--- a/Admin/isatest/isatest-stats Sat Mar 17 12:26:19 2012 +0100
+++ b/Admin/isatest/isatest-stats Sat Mar 17 12:52:40 2012 +0100
@@ -32,7 +32,7 @@
HOL-Isar_Examples
HOL-Lattice
HOL-Library-Codegenerator_Test
- HOL-Matrix
+ HOL-Matrix_LP
HOL-Metis_Examples
HOL-MicroJava
HOL-Mirabelle
--- a/src/HOL/IsaMakefile Sat Mar 17 12:26:19 2012 +0100
+++ b/src/HOL/IsaMakefile Sat Mar 17 12:52:40 2012 +0100
@@ -52,7 +52,7 @@
HOL-Isar_Examples \
HOL-Lattice \
HOL-Library-Codegenerator_Test \
- HOL-Matrix \
+ HOL-Matrix_LP \
HOL-Metis_Examples \
HOL-MicroJava \
HOL-Mirabelle \
@@ -1172,22 +1172,26 @@
@$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL SET_Protocol
-## HOL-Matrix
+## HOL-Matrix_LP
-HOL-Matrix: HOL $(LOG)/HOL-Matrix.gz
+HOL-Matrix_LP: HOL $(LOG)/HOL-Matrix_LP.gz
-$(LOG)/HOL-Matrix.gz: $(OUT)/HOL Matrix/ComputeFloat.thy \
- Matrix/ComputeHOL.thy Matrix/ComputeNumeral.thy \
- Matrix/Compute_Oracle/Compute_Oracle.thy Matrix/Compute_Oracle/am.ML \
- Matrix/Compute_Oracle/am_compiler.ML Matrix/Compute_Oracle/am_ghc.ML \
- Matrix/Compute_Oracle/am_interpreter.ML \
- Matrix/Compute_Oracle/am_sml.ML Matrix/Compute_Oracle/compute.ML \
- Matrix/Compute_Oracle/linker.ML Matrix/Cplex.thy \
- Matrix/CplexMatrixConverter.ML Matrix/Cplex_tools.ML \
- Matrix/FloatSparseMatrixBuilder.ML Matrix/LP.thy Matrix/Matrix.thy \
- Matrix/ROOT.ML Matrix/SparseMatrix.thy Matrix/document/root.tex \
- Matrix/fspmlp.ML Matrix/matrixlp.ML Tools/float_arith.ML
- @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Matrix
+$(LOG)/HOL-Matrix_LP.gz: $(OUT)/HOL Matrix_LP/ComputeFloat.thy \
+ Matrix_LP/ComputeHOL.thy Matrix_LP/ComputeNumeral.thy \
+ Matrix_LP/Compute_Oracle/Compute_Oracle.thy \
+ Matrix_LP/Compute_Oracle/am.ML \
+ Matrix_LP/Compute_Oracle/am_compiler.ML \
+ Matrix_LP/Compute_Oracle/am_ghc.ML \
+ Matrix_LP/Compute_Oracle/am_interpreter.ML \
+ Matrix_LP/Compute_Oracle/am_sml.ML \
+ Matrix_LP/Compute_Oracle/compute.ML \
+ Matrix_LP/Compute_Oracle/linker.ML Matrix_LP/Cplex.thy \
+ Matrix_LP/CplexMatrixConverter.ML Matrix_LP/Cplex_tools.ML \
+ Matrix_LP/FloatSparseMatrixBuilder.ML Matrix_LP/LP.thy \
+ Matrix_LP/Matrix.thy Matrix_LP/ROOT.ML Matrix_LP/SparseMatrix.thy \
+ Matrix_LP/document/root.tex Matrix_LP/fspmlp.ML \
+ Matrix_LP/matrixlp.ML Tools/float_arith.ML
+ @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Matrix_LP
## TLA
@@ -1901,9 +1905,9 @@
$(LOG)/HOL-Lattice $(LOG)/HOL-Lattice.gz \
$(LOG)/HOL-Lex.gz $(LOG)/HOL-Library.gz \
$(LOG)/HOL-Library-Codegenerator_Test.gz \
- $(LOG)/HOL-Main.gz $(LOG)/HOL-Matrix \
- $(LOG)/HOL-Matrix.gz $(LOG)/HOL-Metis_Examples.gz \
- $(LOG)/HOL-MicroJava.gz $(LOG)/HOL-Mirabelle.gz \
+ $(LOG)/HOL-Main.gz $(LOG)/HOL-Matrix_LP.gz \
+ $(LOG)/HOL-Metis_Examples.gz $(LOG)/HOL-MicroJava.gz \
+ $(LOG)/HOL-Mirabelle.gz \
$(LOG)/HOL-Multivariate_Analysis.gz \
$(LOG)/HOL-Mutabelle.gz $(LOG)/HOL-NSA-Examples.gz \
$(LOG)/HOL-NSA.gz $(LOG)/HOL-NanoJava.gz \
--- a/src/HOL/Matrix/ComputeFloat.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,309 +0,0 @@
-(* Title: HOL/Matrix/ComputeFloat.thy
- Author: Steven Obua
-*)
-
-header {* Floating Point Representation of the Reals *}
-
-theory ComputeFloat
-imports Complex_Main "~~/src/HOL/Library/Lattice_Algebras"
-uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML")
-begin
-
-definition int_of_real :: "real \<Rightarrow> int"
- where "int_of_real x = (SOME y. real y = x)"
-
-definition real_is_int :: "real \<Rightarrow> bool"
- where "real_is_int x = (EX (u::int). x = real u)"
-
-lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
- by (auto simp add: real_is_int_def int_of_real_def)
-
-lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
-by (auto simp add: real_is_int_def int_of_real_def)
-
-lemma int_of_real_real[simp]: "int_of_real (real x) = x"
-by (simp add: int_of_real_def)
-
-lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
-by (auto simp add: int_of_real_def real_is_int_def)
-
-lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"
-by (auto simp add: int_of_real_def real_is_int_def)
-
-lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
-apply (subst real_is_int_def2)
-apply (simp add: real_is_int_add_int_of_real real_int_of_real)
-done
-
-lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)"
-by (auto simp add: int_of_real_def real_is_int_def)
-
-lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
-apply (subst real_is_int_def2)
-apply (simp add: int_of_real_sub real_int_of_real)
-done
-
-lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
-by (auto simp add: real_is_int_def)
-
-lemma int_of_real_mult:
- assumes "real_is_int a" "real_is_int b"
- shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
- using assms
- by (auto simp add: real_is_int_def real_of_int_mult[symmetric]
- simp del: real_of_int_mult)
-
-lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
-apply (subst real_is_int_def2)
-apply (simp add: int_of_real_mult)
-done
-
-lemma real_is_int_0[simp]: "real_is_int (0::real)"
-by (simp add: real_is_int_def int_of_real_def)
-
-lemma real_is_int_1[simp]: "real_is_int (1::real)"
-proof -
- have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
- also have "\<dots> = True" by (simp only: real_is_int_real)
- ultimately show ?thesis by auto
-qed
-
-lemma real_is_int_n1: "real_is_int (-1::real)"
-proof -
- have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
- also have "\<dots> = True" by (simp only: real_is_int_real)
- ultimately show ?thesis by auto
-qed
-
-lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
- by (auto simp: real_is_int_def intro!: exI[of _ "number_of x"])
-
-lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
-by (simp add: int_of_real_def)
-
-lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
-proof -
- have 1: "(1::real) = real (1::int)" by auto
- show ?thesis by (simp only: 1 int_of_real_real)
-qed
-
-lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
- unfolding int_of_real_def
- by (intro some_equality)
- (auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
-
-lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
-by (rule zdiv_int)
-
-lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
-by (rule zmod_int)
-
-lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
-by arith
-
-lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
- by auto
-
-lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
- by simp
-
-lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
- by simp
-
-lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
- by simp
-
-lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
- by simp
-
-lemma int_pow_0: "(a::int)^(Numeral0) = 1"
- by simp
-
-lemma int_pow_1: "(a::int)^(Numeral1) = a"
- by simp
-
-lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
- by simp
-
-lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
- by simp
-
-lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
- by simp
-
-lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
- by simp
-
-lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
- by simp
-
-lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
-proof -
- have 1:"((-1)::nat) = 0"
- by simp
- show ?thesis by (simp add: 1)
-qed
-
-lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
- by simp
-
-lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
- by simp
-
-lemma lift_bool: "x \<Longrightarrow> x=True"
- by simp
-
-lemma nlift_bool: "~x \<Longrightarrow> x=False"
- by simp
-
-lemma not_false_eq_true: "(~ False) = True" by simp
-
-lemma not_true_eq_false: "(~ True) = False" by simp
-
-lemmas binarith =
- normalize_bin_simps
- pred_bin_simps succ_bin_simps
- add_bin_simps minus_bin_simps mult_bin_simps
-
-lemma int_eq_number_of_eq:
- "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"
- by (rule eq_number_of_eq)
-
-lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
- by (simp only: iszero_number_of_Pls)
-
-lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
- by simp
-
-lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)"
- by simp
-
-lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)"
- by simp
-
-lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
- unfolding neg_def number_of_is_id by simp
-
-lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
- by simp
-
-lemma int_neg_number_of_Min: "neg (-1::int)"
- by simp
-
-lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)"
- by simp
-
-lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)"
- by simp
-
-lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
- unfolding neg_def number_of_is_id by (simp add: not_less)
-
-lemmas intarithrel =
- int_eq_number_of_eq
- lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0
- lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
- int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq
-
-lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
- by simp
-
-lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
- by simp
-
-lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"
- by simp
-
-lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
- by simp
-
-lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
-
-lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
-
-lemmas powerarith = nat_number_of zpower_number_of_even
- zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
- zpower_Pls zpower_Min
-
-definition float :: "(int \<times> int) \<Rightarrow> real" where
- "float = (\<lambda>(a, b). real a * 2 powr real b)"
-
-lemma float_add_l0: "float (0, e) + x = x"
- by (simp add: float_def)
-
-lemma float_add_r0: "x + float (0, e) = x"
- by (simp add: float_def)
-
-lemma float_add:
- "float (a1, e1) + float (a2, e2) =
- (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) else float (a1*2^(nat (e1-e2))+a2, e2))"
- by (simp add: float_def algebra_simps powr_realpow[symmetric] powr_divide2[symmetric])
-
-lemma float_mult_l0: "float (0, e) * x = float (0, 0)"
- by (simp add: float_def)
-
-lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
- by (simp add: float_def)
-
-lemma float_mult:
- "float (a1, e1) * float (a2, e2) = (float (a1 * a2, e1 + e2))"
- by (simp add: float_def powr_add)
-
-lemma float_minus:
- "- (float (a,b)) = float (-a, b)"
- by (simp add: float_def)
-
-lemma zero_le_float:
- "(0 <= float (a,b)) = (0 <= a)"
- using powr_gt_zero[of 2 "real b", arith]
- by (simp add: float_def zero_le_mult_iff)
-
-lemma float_le_zero:
- "(float (a,b) <= 0) = (a <= 0)"
- using powr_gt_zero[of 2 "real b", arith]
- by (simp add: float_def mult_le_0_iff)
-
-lemma float_abs:
- "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
- using powr_gt_zero[of 2 "real b", arith]
- by (simp add: float_def abs_if mult_less_0_iff)
-
-lemma float_zero:
- "float (0, b) = 0"
- by (simp add: float_def)
-
-lemma float_pprt:
- "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
- by (auto simp add: zero_le_float float_le_zero float_zero)
-
-lemma float_nprt:
- "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
- by (auto simp add: zero_le_float float_le_zero float_zero)
-
-definition lbound :: "real \<Rightarrow> real"
- where "lbound x = min 0 x"
-
-definition ubound :: "real \<Rightarrow> real"
- where "ubound x = max 0 x"
-
-lemma lbound: "lbound x \<le> x"
- by (simp add: lbound_def)
-
-lemma ubound: "x \<le> ubound x"
- by (simp add: ubound_def)
-
-lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"
- by (auto simp: float_def lbound_def)
-
-lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"
- by (auto simp: float_def ubound_def)
-
-lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0
- float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound
-
-(* for use with the compute oracle *)
-lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
-
-use "~~/src/HOL/Tools/float_arith.ML"
-
-end
--- a/src/HOL/Matrix/ComputeHOL.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,187 +0,0 @@
-theory ComputeHOL
-imports Complex_Main "Compute_Oracle/Compute_Oracle"
-begin
-
-lemma Trueprop_eq_eq: "Trueprop X == (X == True)" by (simp add: atomize_eq)
-lemma meta_eq_trivial: "x == y \<Longrightarrow> x == y" by simp
-lemma meta_eq_imp_eq: "x == y \<Longrightarrow> x = y" by auto
-lemma eq_trivial: "x = y \<Longrightarrow> x = y" by auto
-lemma bool_to_true: "x :: bool \<Longrightarrow> x == True" by simp
-lemma transmeta_1: "x = y \<Longrightarrow> y == z \<Longrightarrow> x = z" by simp
-lemma transmeta_2: "x == y \<Longrightarrow> y = z \<Longrightarrow> x = z" by simp
-lemma transmeta_3: "x == y \<Longrightarrow> y == z \<Longrightarrow> x = z" by simp
-
-
-(**** compute_if ****)
-
-lemma If_True: "If True = (\<lambda> x y. x)" by ((rule ext)+,auto)
-lemma If_False: "If False = (\<lambda> x y. y)" by ((rule ext)+, auto)
-
-lemmas compute_if = If_True If_False
-
-(**** compute_bool ****)
-
-lemma bool1: "(\<not> True) = False" by blast
-lemma bool2: "(\<not> False) = True" by blast
-lemma bool3: "(P \<and> True) = P" by blast
-lemma bool4: "(True \<and> P) = P" by blast
-lemma bool5: "(P \<and> False) = False" by blast
-lemma bool6: "(False \<and> P) = False" by blast
-lemma bool7: "(P \<or> True) = True" by blast
-lemma bool8: "(True \<or> P) = True" by blast
-lemma bool9: "(P \<or> False) = P" by blast
-lemma bool10: "(False \<or> P) = P" by blast
-lemma bool11: "(True \<longrightarrow> P) = P" by blast
-lemma bool12: "(P \<longrightarrow> True) = True" by blast
-lemma bool13: "(True \<longrightarrow> P) = P" by blast
-lemma bool14: "(P \<longrightarrow> False) = (\<not> P)" by blast
-lemma bool15: "(False \<longrightarrow> P) = True" by blast
-lemma bool16: "(False = False) = True" by blast
-lemma bool17: "(True = True) = True" by blast
-lemma bool18: "(False = True) = False" by blast
-lemma bool19: "(True = False) = False" by blast
-
-lemmas compute_bool = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10 bool11 bool12 bool13 bool14 bool15 bool16 bool17 bool18 bool19
-
-
-(*** compute_pair ***)
-
-lemma compute_fst: "fst (x,y) = x" by simp
