95 lines
2.9 KiB
Text
95 lines
2.9 KiB
Text
open tactic
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meta definition psimp : tactic unit :=
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do simp_lemmas ← mk_simp_lemmas_core reducible [] [`congr],
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(new_target, Heq) ← target >>= simplify_core failed `eq simp_lemmas,
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assert `Htarget new_target, swap,
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Ht ← get_local `Htarget,
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mk_app `eq.mpr [Heq, Ht] >>= exact,
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triv
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variables (P Q R S : Prop)
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-- Iff
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example : P = P := by psimp
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-- Pi
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example : P → true := by psimp
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example : ∀ (X : Type), X → true := by psimp
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-- Eq
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example : (P = P) = true := by psimp
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example : ((¬ P) = (¬ Q)) = (P = Q) := by psimp
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example : (true = P) = P := by psimp
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example : (false = P) = ¬ P := by psimp
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example : (false = ¬ P) = P := by psimp
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example : (P = true) = P := by psimp
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example : (P = false) = ¬ P := by psimp
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example : (¬ P = false) = P := by psimp
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example : (P = ¬ P) = false := by psimp
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example : (¬ P = P) = false := by psimp
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-- Not
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example : ¬ ¬ P = P := by psimp
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example : ¬ ¬ ¬ P = ¬ P := by psimp
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example : ¬ false = true := by psimp
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example : ¬ true = false := by psimp
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example : (¬ (P = Q)) = ((¬ P) = Q) := by psimp
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-- Ite
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namespace ite
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variable [P_dec : decidable P]
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include P_dec
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example : ite true P Q = P := by psimp
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example : ite false P Q = Q := by psimp
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example : ite P Q Q = Q := by psimp
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-- Classical
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--example : ite (not P) Q R = ite P R Q := by psimp
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-- Instances
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example : ite P (ite P Q R) S = ite P Q S := by psimp
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example : ite P S (ite P Q R) = ite P S R := by psimp
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example : ite P (ite P S (ite P Q R)) S = ite P S S := by psimp
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end ite
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-- And
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example : (P ∧ P) = P := by psimp
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example : (P ∧ P) = (P ∧ P ∧ P) := by psimp
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example : (P ∧ Q) = (Q ∧ P) := by psimp
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example : (P ∧ Q ∧ R) = (R ∧ Q ∧ P) := by psimp
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example : (P ∧ Q ∧ R ∧ P ∧ Q ∧ R) = (R ∧ Q ∧ P) := by psimp
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example : (P ∧ true) = P := by psimp
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example : (true ∧ P ∧ true ∧ P ∧ true) = P := by psimp
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example : (P ∧ false) = false := by psimp
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example : (false ∧ P ∧ false ∧ P ∧ false) = false := by psimp
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example : (P ∧ Q ∧ R ∧ true ∧ ¬ P) = false := by psimp
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-- Or
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example : (P ∨ P) = P := by psimp
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example : (P ∨ P) = (P ∨ P ∨ P) := by psimp
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example : (P ∨ Q) = (Q ∨ P) := by psimp
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example : (P ∨ Q ∨ R) = (R ∨ Q ∨ P) := by psimp
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example : (P ∨ Q ∨ R ∨ P ∨ Q ∨ R) = (R ∨ Q ∨ P) := by psimp
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example : (P ∨ false) = P := by psimp
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example : (false ∨ P ∨ false ∨ P ∨ false) = P := by psimp
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example : (P ∨ true) = true := by psimp
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example : (true ∨ P ∨ true ∨ P ∨ true) = true := by psimp
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example : (P ∨ Q ∨ R ∨ false ∨ ¬ P) = true := by psimp
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-- Contextual
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example : (P = Q) → (P = Q) := by psimp
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example : (P = Q) → ((P ∧ P) = (Q ∧ Q)) := by psimp
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example : (P = (Q ∧ R)) → ((P ∧ P) = (R ∧ P ∧ Q)) := by psimp
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example : (P = (Q ∧ R ∧ Q)) → ((P ∧ P) = (R ∧ P ∧ Q)) := by psimp
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example : (P = (Q ∧ R)) → ((¬ Q ∧ P ∧ ¬ R) = false) := by psimp
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