37 lines
1.3 KiB
Text
37 lines
1.3 KiB
Text
import logic struc.relation
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using relation
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namespace is_equivalence
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inductive class {T : Type} (R : T → T → Type) : Prop :=
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mk : is_reflexive R → is_symmetric R → is_transitive R → class R
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theorem is_reflexive {T : Type} {R : T → T → Type} {C : class R} : is_reflexive R :=
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class_rec (λx y z, x) C
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theorem is_symmetric {T : Type} {R : T → T → Type} {C : class R} : is_symmetric R :=
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class_rec (λx y z, y) C
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theorem is_transitive {T : Type} {R : T → T → Type} {C : class R} : is_transitive R :=
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class_rec (λx y z, z) C
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end is_equivalence
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instance is_equivalence.is_reflexive
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instance is_equivalence.is_symmetric
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instance is_equivalence.is_transitive
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theorem and_inhabited_left {a : Prop} (b : Prop) (Ha : a) : a ∧ b ↔ b :=
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iff_intro (take Hab, and_elim_right Hab) (take Hb, and_intro Ha Hb)
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theorem test (a b c : Prop) (P : Prop → Prop) (H1 : a ↔ b) (H2 : c ∧ a) : c ∧ b :=
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subst_iff H1 H2
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theorem test2 (Q R S : Prop) (H3 : R ↔ Q) (H1 : S) : Q ↔ (S ∧ Q) :=
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iff_symm (and_inhabited_left Q H1)
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theorem test3 (Q R S : Prop) (H3 : R ↔ Q) (H1 : S) : R ↔ (S ∧ Q) :=
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subst_iff (test2 Q R S H3 H1) H3
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theorem test4 (Q R S : Prop) (H3 : R ↔ Q) (H1 : S) : R ↔ (S ∧ Q) :=
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subst_iff (iff_symm (and_inhabited_left Q H1)) H3
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