1(*  Title:      HOL/Semiring_Normalization.thy
2    Author:     Amine Chaieb, TU Muenchen
3*)
4
5section \<open>Semiring normalization\<close>
6
7theory Semiring_Normalization
8imports Numeral_Simprocs
9begin
10
11text \<open>Prelude\<close>
12
13class comm_semiring_1_cancel_crossproduct = comm_semiring_1_cancel +
14  assumes crossproduct_eq: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
15begin
16
17lemma crossproduct_noteq:
18  "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> a * c + b * d \<noteq> a * d + b * c"
19  by (simp add: crossproduct_eq)
20
21lemma add_scale_eq_noteq:
22  "r \<noteq> 0 \<Longrightarrow> a = b \<and> c \<noteq> d \<Longrightarrow> a + r * c \<noteq> b + r * d"
23proof (rule notI)
24  assume nz: "r\<noteq> 0" and cnd: "a = b \<and> c\<noteq>d"
25    and eq: "a + (r * c) = b + (r * d)"
26  have "(0 * d) + (r * c) = (0 * c) + (r * d)"
27    using add_left_imp_eq eq mult_zero_left by (simp add: cnd)
28  then show False using crossproduct_eq [of 0 d] nz cnd by simp
29qed
30
31lemma add_0_iff:
32  "b = b + a \<longleftrightarrow> a = 0"
33  using add_left_imp_eq [of b a 0] by auto
34
35end
36
37subclass (in idom) comm_semiring_1_cancel_crossproduct
38proof
39  fix w x y z
40  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
41  proof
42    assume "w * y + x * z = w * z + x * y"
43    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
44    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
45    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
46    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
47    then show "w = x \<or> y = z" by auto
48  qed (auto simp add: ac_simps)
49qed
50
51instance nat :: comm_semiring_1_cancel_crossproduct
52proof
53  fix w x y z :: nat
54  have aux: "\<And>y z. y < z \<Longrightarrow> w * y + x * z = w * z + x * y \<Longrightarrow> w = x"
55  proof -
56    fix y z :: nat
57    assume "y < z" then have "\<exists>k. z = y + k \<and> k \<noteq> 0" by (intro exI [of _ "z - y"]) auto
58    then obtain k where "z = y + k" and "k \<noteq> 0" by blast
59    assume "w * y + x * z = w * z + x * y"
60    then have "(w * y + x * y) + x * k = (w * y + x * y) + w * k" by (simp add: \<open>z = y + k\<close> algebra_simps)
61    then have "x * k = w * k" by simp
62    then show "w = x" using \<open>k \<noteq> 0\<close> by simp
63  qed
64  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
65    by (auto simp add: neq_iff dest!: aux)
66qed
67
68text \<open>Semiring normalization proper\<close>
69
70ML_file "Tools/semiring_normalizer.ML"
71
72context comm_semiring_1
73begin
74
75lemma semiring_normalization_rules:
76  "(a * m) + (b * m) = (a + b) * m"
77  "(a * m) + m = (a + 1) * m"
78  "m + (a * m) = (a + 1) * m"
79  "m + m = (1 + 1) * m"
80  "0 + a = a"
81  "a + 0 = a"
82  "a * b = b * a"
83  "(a + b) * c = (a * c) + (b * c)"
84  "0 * a = 0"
85  "a * 0 = 0"
86  "1 * a = a"
87  "a * 1 = a"
88  "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
89  "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
90  "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
91  "(lx * ly) * rx = (lx * rx) * ly"
92  "(lx * ly) * rx = lx * (ly * rx)"
93  "lx * (rx * ry) = (lx * rx) * ry"
94  "lx * (rx * ry) = rx * (lx * ry)"
95  "(a + b) + (c + d) = (a + c) + (b + d)"
96  "(a + b) + c = a + (b + c)"
97  "a + (c + d) = c + (a + d)"
98  "(a + b) + c = (a + c) + b"
99  "a + c = c + a"
100  "a + (c + d) = (a + c) + d"
101  "(x ^ p) * (x ^ q) = x ^ (p + q)"
102  "x * (x ^ q) = x ^ (Suc q)"
103  "(x ^ q) * x = x ^ (Suc q)"
104  "x * x = x\<^sup>2"
105  "(x * y) ^ q = (x ^ q) * (y ^ q)"
106  "(x ^ p) ^ q = x ^ (p * q)"
107  "x ^ 0 = 1"
108  "x ^ 1 = x"
109  "x * (y + z) = (x * y) + (x * z)"
110  "x ^ (Suc q) = x * (x ^ q)"
111  "x ^ (2*n) = (x ^ n) * (x ^ n)"
112  by (simp_all add: algebra_simps power_add power2_eq_square
113    power_mult_distrib power_mult del: one_add_one)
114
115local_setup \<open>
116  Semiring_Normalizer.declare @{thm comm_semiring_1_axioms}
117    {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
118      @{thms semiring_normalization_rules}),
119     ring = ([], []),
120     field = ([], []),
121     idom = [],
122     ideal = []}
123\<close>
124
125end
126
127context comm_ring_1
128begin
129
130lemma ring_normalization_rules:
131  "- x = (- 1) * x"
132  "x - y = x + (- y)"
133  by simp_all
134
135local_setup \<open>
136  Semiring_Normalizer.declare @{thm comm_ring_1_axioms}
137    {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
138      @{thms semiring_normalization_rules}),
139      ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
140      field = ([], []),
141      idom = [],
142      ideal = []}
143\<close>
144
145end
146
147context comm_semiring_1_cancel_crossproduct
148begin
149
150local_setup \<open>
151  Semiring_Normalizer.declare @{thm comm_semiring_1_cancel_crossproduct_axioms}
152    {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
153      @{thms semiring_normalization_rules}),
154     ring = ([], []),
155     field = ([], []),
156     idom = @{thms crossproduct_noteq add_scale_eq_noteq},
157     ideal = []}
158\<close>
159
160end
161
162context idom
163begin
164
165local_setup \<open>
166  Semiring_Normalizer.declare @{thm idom_axioms}
167   {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
168      @{thms semiring_normalization_rules}),
169    ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
170    field = ([], []),
171    idom = @{thms crossproduct_noteq add_scale_eq_noteq},
172    ideal = @{thms right_minus_eq add_0_iff}}
173\<close>
174
175end
176
177context field
178begin
179
180local_setup \<open>
181  Semiring_Normalizer.declare @{thm field_axioms}
182   {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
183      @{thms semiring_normalization_rules}),
184    ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
185    field = ([@{term "x / y"}, @{term "inverse x"}], @{thms divide_inverse inverse_eq_divide}),
186    idom = @{thms crossproduct_noteq add_scale_eq_noteq},
187    ideal = @{thms right_minus_eq add_0_iff}}
188\<close>
189
190end
191
192code_identifier
193  code_module Semiring_Normalization \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
194
195end
196