split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
--- a/src/HOL/Groebner_Basis.thy Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Groebner_Basis.thy Fri May 07 15:05:52 2010 +0200
@@ -2,341 +2,14 @@
Author: Amine Chaieb, TU Muenchen
*)
-header {* Semiring normalization and Groebner Bases *}
+header {* Groebner bases *}
theory Groebner_Basis
-imports Numeral_Simprocs Nat_Transfer
+imports Semiring_Normalization
uses
- "Tools/Groebner_Basis/normalizer.ML"
("Tools/Groebner_Basis/groebner.ML")
begin
-subsection {* Semiring normalization *}
-
-setup Normalizer.setup
-
-locale normalizing_semiring =
- fixes add mul pwr r0 r1
- assumes add_a:"(add x (add y z) = add (add x y) z)"
- and add_c: "add x y = add y x" and add_0:"add r0 x = x"
- and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
- and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0"
- and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
- and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
-begin
-
-lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
-proof (induct p)
- case 0
- then show ?case by (auto simp add: pwr_0 mul_1)
-next
- case Suc
- from this [symmetric] show ?case
- by (auto simp add: pwr_Suc mul_1 mul_a)
-qed
-
-lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
-proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
- fix q x y
- assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
- have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
- by (simp add: mul_a)
- also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
- also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
- finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
- mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
-qed
-
-lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
-proof (induct p arbitrary: q)
- case 0
- show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
-next
- case Suc
- thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
-qed
-
-lemma semiring_ops:
- shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
- and "TERM r0" and "TERM r1" .
-
-lemma semiring_rules:
- "add (mul a m) (mul b m) = mul (add a b) m"
- "add (mul a m) m = mul (add a r1) m"
- "add m (mul a m) = mul (add a r1) m"
- "add m m = mul (add r1 r1) m"
- "add r0 a = a"
- "add a r0 = a"
- "mul a b = mul b a"
- "mul (add a b) c = add (mul a c) (mul b c)"
- "mul r0 a = r0"
- "mul a r0 = r0"
- "mul r1 a = a"
- "mul a r1 = a"
- "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
- "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
- "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
- "mul (mul lx ly) rx = mul (mul lx rx) ly"
- "mul (mul lx ly) rx = mul lx (mul ly rx)"
- "mul lx (mul rx ry) = mul (mul lx rx) ry"
- "mul lx (mul rx ry) = mul rx (mul lx ry)"
- "add (add a b) (add c d) = add (add a c) (add b d)"
- "add (add a b) c = add a (add b c)"
- "add a (add c d) = add c (add a d)"
- "add (add a b) c = add (add a c) b"
- "add a c = add c a"
- "add a (add c d) = add (add a c) d"
- "mul (pwr x p) (pwr x q) = pwr x (p + q)"
- "mul x (pwr x q) = pwr x (Suc q)"
- "mul (pwr x q) x = pwr x (Suc q)"
- "mul x x = pwr x 2"
- "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
- "pwr (pwr x p) q = pwr x (p * q)"
- "pwr x 0 = r1"
- "pwr x 1 = x"
- "mul x (add y z) = add (mul x y) (mul x z)"
- "pwr x (Suc q) = mul x (pwr x q)"
- "pwr x (2*n) = mul (pwr x n) (pwr x n)"
- "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
-proof -
- show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
-next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
-next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
-next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
-next show "add r0 a = a" using add_0 by simp
-next show "add a r0 = a" using add_0 add_c by simp
-next show "mul a b = mul b a" using mul_c by simp
-next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
-next show "mul r0 a = r0" using mul_0 by simp
-next show "mul a r0 = r0" using mul_0 mul_c by simp
-next show "mul r1 a = a" using mul_1 by simp
-next show "mul a r1 = a" using mul_1 mul_c by simp
-next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
- using mul_c mul_a by simp
-next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
- using mul_a by simp
-next
- have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
- also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
- finally
- show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
- using mul_c by simp
-next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
-next
- show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
-next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
-next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
-next show "add (add a b) (add c d) = add (add a c) (add b d)"
- using add_c add_a by simp
-next show "add (add a b) c = add a (add b c)" using add_a by simp
-next show "add a (add c d) = add c (add a d)"
- apply (simp add: add_a) by (simp only: add_c)
-next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
-next show "add a c = add c a" by (rule add_c)
-next show "add a (add c d) = add (add a c) d" using add_a by simp
-next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
-next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
-next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
-next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
-next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
-next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
-next show "pwr x 0 = r1" using pwr_0 .
