85 lines
2.8 KiB
Text
85 lines
2.8 KiB
Text
/-
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Copyright (c) 2015 Daniel Selsam. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Daniel Selsam
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-/
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prelude
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import init.logic
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namespace simplifier
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namespace unit_simp
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open eq.ops
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-- TODO(dhs): prove these lemmas elsewhere and only gather the
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-- [simp] attributes here
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variables {A B C : Prop}
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lemma and_imp [simp] : (A ∧ B → C) ↔ (A → B → C) :=
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iff.intro (assume H a b, H (and.intro a b))
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(assume H ab, H (and.left ab) (and.right ab))
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lemma or_imp [simp] : (A ∨ B → C) ↔ ((A → C) ∧ (B → C)) :=
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iff.intro (assume H, and.intro (assume a, H (or.inl a))
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(assume b, H (or.inr b)))
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(assume H ab, and.rec_on H
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(assume Hac Hbc, or.rec_on ab Hac Hbc))
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lemma imp_and [simp] : (A → B ∧ C) ↔ ((A → B) ∧ (A → C)) :=
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iff.intro (assume H, and.intro (assume a, and.left (H a))
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(assume a, and.right (H a)))
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(assume H a, and.rec_on H
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(assume Hab Hac, and.intro (Hab a) (Hac a)))
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-- TODO(dhs, leo): do we want to pre-process away the [iff]s?
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/-
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lemma iff_and_imp [simp] : ((A ↔ B) → C) ↔ (((A → B) ∧ (B → A)) → C) :=
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iff.intro (assume H1 H2, and.rec_on H2 (assume ab ba, H1 (iff.intro ab ba)))
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(assume H1 H2, H1 (and.intro (iff.elim_left H2) (iff.elim_right H2)))
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-/
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lemma a_of_a [simp] : (A → A) ↔ true :=
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iff.intro (assume H, trivial)
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(assume t a, a)
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lemma not_true_of_false [simp] : ¬ true ↔ false :=
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iff.intro (assume H, H trivial)
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(assume f, false.rec (¬ true) f)
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lemma imp_true [simp] : (A → true) ↔ true :=
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iff.intro (assume H, trivial)
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(assume t a, trivial)
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lemma true_imp [simp] : (true → A) ↔ A :=
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iff.intro (assume H, H trivial)
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(assume a t, a)
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lemma fold_not [simp] : (A → false) ↔ ¬ A :=
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iff.intro id id
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lemma false_imp [simp] : (false → A) ↔ true :=
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iff.intro (assume H, trivial)
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(assume t f, false.rec A f)
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lemma ite_and [simp] [A_dec : decidable A] : ite A B C ↔ ((A → B) ∧ (¬ A → C)) :=
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iff.intro (assume H, and.intro (assume a, implies_of_if_pos H a)
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(assume a, implies_of_if_neg H a))
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(assume H, and.rec_on H
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(assume Hab Hnac, decidable.rec_on A_dec
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(assume a,
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assert rw : @decidable.inl A a = A_dec, from
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subsingleton.rec_on (subsingleton_decidable A)
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(assume H, H (@decidable.inl A a) A_dec),
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by rewrite [rw, if_pos a] ; exact Hab a)
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(assume na,
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assert rw : @decidable.inr A na = A_dec, from
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subsingleton.rec_on (subsingleton_decidable A)
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(assume H, H (@decidable.inr A na) A_dec),
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by rewrite [rw, if_neg na] ; exact Hnac na)))
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end unit_simp
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end simplifier
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