# Theory Semiring_Normalization

theory Semiring_Normalization
imports Numeral_Simprocs
```(*  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 ⟷ w = x ∨ y = z"
begin

lemma crossproduct_noteq:
"a ≠ b ∧ c ≠ d ⟷ a * c + b * d ≠ a * d + b * c"

"r ≠ 0 ⟹ a = b ∧ c ≠ d ⟹ a + r * c ≠ b + r * d"
proof (rule notI)
assume nz: "r≠ 0" and cnd: "a = b ∧ c≠d"
and eq: "a + (r * c) = b + (r * d)"
have "(0 * d) + (r * c) = (0 * c) + (r * d)"
then show False using crossproduct_eq [of 0 d] nz cnd by simp
qed

"b = b + a ⟷ 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 ⟷ w = x ∨ 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 ∨ w - x = 0" by (rule divisors_zero)
then show "w = x ∨ y = z" by auto
qed

instance nat :: comm_semiring_1_cancel_crossproduct
proof
fix w x y z :: nat
have aux: "⋀y z. y < z ⟹ w * y + x * z = w * z + x * y ⟹ w = x"
proof -
fix y z :: nat
assume "y < z" then have "∃k. z = y + k ∧ k ≠ 0" by (intro exI [of _ "z - y"]) auto
then obtain k where "z = y + k" and "k ≠ 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 ≠ 0› by simp
qed
show "w * y + x * z = w * z + x * y ⟷ w = x ∨ 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

lemma semiring_normalization_rules:
"(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⇧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)"

local_setup ‹
Semiring_Normalizer.declare @{thm comm_semiring_1_axioms}
{semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
@{thms semiring_normalization_rules}),
ring = ([], []),
field = ([], []),
idom = [],
ideal = []}
›

end

context comm_ring_1
begin

lemma ring_normalization_rules:
"- x = (- 1) * x"
"x - y = x + (- y)"
by simp_all

local_setup ‹
Semiring_Normalizer.declare @{thm comm_ring_1_axioms}
{semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
@{thms semiring_normalization_rules}),
ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
field = ([], []),
idom = [],
ideal = []}
›

end

context comm_semiring_1_cancel_crossproduct
begin

local_setup ‹
Semiring_Normalizer.declare @{thm comm_semiring_1_cancel_crossproduct_axioms}
{semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
@{thms semiring_normalization_rules}),
ring = ([], []),
field = ([], []),
ideal = []}
›

end

context idom
begin

local_setup ‹
Semiring_Normalizer.declare @{thm idom_axioms}
{semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
@{thms semiring_normalization_rules}),
ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
field = ([], []),
›

end

context field
begin

local_setup ‹
Semiring_Normalizer.declare @{thm field_axioms}
{semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
@{thms semiring_normalization_rules}),
ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
field = ([@{term "x / y"}, @{term "inverse x"}], @{thms divide_inverse inverse_eq_divide}),