src/HOL/Semiring_Normalization.thy
author blanchet
Tue Nov 07 15:16:42 2017 +0100 (19 months ago)
changeset 67022 49309fe530fd
parent 66836 4eb431c3f974
child 69593 3dda49e08b9d
permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
     1 (*  Title:      HOL/Semiring_Normalization.thy
     2     Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 section \<open>Semiring normalization\<close>
     6 
     7 theory Semiring_Normalization
     8 imports Numeral_Simprocs
     9 begin
    10 
    11 text \<open>Prelude\<close>
    12 
    13 class 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"
    15 begin
    16 
    17 lemma 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 
    21 lemma add_scale_eq_noteq:
    22   "r \<noteq> 0 \<Longrightarrow> a = b \<and> c \<noteq> d \<Longrightarrow> a + r * c \<noteq> b + r * d"
    23 proof (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
    29 qed
    30 
    31 lemma add_0_iff:
    32   "b = b + a \<longleftrightarrow> a = 0"
    33   using add_left_imp_eq [of b a 0] by auto
    34 
    35 end
    36 
    37 subclass (in idom) comm_semiring_1_cancel_crossproduct
    38 proof
    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)
    49 qed
    50 
    51 instance nat :: comm_semiring_1_cancel_crossproduct
    52 proof
    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)
    66 qed
    67 
    68 text \<open>Semiring normalization proper\<close>
    69 
    70 ML_file "Tools/semiring_normalizer.ML"
    71 
    72 context comm_semiring_1
    73 begin
    74 
    75 lemma 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 
   115 local_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 
   125 end
   126 
   127 context comm_ring_1
   128 begin
   129 
   130 lemma ring_normalization_rules:
   131   "- x = (- 1) * x"
   132   "x - y = x + (- y)"
   133   by simp_all
   134 
   135 local_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 
   145 end
   146 
   147 context comm_semiring_1_cancel_crossproduct
   148 begin
   149 
   150 local_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 
   160 end
   161 
   162 context idom
   163 begin
   164 
   165 local_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 
   175 end
   176 
   177 context field
   178 begin
   179 
   180 local_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 
   190 end
   191 
   192 code_identifier
   193   code_module Semiring_Normalization \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
   194 
   195 end