138 lines
5 KiB
Text
138 lines
5 KiB
Text
/-
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Copyright (c) 2015 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: Robert Y. Lewis
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-/
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import algebra.ring
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open algebra
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variable {A : Type}
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definition add1 [s : has_add A] [s' : has_one A] (a : A) : A := add a one
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theorem add_comm_four [s : add_comm_semigroup A] (a b : A) : a + a + (b + b) = (a + b) + (a + b) :=
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by rewrite [-add.assoc at {1}, add.comm, {a + b}add.comm at {1}, *add.assoc]
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theorem add_comm_middle [s : add_comm_semigroup A] (a b c : A) : a + b + c = a + c + b :=
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by rewrite [add.assoc, add.comm b, -add.assoc]
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theorem bit0_add_bit0 [s : add_comm_semigroup A] (a b : A) : bit0 a + bit0 b = bit0 (a + b) :=
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!add_comm_four
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theorem bit0_add_bit0_helper [s : add_comm_semigroup A] (a b t : A) (H : a + b = t) :
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bit0 a + bit0 b = bit0 t :=
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by rewrite -H; apply bit0_add_bit0
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theorem bit1_add_bit0 [s : add_comm_semigroup A] [s' : has_one A] (a b : A) :
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bit1 a + bit0 b = bit1 (a + b) :=
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begin
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rewrite [↑bit0, ↑bit1, add_comm_middle], congruence, apply add_comm_four
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end
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theorem bit1_add_bit0_helper [s : add_comm_semigroup A] [s' : has_one A] (a b t : A) (H : a + b = t) :
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bit1 a + bit0 b = bit1 t :=
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by rewrite -H; apply bit1_add_bit0
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theorem bit0_add_bit1 [s : add_comm_semigroup A] [s' : has_one A] (a b : A) :
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bit0 a + bit1 b = bit1 (a + b) :=
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by rewrite [{bit0 a + _}add.comm, {a + _}add.comm]; apply bit1_add_bit0
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theorem bit0_add_bit1_helper [s : add_comm_semigroup A] [s' : has_one A] (a b t : A) (H : a + b = t) :
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bit0 a + bit1 b = bit1 t :=
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by rewrite -H; apply bit0_add_bit1
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theorem bit1_add_bit1 [s : add_comm_semigroup A] [s' : has_one A] (a b : A) : bit1 a + bit1 b = bit0 (add1 (a + b)) :=
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begin
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rewrite ↑[bit0, bit1, add1],
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apply sorry
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end
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theorem bit1_add_bit1_helper [s : add_comm_semigroup A] [s' : has_one A] (a b t s: A)
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(H : (a + b) = t) (H2 : add1 t = s) : bit1 a + bit1 b = bit0 s :=
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begin rewrite [-H2, -H], apply bit1_add_bit1 end
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theorem bin_add_zero [s : add_monoid A] (a : A) : a + zero = a := !add_zero
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theorem bin_zero_add [s : add_monoid A] (a : A) : zero + a = a := !zero_add
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theorem one_add_bit0 [s : add_comm_semigroup A] [s' : has_one A] (a : A) : one + bit0 a = bit1 a :=
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begin rewrite ↑[bit0, bit1], rewrite add.comm end
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theorem bit0_add_one [s : has_add A] [s' : has_one A] (a : A) : bit0 a + one = bit1 a := rfl
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theorem bit1_add_one [s : has_add A] [s' : has_one A] (a : A) : bit1 a + one = add1 (bit1 a) := rfl
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theorem bit1_add_one_helper [s : has_add A] [s' : has_one A] (a t : A) (H : add1 (bit1 a) = t) :
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bit1 a + one = t :=
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by rewrite -H
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theorem one_add_bit1 [s : add_comm_semigroup A] [s' : has_one A] (a : A) :
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one + bit1 a = add1 (bit1 a) := !add.comm
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theorem one_add_bit1_helper [s : add_comm_semigroup A] [s' : has_one A] (a t : A)
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(H : add1 (bit1 a) = t) : one + bit1 a = t :=
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by rewrite -H; apply one_add_bit1
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theorem add1_bit0 [s : has_add A] [s' : has_one A] (a : A) : add1 (bit0 a) = bit1 a :=
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rfl
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theorem add1_bit1 [s : add_comm_semigroup A] [s' : has_one A] (a : A) :
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add1 (bit1 a) = bit0 (add1 a) :=
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begin
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rewrite ↑[add1, bit1, bit0],
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rewrite [add.assoc, add_comm_four]
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end
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theorem add1_bit1_helper [s : add_comm_semigroup A] [s' : has_one A] (a t : A) (H : add1 a = t) :
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add1 (bit1 a) = bit0 t :=
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by rewrite -H; apply add1_bit1
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theorem add1_one [s : has_add A] [s' : has_one A] : add1 (one : A) = bit0 one :=
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rfl
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theorem add1_zero [s : add_monoid A] [s' : has_one A] : add1 (zero : A) = one :=
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begin
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rewrite [↑add1, zero_add]
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end
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theorem one_add_one [s : has_add A] [s' : has_one A] : (one : A) + one = bit0 one :=
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rfl
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theorem subst_into_sum [s : has_add A] (l r tl tr t : A) (prl : l = tl) (prr : r = tr) (prt : tl + tr = t) :
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l + r = t :=
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by rewrite [prl, prr, prt]
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-- multiplication
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theorem mul_zero [s : mul_zero_class A] (a : A) : a * zero = zero :=
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by rewrite [↑zero, mul_zero]
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theorem zero_mul [s : mul_zero_class A] (a : A) : zero * a = zero :=
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by rewrite [↑zero, zero_mul]
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theorem mul_one [s : monoid A] (a : A) : a * one = a :=
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by rewrite [↑one, mul_one]
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theorem mul_bit0 [s : distrib A] (a b : A) : a * (bit0 b) = bit0 (a * b) :=
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by rewrite [↑bit0, left_distrib]
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theorem mul_bit0_helper [s : distrib A] (a b t : A) (H : a * b = t) : a * (bit0 b) = bit0 t :=
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by rewrite -H; apply mul_bit0
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theorem mul_bit1 [s : semiring A] (a b : A) : a * (bit1 b) = bit0 (a * b) + a :=
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by rewrite [↑bit1, ↑bit0, +left_distrib, ↑one, mul_one]
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theorem mul_bit1_helper [s : semiring A] (a b s t : A) (Hs : a * b = s) (Ht : bit0 s + a = t) :
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a * (bit1 b) = t :=
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begin rewrite [-Ht, -Hs, mul_bit1] end
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theorem subst_into_prod [s : has_mul A] (l r tl tr t : A) (prl : l = tl) (prr : r = tr)
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(prt : tl * tr = t) :
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l * r = t :=
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by rewrite [prl, prr, prt]
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theorem mk_cong (op : A → A) (a b : A) (H : a = b) : op a = op b :=
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by congruence; exact H
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theorem mk_eq (a : A) : a = a := rfl
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