-lemma compute_snd: "snd (x,y) = y" by simp
-lemma compute_pair_eq: "((a, b) = (c, d)) = (a = c \<and> b = d)" by auto
-
-lemma prod_case_simp: "prod_case f (x,y) = f x y" by simp
-
-lemmas compute_pair = compute_fst compute_snd compute_pair_eq prod_case_simp
-
-(*** compute_option ***)
-
-lemma compute_the: "the (Some x) = x" by simp
-lemma compute_None_Some_eq: "(None = Some x) = False" by auto
-lemma compute_Some_None_eq: "(Some x = None) = False" by auto
-lemma compute_None_None_eq: "(None = None) = True" by auto
-lemma compute_Some_Some_eq: "(Some x = Some y) = (x = y)" by auto
-
-definition option_case_compute :: "'b option \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
- where "option_case_compute opt a f = option_case a f opt"
-
-lemma option_case_compute: "option_case = (\<lambda> a f opt. option_case_compute opt a f)"
- by (simp add: option_case_compute_def)
-
-lemma option_case_compute_None: "option_case_compute None = (\<lambda> a f. a)"
- apply (rule ext)+
- apply (simp add: option_case_compute_def)
- done
-
-lemma option_case_compute_Some: "option_case_compute (Some x) = (\<lambda> a f. f x)"
- apply (rule ext)+
- apply (simp add: option_case_compute_def)
- done
-
-lemmas compute_option_case = option_case_compute option_case_compute_None option_case_compute_Some
-
-lemmas compute_option = compute_the compute_None_Some_eq compute_Some_None_eq compute_None_None_eq compute_Some_Some_eq compute_option_case
-
-(**** compute_list_length ****)
-
-lemma length_cons:"length (x#xs) = 1 + (length xs)"
- by simp
-
-lemma length_nil: "length [] = 0"
- by simp
-
-lemmas compute_list_length = length_nil length_cons
-
-(*** compute_list_case ***)
-
-definition list_case_compute :: "'b list \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'a) \<Rightarrow> 'a"
- where "list_case_compute l a f = list_case a f l"
-
-lemma list_case_compute: "list_case = (\<lambda> (a::'a) f (l::'b list). list_case_compute l a f)"
- apply (rule ext)+
- apply (simp add: list_case_compute_def)
- done
-
-lemma list_case_compute_empty: "list_case_compute ([]::'b list) = (\<lambda> (a::'a) f. a)"
- apply (rule ext)+
- apply (simp add: list_case_compute_def)
- done
-
-lemma list_case_compute_cons: "list_case_compute (u#v) = (\<lambda> (a::'a) f. (f (u::'b) v))"
- apply (rule ext)+
- apply (simp add: list_case_compute_def)
- done
-
-lemmas compute_list_case = list_case_compute list_case_compute_empty list_case_compute_cons
-
-(*** compute_list_nth ***)
-(* Of course, you will need computation with nats for this to work \<dots> *)
-
-lemma compute_list_nth: "((x#xs) ! n) = (if n = 0 then x else (xs ! (n - 1)))"
- by (cases n, auto)
-
-(*** compute_list ***)
-
-lemmas compute_list = compute_list_case compute_list_length compute_list_nth
-
-(*** compute_let ***)
-
-lemmas compute_let = Let_def
-
-(***********************)
-(* Everything together *)
-(***********************)
-
-lemmas compute_hol = compute_if compute_bool compute_pair compute_option compute_list compute_let
-
-ML {*
-signature ComputeHOL =
-sig
- val prep_thms : thm list -> thm list
- val to_meta_eq : thm -> thm
- val to_hol_eq : thm -> thm
- val symmetric : thm -> thm
- val trans : thm -> thm -> thm
-end
-
-structure ComputeHOL : ComputeHOL =
-struct
-
-local
-fun lhs_of eq = fst (Thm.dest_equals (cprop_of eq));
-in
-fun rewrite_conv [] ct = raise CTERM ("rewrite_conv", [ct])
- | rewrite_conv (eq :: eqs) ct =
- Thm.instantiate (Thm.match (lhs_of eq, ct)) eq
- handle Pattern.MATCH => rewrite_conv eqs ct;
-end
-
-val convert_conditions = Conv.fconv_rule (Conv.prems_conv ~1 (Conv.try_conv (rewrite_conv [@{thm "Trueprop_eq_eq"}])))
-
-val eq_th = @{thm "HOL.eq_reflection"}
-val meta_eq_trivial = @{thm "ComputeHOL.meta_eq_trivial"}
-val bool_to_true = @{thm "ComputeHOL.bool_to_true"}
-
-fun to_meta_eq th = eq_th OF [th] handle THM _ => meta_eq_trivial OF [th] handle THM _ => bool_to_true OF [th]
-
-fun to_hol_eq th = @{thm "meta_eq_imp_eq"} OF [th] handle THM _ => @{thm "eq_trivial"} OF [th]
-
-fun prep_thms ths = map (convert_conditions o to_meta_eq) ths
-
-fun symmetric th = @{thm "HOL.sym"} OF [th] handle THM _ => @{thm "Pure.symmetric"} OF [th]
-
-local
- val trans_HOL = @{thm "HOL.trans"}
- val trans_HOL_1 = @{thm "ComputeHOL.transmeta_1"}
- val trans_HOL_2 = @{thm "ComputeHOL.transmeta_2"}
- val trans_HOL_3 = @{thm "ComputeHOL.transmeta_3"}
- fun tr [] th1 th2 = trans_HOL OF [th1, th2]
- | tr (t::ts) th1 th2 = (t OF [th1, th2] handle THM _ => tr ts th1 th2)
-in
- fun trans th1 th2 = tr [trans_HOL, trans_HOL_1, trans_HOL_2, trans_HOL_3] th1 th2
-end
-
-end
-*}
-
-end
--- a/src/HOL/Matrix/ComputeNumeral.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,189 +0,0 @@
-theory ComputeNumeral
-imports ComputeHOL ComputeFloat
-begin
-
-(* normalization of bit strings *)
-lemmas bitnorm = normalize_bin_simps
-
-(* neg for bit strings *)
-lemma neg1: "neg Int.Pls = False" by (simp add: Int.Pls_def)
-lemma neg2: "neg Int.Min = True" apply (subst Int.Min_def) by auto
-lemma neg3: "neg (Int.Bit0 x) = neg x" apply (simp add: neg_def) apply (subst Bit0_def) by auto
-lemma neg4: "neg (Int.Bit1 x) = neg x" apply (simp add: neg_def) apply (subst Bit1_def) by auto
-lemmas bitneg = neg1 neg2 neg3 neg4
-
-(* iszero for bit strings *)
-lemma iszero1: "iszero Int.Pls = True" by (simp add: Int.Pls_def iszero_def)
-lemma iszero2: "iszero Int.Min = False" apply (subst Int.Min_def) apply (subst iszero_def) by simp
-lemma iszero3: "iszero (Int.Bit0 x) = iszero x" apply (subst Int.Bit0_def) apply (subst iszero_def)+ by auto
-lemma iszero4: "iszero (Int.Bit1 x) = False" apply (subst Int.Bit1_def) apply (subst iszero_def)+ apply simp by arith
-lemmas bitiszero = iszero1 iszero2 iszero3 iszero4
-
-(* lezero for bit strings *)
-definition "lezero x \<longleftrightarrow> x \<le> 0"
-lemma lezero1: "lezero Int.Pls = True" unfolding Int.Pls_def lezero_def by auto
-lemma lezero2: "lezero Int.Min = True" unfolding Int.Min_def lezero_def by auto
-lemma lezero3: "lezero (Int.Bit0 x) = lezero x" unfolding Int.Bit0_def lezero_def by auto
-lemma lezero4: "lezero (Int.Bit1 x) = neg x" unfolding Int.Bit1_def lezero_def neg_def by auto
-lemmas bitlezero = lezero1 lezero2 lezero3 lezero4
-
-(* equality for bit strings *)
-lemmas biteq = eq_bin_simps
-
-(* x < y for bit strings *)
-lemmas bitless = less_bin_simps
-
-(* x \<le> y for bit strings *)
-lemmas bitle = le_bin_simps
-
-(* succ for bit strings *)
-lemmas bitsucc = succ_bin_simps
-
-(* pred for bit strings *)
-lemmas bitpred = pred_bin_simps
-
-(* unary minus for bit strings *)
-lemmas bituminus = minus_bin_simps
-
-(* addition for bit strings *)
-lemmas bitadd = add_bin_simps
-
-(* multiplication for bit strings *)
-lemma mult_Pls_right: "x * Int.Pls = Int.Pls" by (simp add: Pls_def)
-lemma mult_Min_right: "x * Int.Min = - x" by (subst mult_commute) simp
-lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0)
-lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp
-lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)"
- unfolding Bit0_def Bit1_def by (simp add: algebra_simps)
-lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1
-
-lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul
-
-definition "nat_norm_number_of (x::nat) = x"
-
-lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)"
- apply (simp add: nat_norm_number_of_def)
- unfolding lezero_def iszero_def neg_def
- apply (simp add: numeral_simps)
- done
-
-(* Normalization of nat literals *)
-lemma natnorm0: "(0::nat) = number_of (Int.Pls)" by auto
-lemma natnorm1: "(1 :: nat) = number_of (Int.Bit1 Int.Pls)" by auto
-lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of
-
-(* Suc *)
-lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Int.succ x))" by (auto simp add: number_of_is_id)
-
-(* Addition for nat *)
-lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by auto
-
-(* Subtraction for nat *)
-lemma natsub: "(number_of x) - ((number_of y)::nat) =
- (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))"
- unfolding nat_norm_number_of
- by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def)
-
-(* Multiplication for nat *)
-lemma natmul: "(number_of x) * ((number_of y)::nat) =
- (if neg x then 0 else (if neg y then 0 else number_of (x * y)))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by (simp add: nat_mult_distrib)
-
-lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \<and> lezero y) \<or> (x = y))"
- by (auto simp add: iszero_def lezero_def neg_def number_of_is_id)
-
-lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \<and> (\<not> (lezero y)))"
- by (simp add: lezero_def numeral_simps not_le)
-
-lemma natle: "(((number_of x)::nat) \<le> (number_of y)) = (y < x \<longrightarrow> lezero x)"
- by (auto simp add: number_of_is_id lezero_def nat_number_of_def)
-
-fun natfac :: "nat \<Rightarrow> nat"
- where "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))"
-
-lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps
-
-lemma number_eq: "(((number_of x)::'a::{number_ring, linordered_idom}) = (number_of y)) = (x = y)"
- unfolding number_of_eq
- apply simp
- done
-
-lemma number_le: "(((number_of x)::'a::{number_ring, linordered_idom}) \<le> (number_of y)) = (x \<le> y)"
- unfolding number_of_eq
- apply simp
- done
-
-lemma number_less: "(((number_of x)::'a::{number_ring, linordered_idom}) < (number_of y)) = (x < y)"
- unfolding number_of_eq
- apply simp
- done
-
-lemma number_diff: "((number_of x)::'a::{number_ring, linordered_idom}) - number_of y = number_of (x + (- y))"
- apply (subst diff_number_of_eq)
- apply simp
- done
-
-lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric]
-
-lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less
-
-lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)"
- by (simp only: real_of_nat_number_of number_of_is_id)
-
-lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)"
- by simp
-
-lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of
-
-lemmas zpowerarith = zpower_number_of_even
- zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
- zpower_Pls zpower_Min
-
-(* div, mod *)
-
-lemma adjust: "adjust b (q, r) = (if 0 \<le> r - b then (2 * q + 1, r - b) else (2 * q, r))"
- by (auto simp only: adjust_def)
-
-lemma divmod: "divmod_int a b = (if 0\<le>a then
- if 0\<le>b then posDivAlg a b
- else if a=0 then (0, 0)
- else apsnd uminus (negDivAlg (-a) (-b))
- else
- if 0<b then negDivAlg a b
- else apsnd uminus (posDivAlg (-a) (-b)))"
- by (auto simp only: divmod_int_def)
-
-lemmas compute_div_mod = div_int_def mod_int_def divmod adjust apsnd_def map_pair_def posDivAlg.simps negDivAlg.simps
-
-
-
-(* collecting all the theorems *)
-
-lemma even_Pls: "even (Int.Pls) = True"
- apply (unfold Pls_def even_def)
- by simp
-
-lemma even_Min: "even (Int.Min) = False"
- apply (unfold Min_def even_def)
- by simp
-
-lemma even_B0: "even (Int.Bit0 x) = True"
- apply (unfold Bit0_def)
- by simp
-
-lemma even_B1: "even (Int.Bit1 x) = False"
- apply (unfold Bit1_def)
- by simp
-
-lemma even_number_of: "even ((number_of w)::int) = even w"
- by (simp only: number_of_is_id)
-
-lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of
-
-lemmas compute_numeral = compute_if compute_let compute_pair compute_bool
- compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even
-
-end
--- a/src/HOL/Matrix/Compute_Oracle/Compute_Oracle.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,11 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/Compute_Oracle.thy
- Author: Steven Obua, TU Munich
-
-Steven Obua's evaluator.
-*)
-
-theory Compute_Oracle imports HOL
-uses "am.ML" "am_compiler.ML" "am_interpreter.ML" "am_ghc.ML" "am_sml.ML" "report.ML" "compute.ML" "linker.ML"
-begin
-
-end
\ No newline at end of file
--- a/src/HOL/Matrix/Compute_Oracle/am.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,71 +0,0 @@
-signature ABSTRACT_MACHINE =
-sig
-
-datatype term = Var of int | Const of int | App of term * term | Abs of term | Computed of term
-
-datatype pattern = PVar | PConst of int * (pattern list)
-
-datatype guard = Guard of term * term
-
-type program
-
-exception Compile of string;
-
-(* The de-Bruijn index 0 occurring on the right hand side refers to the LAST pattern variable, when traversing the pattern from left to right,
- 1 to the second last, and so on. *)
-val compile : (guard list * pattern * term) list -> program
-
-exception Run of string;
-val run : program -> term -> term
-
-(* Utilities *)
-
-val check_freevars : int -> term -> bool
-val forall_consts : (int -> bool) -> term -> bool
-val closed : term -> bool
-val erase_Computed : term -> term
-
-end
-
-structure AbstractMachine : ABSTRACT_MACHINE =
-struct
-
-datatype term = Var of int | Const of int | App of term * term | Abs of term | Computed of term
-
-datatype pattern = PVar | PConst of int * (pattern list)
-
-datatype guard = Guard of term * term
-
-type program = unit
-
-exception Compile of string;
-
-fun erase_Computed (Computed t) = erase_Computed t
- | erase_Computed (App (t1, t2)) = App (erase_Computed t1, erase_Computed t2)
- | erase_Computed (Abs t) = Abs (erase_Computed t)
- | erase_Computed t = t
-
-(*Returns true iff at most 0 .. (free-1) occur unbound. therefore
- check_freevars 0 t iff t is closed*)
-fun check_freevars free (Var x) = x < free
- | check_freevars free (Const _) = true
- | check_freevars free (App (u, v)) = check_freevars free u andalso check_freevars free v
- | check_freevars free (Abs m) = check_freevars (free+1) m
- | check_freevars free (Computed t) = check_freevars free t
-
-fun forall_consts pred (Const c) = pred c
- | forall_consts pred (Var _) = true
- | forall_consts pred (App (u,v)) = forall_consts pred u
- andalso forall_consts pred v
- | forall_consts pred (Abs m) = forall_consts pred m
- | forall_consts pred (Computed t) = forall_consts pred t
-
-fun closed t = check_freevars 0 t
-
-fun compile _ = raise Compile "abstract machine stub"
-
