--- a/NEWS Thu May 06 11:08:19 2010 -0700
+++ b/NEWS Fri May 07 09:59:59 2010 +0200
@@ -140,6 +140,8 @@
*** HOL ***
+* Dropped theorem duplicate comp_arith; use semiring_norm instead. INCOMPATIBILITY.
+
* Theory 'Finite_Set': various folding_* locales facilitate the application
of the various fold combinators on finite sets.
--- a/src/HOL/Fields.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Fields.thy Fri May 07 09:59:59 2010 +0200
@@ -234,6 +234,18 @@
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
by (simp add: eq_commute [of 1])
+lemma times_divide_times_eq:
+ "(x / y) * (z / w) = (x * z) / (y * w)"
+ by simp
+
+lemma add_frac_num:
+ "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
+ by (simp add: add_divide_distrib)
+
+lemma add_num_frac:
+ "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
+ by (simp add: add_divide_distrib add.commute)
+
end
--- a/src/HOL/Groebner_Basis.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Groebner_Basis.thy Fri May 07 09:59:59 2010 +0200
@@ -5,20 +5,17 @@
header {* Semiring normalization and Groebner Bases *}
theory Groebner_Basis
-imports Numeral_Simprocs
+imports Numeral_Simprocs Nat_Transfer
uses
- "Tools/Groebner_Basis/misc.ML"
- "Tools/Groebner_Basis/normalizer_data.ML"
- ("Tools/Groebner_Basis/normalizer.ML")
+ "Tools/Groebner_Basis/normalizer.ML"
("Tools/Groebner_Basis/groebner.ML")
begin
subsection {* Semiring normalization *}
-setup NormalizerData.setup
+setup Normalizer.setup
-
-locale gb_semiring =
+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"
@@ -59,9 +56,6 @@
thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
qed
-
-subsubsection {* Declaring the abstract theory *}
-
lemma semiring_ops:
shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
and "TERM r0" and "TERM r1" .
@@ -156,71 +150,21 @@
qed
-lemmas gb_semiring_axioms' =
- gb_semiring_axioms [normalizer
+lemmas normalizing_semiring_axioms' =
+ normalizing_semiring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules]
end
-interpretation class_semiring: gb_semiring
- "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
- proof qed (auto simp add: algebra_simps)
-
-lemmas nat_arith =
- add_nat_number_of
- diff_nat_number_of
- mult_nat_number_of
- eq_nat_number_of
- less_nat_number_of
-
-lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
- by simp
-
-lemmas comp_arith =
- Let_def arith_simps nat_arith rel_simps neg_simps if_False
- if_True add_0 add_Suc add_number_of_left mult_number_of_left
- numeral_1_eq_1[symmetric] Suc_eq_plus1
- numeral_0_eq_0[symmetric] numerals[symmetric]
- iszero_simps not_iszero_Numeral1
-
-lemmas semiring_norm = comp_arith
-
-ML {*
-local
-
-open Conv;
+sublocale comm_semiring_1
+ < normalizing!: normalizing_semiring plus times power zero one
+proof
+qed (simp_all add: algebra_simps)
-fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
-
-fun int_of_rat x =
- (case Rat.quotient_of_rat x of (i, 1) => i
- | _ => error "int_of_rat: bad int");
-
-val numeral_conv =
- Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
- Simplifier.rewrite (HOL_basic_ss addsimps
- (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
-
-in
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
-fun normalizer_funs key =
- NormalizerData.funs key
- {is_const = fn phi => numeral_is_const,
- dest_const = fn phi => fn ct =>
- Rat.rat_of_int (snd
- (HOLogic.dest_number (Thm.term_of ct)
- handle TERM _ => error "ring_dest_const")),
- mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
- conv = fn phi => K numeral_conv}
-
-end
-*}
-
-declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
-
-
-locale gb_ring = gb_semiring +
+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"
@@ -231,8 +175,8 @@
lemmas ring_rules = neg_mul sub_add
-lemmas gb_ring_axioms' =
- gb_ring_axioms [normalizer
+lemmas normalizing_ring_axioms' =
+ normalizing_ring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
@@ -240,23 +184,14 @@
end
-
-interpretation class_ring: gb_ring "op +" "op *" "op ^"
- "0::'a::{comm_semiring_1,number_ring}" 1 "op -" "uminus"
- proof qed simp_all
-
-
-declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
+sublocale comm_ring_1
+ < normalizing!: normalizing_ring plus times power zero one minus uminus
+proof
+qed (simp_all add: diff_minus)
-use "Tools/Groebner_Basis/normalizer.ML"
-
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
-method_setup sring_norm = {*
- Scan.succeed (SIMPLE_METHOD' o Normalizer.semiring_normalize_tac)
-*} "semiring normalizer"
-
-
-locale gb_field = gb_ring +
+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)"
@@ -267,8 +202,8 @@
lemmas field_rules = divide_inverse inverse_divide
-lemmas gb_field_axioms' =
- gb_field_axioms [normalizer
+lemmas normalizing_field_axioms' =
+ normalizing_field_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
@@ -278,10 +213,7 @@
end
-
-subsection {* Groebner Bases *}
-
-locale semiringb = gb_semiring +
+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"
@@ -313,22 +245,23 @@
thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
qed
-declare gb_semiring_axioms' [normalizer del]
+declare normalizing_semiring_axioms' [normalizer del]
-lemmas semiringb_axioms' = semiringb_axioms [normalizer
- semiring ops: semiring_ops
- semiring rules: semiring_rules
- idom rules: noteq_reduce add_scale_eq_noteq]
+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 ringb = semiringb + gb_ring +
+locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
begin
-declare gb_ring_axioms' [normalizer del]
+declare normalizing_ring_axioms' [normalizer del]
-lemmas ringb_axioms' = ringb_axioms [normalizer
+lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
@@ -338,33 +271,24 @@
end
-
-lemma no_zero_divirors_neq0:
- assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
- and ab: "a*b = 0" shows "b = 0"
-proof -
- { assume bz: "b \<noteq> 0"
- from no_zero_divisors [OF az bz] ab have False by blast }
- thus "b = 0" by blast
-qed
+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)
-interpretation class_ringb: ringb
- "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
-proof(unfold_locales, simp add: algebra_simps, auto)
- fix w x y z ::"'a::{idom,number_ring}"
- assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
- hence ynz': "y - z \<noteq> 0" by simp
- from p have "w * y + x* z - w*z - x*y = 0" by simp
- hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
- hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
- with no_zero_divirors_neq0 [OF ynz']
- have "w - x = 0" by blast
- thus "w = x" by simp
-qed
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
-declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
-
-interpretation natgb: semiringb
+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"
@@ -386,14 +310,14 @@
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
qed
-declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
+declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
-locale fieldgb = ringb + gb_field
+locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
begin
-declare gb_field_axioms' [normalizer del]
+declare normalizing_field_axioms' [normalizer del]
-lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
+lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
@@ -405,8 +329,18 @@
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)
+
lemma dnf:
"(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
"(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
@@ -423,23 +357,16 @@
"P \<equiv> False \<Longrightarrow> \<not> P"
"\<not> P \<Longrightarrow> (P \<equiv> False)"
by auto
-use "Tools/Groebner_Basis/groebner.ML"
-method_setup algebra =
-{*
-let
- fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
- val addN = "add"
- val delN = "del"