-next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
-next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
-next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
-next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
-next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
- by (simp add: nat_number' pwr_Suc mul_pwr)
-qed
-
-
-lemmas normalizing_semiring_axioms' =
- normalizing_semiring_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules]
-
-end
-
-sublocale comm_semiring_1
- < normalizing!: normalizing_semiring plus times power zero one
-proof
-qed (simp_all add: algebra_simps)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
-
-locale normalizing_ring = normalizing_semiring +
- fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
- and neg :: "'a \<Rightarrow> 'a"
- assumes neg_mul: "neg x = mul (neg r1) x"
- and sub_add: "sub x y = add x (neg y)"
-begin
-
-lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
-
-lemmas ring_rules = neg_mul sub_add
-
-lemmas normalizing_ring_axioms' =
- normalizing_ring_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- ring ops: ring_ops
- ring rules: ring_rules]
-
-end
-
-sublocale comm_ring_1
- < normalizing!: normalizing_ring plus times power zero one minus uminus
-proof
-qed (simp_all add: diff_minus)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
-
-locale normalizing_field = normalizing_ring +
- fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
- and inverse:: "'a \<Rightarrow> 'a"
- assumes divide_inverse: "divide x y = mul x (inverse y)"
- and inverse_divide: "inverse x = divide r1 x"
-begin
-
-lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
-
-lemmas field_rules = divide_inverse inverse_divide
-
-lemmas normalizing_field_axioms' =
- normalizing_field_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- ring ops: ring_ops
- ring rules: ring_rules
- field ops: field_ops
- field rules: field_rules]
-
-end
-
-locale normalizing_semiring_cancel = normalizing_semiring +
- assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
- and add_mul_solve: "add (mul w y) (mul x z) =
- add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
-begin
-
-lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
-proof-
- have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
- also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
- using add_mul_solve by blast
- finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
- by simp
-qed
-
-lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
- \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
-proof(clarify)
- assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
- and eq: "add b (mul r c) = add b (mul r d)"
- hence "mul r c = mul r d" using cnd add_cancel by simp
- hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
- using mul_0 add_cancel by simp
- thus "False" using add_mul_solve nz cnd by simp
-qed
-
-lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
-proof-
- have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
- thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
-qed
-
-declare normalizing_semiring_axioms' [normalizer del]
-
-lemmas normalizing_semiring_cancel_axioms' =
- normalizing_semiring_cancel_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- idom rules: noteq_reduce add_scale_eq_noteq]
-
-end
-
-locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
- assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
-begin
-
-declare normalizing_ring_axioms' [normalizer del]
-
-lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- ring ops: ring_ops
- ring rules: ring_rules
- idom rules: noteq_reduce add_scale_eq_noteq
- ideal rules: subr0_iff add_r0_iff]
-
-end
-
-sublocale idom
- < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
-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: add_ac)
-qed (simp_all add: algebra_simps)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
-
-interpretation normalizing_nat!: normalizing_semiring_cancel
- "op +" "op *" "op ^" "0::nat" "1"
-proof (unfold_locales, simp add: algebra_simps)
- fix w x y z ::"nat"
- { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
- hence "y < z \<or> y > z" by arith
- moreover {
- assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
- then obtain k where kp: "k>0" and yz:"z = y + k" by blast
- from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
- hence "x*k = w*k" by simp
- hence "w = x" using kp by simp }
- moreover {
- assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
- then obtain k where kp: "k>0" and yz:"y = z + k" by blast
- from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
- hence "w*k = x*k" by simp
- hence "w = x" using kp by simp }
- ultimately have "w=x" by blast }
- thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
-qed
-
-declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
-
-locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
-begin
-
-declare normalizing_field_axioms' [normalizer del]
-
-lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- ring ops: ring_ops
- ring rules: ring_rules
- field ops: field_ops
- field rules: field_rules
- idom rules: noteq_reduce add_scale_eq_noteq
- ideal rules: subr0_iff add_r0_iff]
-
-end
-
-sublocale field
- < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
-proof
-qed (simp_all add: divide_inverse)
-
-declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
-
-
subsection {* Groebner Bases *}