-exception Run of string;
-
-fun run _ _ = raise Run "abstract machine stub"
-
-end
--- a/src/HOL/Matrix/Compute_Oracle/am_compiler.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,208 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/am_compiler.ML
- Author: Steven Obua
-*)
-
-signature COMPILING_AM =
-sig
- include ABSTRACT_MACHINE
-
- val set_compiled_rewriter : (term -> term) -> unit
- val list_nth : 'a list * int -> 'a
- val list_map : ('a -> 'b) -> 'a list -> 'b list
-end
-
-structure AM_Compiler : COMPILING_AM = struct
-
-val list_nth = List.nth;
-val list_map = map;
-
-open AbstractMachine;
-
-val compiled_rewriter = Unsynchronized.ref (NONE:(term -> term)Option.option)
-
-fun set_compiled_rewriter r = (compiled_rewriter := SOME r)
-
-type program = (term -> term)
-
-fun count_patternvars PVar = 1
- | count_patternvars (PConst (_, ps)) =
- List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
-
-fun print_rule (p, t) =
- let
- fun str x = string_of_int x
- fun print_pattern n PVar = (n+1, "x"^(str n))
- | print_pattern n (PConst (c, [])) = (n, "c"^(str c))
- | print_pattern n (PConst (c, args)) =
- let
- val h = print_pattern n (PConst (c,[]))
- in
- print_pattern_list h args
- end
- and print_pattern_list r [] = r
- | print_pattern_list (n, p) (t::ts) =
- let
- val (n, t) = print_pattern n t
- in
- print_pattern_list (n, "App ("^p^", "^t^")") ts
- end
-
- val (n, pattern) = print_pattern 0 p
- val pattern =
- if exists_string Symbol.is_ascii_blank pattern then "(" ^ pattern ^")"
- else pattern
-
- fun print_term d (Var x) = "Var " ^ str x
- | print_term d (Const c) = "c" ^ str c
- | print_term d (App (a,b)) = "App (" ^ print_term d a ^ ", " ^ print_term d b ^ ")"
- | print_term d (Abs c) = "Abs (" ^ print_term (d + 1) c ^ ")"
- | print_term d (Computed c) = print_term d c
-
- fun listvars n = if n = 0 then "x0" else "x"^(str n)^", "^(listvars (n-1))
-
- val term = print_term 0 t
- val term =
- if n > 0 then "Closure (["^(listvars (n-1))^"], "^term^")"
- else "Closure ([], "^term^")"
-
- in
- " | weak_reduce (false, stack, "^pattern^") = Continue (false, stack, "^term^")"
- end
-
-fun constants_of PVar = []
- | constants_of (PConst (c, ps)) = c :: maps constants_of ps
-
-fun constants_of_term (Var _) = []
- | constants_of_term (Abs m) = constants_of_term m
- | constants_of_term (App (a,b)) = (constants_of_term a)@(constants_of_term b)
- | constants_of_term (Const c) = [c]
- | constants_of_term (Computed c) = constants_of_term c
-
-fun load_rules sname name prog =
- let
- val buffer = Unsynchronized.ref ""
- fun write s = (buffer := (!buffer)^s)
- fun writeln s = (write s; write "\n")
- fun writelist [] = ()
- | writelist (s::ss) = (writeln s; writelist ss)
- fun str i = string_of_int i
- val _ = writelist [
- "structure "^name^" = struct",
- "",
- "datatype term = Dummy | App of term * term | Abs of term | Var of int | Const of int | Closure of term list * term"]
- val constants = distinct (op =) (maps (fn (p, r) => ((constants_of p)@(constants_of_term r))) prog)
- val _ = map (fn x => write (" | c"^(str x))) constants
- val _ = writelist [
- "",
- "datatype stack = SEmpty | SAppL of term * stack | SAppR of term * stack | SAbs of stack",
- "",
- "type state = bool * stack * term",
- "",
- "datatype loopstate = Continue of state | Stop of stack * term",
- "",
- "fun proj_C (Continue s) = s",
- " | proj_C _ = raise Match",
- "",
- "fun proj_S (Stop s) = s",
- " | proj_S _ = raise Match",
- "",
- "fun cont (Continue _) = true",
- " | cont _ = false",
- "",
- "fun do_reduction reduce p =",
- " let",
- " val s = Unsynchronized.ref (Continue p)",
- " val _ = while cont (!s) do (s := reduce (proj_C (!s)))",
- " in",
- " proj_S (!s)",
- " end",
- ""]
-
- val _ = writelist [
- "fun weak_reduce (false, stack, Closure (e, App (a, b))) = Continue (false, SAppL (Closure (e, b), stack), Closure (e, a))",
- " | weak_reduce (false, SAppL (b, stack), Closure (e, Abs m)) = Continue (false, stack, Closure (b::e, m))",
- " | weak_reduce (false, stack, c as Closure (e, Abs m)) = Continue (true, stack, c)",
- " | weak_reduce (false, stack, Closure (e, Var n)) = Continue (false, stack, case "^sname^".list_nth (e, n) of Dummy => Var n | r => r)",
- " | weak_reduce (false, stack, Closure (e, c)) = Continue (false, stack, c)"]
- val _ = writelist (map print_rule prog)
- val _ = writelist [
- " | weak_reduce (false, stack, clos) = Continue (true, stack, clos)",
- " | weak_reduce (true, SAppR (a, stack), b) = Continue (false, stack, App (a,b))",
- " | weak_reduce (true, s as (SAppL (b, stack)), a) = Continue (false, SAppR (a, stack), b)",
- " | weak_reduce (true, stack, c) = Stop (stack, c)",
- "",
- "fun strong_reduce (false, stack, Closure (e, Abs m)) =",
- " let",
- " val (stack', wnf) = do_reduction weak_reduce (false, SEmpty, Closure (Dummy::e, m))",
- " in",
- " case stack' of",
- " SEmpty => Continue (false, SAbs stack, wnf)",
- " | _ => raise ("^sname^".Run \"internal error in strong: weak failed\")",
- " end",
- " | strong_reduce (false, stack, clos as (App (u, v))) = Continue (false, SAppL (v, stack), u)",
- " | strong_reduce (false, stack, clos) = Continue (true, stack, clos)",
- " | strong_reduce (true, SAbs stack, m) = Continue (false, stack, Abs m)",
- " | strong_reduce (true, SAppL (b, stack), a) = Continue (false, SAppR (a, stack), b)",
- " | strong_reduce (true, SAppR (a, stack), b) = Continue (true, stack, App (a, b))",
- " | strong_reduce (true, stack, clos) = Stop (stack, clos)",
- ""]
-
- val ic = "(case c of "^(implode (map (fn c => (str c)^" => c"^(str c)^" | ") constants))^" _ => Const c)"
- val _ = writelist [
- "fun importTerm ("^sname^".Var x) = Var x",
- " | importTerm ("^sname^".Const c) = "^ic,
- " | importTerm ("^sname^".App (a, b)) = App (importTerm a, importTerm b)",
- " | importTerm ("^sname^".Abs m) = Abs (importTerm m)",
- ""]
-
- fun ec c = " | exportTerm c"^(str c)^" = "^sname^".Const "^(str c)
- val _ = writelist [
- "fun exportTerm (Var x) = "^sname^".Var x",
- " | exportTerm (Const c) = "^sname^".Const c",
- " | exportTerm (App (a,b)) = "^sname^".App (exportTerm a, exportTerm b)",
- " | exportTerm (Abs m) = "^sname^".Abs (exportTerm m)",
- " | exportTerm (Closure (closlist, clos)) = raise ("^sname^".Run \"internal error, cannot export Closure\")",
- " | exportTerm Dummy = raise ("^sname^".Run \"internal error, cannot export Dummy\")"]
- val _ = writelist (map ec constants)
-
- val _ = writelist [
- "",
- "fun rewrite t = ",
- " let",
- " val (stack, wnf) = do_reduction weak_reduce (false, SEmpty, Closure ([], importTerm t))",
- " in",
- " case stack of ",
- " SEmpty => (case do_reduction strong_reduce (false, SEmpty, wnf) of",
- " (SEmpty, snf) => exportTerm snf",
- " | _ => raise ("^sname^".Run \"internal error in rewrite: strong failed\"))",
- " | _ => (raise ("^sname^".Run \"internal error in rewrite: weak failed\"))",
- " end",
- "",
- "val _ = "^sname^".set_compiled_rewriter rewrite",
- "",
- "end;"]
-
- in