- val any_keyword = keyword addN || keyword delN
- val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-in
- ((Scan.optional (keyword addN |-- thms) []) --
- (Scan.optional (keyword delN |-- thms) [])) >>
- (fn (add_ths, del_ths) => fn ctxt =>
- SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
-end
-*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
+ML {*
+structure Algebra_Simplification = Named_Thms(
+ val name = "algebra"
+ val description = "pre-simplification rules for algebraic methods"
+)
+*}
+
+setup Algebra_Simplification.setup
+
declare dvd_def[algebra]
declare dvd_eq_mod_eq_0[symmetric, algebra]
declare mod_div_trivial[algebra]
@@ -468,222 +395,9 @@
declare zmod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]
-subsection{* Groebner Bases for fields *}
-
-interpretation class_fieldgb:
- fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
-
-lemma divide_Numeral1: "(x::'a::{field, number_ring}) / Numeral1 = x" by simp
-lemma divide_Numeral0: "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
- by simp
-lemma mult_frac_frac: "((x::'a::field_inverse_zero) / y) * (z / w) = (x*z) / (y*w)"
- by simp
-lemma mult_frac_num: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y"
- by simp
-lemma mult_num_frac: "((x::'a::field_inverse_zero) / y) * z = (x*z) / y"
- by simp
-
-lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
-
-lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::field_inverse_zero) / y + z = (x + z*y) / y"
- by (simp add: add_divide_distrib)
-lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::field_inverse_zero) / y = (x + z*y) / y"
- by (simp add: add_divide_distrib)
-
-ML {*
-let open Conv
-in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute})))) (@{thm field_divide_inverse} RS sym)
-end
-*}
-
-ML{*
-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
+use "Tools/Groebner_Basis/groebner.ML"
- 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 \<le> ?c"},
- @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
- @{cpat "?c \<le> (?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 = []}
-
-local
-open Conv
-in
-
-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 "mult_frac_frac"},
- @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
- @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
- @{thm "times_divide_eq_left"}, @{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},
- fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))
- (@{thm field_divide_inverse} RS sym)]
-
-val comp_conv = (Simplifier.rewrite
-(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
- 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
-
-fun numeral_is_const ct =
- case term_of ct of
- Const (@{const_name Rings.divide},_) $ a $ b =>
- can HOLogic.dest_number a andalso can HOLogic.dest_number b
- | Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t
- | t => can HOLogic.dest_number t
-
-fun dest_const ct = ((case term_of ct of
- Const (@{const_name Rings.divide},_) $ a $ b=>
- Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
- | Const (@{const_name Rings.inverse},_)$t =>
- Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
- | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
- handle TERM _ => error "ring_dest_const")
-
-fun mk_const phi cT x =
- let val (a, b) = Rat.quotient_of_rat x
- in if b = 1 then Numeral.mk_cnumber cT a
- else Thm.capply
- (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
- (Numeral.mk_cnumber cT a))
- (Numeral.mk_cnumber cT b)
- end
-
-in
- val field_comp_conv = comp_conv;
- val fieldgb_declaration =
- NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
- {is_const = K numeral_is_const,
- dest_const = K dest_const,
- mk_const = mk_const,
- conv = K (K comp_conv)}
-end;
-*}
-
-declaration fieldgb_declaration
+method_setup algebra = Groebner.algebra_method
+ "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
end
--- a/src/HOL/Int.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Int.thy Fri May 07 09:59:59 2010 +0200
@@ -1063,20 +1063,24 @@
text {* First version by Norbert Voelker *}
-definition (*for simplifying equalities*)
- iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
-where
+definition (*for simplifying equalities*) iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" where
"iszero z \<longleftrightarrow> z = 0"
lemma iszero_0: "iszero 0"
-by (simp add: iszero_def)
-
-lemma not_iszero_1: "~ iszero 1"
-by (simp add: iszero_def eq_commute)
+ by (simp add: iszero_def)
+
+lemma iszero_Numeral0: "iszero (Numeral0 :: 'a::number_ring)"
+ by (simp add: iszero_0)
+
+lemma not_iszero_1: "\<not> iszero 1"
+ by (simp add: iszero_def)
+
+lemma not_iszero_Numeral1: "\<not> iszero (Numeral1 :: 'a::number_ring)"
+ by (simp add: not_iszero_1)
lemma eq_number_of_eq [simp]:
"((number_of x::'a::number_ring) = number_of y) =
- iszero (number_of (x + uminus y) :: 'a)"
+ iszero (number_of (x + uminus y) :: 'a)"
unfolding iszero_def number_of_add number_of_minus
by (simp add: algebra_simps)
@@ -2021,6 +2025,14 @@
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
+lemma divide_Numeral1:
+ "(x::'a::{field, number_ring}) / Numeral1 = x"
+ by simp
+
+lemma divide_Numeral0:
+ "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
+ by simp
+
subsection {* The divides relation *}
--- a/src/HOL/IsaMakefile Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/IsaMakefile Fri May 07 09:59:59 2010 +0200
@@ -284,9 +284,7 @@
Tools/ATP_Manager/atp_manager.ML \
Tools/ATP_Manager/atp_systems.ML \
Tools/Groebner_Basis/groebner.ML \
- Tools/Groebner_Basis/misc.ML \
Tools/Groebner_Basis/normalizer.ML \
- Tools/Groebner_Basis/normalizer_data.ML \
Tools/choice_specification.ML \
Tools/int_arith.ML \
Tools/list_code.ML \
--- a/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Fri May 07 09:59:59 2010 +0200
@@ -1195,7 +1195,7 @@
fun real_nonlinear_prover proof_method ctxt =
let
val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
@@ -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 field_comp_conv) th, RealArith.Trivial)
+ (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Normalizer.field_comp_conv) th, RealArith.Trivial)
end)
handle Failure _ =>
(let val proof =
@@ -1310,7 +1310,7 @@
fun real_nonlinear_subst_prover prover ctxt =
let
val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
--- a/src/HOL/Library/normarith.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Library/normarith.ML Fri May 07 09:59:59 2010 +0200
@@ -167,8 +167,8 @@
(* FIXME : Should be computed statically!! *)
val real_poly_conv =
Normalizer.semiring_normalize_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
- in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (field_comp_conv then_conv real_poly_conv)))
+ (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)))
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 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 field_comp_conv);
+ 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_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 field_comp_conv) else_conv
+ ((apply_pth5 then_conv arg1_conv Normalizer.field_comp_conv) else_conv
(apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
fun vector_add_conv ct = apply_pth7 ct
@@ -278,7 +278,7 @@
(* FIXME: Should be computed statically!!*)
val real_poly_conv =
Normalizer.semiring_normalize_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
val sources = map (Thm.dest_arg o Thm.dest_arg1 o concl) nubs
val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) []
val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
@@ -384,7 +384,7 @@
let
val real_poly_neg_conv = #neg
(Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
+ (the (Normalizer.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
val (th1,th2) = conj_pair(rawrule th)
