# Theory Semiring_Normalization

Up to index of Isabelle/HOL-Proofs

theory Semiring_Normalization
imports Numeral_Simprocs
`(*  Title:      HOL/Semiring_Normalization.thy    Author:     Amine Chaieb, TU Muenchen*)header {* Semiring normalization *}theory Semiring_Normalizationimports Numeral_Simprocs Nat_TransferbeginML_file "Tools/semiring_normalizer.ML"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"beginlemma crossproduct_noteq:  "a ≠ b ∧ c ≠ d <-> a * c + b * d ≠ a * d + b * c"  by (simp add: crossproduct_eq)lemma add_scale_eq_noteq:  "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)"    using add_imp_eq eq mult_zero_left by (simp add: cnd)  then show False using crossproduct_eq [of 0 d] nz cnd by simpqedlemma add_0_iff:  "b = b + a <-> a = 0"  using add_imp_eq [of b a 0] by autoendsubclass (in idom) comm_semiring_1_cancel_crossproductproof  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 (auto simp add: add_ac)qedinstance nat :: comm_semiring_1_cancel_crossproductproof  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)qedtext {* Semiring normalization proper *}setup Semiring_Normalizer.setupcontext comm_semiring_1beginlemma normalizing_semiring_ops:  shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)"    and "TERM 0" and "TERM 1" .lemma normalizing_semiring_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)"  "x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))"  by (simp_all add: algebra_simps power_add power2_eq_square    power_mult_distrib power_mult del: one_add_one)lemmas normalizing_comm_semiring_1_axioms =  comm_semiring_1_axioms [normalizer    semiring ops: normalizing_semiring_ops    semiring rules: normalizing_semiring_rules]declaration  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}endcontext comm_ring_1beginlemma normalizing_ring_ops: shows "TERM (x- y)" and "TERM (- x)" .lemma normalizing_ring_rules:  "- x = (- 1) * x"  "x - y = x + (- y)"  by (simp_all add: diff_minus)lemmas normalizing_comm_ring_1_axioms =  comm_ring_1_axioms [normalizer    semiring ops: normalizing_semiring_ops    semiring rules: normalizing_semiring_rules    ring ops: normalizing_ring_ops    ring rules: normalizing_ring_rules]declaration  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}endcontext comm_semiring_1_cancel_crossproductbegindeclare  normalizing_comm_semiring_1_axioms [normalizer del]lemmas  normalizing_comm_semiring_1_cancel_crossproduct_axioms =  comm_semiring_1_cancel_crossproduct_axioms [normalizer    semiring ops: normalizing_semiring_ops    semiring rules: normalizing_semiring_rules    idom rules: crossproduct_noteq add_scale_eq_noteq]declaration  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_crossproduct_axioms} *}endcontext idombegindeclare normalizing_comm_ring_1_axioms [normalizer del]lemmas normalizing_idom_axioms = idom_axioms [normalizer  semiring ops: normalizing_semiring_ops  semiring rules: normalizing_semiring_rules  ring ops: normalizing_ring_ops  ring rules: normalizing_ring_rules  idom rules: crossproduct_noteq add_scale_eq_noteq  ideal rules: right_minus_eq add_0_iff]declaration  {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}endcontext fieldbeginlemma normalizing_field_ops:  shows "TERM (x / y)" and "TERM (inverse x)" .lemmas normalizing_field_rules = divide_inverse inverse_eq_dividelemmas normalizing_field_axioms =  field_axioms [normalizer    semiring ops: normalizing_semiring_ops    semiring rules: normalizing_semiring_rules    ring ops: normalizing_ring_ops    ring rules: normalizing_ring_rules    field ops: normalizing_field_ops    field rules: normalizing_field_rules    idom rules: crossproduct_noteq add_scale_eq_noteq    ideal rules: right_minus_eq add_0_iff]declaration  {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}endhide_fact (open) normalizing_comm_semiring_1_axioms  normalizing_comm_semiring_1_cancel_crossproduct_axioms normalizing_semiring_ops normalizing_semiring_ruleshide_fact (open) normalizing_comm_ring_1_axioms  normalizing_idom_axioms normalizing_ring_ops normalizing_ring_ruleshide_fact (open) normalizing_field_axioms normalizing_field_ops normalizing_field_rulescode_modulename SML  Semiring_Normalization Arithcode_modulename OCaml  Semiring_Normalization Arithcode_modulename Haskell  Semiring_Normalization Arithend`