- compiled_rewriter := NONE;
- use_text ML_Env.local_context (1, "") false (!buffer);
- case !compiled_rewriter of
- NONE => raise (Compile "cannot communicate with compiled function")
- | SOME r => (compiled_rewriter := NONE; r)
- end
-
-fun compile eqs =
- let
- val _ = if exists (fn (a,_,_) => not (null a)) eqs then raise Compile ("cannot deal with guards") else ()
- val eqs = map (fn (_,b,c) => (b,c)) eqs
- fun check (p, r) = if check_freevars (count_patternvars p) r then () else raise Compile ("unbound variables in rule")
- val _ = map (fn (p, r) =>
- (check (p, r);
- case p of PVar => raise (Compile "pattern is just a variable") | _ => ())) eqs
- in
- load_rules "AM_Compiler" "AM_compiled_code" eqs
- end
-
-fun run prog t = prog t
-
-end
-
--- a/src/HOL/Matrix/Compute_Oracle/am_ghc.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,324 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/am_ghc.ML
- Author: Steven Obua
-*)
-
-structure AM_GHC : ABSTRACT_MACHINE =
-struct
-
-open AbstractMachine;
-
-type program = string * string * (int Inttab.table)
-
-fun count_patternvars PVar = 1
- | count_patternvars (PConst (_, ps)) =
- List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
-
-fun update_arity arity code a =
- (case Inttab.lookup arity code of
- NONE => Inttab.update_new (code, a) arity
- | SOME (a': int) => if a > a' then Inttab.update (code, a) arity else arity)
-
-(* We have to find out the maximal arity of each constant *)
-fun collect_pattern_arity PVar arity = arity
- | collect_pattern_arity (PConst (c, args)) arity = fold collect_pattern_arity args (update_arity arity c (length args))
-
-local
-fun collect applevel (Var _) arity = arity
- | collect applevel (Const c) arity = update_arity arity c applevel
- | collect applevel (Abs m) arity = collect 0 m arity
- | collect applevel (App (a,b)) arity = collect 0 b (collect (applevel + 1) a arity)
-in
-fun collect_term_arity t arity = collect 0 t arity
-end
-
-fun nlift level n (Var m) = if m < level then Var m else Var (m+n)
- | nlift level n (Const c) = Const c
- | nlift level n (App (a,b)) = App (nlift level n a, nlift level n b)
- | nlift level n (Abs b) = Abs (nlift (level+1) n b)
-
-fun rep n x = if n = 0 then [] else x::(rep (n-1) x)
-
-fun adjust_rules rules =
- let
- val arity = fold (fn (p, t) => fn arity => collect_term_arity t (collect_pattern_arity p arity)) rules Inttab.empty
- fun arity_of c = the (Inttab.lookup arity c)
- fun adjust_pattern PVar = PVar
- | adjust_pattern (C as PConst (c, args)) = if (length args <> arity_of c) then raise Compile ("Constant inside pattern must have maximal arity") else C
- fun adjust_rule (PVar, _) = raise Compile ("pattern may not be a variable")
- | adjust_rule (rule as (p as PConst (c, args),t)) =
- let
- val _ = if not (check_freevars (count_patternvars p) t) then raise Compile ("unbound variables on right hand side") else ()
- val args = map adjust_pattern args
- val len = length args
- val arity = arity_of c
- fun lift level n (Var m) = if m < level then Var m else Var (m+n)
- | lift level n (Const c) = Const c
- | lift level n (App (a,b)) = App (lift level n a, lift level n b)
- | lift level n (Abs b) = Abs (lift (level+1) n b)
- val lift = lift 0
- fun adjust_term n t = if n=0 then t else adjust_term (n-1) (App (t, Var (n-1)))
- in
- if len = arity then
- rule
- else if arity >= len then
- (PConst (c, args @ (rep (arity-len) PVar)), adjust_term (arity-len) (lift (arity-len) t))
- else (raise Compile "internal error in adjust_rule")
- end
- in
- (arity, map adjust_rule rules)
- end
-
-fun print_term arity_of n =
-let
- fun str x = string_of_int x
- fun protect_blank s = if exists_string Symbol.is_ascii_blank s then "(" ^ s ^")" else s
-
- fun print_apps d f [] = f
- | print_apps d f (a::args) = print_apps d ("app "^(protect_blank f)^" "^(protect_blank (print_term d a))) args
- and print_call d (App (a, b)) args = print_call d a (b::args)
- | print_call d (Const c) args =
- (case arity_of c of
- NONE => print_apps d ("Const "^(str c)) args
- | SOME a =>
- let
- val len = length args
- in
- if a <= len then
- let
- val s = "c"^(str c)^(implode (map (fn t => " "^(protect_blank (print_term d t))) (List.take (args, a))))
- in
- print_apps d s (List.drop (args, a))
- end
- else
- let
- fun mk_apps n t = if n = 0 then t else mk_apps (n-1) (App (t, Var (n-1)))
- fun mk_lambdas n t = if n = 0 then t else mk_lambdas (n-1) (Abs t)
- fun append_args [] t = t
- | append_args (c::cs) t = append_args cs (App (t, c))
- in
- print_term d (mk_lambdas (a-len) (mk_apps (a-len) (nlift 0 (a-len) (append_args args (Const c)))))
- end
- end)
- | print_call d t args = print_apps d (print_term d t) args
- and print_term d (Var x) = if x < d then "b"^(str (d-x-1)) else "x"^(str (n-(x-d)-1))
- | print_term d (Abs c) = "Abs (\\b"^(str d)^" -> "^(print_term (d + 1) c)^")"
- | print_term d t = print_call d t []
-in
- print_term 0
-end
-
-fun print_rule arity_of (p, t) =
- let
- fun str x = string_of_int x
- fun print_pattern top n PVar = (n+1, "x"^(str n))
- | print_pattern top n (PConst (c, [])) = (n, (if top then "c" else "C")^(str c))
- | print_pattern top n (PConst (c, args)) =
- let
- val (n,s) = print_pattern_list (n, (if top then "c" else "C")^(str c)) args
- in
- (n, if top then s else "("^s^")")
- end
- and print_pattern_list r [] = r
- | print_pattern_list (n, p) (t::ts) =
- let
- val (n, t) = print_pattern false n t
- in
- print_pattern_list (n, p^" "^t) ts
- end
- val (n, pattern) = print_pattern true 0 p
- in
- pattern^" = "^(print_term arity_of n t)
- end
-
-fun group_rules rules =
- let
- fun add_rule (r as (PConst (c,_), _)) groups =
- let
- val rs = (case Inttab.lookup groups c of NONE => [] | SOME rs => rs)
- in
- Inttab.update (c, r::rs) groups
- end
- | add_rule _ _ = raise Compile "internal error group_rules"
- in
- fold_rev add_rule rules Inttab.empty
- end
-
-fun haskell_prog name rules =
- let
- val buffer = Unsynchronized.ref ""
- fun write s = (buffer := (!buffer)^s)
- fun writeln s = (write s; write "\n")
- fun writelist [] = ()
- | writelist (s::ss) = (writeln s; writelist ss)
- fun str i = string_of_int i
- val (arity, rules) = adjust_rules rules
- val rules = group_rules rules
- val constants = Inttab.keys arity
- fun arity_of c = Inttab.lookup arity c
- fun rep_str s n = implode (rep n s)
- fun indexed s n = s^(str n)
- fun section n = if n = 0 then [] else (section (n-1))@[n-1]
- fun make_show c =
- let
- val args = section (the (arity_of c))
- in
- " show ("^(indexed "C" c)^(implode (map (indexed " a") args))^") = "
- ^"\""^(indexed "C" c)^"\""^(implode (map (fn a => "++(show "^(indexed "a" a)^")") args))
- end
- fun default_case c =
- let
- val args = implode (map (indexed " x") (section (the (arity_of c))))
- in
- (indexed "c" c)^args^" = "^(indexed "C" c)^args
- end
- val _ = writelist [
- "module "^name^" where",
- "",
- "data Term = Const Integer | App Term Term | Abs (Term -> Term)",