in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc
end
@@ -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 field_comp_conv
+ then_conv Normalizer.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 Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Library/positivstellensatz.ML Fri May 07 09:59:59 2010 +0200
@@ -748,10 +748,10 @@
fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
val {add,mul,neg,pow,sub,main} =
Normalizer.semiring_normalizers_ord_wrapper ctxt
- (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
in gen_real_arith ctxt
- (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
+ (cterm_of_rat, Normalizer.field_comp_conv, Normalizer.field_comp_conv, Normalizer.field_comp_conv,
main,neg,add,mul, prover)
end;
--- a/src/HOL/Metis_Examples/BT.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Metis_Examples/BT.thy Fri May 07 09:59:59 2010 +0200
@@ -88,7 +88,7 @@
case Lf thus ?case by (metis reflect.simps(1))
next
case (Br a t1 t2) thus ?case
- by (metis class_semiring.semiring_rules(24) n_nodes.simps(2) reflect.simps(2))
+ by (metis normalizing.semiring_rules(24) n_nodes.simps(2) reflect.simps(2))
qed
declare [[ atp_problem_prefix = "BT__depth_reflect" ]]
--- a/src/HOL/Metis_Examples/BigO.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Metis_Examples/BigO.thy Fri May 07 09:59:59 2010 +0200
@@ -41,7 +41,7 @@
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_ge_zero)
- have F2: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis class_semiring.mul_1)
+ have F2: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis normalizing.mul_1)
have F3: "\<forall>x\<^isub>1 x\<^isub>3. x\<^isub>3 \<le> \<bar>h x\<^isub>1\<bar> \<longrightarrow> x\<^isub>3 \<le> c * \<bar>f x\<^isub>1\<bar>" by (metis A1 order_trans)
have F4: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
by (metis abs_mult)
@@ -70,7 +70,7 @@
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
- have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis class_semiring.mul_1)
+ have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis normalizing.mul_1)
have F2: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
by (metis abs_mult)
have "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_mult_pos abs_one)
@@ -92,7 +92,7 @@
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
- have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis class_semiring.mul_1)
+ have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis normalizing.mul_1)
have F2: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1" by (metis abs_mult_pos)
hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_one)
hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans)
@@ -111,7 +111,7 @@
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
- have "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis class_semiring.mul_1)
+ have "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis normalizing.mul_1)
hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>"
by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
@@ -145,12 +145,12 @@
declare [[ atp_problem_prefix = "BigO__bigo_refl" ]]
lemma bigo_refl [intro]: "f : O(f)"
apply (auto simp add: bigo_def)
-by (metis class_semiring.mul_1 order_refl)
+by (metis normalizing.mul_1 order_refl)
declare [[ atp_problem_prefix = "BigO__bigo_zero" ]]
lemma bigo_zero: "0 : O(g)"
apply (auto simp add: bigo_def func_zero)
-by (metis class_semiring.mul_0 order_refl)
+by (metis normalizing.mul_0 order_refl)
lemma bigo_zero2: "O(%x.0) = {%x.0}"
apply (auto simp add: bigo_def)
@@ -307,7 +307,7 @@
apply (auto simp add: diff_minus fun_Compl_def func_plus)
prefer 2
apply (drule_tac x = x in spec)+
- apply (metis add_right_mono class_semiring.semiring_rules(24) diff_add_cancel diff_minus_eq_add le_less order_trans)
+ apply (metis add_right_mono normalizing.semiring_rules(24) diff_add_cancel diff_minus_eq_add le_less order_trans)
proof -
fix x :: 'a
assume "\<forall>x. lb x \<le> f x"
@@ -318,13 +318,13 @@
lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
apply (unfold bigo_def)
apply auto
-by (metis class_semiring.mul_1 order_refl)
+by (metis normalizing.mul_1 order_refl)
declare [[ atp_problem_prefix = "BigO__bigo_abs2" ]]
lemma bigo_abs2: "f =o O(%x. abs(f x))"
apply (unfold bigo_def)
apply auto
-by (metis class_semiring.mul_1 order_refl)
+by (metis normalizing.mul_1 order_refl)
lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
proof -
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Fri May 07 09:59:59 2010 +0200
@@ -1877,7 +1877,7 @@
using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
apply (rule mult_left_le_imp_le[of "1 - u"])
- unfolding class_semiring.mul_a using `u<1` by auto
+ unfolding normalizing.mul_a using `u<1` by auto
thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
using as unfolding scaleR_scaleR by auto qed auto
thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
@@ -2231,7 +2231,7 @@
apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
fix z assume z:"z\<in>{x - ?d..x + ?d}"
have e:"e = setsum (\<lambda>i. d) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
- by (metis eq_divide_imp mult_frac_num real_dimindex_gt_0 real_eq_of_nat real_less_def real_mult_commute)
+ by (metis eq_divide_imp times_divide_eq_left real_dimindex_gt_0 real_eq_of_nat real_less_def real_mult_commute)
show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
using z[unfolded mem_interval] apply(erule_tac x=i in allE) by auto qed
hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
--- a/src/HOL/Multivariate_Analysis/Derivative.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Fri May 07 09:59:59 2010 +0200
@@ -698,7 +698,7 @@
unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
then guess x .. note x=this
show ?thesis proof(cases "f a = f b")
- case False have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add:class_semiring.semiring_rules)
+ case False have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add:normalizing.semiring_rules)
also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x unfolding inner_simps by auto
also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
@@ -810,7 +810,7 @@
guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
- also have "\<dots> \<le> e * norm(z - y)" unfolding mult_frac_num pos_divide_le_eq[OF `B>0`]
+ also have "\<dots> \<le> e * norm(z - y)" unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
--- a/src/HOL/Multivariate_Analysis/Integration.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Fri May 07 09:59:59 2010 +0200
@@ -2533,7 +2533,7 @@
show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
- note this[unfolded real_scaleR_def real_norm_def class_semiring.semiring_rules, of k c d] note le_less_trans[OF this d(2)]
+ note this[unfolded real_scaleR_def real_norm_def normalizing.semiring_rules, of k c d] note le_less_trans[OF this d(2)]
from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})) < e"
apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
@@ -4723,7 +4723,7 @@
have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
\<longrightarrow> norm(ig) < dia + e"
proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
- apply(subst real_sum_of_halves[of e,THEN sym]) unfolding class_semiring.add_a
+ apply(subst real_sum_of_halves[of e,THEN sym]) unfolding normalizing.add_a
apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
qed note norm=this[rule_format]
--- a/src/HOL/Nat_Numeral.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Nat_Numeral.thy Fri May 07 09:59:59 2010 +0200
@@ -319,6 +319,10 @@
lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
by (simp add: nat_number_of_def)
+lemma Numeral1_eq1_nat:
+ "(1::nat) = Numeral1"
+ by simp
+
lemma numeral_1_eq_Suc_0 [code_post]: "Numeral1 = Suc 0"
by (simp only: nat_numeral_1_eq_1 One_nat_def)
