src/HOL/Semiring_Normalization.thy

tuned proof;

(*  Title:      HOL/Semiring_Normalization.thy
    Author:     Amine Chaieb, TU Muenchen
*)

section {* Semiring normalization *}

theory Semiring_Normalization
imports Numeral_Simprocs Nat_Transfer
begin

text {* Prelude *}

class comm_semiring_1_cancel_crossproduct = comm_semiring_1_cancel +
  assumes crossproduct_eq: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
begin

lemma crossproduct_noteq:
  "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> a * c + b * d \<noteq> a * d + b * c"
  by (simp add: crossproduct_eq)

lemma add_scale_eq_noteq:
  "r \<noteq> 0 \<Longrightarrow> a = b \<and> c \<noteq> d \<Longrightarrow> a + r * c \<noteq> b + r * d"
proof (rule notI)
  assume nz: "r\<noteq> 0" and cnd: "a = b \<and> c\<noteq>d"
    and eq: "a + (r * c) = b + (r * d)"
  have "(0 * d) + (r * c) = (0 * c) + (r * d)"
    using add_left_imp_eq eq mult_zero_left by (simp add: cnd)
  then show False using crossproduct_eq [of 0 d] nz cnd by simp
qed

lemma add_0_iff:
  "b = b + a \<longleftrightarrow> a = 0"
  using add_left_imp_eq [of b a 0] by auto

end

subclass (in idom) comm_semiring_1_cancel_crossproduct
proof
  fix w x y z
  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
  proof
    assume "w * y + x * z = w * z + x * y"
    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
    then show "w = x \<or> y = z" by auto
  qed (auto simp add: ac_simps)
qed

instance nat :: comm_semiring_1_cancel_crossproduct
proof
  fix w x y z :: nat
  have aux: "\<And>y z. y < z \<Longrightarrow> w * y + x * z = w * z + x * y \<Longrightarrow> w = x"
  proof -
    fix y z :: nat
    assume "y < z" then have "\<exists>k. z = y + k \<and> k \<noteq> 0" by (intro exI [of _ "z - y"]) auto
    then obtain k where "z = y + k" and "k \<noteq> 0" by blast
    assume "w * y + x * z = w * z + x * y"
    then have "(w * y + x * y) + x * k = (w * y + x * y) + w * k" by (simp add: `z = y + k` algebra_simps)
    then have "x * k = w * k" by simp
    then show "w = x" using `k \<noteq> 0` by simp
  qed
  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
    by (auto simp add: neq_iff dest!: aux)
qed

text {* Semiring normalization proper *}

ML_file "Tools/semiring_normalizer.ML"

context comm_semiring_1
begin

declaration \<open>
let
  val rules = @{lemma
    "(a * m) + (b * m) = (a + b) * m"
    "(a * m) + m = (a + 1) * m"
    "m + (a * m) = (a + 1) * m"
    "m + m = (1 + 1) * m"
    "0 + a = a"
    "a + 0 = a"
    "a * b = b * a"
    "(a + b) * c = (a * c) + (b * c)"
    "0 * a = 0"
    "a * 0 = 0"
    "1 * a = a"
    "a * 1 = a"
    "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
    "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
    "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
    "(lx * ly) * rx = (lx * rx) * ly"
    "(lx * ly) * rx = lx * (ly * rx)"
    "lx * (rx * ry) = (lx * rx) * ry"
    "lx * (rx * ry) = rx * (lx * ry)"
    "(a + b) + (c + d) = (a + c) + (b + d)"
    "(a + b) + c = a + (b + c)"
    "a + (c + d) = c + (a + d)"
    "(a + b) + c = (a + c) + b"
    "a + c = c + a"
    "a + (c + d) = (a + c) + d"
    "(x ^ p) * (x ^ q) = x ^ (p + q)"
    "x * (x ^ q) = x ^ (Suc q)"
    "(x ^ q) * x = x ^ (Suc q)"
    "x * x = x\<^sup>2"
    "(x * y) ^ q = (x ^ q) * (y ^ q)"
    "(x ^ p) ^ q = x ^ (p * q)"
    "x ^ 0 = 1"
    "x ^ 1 = x"
    "x * (y + z) = (x * y) + (x * z)"
    "x ^ (Suc q) = x * (x ^ q)"
    "x ^ (2*n) = (x ^ n) * (x ^ n)"
    by (simp_all add: algebra_simps power_add power2_eq_square
      power_mult_distrib power_mult del: one_add_one)}
in
  Semiring_Normalizer.declare @{thm comm_semiring_1_axioms}
    {semiring = ([@{cpat "?x + ?y"}, @{cpat "?x * ?y"}, @{cpat "?x ^ ?n"}, @{cpat 0}, @{cpat 1}],
      rules), ring = ([], []), field = ([], []), idom = [], ideal = []}
end\<close>

end

context comm_ring_1
begin

declaration \<open>
let
  val rules = @{lemma
    "- x = (- 1) * x"
    "x - y = x + (- y)"
    by simp_all}
in
  Semiring_Normalizer.declare @{thm comm_ring_1_axioms}
    {semiring = Semiring_Normalizer.the_semiring @{context} @{thm comm_semiring_1_axioms},
      ring = ([@{cpat "?x - ?y"}, @{cpat "- ?x"}], rules), field = ([], []), idom = [], ideal = []}
end\<close>

end

context comm_semiring_1_cancel_crossproduct
begin

declaration \<open>Semiring_Normalizer.declare @{thm comm_semiring_1_cancel_crossproduct_axioms}
  {semiring = Semiring_Normalizer.the_semiring @{context} @{thm comm_semiring_1_axioms},
    ring = ([], []), field = ([], []), idom = @{thms crossproduct_noteq add_scale_eq_noteq}, ideal = []}\<close>

end

context idom
begin

declaration \<open>Semiring_Normalizer.declare @{thm idom_axioms}
  {semiring = Semiring_Normalizer.the_semiring @{context} @{thm comm_ring_1_axioms},
    ring = Semiring_Normalizer.the_ring @{context} @{thm comm_ring_1_axioms},
    field = ([], []), idom = @{thms crossproduct_noteq add_scale_eq_noteq},
    ideal = @{thms right_minus_eq add_0_iff}}\<close>

end

context field
begin

declaration \<open>Semiring_Normalizer.declare @{thm field_axioms}
  {semiring = Semiring_Normalizer.the_semiring @{context} @{thm idom_axioms},
    ring = Semiring_Normalizer.the_ring @{context} @{thm idom_axioms},
    field = ([@{cpat "?x / ?y"}, @{cpat "inverse ?x"}], @{thms divide_inverse inverse_eq_divide}),
    idom = Semiring_Normalizer.the_idom @{context} @{thm idom_axioms},
    ideal = Semiring_Normalizer.the_ideal @{context} @{thm idom_axioms}}\<close>

end

code_identifier
  code_module Semiring_Normalization \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

end