- " "^(implode (map (fn c => " | C"^(str c)^(rep_str " Term" (the (arity_of c)))) constants)),
- "",
- "instance Show Term where"]
- val _ = writelist (map make_show constants)
- val _ = writelist [
- " show (Const c) = \"c\"++(show c)",
- " show (App a b) = \"A\"++(show a)++(show b)",
- " show (Abs _) = \"L\"",
- ""]
- val _ = writelist [
- "app (Abs a) b = a b",
- "app a b = App a b",
- "",
- "calc s c = writeFile s (show c)",
- ""]
- fun list_group c = (writelist (case Inttab.lookup rules c of
- NONE => [default_case c, ""]
- | SOME (rs as ((PConst (_, []), _)::rs')) =>
- if not (null rs') then raise Compile "multiple declaration of constant"
- else (map (print_rule arity_of) rs) @ [""]
- | SOME rs => (map (print_rule arity_of) rs) @ [default_case c, ""]))
- val _ = map list_group constants
- in
- (arity, !buffer)
- end
-
-val guid_counter = Unsynchronized.ref 0
-fun get_guid () =
- let
- val c = !guid_counter
- val _ = guid_counter := !guid_counter + 1
- in
- string_of_int (Time.toMicroseconds (Time.now ())) ^ string_of_int c
- end
-
-fun tmp_file s = Path.implode (Path.expand (File.tmp_path (Path.basic s)));
-
-fun writeTextFile name s = File.write (Path.explode name) s
-
-fun fileExists name = ((OS.FileSys.fileSize name; true) handle OS.SysErr _ => false)
-
-fun compile eqs =
- let
- val _ = if exists (fn (a,_,_) => not (null a)) eqs then raise Compile ("cannot deal with guards") else ()
- val eqs = map (fn (_,b,c) => (b,c)) eqs
- val guid = get_guid ()
- val module = "AMGHC_Prog_"^guid
- val (arity, source) = haskell_prog module eqs
- val module_file = tmp_file (module^".hs")
- val object_file = tmp_file (module^".o")
- val _ = writeTextFile module_file source
- val _ = Isabelle_System.bash ("exec \"$ISABELLE_GHC\" -c " ^ module_file)
- val _ =
- if not (fileExists object_file) then
- raise Compile ("Failure compiling haskell code (ISABELLE_GHC='" ^ getenv "ISABELLE_GHC" ^ "')")
- else ()
- in
- (guid, module_file, arity)
- end
-
-fun readResultFile name = File.read (Path.explode name)
-
-fun parse_result arity_of result =
- let
- val result = String.explode result
- fun shift NONE x = SOME x
- | shift (SOME y) x = SOME (y*10 + x)
- fun parse_int' x (#"0"::rest) = parse_int' (shift x 0) rest
- | parse_int' x (#"1"::rest) = parse_int' (shift x 1) rest
- | parse_int' x (#"2"::rest) = parse_int' (shift x 2) rest
- | parse_int' x (#"3"::rest) = parse_int' (shift x 3) rest
- | parse_int' x (#"4"::rest) = parse_int' (shift x 4) rest
- | parse_int' x (#"5"::rest) = parse_int' (shift x 5) rest
- | parse_int' x (#"6"::rest) = parse_int' (shift x 6) rest
- | parse_int' x (#"7"::rest) = parse_int' (shift x 7) rest
- | parse_int' x (#"8"::rest) = parse_int' (shift x 8) rest
- | parse_int' x (#"9"::rest) = parse_int' (shift x 9) rest
- | parse_int' x rest = (x, rest)
- fun parse_int rest = parse_int' NONE rest
-
- fun parse (#"C"::rest) =
- (case parse_int rest of
- (SOME c, rest) =>
- let
- val (args, rest) = parse_list (the (arity_of c)) rest
- fun app_args [] t = t
- | app_args (x::xs) t = app_args xs (App (t, x))
- in
- (app_args args (Const c), rest)
- end
- | (NONE, _) => raise Run "parse C")
- | parse (#"c"::rest) =
- (case parse_int rest of
- (SOME c, rest) => (Const c, rest)
- | _ => raise Run "parse c")
- | parse (#"A"::rest) =
- let
- val (a, rest) = parse rest
- val (b, rest) = parse rest
- in
- (App (a,b), rest)
- end
- | parse (#"L"::_) = raise Run "there may be no abstraction in the result"
- | parse _ = raise Run "invalid result"
- and parse_list n rest =
- if n = 0 then
- ([], rest)
- else
- let
- val (x, rest) = parse rest
- val (xs, rest) = parse_list (n-1) rest
- in
- (x::xs, rest)
- end
- val (parsed, rest) = parse result
- fun is_blank (#" "::rest) = is_blank rest
- | is_blank (#"\n"::rest) = is_blank rest
- | is_blank [] = true
- | is_blank _ = false
- in
- if is_blank rest then parsed else raise Run "non-blank suffix in result file"
- end
-
-fun run (guid, module_file, arity) t =
- let
- val _ = if check_freevars 0 t then () else raise Run ("can only compute closed terms")
- fun arity_of c = Inttab.lookup arity c
- val callguid = get_guid()
- val module = "AMGHC_Prog_"^guid
- val call = module^"_Call_"^callguid
- val result_file = tmp_file (module^"_Result_"^callguid^".txt")
- val call_file = tmp_file (call^".hs")
- val term = print_term arity_of 0 t
- val call_source = "module "^call^" where\n\nimport "^module^"\n\ncall = "^module^".calc \""^result_file^"\" ("^term^")"
- val _ = writeTextFile call_file call_source
- val _ = Isabelle_System.bash ("exec \"$ISABELLE_GHC\" -e \""^call^".call\" "^module_file^" "^call_file)
- val result = readResultFile result_file handle IO.Io _ =>
- raise Run ("Failure running haskell compiler (ISABELLE_GHC='" ^ getenv "ISABELLE_GHC" ^ "')")
- val t' = parse_result arity_of result
- val _ = OS.FileSys.remove call_file
- val _ = OS.FileSys.remove result_file
- in
- t'
- end
-
-end
-
--- a/src/HOL/Matrix/Compute_Oracle/am_interpreter.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,211 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/am_interpreter.ML
- Author: Steven Obua
-*)
-
-signature AM_BARRAS =
-sig
- include ABSTRACT_MACHINE
- val max_reductions : int option Unsynchronized.ref
-end
-
-structure AM_Interpreter : AM_BARRAS = struct
-
-open AbstractMachine;
-
-datatype closure = CDummy | CVar of int | CConst of int
- | CApp of closure * closure | CAbs of closure
- | Closure of (closure list) * closure
-
-structure prog_struct = Table(type key = int*int val ord = prod_ord int_ord int_ord);
-
-datatype program = Program of ((pattern * closure * (closure*closure) list) list) prog_struct.table
-
-datatype stack = SEmpty | SAppL of closure * stack | SAppR of closure * stack | SAbs of stack
-
-fun clos_of_term (Var x) = CVar x
- | clos_of_term (Const c) = CConst c
- | clos_of_term (App (u, v)) = CApp (clos_of_term u, clos_of_term v)
- | clos_of_term (Abs u) = CAbs (clos_of_term u)
- | clos_of_term (Computed t) = clos_of_term t
-
-fun term_of_clos (CVar x) = Var x
- | term_of_clos (CConst c) = Const c
- | term_of_clos (CApp (u, v)) = App (term_of_clos u, term_of_clos v)
- | term_of_clos (CAbs u) = Abs (term_of_clos u)
- | term_of_clos (Closure _) = raise (Run "internal error: closure in normalized term found")
- | term_of_clos CDummy = raise (Run "internal error: dummy in normalized term found")
-
-fun resolve_closure closures (CVar x) = (case nth closures x of CDummy => CVar x | r => r)
- | resolve_closure closures (CConst c) = CConst c
- | resolve_closure closures (CApp (u, v)) = CApp (resolve_closure closures u, resolve_closure closures v)
- | resolve_closure closures (CAbs u) = CAbs (resolve_closure (CDummy::closures) u)
- | resolve_closure closures (CDummy) = raise (Run "internal error: resolve_closure applied to CDummy")
- | resolve_closure closures (Closure (e, u)) = resolve_closure e u
-
-fun resolve_closure' c = resolve_closure [] c
-
-fun resolve_stack tm SEmpty = tm
- | resolve_stack tm (SAppL (c, s)) = resolve_stack (CApp (tm, resolve_closure' c)) s