@@ -687,6 +691,20 @@
lemmas nat_number' =
nat_number_of_Bit0 nat_number_of_Bit1
+lemmas nat_arith =
+ add_nat_number_of
+ diff_nat_number_of
+ mult_nat_number_of
+ eq_nat_number_of
+ less_nat_number_of
+
+lemmas semiring_norm =
+ Let_def arith_simps nat_arith rel_simps neg_simps if_False
+ if_True add_0 add_Suc add_number_of_left mult_number_of_left
+ numeral_1_eq_1 [symmetric] Suc_eq_plus1
+ numeral_0_eq_0 [symmetric] numerals [symmetric]
+ iszero_simps not_iszero_Numeral1
+
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
by (fact Let_def)
--- a/src/HOL/Parity.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Parity.thy Fri May 07 09:59:59 2010 +0200
@@ -229,7 +229,7 @@
lemma zero_le_odd_power: "odd n ==>
(0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
-apply (metis field_power_not_zero no_zero_divirors_neq0 order_antisym_conv zero_le_square)
+apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
done
lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
--- a/src/HOL/Probability/Lebesgue.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Probability/Lebesgue.thy Fri May 07 09:59:59 2010 +0200
@@ -938,17 +938,17 @@
proof safe
fix t assume t: "t \<in> space M"
{ fix m n :: nat assume "m \<le> n"
- hence *: "(2::real)^n = 2^m * 2^(n - m)" unfolding class_semiring.mul_pwr by auto
+ hence *: "(2::real)^n = 2^m * 2^(n - m)" unfolding normalizing.mul_pwr by auto
have "real (natfloor (f t * 2^m) * natfloor (2^(n-m))) \<le> real (natfloor (f t * 2 ^ n))"
apply (subst *)
- apply (subst class_semiring.mul_a)
+ apply (subst normalizing.mul_a)
apply (subst real_of_nat_le_iff)
apply (rule le_mult_natfloor)
using nonneg[OF t] by (auto intro!: mult_nonneg_nonneg)
hence "real (natfloor (f t * 2^m)) * 2^n \<le> real (natfloor (f t * 2^n)) * 2^m"
apply (subst *)
- apply (subst (3) class_semiring.mul_c)
- apply (subst class_semiring.mul_a)
+ apply (subst (3) normalizing.mul_c)
+ apply (subst normalizing.mul_a)
by (auto intro: mult_right_mono simp: natfloor_power real_of_nat_power[symmetric]) }
thus "incseq (\<lambda>n. ?u n t)" unfolding u_at_t[OF t] unfolding incseq_def
by (auto simp add: le_divide_eq divide_le_eq less_divide_eq)
--- a/src/HOL/Rings.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Rings.thy Fri May 07 09:59:59 2010 +0200
@@ -183,9 +183,21 @@
end
-
class no_zero_divisors = zero + times +
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
+begin
+
+lemma divisors_zero:
+ assumes "a * b = 0"
+ shows "a = 0 \<or> b = 0"
+proof (rule classical)
+ assume "\<not> (a = 0 \<or> b = 0)"
+ then have "a \<noteq> 0" and "b \<noteq> 0" by auto
+ with no_zero_divisors have "a * b \<noteq> 0" by blast
+ with assms show ?thesis by simp
+qed
+
+end
class semiring_1_cancel = semiring + cancel_comm_monoid_add
+ zero_neq_one + monoid_mult
--- a/src/HOL/Tools/Groebner_Basis/groebner.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Tools/Groebner_Basis/groebner.ML Fri May 07 09:59:59 2010 +0200
@@ -9,19 +9,20 @@
vars: cterm list, semiring: cterm list * thm list, ideal : thm list} ->
(cterm -> Rat.rat) -> (Rat.rat -> cterm) ->
conv -> conv ->
- {ring_conv : conv,
- simple_ideal: (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list),
- multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
- poly_eq_ss: simpset, unwind_conv : conv}
- val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic
- val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic
- val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic
+ {ring_conv : conv,
+ simple_ideal: (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list),
+ multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
+ poly_eq_ss: simpset, unwind_conv : conv}
+ val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic
+ val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic
+ val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic
+ val algebra_method: (Proof.context -> Method.method) context_parser
end
structure Groebner : GROEBNER =
struct
-open Conv Normalizer Drule Thm;
+open Conv Drule Thm;
fun is_comb ct =
(case Thm.term_of ct of
@@ -50,11 +51,11 @@
val lcm_rat = fn x => fn y => Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
val (eqF_intr, eqF_elim) =
- let val [th1,th2] = thms "PFalse"
+ let val [th1,th2] = @{thms PFalse}
in (fn th => th COMP th2, fn th => th COMP th1) end;
val (PFalse, PFalse') =
- let val PFalse_eq = nth (thms "simp_thms") 13
+ let val PFalse_eq = nth @{thms simp_thms} 13
in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
@@ -398,7 +399,7 @@
compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
| _ => rfn tm ;
-val notnotD = @{thm "notnotD"};
+val notnotD = @{thm notnotD};
fun mk_binop ct x y = capply (capply ct x) y
val mk_comb = capply;
@@ -440,10 +441,10 @@
| _ => false;
val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
-val bool_simps = @{thms "bool_simps"};
-val nnf_simps = @{thms "nnf_simps"};
+val bool_simps = @{thms bool_simps};
+val nnf_simps = @{thms nnf_simps};
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
-val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms "weak_dnf_simps"});
+val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms weak_dnf_simps});
val initial_conv =
Simplifier.rewrite
(HOL_basic_ss addsimps nnf_simps
@@ -947,29 +948,31 @@
case try (find_term 0) form of
NONE => NONE
| SOME tm =>
- (case NormalizerData.match ctxt tm of
+ (case Normalizer.match ctxt tm of
NONE => NONE
| SOME (res as (theory, {is_const, dest_const,
mk_const, conv = ring_eq_conv})) =>
SOME (ring_and_ideal_conv theory
dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
- (semiring_normalize_wrapper ctxt res)))
+ (Normalizer.semiring_normalize_wrapper ctxt res)))
fun ring_solve ctxt form =
(case try (find_term 0 (* FIXME !? *)) form of
NONE => reflexive form
| SOME tm =>
- (case NormalizerData.match ctxt tm of
+ (case Normalizer.match ctxt tm of
NONE => reflexive form
| SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
#ring_conv (ring_and_ideal_conv theory
dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
- (semiring_normalize_wrapper ctxt res)) form));
+ (Normalizer.semiring_normalize_wrapper ctxt res)) form));
+
+fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
+ (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
fun ring_tac add_ths del_ths ctxt =
Object_Logic.full_atomize_tac
- THEN' asm_full_simp_tac
- (Simplifier.context ctxt (fst (NormalizerData.get ctxt)) delsimps del_ths addsimps add_ths)
+ THEN' presimplify ctxt add_ths del_ths
THEN' CSUBGOAL (fn (p, i) =>
rtac (let val form = Object_Logic.dest_judgment p
in case get_ring_ideal_convs ctxt form of
@@ -988,8 +991,7 @@
| exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
in
fun ideal_tac add_ths del_ths ctxt =
- asm_full_simp_tac
- (Simplifier.context ctxt (fst (NormalizerData.get ctxt)) delsimps del_ths addsimps add_ths)
+ presimplify ctxt add_ths del_ths
THEN'
CSUBGOAL (fn (p, i) =>
case get_ring_ideal_convs ctxt p of
@@ -1023,6 +1025,21 @@
fun algebra_tac add_ths del_ths ctxt i =
ring_tac add_ths del_ths ctxt i ORELSE ideal_tac add_ths del_ths ctxt i
-
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+val addN = "add"
+val delN = "del"
+val any_keyword = keyword addN || keyword delN
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+
+in
+
+val algebra_method = ((Scan.optional (keyword addN |-- thms) []) --
+ (Scan.optional (keyword delN |-- thms) [])) >>
+ (fn (add_ths, del_ths) => fn ctxt =>
+ SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
end;
+
+end;
--- a/src/HOL/Tools/Groebner_Basis/misc.ML Thu May 06 11:08:19 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,29 +0,0 @@
-(* Title: HOL/Tools/Groebner_Basis/misc.ML
- ID: $Id$
- Author: Amine Chaieb, TU Muenchen
-
-Very basic stuff for cterms.