lemmas bool_simps = simp_thms(1-34)
@@ -367,6 +40,11 @@
setup Algebra_Simplification.setup
+use "Tools/Groebner_Basis/groebner.ML"
+
+method_setup algebra = Groebner.algebra_method
+ "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
+
declare dvd_def[algebra]
declare dvd_eq_mod_eq_0[symmetric, algebra]
declare mod_div_trivial[algebra]
@@ -395,9 +73,4 @@
declare zmod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]
-use "Tools/Groebner_Basis/groebner.ML"
-
-method_setup algebra = Groebner.algebra_method
- "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
-
end
--- a/src/HOL/IsaMakefile Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/IsaMakefile Fri May 07 15:05:52 2010 +0200
@@ -271,6 +271,7 @@
Random.thy \
Random_Sequence.thy \
Recdef.thy \
+ Semiring_Normalization.thy \
SetInterval.thy \
Sledgehammer.thy \
String.thy \
--- a/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Fri May 07 15:05:52 2010 +0200
@@ -1222,7 +1222,7 @@
in
(let val th = tryfind trivial_axiom (keq @ klep @ kltp)
in
- (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Normalizer.field_comp_conv) th, RealArith.Trivial)
+ (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Numeral_Simprocs.field_comp_conv) th, RealArith.Trivial)
end)
handle Failure _ =>
(let val proof =
--- a/src/HOL/Library/normarith.ML Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/normarith.ML Fri May 07 15:05:52 2010 +0200
@@ -168,7 +168,7 @@
val real_poly_conv =
Normalizer.semiring_normalize_wrapper ctxt
(the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
- in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Normalizer.field_comp_conv then_conv real_poly_conv)))
+ in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Numeral_Simprocs.field_comp_conv then_conv real_poly_conv)))
end;
fun absc cv ct = case term_of ct of
@@ -190,8 +190,8 @@
val apply_pth5 = rewr_conv @{thm pth_5};
val apply_pth6 = rewr_conv @{thm pth_6};
val apply_pth7 = rewrs_conv @{thms pth_7};
- val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Normalizer.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
- val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Normalizer.field_comp_conv);
+ val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Numeral_Simprocs.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
+ val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Numeral_Simprocs.field_comp_conv);
val apply_ptha = rewr_conv @{thm pth_a};
val apply_pthb = rewrs_conv @{thms pth_b};
val apply_pthc = rewrs_conv @{thms pth_c};
@@ -204,7 +204,7 @@
| _ => error "headvector: non-canonical term"
fun vector_cmul_conv ct =
- ((apply_pth5 then_conv arg1_conv Normalizer.field_comp_conv) else_conv
+ ((apply_pth5 then_conv arg1_conv Numeral_Simprocs.field_comp_conv) else_conv
(apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
fun vector_add_conv ct = apply_pth7 ct
@@ -396,7 +396,7 @@
fun init_conv ctxt =
Simplifier.rewrite (Simplifier.context ctxt
(HOL_basic_ss addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
- then_conv Normalizer.field_comp_conv
+ then_conv Numeral_Simprocs.field_comp_conv
then_conv nnf_conv
fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
--- a/src/HOL/Library/positivstellensatz.ML Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/positivstellensatz.ML Fri May 07 15:05:52 2010 +0200
@@ -751,7 +751,7 @@
(the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
in gen_real_arith ctxt
- (cterm_of_rat, Normalizer.field_comp_conv, Normalizer.field_comp_conv, Normalizer.field_comp_conv,
+ (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
main,neg,add,mul, prover)
end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Semiring_Normalization.thy Fri May 07 15:05:52 2010 +0200
@@ -0,0 +1,336 @@
+(* Title: HOL/Semiring_Normalization.thy
+ Author: Amine Chaieb, TU Muenchen
+*)
+
+header {* Semiring normalization *}
+
+theory Semiring_Normalization
+imports Numeral_Simprocs Nat_Transfer
+uses
+ "Tools/Groebner_Basis/normalizer.ML"
+begin
+
+setup Normalizer.setup
+
+locale normalizing_semiring =
+ fixes add mul pwr r0 r1
+ assumes add_a:"(add x (add y z) = add (add x y) z)"
+ and add_c: "add x y = add y x" and add_0:"add r0 x = x"
+ and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
+ and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0"
+ and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
+ and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
+begin
+
+lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
+proof (induct p)
+ case 0
+ then show ?case by (auto simp add: pwr_0 mul_1)
+next
+ case Suc
+ from this [symmetric] show ?case
+ by (auto simp add: pwr_Suc mul_1 mul_a)
+qed
+
+lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
+ fix q x y
+ assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+ have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
+ by (simp add: mul_a)
+ also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
+ also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
+ finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
+ mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
+qed
+
+lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
+proof (induct p arbitrary: q)
+ case 0
+ show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
+next
+ case Suc
+ thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
+qed
+
+lemma semiring_ops:
+ shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
+ and "TERM r0" and "TERM r1" .
