155 lines
5.7 KiB
Text
155 lines
5.7 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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import logic cast
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axiom prop_complete (a : Prop) : a = true ∨ a = false
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theorem case (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a
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:= or_elim (prop_complete a)
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(assume Ht : a = true, subst (symm Ht) H1)
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(assume Hf : a = false, subst (symm Hf) H2)
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theorem em (a : Prop) : a ∨ ¬a
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:= or_elim (prop_complete a)
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(assume Ht : a = true, or_intro_left (¬ a) (eqt_elim Ht))
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(assume Hf : a = false, or_intro_right a (eqf_elim Hf))
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theorem prop_complete_swapped (a : Prop) : a = false ∨ a = true
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:= case (λ x, x = false ∨ x = true)
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(or_intro_right (true = false) (refl true))
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(or_intro_left (false = true) (refl false))
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a
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theorem not_true : (¬true) = false
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:= have aux : ¬ (¬true) = true, from
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not_intro (assume H : (¬true) = true,
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absurd_not_true (subst (symm H) trivial)),
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resolve_right (prop_complete (¬true)) aux
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theorem not_false : (¬false) = true
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:= have aux : ¬ (¬false) = false, from
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not_intro (assume H : (¬false) = false,
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subst H not_false_trivial),
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resolve_right (prop_complete_swapped (¬ false)) aux
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theorem not_not_eq (a : Prop) : (¬¬a) = a
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:= case (λ x, (¬¬x) = x)
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(calc (¬¬true) = (¬false) : { not_true }
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... = true : not_false)
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(calc (¬¬false) = (¬true) : { not_false }
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... = false : not_true)
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a
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theorem not_not_elim {a : Prop} (H : ¬¬a) : a
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:= (not_not_eq a) ◂ H
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theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b
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:= or_elim (prop_complete a)
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(assume Hat, or_elim (prop_complete b)
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(assume Hbt, trans Hat (symm Hbt))
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(assume Hbf, false_elim (a = b) (subst Hbf (Hab (eqt_elim Hat)))))
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(assume Haf, or_elim (prop_complete b)
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(assume Hbt, false_elim (a = b) (subst Haf (Hba (eqt_elim Hbt))))
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(assume Hbf, trans Haf (symm Hbf)))
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theorem iff_to_eq {a b : Prop} (H : a ↔ b) : a = b
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:= iff_elim (assume H1 H2, propext H1 H2) H
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theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b)
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:= propext
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(assume H, iff_to_eq H)
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(assume H, eq_to_iff H)
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theorem eqt_intro {a : Prop} (H : a) : a = true
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:= propext (assume H1 : a, trivial)
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(assume H2 : true, H)
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theorem eqf_intro {a : Prop} (H : ¬a) : a = false
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:= propext (assume H1 : a, absurd H1 H)
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(assume H2 : false, false_elim a H2)
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theorem by_contradiction {a : Prop} (H : ¬a → false) : a
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:= or_elim (em a) (assume H1 : a, H1) (assume H1 : ¬a, false_elim a (H H1))
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theorem a_neq_a {A : Type} (a : A) : (a ≠ a) = false
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:= propext (assume H, a_neq_a_elim H)
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(assume H, false_elim (a ≠ a) H)
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theorem eq_id {A : Type} (a : A) : (a = a) = true
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:= eqt_intro (refl a)
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theorem heq_id {A : Type} (a : A) : (a == a) = true
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:= eqt_intro (hrefl a)
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theorem not_or (a b : Prop) : (¬ (a ∨ b)) = (¬ a ∧ ¬ b)
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:= propext
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(assume H, or_elim (em a)
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(assume Ha, absurd_elim (¬ a ∧ ¬ b) (or_intro_left b Ha) H)
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(assume Hna, or_elim (em b)
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(assume Hb, absurd_elim (¬ a ∧ ¬ b) (or_intro_right a Hb) H)
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(assume Hnb, and_intro Hna Hnb)))
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(assume (H : ¬ a ∧ ¬ b), not_intro (assume (N : a ∨ b),
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or_elim N
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(assume Ha, absurd Ha (and_elim_left H))
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(assume Hb, absurd Hb (and_elim_right H))))
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theorem not_and (a b : Prop) : (¬ (a ∧ b)) = (¬ a ∨ ¬ b)
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:= propext
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(assume H, or_elim (em a)
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(assume Ha, or_elim (em b)
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(assume Hb, absurd_elim (¬ a ∨ ¬ b) (and_intro Ha Hb) H)
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(assume Hnb, or_intro_right (¬ a) Hnb))
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(assume Hna, or_intro_left (¬ b) Hna))
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(assume (H : ¬ a ∨ ¬ b), not_intro (assume (N : a ∧ b),
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or_elim H
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(assume Hna, absurd (and_elim_left N) Hna)
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(assume Hnb, absurd (and_elim_right N) Hnb)))
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theorem imp_or (a b : Prop) : (a → b) = (¬ a ∨ b)
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:= propext
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(assume H : a → b, (or_elim (em a)
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(assume Ha : a, or_intro_right (¬ a) (H Ha))
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(assume Hna : ¬ a, or_intro_left b Hna)))
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(assume (H : ¬ a ∨ b) (Ha : a),
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resolve_right H ((symm (not_not_eq a)) ◂ Ha))
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theorem not_implies (a b : Prop) : (¬ (a → b)) = (a ∧ ¬b)
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:= calc (¬ (a → b)) = (¬(¬a ∨ b)) : {imp_or a b}
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... = (¬¬a ∧ ¬b) : not_or (¬ a) b
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... = (a ∧ ¬b) : {not_not_eq a}
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theorem a_eq_not_a (a : Prop) : (a = ¬a) = false
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:= propext
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(assume H, or_elim (em a)
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(assume Ha, absurd Ha (subst H Ha))
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(assume Hna, absurd (subst (symm H) Hna) Hna))
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(assume H, false_elim (a = ¬ a) H)
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theorem true_eq_false : (true = false) = false
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:= subst not_true (a_eq_not_a true)
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theorem false_eq_true : (false = true) = false
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:= subst not_false (a_eq_not_a false)
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theorem a_eq_true (a : Prop) : (a = true) = a
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:= propext (assume H, eqt_elim H) (assume H, eqt_intro H)
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theorem a_eq_false (a : Prop) : (a = false) = ¬a
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:= propext (assume H, eqf_elim H) (assume H, eqf_intro H)
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theorem not_exists_forall {A : Type} {P : A → Prop} (H : ¬∃x, P x) : ∀x, ¬P x
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:= take x, or_elim (em (P x))
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(assume Hp : P x, absurd_elim (¬P x) (exists_intro x Hp) H)
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(assume Hn : ¬P x, Hn)
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theorem not_forall_exists {A : Type} {P : A → Prop} (H : ¬∀x, P x) : ∃x, ¬P x
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:= by_contradiction (assume H1 : ¬∃ x, ¬P x,
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have H2 : ∀x, ¬¬P x, from not_exists_forall H1,
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have H3 : ∀x, P x, from take x, not_not_elim (H2 x),
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absurd H3 H)
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theorem peirce (a b : Prop) : ((a → b) → a) → a
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:= assume H, by_contradiction (assume Hna : ¬a,
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have Hnna : ¬¬a, from not_implies_left (mt H Hna),
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absurd (not_not_elim Hnna) Hna)
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