123 lines
3.7 KiB
Text
123 lines
3.7 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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import algebra.ring algebra.numeral
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namespace simplifier
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namespace empty
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end empty
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namespace iff
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attribute eq_self_iff_true [simp]
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end iff
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-- TODO(dhs): make these [simp] rules in the global namespace
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namespace neg_helper
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open algebra
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variables {A : Type} [s : ring A] (a b : A)
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include s
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lemma neg_mul1 : (- a) * b = - (a * b) := eq.symm !algebra.neg_mul_eq_neg_mul
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lemma neg_mul2 : a * (- b) = - (a * b) := eq.symm !algebra.neg_mul_eq_mul_neg
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lemma sub_def : a - b = a + (- b) := rfl
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end neg_helper
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namespace neg
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attribute neg_helper.neg_mul1 [simp]
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attribute neg_helper.neg_mul2 [simp]
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attribute neg_helper.sub_def [simp]
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attribute algebra.neg_neg [simp]
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end neg
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namespace unit
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attribute algebra.zero_add [simp]
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attribute algebra.add_zero [simp]
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attribute algebra.zero_mul [simp]
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attribute algebra.mul_zero [simp]
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attribute algebra.one_mul [simp]
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attribute algebra.mul_one [simp]
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end unit
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namespace ac
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export simplifier.iff simplifier.neg simplifier.unit
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attribute algebra.add.assoc [simp]
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attribute algebra.add.comm [simp]
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attribute algebra.add.left_comm [simp]
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attribute algebra.mul.left_comm [simp]
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attribute algebra.mul.comm [simp]
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attribute algebra.mul.assoc [simp]
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end ac
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namespace distrib
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attribute algebra.left_distrib [simp]
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attribute algebra.right_distrib [simp]
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end distrib
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namespace som
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export simplifier.ac simplifier.distrib
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end som
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namespace numeral
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-- TODO(dhs): remove `add1` from the original lemmas and delete this
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namespace numeral_helper
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open algebra
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theorem bit1_add_bit1 {A : Type} [s : add_comm_semigroup A]
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[s' : has_one A] (a b : A) : bit1 a + bit1 b = bit0 ((a + b) + 1)
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:= norm_num.bit1_add_bit1 a b
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theorem bit1_add_one {A : Type} [s : add_comm_semigroup A] [s' : has_one A] (a : A)
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: bit1 a + one = bit0 (a + 1) := norm_num.add1_bit1 a
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theorem one_add_bit1 {A : Type} [s : add_comm_semigroup A] [s' : has_one A] (a : A)
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: one + bit1 a = bit0 (a + 1) := by rewrite [!add.comm, bit1_add_one]
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lemma one_add_bit0 [simp] {A : Type} [s : add_comm_semigroup A] [s' : has_one A] (a : A)
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: 1 + bit0 a = bit1 a := norm_num.one_add_bit0 a
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lemma bit0_add_one [simp] {A : Type} [s : add_comm_semigroup A] [s' : has_one A] (a : A)
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: bit0 a + 1 = bit1 a := norm_num.bit0_add_one a
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lemma mul_bit0_helper0 [simp] {A : Type} [s : comm_ring A] (a b : A)
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: bit0 a * bit0 b = bit0 (bit0 a * b) := norm_num.mul_bit0_helper (bit0 a) b (bit0 a * b) rfl
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lemma mul_bit0_helper1 [simp] {A : Type} [s : comm_ring A] (a b : A)
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: bit1 a * bit0 b = bit0 (bit1 a * b) := norm_num.mul_bit0_helper (bit1 a) b (bit1 a * b) rfl
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lemma mul_bit1_helper0 [simp] {A : Type} [s : comm_ring A] (a b : A)
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: bit0 a * bit1 b = bit0 (bit0 a * b) + bit0 a := norm_num.mul_bit1_helper (bit0 a) b (bit0 a * b) (bit0 (bit0 a * b) + bit0 a) rfl rfl
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lemma mul_bit1_helper1 [simp] {A : Type} [s : comm_ring A] (a b : A)
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: bit1 a * bit1 b = bit0 (bit1 a * b) + bit1 a := norm_num.mul_bit1_helper (bit1 a) b (bit1 a * b) (bit0 (bit1 a * b) + bit1 a) rfl rfl
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end numeral_helper
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export simplifier.ac
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attribute norm_num.bit0_add_bit0 [simp]
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attribute numeral_helper.bit1_add_one [simp]
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attribute norm_num.bit1_add_bit0 [simp]
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attribute numeral_helper.bit1_add_bit1 [simp]
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attribute norm_num.bit0_add_bit1 [simp]
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attribute numeral_helper.one_add_bit1 [simp]
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attribute norm_num.one_add_one [simp]
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attribute numeral_helper.one_add_bit0 [simp]
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attribute numeral_helper.bit0_add_one [simp]
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attribute numeral_helper.mul_bit0_helper0 [simp]
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attribute numeral_helper.mul_bit0_helper1 [simp]
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attribute numeral_helper.mul_bit1_helper0 [simp]
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attribute numeral_helper.mul_bit1_helper1 [simp]
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end numeral
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end simplifier
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