+
+lemma semiring_rules:
+ "add (mul a m) (mul b m) = mul (add a b) m"
+ "add (mul a m) m = mul (add a r1) m"
+ "add m (mul a m) = mul (add a r1) m"
+ "add m m = mul (add r1 r1) m"
+ "add r0 a = a"
+ "add a r0 = a"
+ "mul a b = mul b a"
+ "mul (add a b) c = add (mul a c) (mul b c)"
+ "mul r0 a = r0"
+ "mul a r0 = r0"
+ "mul r1 a = a"
+ "mul a r1 = a"
+ "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
+ "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
+ "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
+ "mul (mul lx ly) rx = mul (mul lx rx) ly"
+ "mul (mul lx ly) rx = mul lx (mul ly rx)"
+ "mul lx (mul rx ry) = mul (mul lx rx) ry"
+ "mul lx (mul rx ry) = mul rx (mul lx ry)"
+ "add (add a b) (add c d) = add (add a c) (add b d)"
+ "add (add a b) c = add a (add b c)"
+ "add a (add c d) = add c (add a d)"
+ "add (add a b) c = add (add a c) b"
+ "add a c = add c a"
+ "add a (add c d) = add (add a c) d"
+ "mul (pwr x p) (pwr x q) = pwr x (p + q)"
+ "mul x (pwr x q) = pwr x (Suc q)"
+ "mul (pwr x q) x = pwr x (Suc q)"
+ "mul x x = pwr x 2"
+ "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+ "pwr (pwr x p) q = pwr x (p * q)"
+ "pwr x 0 = r1"
+ "pwr x 1 = x"
+ "mul x (add y z) = add (mul x y) (mul x z)"
+ "pwr x (Suc q) = mul x (pwr x q)"
+ "pwr x (2*n) = mul (pwr x n) (pwr x n)"
+ "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
+proof -
+ show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
+next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
+next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
+next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
+next show "add r0 a = a" using add_0 by simp
+next show "add a r0 = a" using add_0 add_c by simp
+next show "mul a b = mul b a" using mul_c by simp
+next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
+next show "mul r0 a = r0" using mul_0 by simp
+next show "mul a r0 = r0" using mul_0 mul_c by simp
+next show "mul r1 a = a" using mul_1 by simp
+next show "mul a r1 = a" using mul_1 mul_c by simp
+next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
+ using mul_c mul_a by simp
+next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
+ using mul_a by simp
+next
+ have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
+ also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
+ finally
+ show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
+ using mul_c by simp
+next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
+next
+ show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
+next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
+next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
+next show "add (add a b) (add c d) = add (add a c) (add b d)"
+ using add_c add_a by simp
+next show "add (add a b) c = add a (add b c)" using add_a by simp
+next show "add a (add c d) = add c (add a d)"
+ apply (simp add: add_a) by (simp only: add_c)
+next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
+next show "add a c = add c a" by (rule add_c)
+next show "add a (add c d) = add (add a c) d" using add_a by simp
+next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
+next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
+next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
+next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
+next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
+next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
+next show "pwr x 0 = r1" using pwr_0 .