-*)
-
-structure Misc =
-struct
-
-fun is_comb ct =
- (case Thm.term_of ct of
- _ $ _ => true
- | _ => false);
-
-val concl = Thm.cprop_of #> Thm.dest_arg;
-
-fun is_binop ct ct' =
- (case Thm.term_of ct' of
- c $ _ $ _ => term_of ct aconv c
- | _ => false);
-
-fun dest_binop ct ct' =
- if is_binop ct ct' then Thm.dest_binop ct'
- else raise CTERM ("dest_binop: bad binop", [ct, ct'])
-
-fun inst_thm inst = Thm.instantiate ([], inst);
-
-end;
--- a/src/HOL/Tools/Groebner_Basis/normalizer.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Tools/Groebner_Basis/normalizer.ML Fri May 07 09:59:59 2010 +0200
@@ -1,30 +1,376 @@
(* Title: HOL/Tools/Groebner_Basis/normalizer.ML
Author: Amine Chaieb, TU Muenchen
+
+Normalization of expressions in semirings.
*)
signature NORMALIZER =
sig
- val semiring_normalize_conv : Proof.context -> conv
- val semiring_normalize_ord_conv : Proof.context -> (cterm -> cterm -> bool) -> conv
- val semiring_normalize_tac : Proof.context -> int -> tactic
- val semiring_normalize_wrapper : Proof.context -> NormalizerData.entry -> conv
- val semiring_normalizers_ord_wrapper :
- Proof.context -> NormalizerData.entry -> (cterm -> cterm -> bool) ->
+ type entry
+ val get: Proof.context -> (thm * entry) list
+ val match: Proof.context -> cterm -> entry option
+ val del: attribute
+ val add: {semiring: cterm list * thm list, ring: cterm list * thm list,
+ field: cterm list * thm list, idom: thm list, ideal: thm list} -> attribute
+ val funs: thm -> {is_const: morphism -> cterm -> bool,
+ dest_const: morphism -> cterm -> Rat.rat,
+ mk_const: morphism -> ctyp -> Rat.rat -> cterm,
+ conv: morphism -> Proof.context -> cterm -> thm} -> declaration
+ val semiring_funs: thm -> declaration
+ val field_funs: thm -> declaration
+
+ val semiring_normalize_conv: Proof.context -> conv
+ val semiring_normalize_ord_conv: Proof.context -> (cterm -> cterm -> bool) -> conv
+ val semiring_normalize_wrapper: Proof.context -> entry -> conv
+ val semiring_normalize_ord_wrapper: Proof.context -> entry
+ -> (cterm -> cterm -> bool) -> conv
+ val semiring_normalizers_conv: cterm list -> cterm list * thm list
+ -> cterm list * thm list -> cterm list * thm list ->
+ (cterm -> bool) * conv * conv * conv -> (cterm -> cterm -> bool) ->
+ {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
+ val semiring_normalizers_ord_wrapper: Proof.context -> entry ->
+ (cterm -> cterm -> bool) ->
{add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
- val semiring_normalize_ord_wrapper : Proof.context -> NormalizerData.entry ->
- (cterm -> cterm -> bool) -> conv
- val semiring_normalizers_conv :
- cterm list -> cterm list * thm list -> cterm list * thm list -> cterm list * thm list ->
- (cterm -> bool) * conv * conv * conv -> (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
structure Normalizer: NORMALIZER =
struct
-open Conv;
+(** 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 **)
-(* Very basic stuff for terms *)
+type entry =
+ {vars: cterm list,
+ semiring: cterm list * thm list,
+ ring: cterm list * thm list,
+ field: cterm list * thm list,
+ idom: thm list,
+ ideal: thm list} *
+ {is_const: cterm -> bool,
+ dest_const: cterm -> Rat.rat,
+ mk_const: ctyp -> Rat.rat -> cterm,
+ conv: Proof.context -> cterm -> thm};
+
+structure Data = Generic_Data
+(
+ type T = (thm * entry) list;
+ val empty = [];
+ val extend = I;
+ val merge = AList.merge Thm.eq_thm (K true);
+);
+
+val get = Data.get o Context.Proof;
+
+fun match ctxt tm =
+ let
+ fun match_inst
+ ({vars, semiring = (sr_ops, sr_rules),
+ ring = (r_ops, r_rules), field = (f_ops, f_rules), idom, ideal},
+ fns as {is_const, dest_const, mk_const, conv}) pat =
+ let
+ fun h instT =
+ let
+ val substT = Thm.instantiate (instT, []);
+ val substT_cterm = Drule.cterm_rule substT;
+
+ val vars' = map substT_cterm vars;
+ val semiring' = (map substT_cterm sr_ops, map substT sr_rules);
+ val ring' = (map substT_cterm r_ops, map substT r_rules);
+ val field' = (map substT_cterm f_ops, map substT f_rules);
+ val idom' = map substT idom;
+ val ideal' = map substT ideal;
+
+ val result = ({vars = vars', semiring = semiring',
+ ring = ring', field = field', idom = idom', ideal = ideal'}, fns);
+ in SOME result end
+ in (case try Thm.match (pat, tm) of
+ NONE => NONE
+ | SOME (instT, _) => h instT)
+ end;
+
+ fun match_struct (_,
+ entry as ({semiring = (sr_ops, _), ring = (r_ops, _), field = (f_ops, _), ...}, _): entry) =
+ get_first (match_inst entry) (sr_ops @ r_ops @ f_ops);
+ in get_first match_struct (get ctxt) end;
+
+
+(* logical content *)
+
+val semiringN = "semiring";
+val ringN = "ring";
+val idomN = "idom";
+val idealN = "ideal";
+val fieldN = "field";
+
+fun undefined _ = raise Match;
+
+val del = Thm.declaration_attribute (Data.map o AList.delete Thm.eq_thm);
+
+fun add {semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules),
+ field = (f_ops, f_rules), idom, ideal} =
+ Thm.declaration_attribute (fn key => fn context => context |> Data.map
+ let
+ val ctxt = Context.proof_of context;
+
+ fun check kind name xs n =
+ null xs orelse length xs = n orelse
+ error ("Expected " ^ string_of_int n ^ " " ^ kind ^ " for " ^ name);
+ val check_ops = check "operations";
+ val check_rules = check "rules";
+
+ val _ =
+ check_ops semiringN sr_ops 5 andalso
+ check_rules semiringN sr_rules 37 andalso
+ check_ops ringN r_ops 2 andalso
+ check_rules ringN r_rules 2 andalso
+ check_ops fieldN f_ops 2 andalso
+ check_rules fieldN f_rules 2 andalso
+ check_rules idomN idom 2;
+
+ val mk_meta = Local_Defs.meta_rewrite_rule ctxt;
+ val sr_rules' = map mk_meta sr_rules;
+ val r_rules' = map mk_meta r_rules;
+ val f_rules' = map mk_meta f_rules;
+
+ fun rule i = nth sr_rules' (i - 1);
+
+ val (cx, cy) = Thm.dest_binop (hd sr_ops);
+ val cz = rule 34 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
+ val cn = rule 36 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
+ val ((clx, crx), (cly, cry)) =
+ rule 13 |> Thm.rhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
+ val ((ca, cb), (cc, cd)) =
+ rule 20 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
+ val cm = rule 1 |> Thm.rhs_of |> Thm.dest_arg;
+ val (cp, cq) = rule 26 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_arg;
+
+ val vars = [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry];
+ val semiring = (sr_ops, sr_rules');
+ val ring = (r_ops, r_rules');
+ val field = (f_ops, f_rules');
+ val ideal' = map (symmetric o mk_meta) ideal
+ in
+ AList.delete Thm.eq_thm key #>
+ cons (key, ({vars = vars, semiring = semiring,
