2 H1
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theorem inhabited_Bool : inhabited Bool
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:= inhabited_intro true
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theorem inhabited_fun (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B)
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:= inhabited_elim H (take (b : B), inhabited_intro (λ a : A, b))
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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_eq {A : Type} (H : A = A) (a : A) : cast H a = a
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:= calc cast H a = cast (refl A) a : refl (cast H a) -- by proof irrelevance
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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_type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
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:= exists_elim H (λ H Hw, H)
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theorem 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 to_eq {A : Type} {a b : A} (H : a == b) : a = b
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:= exists_elim H (λ (H : A = A) (Hw : cast H a = b),
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calc a = cast H a : symm (cast_eq H a)
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... = b : Hw)
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theorem heq_refl {A : Type} (a : A) : a == a
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:= to_heq (refl a)
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theorem heqt_elim {a : Bool} (H : a == true) : a
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:= eqt_elim (to_eq H)
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