+next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
+next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
+next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
+next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
+next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
+ by (simp add: nat_number' pwr_Suc mul_pwr)
+qed
+
+
+lemmas normalizing_semiring_axioms' =
+ normalizing_semiring_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules]
+
+end
+
+sublocale comm_semiring_1
+ < normalizing!: normalizing_semiring plus times power zero one
+proof
+qed (simp_all add: algebra_simps)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
+
+locale normalizing_ring = normalizing_semiring +
+ fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+ and neg :: "'a \<Rightarrow> 'a"
+ assumes neg_mul: "neg x = mul (neg r1) x"
+ and sub_add: "sub x y = add x (neg y)"
+begin
+
+lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
+
+lemmas ring_rules = neg_mul sub_add
+
+lemmas normalizing_ring_axioms' =
+ normalizing_ring_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules
+ ring ops: ring_ops
+ ring rules: ring_rules]
+
+end
+
+sublocale comm_ring_1
+ < normalizing!: normalizing_ring plus times power zero one minus uminus
+proof
+qed (simp_all add: diff_minus)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
+
+locale normalizing_field = normalizing_ring +
+ fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+ and inverse:: "'a \<Rightarrow> 'a"
+ assumes divide_inverse: "divide x y = mul x (inverse y)"
+ and inverse_divide: "inverse x = divide r1 x"
+begin
+
+lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
+
+lemmas field_rules = divide_inverse inverse_divide
+
+lemmas normalizing_field_axioms' =
+ normalizing_field_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules
+ ring ops: ring_ops
+ ring rules: ring_rules
+ field ops: field_ops
+ field rules: field_rules]
+
+end
+
+locale normalizing_semiring_cancel = normalizing_semiring +
+ assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
+ and add_mul_solve: "add (mul w y) (mul x z) =
+ add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
+begin
+
+lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+proof-
+ have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
+ also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+ using add_mul_solve by blast
+ finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+ by simp
+qed
+
+lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
+ \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
+proof(clarify)
+ assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
+ and eq: "add b (mul r c) = add b (mul r d)"
+ hence "mul r c = mul r d" using cnd add_cancel by simp
+ hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
+ using mul_0 add_cancel by simp
+ thus "False" using add_mul_solve nz cnd by simp
+qed
+
+lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
+proof-
+ have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
+ thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
+qed
+
+declare normalizing_semiring_axioms' [normalizer del]
+
+lemmas normalizing_semiring_cancel_axioms' =
+ normalizing_semiring_cancel_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules
+ idom rules: noteq_reduce add_scale_eq_noteq]
+
+end
+
+locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
+ assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
+begin
+
+declare normalizing_ring_axioms' [normalizer del]
+
+lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules
+ ring ops: ring_ops
+ ring rules: ring_rules
+ idom rules: noteq_reduce add_scale_eq_noteq
+ ideal rules: subr0_iff add_r0_iff]
+
+end
+
+sublocale idom
+ < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
+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: add_ac)
+qed (simp_all add: algebra_simps)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
+
+interpretation normalizing_nat!: normalizing_semiring_cancel
+ "op +" "op *" "op ^" "0::nat" "1"
+proof (unfold_locales, simp add: algebra_simps)
+ fix w x y z ::"nat"
+ { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
+ hence "y < z \<or> y > z" by arith
+ moreover {
+ assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
+ then obtain k where kp: "k>0" and yz:"z = y + k" by blast
+ from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
+ hence "x*k = w*k" by simp
+ hence "w = x" using kp by simp }
+ moreover {
+ assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
+ then obtain k where kp: "k>0" and yz:"y = z + k" by blast
+ from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
+ hence "w*k = x*k" by simp
+ hence "w = x" using kp by simp }
+ ultimately have "w=x" by blast }
+ thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
+qed
+
+declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
+
+locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
+begin
+
+declare normalizing_field_axioms' [normalizer del]
+
+lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
+ semiring ops: semiring_ops
+ semiring rules: semiring_rules
+ ring ops: ring_ops
+ ring rules: ring_rules
+ field ops: field_ops