+ ring = ring, field = field, idom = idom, ideal = ideal'},
+ {is_const = undefined, dest_const = undefined, mk_const = undefined,
+ conv = undefined}))
+ end);
+
+
+(* extra-logical functions *)
+
+fun funs raw_key {is_const, dest_const, mk_const, conv} phi =
+ Data.map (fn data =>
+ let
+ val key = Morphism.thm phi raw_key;
+ val _ = AList.defined Thm.eq_thm data key orelse
+ raise THM ("No data entry for structure key", 0, [key]);
+ val fns = {is_const = is_const phi, dest_const = dest_const phi,
+ mk_const = mk_const phi, conv = conv phi};
+ in AList.map_entry Thm.eq_thm key (apsnd (K fns)) data end);
+
+fun semiring_funs key = funs key
+ {is_const = fn phi => can HOLogic.dest_number o Thm.term_of,
+ dest_const = fn phi => fn ct =>
+ Rat.rat_of_int (snd
+ (HOLogic.dest_number (Thm.term_of ct)
+ handle TERM _ => error "ring_dest_const")),
+ mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT
+ (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int"),
+ conv = fn phi => fn _ => Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm})
+ then_conv Simplifier.rewrite (HOL_basic_ss addsimps
+ (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}))};
+
+fun field_funs key =
+ let
+ fun numeral_is_const ct =
+ case term_of ct of
+ Const (@{const_name Rings.divide},_) $ a $ b =>
+ can HOLogic.dest_number a andalso can HOLogic.dest_number b
+ | Const (@{const_name Rings.inverse},_)$t => can HOLogic.dest_number t
+ | t => can HOLogic.dest_number t
+ fun dest_const ct = ((case term_of ct of
+ Const (@{const_name Rings.divide},_) $ a $ b=>
+ Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | Const (@{const_name Rings.inverse},_)$t =>
+ Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
+ handle TERM _ => error "ring_dest_const")
+ fun mk_const phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Numeral.mk_cnumber cT a
+ else Thm.capply
+ (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+ (Numeral.mk_cnumber cT a))
+ (Numeral.mk_cnumber cT b)
+ end
+ in funs key
+ {is_const = K numeral_is_const,
+ dest_const = K dest_const,
+ mk_const = mk_const,
+ conv = K (K field_comp_conv)}
+ end;
+
+
+
+(** auxiliary **)
fun is_comb ct =
(case Thm.term_of ct of
@@ -55,6 +401,7 @@
val natarith = [@{thm "add_nat_number_of"}, @{thm "diff_nat_number_of"},
@{thm "mult_nat_number_of"}, @{thm "eq_nat_number_of"},
@{thm "less_nat_number_of"}];
+
val nat_add_conv =
zerone_conv
(Simplifier.rewrite
@@ -64,13 +411,15 @@
@{thm add_number_of_left}, @{thm Suc_eq_plus1}]
@ map (fn th => th RS sym) @{thms numerals}));
-val nat_mul_conv = nat_add_conv;
val zeron_tm = @{cterm "0::nat"};
val onen_tm = @{cterm "1::nat"};
val true_tm = @{cterm "True"};
-(* The main function! *)
+(** normalizing conversions **)
+
+(* core conversion *)
+
fun semiring_normalizers_conv vars (sr_ops, sr_rules) (r_ops, r_rules) (f_ops, f_rules)
(is_semiring_constant, semiring_add_conv, semiring_mul_conv, semiring_pow_conv) =
let
@@ -182,7 +531,7 @@
then
let val th1 = inst_thm [(cx,l),(cp,r),(cq,ntm)] pthm_34
val (l,r) = Thm.dest_comb(concl th1)
- in transitive th1 (Drule.arg_cong_rule l (nat_mul_conv r))
+ in transitive th1 (Drule.arg_cong_rule l (nat_add_conv r))
end
else
if opr aconvc mul_tm
@@ -563,7 +912,7 @@
let val (l,r) = Thm.dest_comb tm in
if not (l aconvc neg_tm) then raise CTERM ("polynomial_neg_conv",[tm]) else
let val th1 = inst_thm [(cx',r)] neg_mul
- val th2 = transitive th1 (arg1_conv semiring_mul_conv (concl th1))
+ val th2 = transitive th1 (Conv.arg1_conv semiring_mul_conv (concl th1))
in transitive th2 (polynomial_monomial_mul_conv (concl th2))
end
end;
@@ -606,7 +955,7 @@
then
let val th1 = combination (Drule.arg_cong_rule opr (polynomial_conv l))
(polynomial_conv r)
- val th2 = (rewr_conv divide_inverse then_conv polynomial_mul_conv)
+ val th2 = (Conv.rewr_conv divide_inverse then_conv polynomial_mul_conv)
(Thm.rhs_of th1)
in transitive th1 th2
end
@@ -638,11 +987,14 @@
fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS;
+
+(* various normalizing conversions *)
+
fun semiring_normalizers_ord_wrapper ctxt ({vars, semiring, ring, field, idom, ideal},
{conv, dest_const, mk_const, is_const}) ord =
let
val pow_conv =
- arg_conv (Simplifier.rewrite nat_exp_ss)
+ Conv.arg_conv (Simplifier.rewrite nat_exp_ss)
then_conv Simplifier.rewrite
(HOL_basic_ss addsimps [nth (snd semiring) 31, nth (snd semiring) 34])
then_conv conv ctxt
@@ -656,14 +1008,57 @@
semiring_normalize_ord_wrapper ctxt data simple_cterm_ord;
fun semiring_normalize_ord_conv ctxt ord tm =
- (case NormalizerData.match ctxt tm of
+ (case match ctxt tm of
NONE => reflexive tm
| SOME res => semiring_normalize_ord_wrapper ctxt res ord tm);
-
fun semiring_normalize_conv ctxt = semiring_normalize_ord_conv ctxt simple_cterm_ord;
-fun semiring_normalize_tac ctxt = SUBGOAL (fn (goal, i) =>
- rtac (semiring_normalize_conv ctxt
- (cterm_of (ProofContext.theory_of ctxt) (fst (Logic.dest_equals goal)))) i);
+
+(** Isar setup **)
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
+fun keyword2 k1 k2 = Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.colon) >> K ();
+fun keyword3 k1 k2 k3 =
+ Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.$$$ k3 -- Args.colon) >> K ();
+
+val opsN = "ops";
+val rulesN = "rules";
+
+val normN = "norm";
+val constN = "const";
+val delN = "del";
+
+val any_keyword =
+ keyword2 semiringN opsN || keyword2 semiringN rulesN ||
+ keyword2 ringN opsN || keyword2 ringN rulesN ||
+ keyword2 fieldN opsN || keyword2 fieldN rulesN ||
+ keyword2 idomN rulesN || keyword2 idealN rulesN;
+
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+val terms = thms >> map Drule.dest_term;
+
+fun optional scan = Scan.optional scan [];
+
+in
+
+val setup =
+ Attrib.setup @{binding normalizer}
+ (Scan.lift (Args.$$$ delN >> K del) ||
+ ((keyword2 semiringN opsN |-- terms) --
+ (keyword2 semiringN rulesN |-- thms)) --
+ (optional (keyword2 ringN opsN |-- terms) --
+ optional (keyword2 ringN rulesN |-- thms)) --
+ (optional (keyword2 fieldN opsN |-- terms) --
+ optional (keyword2 fieldN rulesN |-- thms)) --
+ optional (keyword2 idomN rulesN |-- thms) --
+ optional (keyword2 idealN rulesN |-- thms)
+ >> (fn ((((sr, r), f), id), idl) =>
+ add {semiring = sr, ring = r, field = f, idom = id, ideal = idl}))
+ "semiring normalizer data";
+
end;
+
+end;
--- a/src/HOL/Tools/Groebner_Basis/normalizer_data.ML Thu May 06 11:08:19 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,227 +0,0 @@
-(* Title: HOL/Tools/Groebner_Basis/normalizer_data.ML
- ID: $Id$
- Author: Amine Chaieb, TU Muenchen
-
-Ring normalization data.
-*)
-
-signature NORMALIZER_DATA =
-sig
- type entry
- val get: Proof.context -> simpset * (thm * entry) list
- val match: Proof.context -> cterm -> entry option
- val del: attribute
- val add: {semiring: cterm list * thm list, ring: cterm list * thm list, field: cterm list * thm list, idom: thm list, ideal: thm list}
- -> attribute
- val funs: thm -> {is_const: morphism -> cterm -> bool,
- dest_const: morphism -> cterm -> Rat.rat,
- mk_const: morphism -> ctyp -> Rat.rat -> cterm,
- conv: morphism -> Proof.context -> cterm -> thm} -> declaration
- val setup: theory -> theory
-end;
-
-structure NormalizerData: NORMALIZER_DATA =
-struct
-
-(* data *)
-
-type entry =
- {vars: cterm list,
- semiring: cterm list * thm list,
- ring: cterm list * thm list,
- field: cterm list * thm list,
- idom: thm list,
- ideal: thm list} *
- {is_const: cterm -> bool,
- dest_const: cterm -> Rat.rat,
- mk_const: ctyp -> Rat.rat -> cterm,
- conv: Proof.context -> cterm -> thm};
-
-val eq_key = Thm.eq_thm;
-fun eq_data arg = eq_fst eq_key arg;
-
-structure Data = Generic_Data
-(
- type T = simpset * (thm * entry) list;
- val empty = (HOL_basic_ss, []);
- val extend = I;
- fun merge ((ss, e), (ss', e')) : T =
- (merge_ss (ss, ss'), AList.merge eq_key (K true) (e, e'));
-);
-
-val get = Data.get o Context.Proof;
-
-
-(* match data *)
-
-fun match ctxt tm =
- let
- fun match_inst
- ({vars, semiring = (sr_ops, sr_rules),
- ring = (r_ops, r_rules), field = (f_ops, f_rules), idom, ideal},
- fns as {is_const, dest_const, mk_const, conv}) pat =
- let
- fun h instT =
- let
- val substT = Thm.instantiate (instT, []);
- val substT_cterm = Drule.cterm_rule substT;
-
- val vars' = map substT_cterm vars;
- val semiring' = (map substT_cterm sr_ops, map substT sr_rules);
- val ring' = (map substT_cterm r_ops, map substT r_rules);
- val field' = (map substT_cterm f_ops, map substT f_rules);
- val idom' = map substT idom;
- val ideal' = map substT ideal;
-
- val result = ({vars = vars', semiring = semiring',
- ring = ring', field = field', idom = idom', ideal = ideal'}, fns);
- in SOME result end
- in (case try Thm.match (pat, tm) of
- NONE => NONE
- | SOME (instT, _) => h instT)
- end;
-
- fun match_struct (_,
- entry as ({semiring = (sr_ops, _), ring = (r_ops, _), field = (f_ops, _), ...}, _): entry) =
- get_first (match_inst entry) (sr_ops @ r_ops @ f_ops);
- in get_first match_struct (snd (get ctxt)) end;
-
-
-(* logical content *)
-
-val semiringN = "semiring";
-val ringN = "ring";
-val idomN = "idom";
-val idealN = "ideal";
-val fieldN = "field";
-
-fun undefined _ = raise Match;
-
-fun del_data key = apsnd (remove eq_data (key, []));
-
-val del = Thm.declaration_attribute (Data.map o del_data);
-val add_ss = Thm.declaration_attribute
- (fn th => Data.map (fn (ss,data) => (ss addsimps [th], data)));
-
-val del_ss = Thm.declaration_attribute
- (fn th => Data.map (fn (ss,data) => (ss delsimps [th], data)));
-
-fun add {semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules),
- field = (f_ops, f_rules), idom, ideal} =
- Thm.declaration_attribute (fn key => fn context => context |> Data.map
- let
- val ctxt = Context.proof_of context;
-
- fun check kind name xs n =
- null xs orelse length xs = n orelse
- error ("Expected " ^ string_of_int n ^ " " ^ kind ^ " for " ^ name);
- val check_ops = check "operations";
- val check_rules = check "rules";
-
- val _ =
- check_ops semiringN sr_ops 5 andalso
- check_rules semiringN sr_rules 37 andalso
- check_ops ringN r_ops 2 andalso
- check_rules ringN r_rules 2 andalso
- check_ops fieldN f_ops 2 andalso
- check_rules fieldN f_rules 2 andalso
- check_rules idomN idom 2;
-
- val mk_meta = Local_Defs.meta_rewrite_rule ctxt;
- val sr_rules' = map mk_meta sr_rules;
- val r_rules' = map mk_meta r_rules;
- val f_rules' = map mk_meta f_rules;
-
- fun rule i = nth sr_rules' (i - 1);
-
- val (cx, cy) = Thm.dest_binop (hd sr_ops);
- val cz = rule 34 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
- val cn = rule 36 |> Thm.rhs_of |> Thm.dest_arg |> Thm.dest_arg;
- val ((clx, crx), (cly, cry)) =
- rule 13 |> Thm.rhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
- val ((ca, cb), (cc, cd)) =
- rule 20 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_binop;
- val cm = rule 1 |> Thm.rhs_of |> Thm.dest_arg;
- val (cp, cq) = rule 26 |> Thm.lhs_of |> Thm.dest_binop |> pairself Thm.dest_arg;
-
- val vars = [ca, cb, cc, cd, cm, cn, cp, cq, cx, cy, cz, clx, crx, cly, cry];
- val semiring = (sr_ops, sr_rules');
- val ring = (r_ops, r_rules');
- val field = (f_ops, f_rules');
- val ideal' = map (symmetric o mk_meta) ideal
- in
- del_data key #>
- apsnd (cons (key, ({vars = vars, semiring = semiring,
- ring = ring, field = field, idom = idom, ideal = ideal'},
- {is_const = undefined, dest_const = undefined, mk_const = undefined,
- conv = undefined})))
- end);
-
-
-(* extra-logical functions *)
-
-fun funs raw_key {is_const, dest_const, mk_const, conv} phi =
- (Data.map o apsnd) (fn data =>
- let
- val key = Morphism.thm phi raw_key;
- val _ = AList.defined eq_key data key orelse
- raise THM ("No data entry for structure key", 0, [key]);