+ field rules: field_rules
+ idom rules: noteq_reduce add_scale_eq_noteq
+ ideal rules: subr0_iff add_r0_iff]
+
+end
+
+sublocale field
+ < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
+proof
+qed (simp_all add: divide_inverse)
+
+declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
+
+end
--- a/src/HOL/Tools/Groebner_Basis/normalizer.ML Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Tools/Groebner_Basis/normalizer.ML Fri May 07 15:05:52 2010 +0200
@@ -31,7 +31,6 @@
val semiring_normalizers_ord_wrapper: Proof.context -> entry ->
(cterm -> cterm -> bool) ->
{add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
- val field_comp_conv: conv
val setup: theory -> theory
end
@@ -41,156 +40,6 @@
(** some conversion **)
-local
- val zr = @{cpat "0"}
- val zT = ctyp_of_term zr
- val geq = @{cpat "op ="}
- val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
- val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
- val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
- val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
-
- fun prove_nz ss T t =
- let
- val z = instantiate_cterm ([(zT,T)],[]) zr
- val eq = instantiate_cterm ([(eqT,T)],[]) geq
- val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
- (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
- (Thm.capply (Thm.capply eq t) z)))
- in equal_elim (symmetric th) TrueI
- end
-
- fun proc phi ss ct =
- let
- val ((x,y),(w,z)) =
- (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
- val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
- val T = ctyp_of_term x
- val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
- val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
- in SOME (implies_elim (implies_elim th y_nz) z_nz)
- end
- handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun proc2 phi ss ct =
- let
- val (l,r) = Thm.dest_binop ct
- val T = ctyp_of_term l
- in (case (term_of l, term_of r) of
- (Const(@{const_name Rings.divide},_)$_$_, _) =>
- let val (x,y) = Thm.dest_binop l val z = r
- val _ = map (HOLogic.dest_number o term_of) [x,y,z]
- val ynz = prove_nz ss T y
- in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
- end
- | (_, Const (@{const_name Rings.divide},_)$_$_) =>
- let val (x,y) = Thm.dest_binop r val z = l
- val _ = map (HOLogic.dest_number o term_of) [x,y,z]
- val ynz = prove_nz ss T y
- in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
- end
- | _ => NONE)
- end
- handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
- | is_number t = can HOLogic.dest_number t
-
- val is_number = is_number o term_of
-
- fun proc3 phi ss ct =
- (case term_of ct of
- Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
- in SOME (mk_meta_eq th) end
- | Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
- in SOME (mk_meta_eq th) end
- | Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
- in SOME (mk_meta_eq th) end
- | _ => NONE)
- handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
-
-val add_frac_frac_simproc =
- make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
- name = "add_frac_frac_simproc",
- proc = proc, identifier = []}
-
-val add_frac_num_simproc =
- make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
- name = "add_frac_num_simproc",
- proc = proc2, identifier = []}
-
-val ord_frac_simproc =
- make_simproc
- {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
- @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
- @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
- @{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
- @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
- @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
- name = "ord_frac_simproc", proc = proc3, identifier = []}
-
-val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
- @{thm "divide_Numeral1"},
- @{thm "divide_zero"}, @{thm "divide_Numeral0"},
- @{thm "divide_divide_eq_left"},
- @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
- @{thm "times_divide_times_eq"},
- @{thm "divide_divide_eq_right"},
- @{thm "diff_def"}, @{thm "minus_divide_left"},
- @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
- @{thm field_divide_inverse} RS sym, @{thm inverse_divide},
- Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))
- (@{thm field_divide_inverse} RS sym)]
-
-in
-
-val field_comp_conv = (Simplifier.rewrite
-(HOL_basic_ss addsimps @{thms "semiring_norm"}
- addsimps ths addsimps @{thms simp_thms}
- addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
- addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
- ord_frac_simproc]
- addcongs [@{thm "if_weak_cong"}]))
-then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
- [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
-
-end
(** data **)
@@ -365,7 +214,7 @@
{is_const = K numeral_is_const,
dest_const = K dest_const,
mk_const = mk_const,
- conv = K (K field_comp_conv)}
+ conv = K (K Numeral_Simprocs.field_comp_conv)}
end;
--- a/src/HOL/Tools/numeral_simprocs.ML Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Tools/numeral_simprocs.ML Fri May 07 15:05:52 2010 +0200
@@ -1,7 +1,7 @@
(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
-Simprocs for the integer numerals.
+Simprocs for the (integer) numerals.
*)
(*To quote from Provers/Arith/cancel_numeral_factor.ML:
@@ -24,6 +24,7 @@
val field_combine_numerals: simproc
val field_cancel_numeral_factors: simproc list
val num_ss: simpset
+ val field_comp_conv: conv
end;
structure Numeral_Simprocs : NUMERAL_SIMPROCS =
@@ -602,6 +603,157 @@
"(l::'a::field_inverse_zero) / (m * n)"],
K DivideCancelFactor.proc)];
+local
+ val zr = @{cpat "0"}
+ val zT = ctyp_of_term zr
+ val geq = @{cpat "op ="}
+ val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
+ val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
+ val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
+ val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
+
+ fun prove_nz ss T t =
+ let
+ val z = instantiate_cterm ([(zT,T)],[]) zr
+ val eq = instantiate_cterm ([(eqT,T)],[]) geq
+ val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
+ (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
+ (Thm.capply (Thm.capply eq t) z)))
+ in equal_elim (symmetric th) TrueI
+ end
+
+ fun proc phi ss ct =
+ let
+ val ((x,y),(w,z)) =
+ (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
+ val T = ctyp_of_term x
+ val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
+ val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
+ in SOME (implies_elim (implies_elim th y_nz) z_nz)
+ end
+ handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun proc2 phi ss ct =
+ let
+ val (l,r) = Thm.dest_binop ct
+ val T = ctyp_of_term l
+ in (case (term_of l, term_of r) of
+ (Const(@{const_name Rings.divide},_)$_$_, _) =>
+ let val (x,y) = Thm.dest_binop l val z = r
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z]
+ val ynz = prove_nz ss T y
+ in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
+ end
+ | (_, Const (@{const_name Rings.divide},_)$_$_) =>
+ let val (x,y) = Thm.dest_binop r val z = l
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z]
+ val ynz = prove_nz ss T y
+ in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
+ end
+ | _ => NONE)
+ end
+ handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
+ | is_number t = can HOLogic.dest_number t
+
+ val is_number = is_number o term_of
+
+ fun proc3 phi ss ct =
+ (case term_of ct of
+ Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | _ => NONE)
+ handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
+
+val add_frac_frac_simproc =
+ make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
+ name = "add_frac_frac_simproc",
+ proc = proc, identifier = []}
+
+val add_frac_num_simproc =
+ make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
+ name = "add_frac_num_simproc",
+ proc = proc2, identifier = []}
+
+val ord_frac_simproc =
+ make_simproc
+ {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
+ @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
+ @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
+ @{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
+ @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
+ @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
+ name = "ord_frac_simproc", proc = proc3, identifier = []}
+
+val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
+ @{thm "divide_Numeral1"},
+ @{thm "divide_zero"}, @{thm "divide_Numeral0"},
+ @{thm "divide_divide_eq_left"},
+ @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
+ @{thm "times_divide_times_eq"},
+ @{thm "divide_divide_eq_right"},
+ @{thm "diff_def"}, @{thm "minus_divide_left"},
+ @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
+ @{thm field_divide_inverse} RS sym, @{thm inverse_divide},
+ Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))
+ (@{thm field_divide_inverse} RS sym)]
+
+in
+
+val field_comp_conv = (Simplifier.rewrite
+(HOL_basic_ss addsimps @{thms "semiring_norm"}
+ addsimps ths addsimps @{thms simp_thms}
+ addsimprocs field_cancel_numeral_factors
+ addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
+ ord_frac_simproc]
+ addcongs [@{thm "if_weak_cong"}]))
+then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
+ [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
+
+end
+
end;
Addsimprocs Numeral_Simprocs.cancel_numerals;