- val fns = {is_const = is_const phi, dest_const = dest_const phi,
- mk_const = mk_const phi, conv = conv phi};
- in AList.map_entry eq_key key (apsnd (K fns)) data end);
-
-
-(* concrete syntax *)
-
-local
-
-fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
-fun keyword2 k1 k2 = Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.colon) >> K ();
-fun keyword3 k1 k2 k3 =
- Scan.lift (Args.$$$ k1 -- Args.$$$ k2 -- Args.$$$ k3 -- Args.colon) >> K ();
-
-val opsN = "ops";
-val rulesN = "rules";
-
-val normN = "norm";
-val constN = "const";
-val delN = "del";
-
-val any_keyword =
- keyword2 semiringN opsN || keyword2 semiringN rulesN ||
- keyword2 ringN opsN || keyword2 ringN rulesN ||
- keyword2 fieldN opsN || keyword2 fieldN rulesN ||
- keyword2 idomN rulesN || keyword2 idealN rulesN;
-
-val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-val terms = thms >> map Drule.dest_term;
-
-fun optional scan = Scan.optional scan [];
-
-in
-
-val normalizer_setup =
- Attrib.setup @{binding normalizer}
- (Scan.lift (Args.$$$ delN >> K del) ||
- ((keyword2 semiringN opsN |-- terms) --
- (keyword2 semiringN rulesN |-- thms)) --
- (optional (keyword2 ringN opsN |-- terms) --
- optional (keyword2 ringN rulesN |-- thms)) --
- (optional (keyword2 fieldN opsN |-- terms) --
- optional (keyword2 fieldN rulesN |-- thms)) --
- optional (keyword2 idomN rulesN |-- thms) --
- optional (keyword2 idealN rulesN |-- thms)
- >> (fn ((((sr, r), f), id), idl) =>
- add {semiring = sr, ring = r, field = f, idom = id, ideal = idl}))
- "semiring normalizer data";
-
-end;
-
-
-(* theory setup *)
-
-val setup =
- normalizer_setup #>
- Attrib.setup @{binding algebra} (Attrib.add_del add_ss del_ss) "pre-simplification for algebra";
-
-end;
--- a/src/HOL/Tools/Qelim/cooper.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Tools/Qelim/cooper.ML Fri May 07 09:59:59 2010 +0200
@@ -3,7 +3,7 @@
*)
signature COOPER =
- sig
+sig
val cooper_conv : Proof.context -> conv
exception COOPER of string * exn
end;
@@ -12,7 +12,6 @@
struct
open Conv;
-open Normalizer;
exception COOPER of string * exn;
fun simp_thms_conv ctxt =
--- a/src/HOL/Tools/Qelim/cooper_data.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Tools/Qelim/cooper_data.ML Fri May 07 09:59:59 2010 +0200
@@ -1,5 +1,4 @@
(* Title: HOL/Tools/Qelim/cooper_data.ML
- ID: $Id$
Author: Amine Chaieb, TU Muenchen
*)
@@ -16,8 +15,7 @@
struct
type entry = simpset * (term list);
-val start_ss = HOL_ss (* addsimps @{thms "Groebner_Basis.comp_arith"}
- addcongs [if_weak_cong, @{thm "let_weak_cong"}];*)
+
val allowed_consts =
[@{term "op + :: int => _"}, @{term "op + :: nat => _"},
@{term "op - :: int => _"}, @{term "op - :: nat => _"},
@@ -47,7 +45,7 @@
structure Data = Generic_Data
(
type T = simpset * term list;
- val empty = (start_ss, allowed_consts);
+ val empty = (HOL_ss, allowed_consts);
val extend = I;
fun merge ((ss1, ts1), (ss2, ts2)) =
(merge_ss (ss1, ss2), Library.merge (op aconv) (ts1, ts2));
@@ -64,7 +62,7 @@
(ss delsimps [th], subtract (op aconv) ts' ts )))
-(* concrete syntax *)
+(* theory setup *)
local
@@ -79,16 +77,11 @@
in
-val presburger_setup =
+val setup =
Attrib.setup @{binding presburger}
((Scan.lift (Args.$$$ "del") |-- optional (keyword constsN |-- terms)) >> del ||
optional (keyword constsN |-- terms) >> add) "Cooper data";
end;
-
-(* theory setup *)
-
-val setup = presburger_setup;
-
end;
--- a/src/HOL/Tools/Qelim/presburger.ML Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/Tools/Qelim/presburger.ML Fri May 07 09:59:59 2010 +0200
@@ -11,7 +11,7 @@
struct
open Conv;
-val comp_ss = HOL_ss addsimps @{thms "Groebner_Basis.comp_arith"};
+val comp_ss = HOL_ss addsimps @{thms semiring_norm};
fun strip_objimp ct =
(case Thm.term_of ct of
--- a/src/HOL/ex/Groebner_Examples.thy Thu May 06 11:08:19 2010 -0700
+++ b/src/HOL/ex/Groebner_Examples.thy Fri May 07 09:59:59 2010 +0200
@@ -10,18 +10,30 @@
subsection {* Basic examples *}
-schematic_lemma "3 ^ 3 == (?X::'a::{number_ring})"
- by sring_norm
+lemma
+ fixes x :: int
+ shows "x ^ 3 = x ^ 3"
+ apply (tactic {* ALLGOALS (CONVERSION
+ (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_conv @{context})))) *})
+ by (rule refl)
-schematic_lemma "(x - (-2))^5 == ?X::int"
- by sring_norm
+lemma
+ fixes x :: int
+ shows "(x - (-2))^5 = x ^ 5 + (10 * x ^ 4 + (40 * x ^ 3 + (80 * x\<twosuperior> + (80 * x + 32))))"
+ apply (tactic {* ALLGOALS (CONVERSION
+ (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_conv @{context})))) *})
+ by (rule refl)
-schematic_lemma "(x - (-2))^5 * (y - 78) ^ 8 == ?X::int"
- by sring_norm
+schematic_lemma
+ fixes x :: int
+ shows "(x - (-2))^5 * (y - 78) ^ 8 = ?X"
+ apply (tactic {* ALLGOALS (CONVERSION
+ (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_conv @{context})))) *})
+ by (rule refl)
lemma "((-3) ^ (Suc (Suc (Suc 0)))) == (X::'a::{number_ring})"
apply (simp only: power_Suc power_0)
- apply (simp only: comp_arith)
+ apply (simp only: semiring_norm)
oops
lemma "((x::int) + y)^3 - 1 = (x - z)^2 - 10 \<Longrightarrow> x = z + 3 \<Longrightarrow> x = - y"