- | resolve_stack tm (SAppR (c, s)) = resolve_stack (CApp (resolve_closure' c, tm)) s
- | resolve_stack tm (SAbs s) = resolve_stack (CAbs tm) s
-
-fun resolve (stack, closure) =
- let
- val _ = writeln "start resolving"
- val t = resolve_stack (resolve_closure' closure) stack
- val _ = writeln "finished resolving"
- in
- t
- end
-
-fun strip_closure args (CApp (a,b)) = strip_closure (b::args) a
- | strip_closure args x = (x, args)
-
-fun len_head_of_closure n (CApp (a, _)) = len_head_of_closure (n+1) a
- | len_head_of_closure n x = (n, x)
-
-
-(* earlier occurrence of PVar corresponds to higher de Bruijn index *)
-fun pattern_match args PVar clos = SOME (clos::args)
- | pattern_match args (PConst (c, patterns)) clos =
- let
- val (f, closargs) = strip_closure [] clos
- in
- case f of
- CConst d =>
- if c = d then
- pattern_match_list args patterns closargs
- else
- NONE
- | _ => NONE
- end
-and pattern_match_list args [] [] = SOME args
- | pattern_match_list args (p::ps) (c::cs) =
- (case pattern_match args p c of
- NONE => NONE
- | SOME args => pattern_match_list args ps cs)
- | pattern_match_list _ _ _ = NONE
-
-fun count_patternvars PVar = 1
- | count_patternvars (PConst (_, ps)) = List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
-
-fun pattern_key (PConst (c, ps)) = (c, length ps)
- | pattern_key _ = raise (Compile "pattern reduces to variable")
-
-(*Returns true iff at most 0 .. (free-1) occur unbound. therefore
- check_freevars 0 t iff t is closed*)
-fun check_freevars free (Var x) = x < free
- | check_freevars free (Const _) = true
- | check_freevars free (App (u, v)) = check_freevars free u andalso check_freevars free v
- | check_freevars free (Abs m) = check_freevars (free+1) m
- | check_freevars free (Computed t) = check_freevars free t
-
-fun compile eqs =
- let
- fun check p r = if check_freevars p r then () else raise Compile ("unbound variables in rule")
- fun check_guard p (Guard (a,b)) = (check p a; check p b)
- fun clos_of_guard (Guard (a,b)) = (clos_of_term a, clos_of_term b)
- val eqs = map (fn (guards, p, r) => let val pcount = count_patternvars p val _ = map (check_guard pcount) (guards) val _ = check pcount r in
- (pattern_key p, (p, clos_of_term r, map clos_of_guard guards)) end) eqs
- fun merge (k, a) table = prog_struct.update (k, case prog_struct.lookup table k of NONE => [a] | SOME l => a::l) table
- val p = fold merge eqs prog_struct.empty
- in
- Program p
- end
-
-
-type state = bool * program * stack * closure
-
-datatype loopstate = Continue of state | Stop of stack * closure
-
-fun proj_C (Continue s) = s
- | proj_C _ = raise Match
-
-exception InterruptedExecution of stack * closure
-
-fun proj_S (Stop s) = s
- | proj_S (Continue (_,_,s,c)) = (s,c)
-
-fun cont (Continue _) = true
- | cont _ = false
-
-val max_reductions = Unsynchronized.ref (NONE : int option)
-
-fun do_reduction reduce p =
- let
- val s = Unsynchronized.ref (Continue p)
- val counter = Unsynchronized.ref 0
- val _ = case !max_reductions of
- NONE => while cont (!s) do (s := reduce (proj_C (!s)))
- | SOME m => while cont (!s) andalso (!counter < m) do (s := reduce (proj_C (!s)); counter := (!counter) + 1)
- in
- case !max_reductions of
- SOME m => if !counter >= m then raise InterruptedExecution (proj_S (!s)) else proj_S (!s)
- | NONE => proj_S (!s)
- end
-
-fun match_rules prog n [] clos = NONE
- | match_rules prog n ((p,eq,guards)::rs) clos =
- case pattern_match [] p clos of
- NONE => match_rules prog (n+1) rs clos
- | SOME args => if forall (guard_checks prog args) guards then SOME (Closure (args, eq)) else match_rules prog (n+1) rs clos
-and guard_checks prog args (a,b) = (simp prog (Closure (args, a)) = simp prog (Closure (args, b)))
-and match_closure (p as (Program prog)) clos =
- case len_head_of_closure 0 clos of
- (len, CConst c) =>
- (case prog_struct.lookup prog (c, len) of
- NONE => NONE
- | SOME rules => match_rules p 0 rules clos)
- | _ => NONE
-
-and weak_reduce (false, prog, stack, Closure (e, CApp (a, b))) = Continue (false, prog, SAppL (Closure (e, b), stack), Closure (e, a))
- | weak_reduce (false, prog, SAppL (b, stack), Closure (e, CAbs m)) = Continue (false, prog, stack, Closure (b::e, m))
- | weak_reduce (false, prog, stack, Closure (e, CVar n)) = Continue (false, prog, stack, case nth e n of CDummy => CVar n | r => r)
- | weak_reduce (false, prog, stack, Closure (_, c as CConst _)) = Continue (false, prog, stack, c)
- | weak_reduce (false, prog, stack, clos) =
- (case match_closure prog clos of
- NONE => Continue (true, prog, stack, clos)
- | SOME r => Continue (false, prog, stack, r))
- | weak_reduce (true, prog, SAppR (a, stack), b) = Continue (false, prog, stack, CApp (a,b))
- | weak_reduce (true, prog, SAppL (b, stack), a) = Continue (false, prog, SAppR (a, stack), b)
- | weak_reduce (true, prog, stack, c) = Stop (stack, c)
-
-and strong_reduce (false, prog, stack, Closure (e, CAbs m)) =
- (let
- val (stack', wnf) = do_reduction weak_reduce (false, prog, SEmpty, Closure (CDummy::e, m))
- in
- case stack' of
- SEmpty => Continue (false, prog, SAbs stack, wnf)
- | _ => raise (Run "internal error in strong: weak failed")
- end handle InterruptedExecution state => raise InterruptedExecution (stack, resolve state))
- | strong_reduce (false, prog, stack, CApp (u, v)) = Continue (false, prog, SAppL (v, stack), u)
- | strong_reduce (false, prog, stack, clos) = Continue (true, prog, stack, clos)
- | strong_reduce (true, prog, SAbs stack, m) = Continue (false, prog, stack, CAbs m)
- | strong_reduce (true, prog, SAppL (b, stack), a) = Continue (false, prog, SAppR (a, stack), b)
- | strong_reduce (true, prog, SAppR (a, stack), b) = Continue (true, prog, stack, CApp (a, b))
- | strong_reduce (true, prog, stack, clos) = Stop (stack, clos)
-
-and simp prog t =
- (let
- val (stack, wnf) = do_reduction weak_reduce (false, prog, SEmpty, t)
- in
- case stack of
- SEmpty => (case do_reduction strong_reduce (false, prog, SEmpty, wnf) of
- (SEmpty, snf) => snf
- | _ => raise (Run "internal error in run: strong failed"))
- | _ => raise (Run "internal error in run: weak failed")
- end handle InterruptedExecution state => resolve state)
-
-
-fun run prog t =
- (let
- val (stack, wnf) = do_reduction weak_reduce (false, prog, SEmpty, Closure ([], clos_of_term t))
- in
- case stack of
- SEmpty => (case do_reduction strong_reduce (false, prog, SEmpty, wnf) of
- (SEmpty, snf) => term_of_clos snf
- | _ => raise (Run "internal error in run: strong failed"))
- | _ => raise (Run "internal error in run: weak failed")
- end handle InterruptedExecution state => term_of_clos (resolve state))
-
-end
--- a/src/HOL/Matrix/Compute_Oracle/am_sml.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,517 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/am_sml.ML
- Author: Steven Obua
-
-TODO: "parameterless rewrite cannot be used in pattern": In a lot of
-cases it CAN be used, and these cases should be handled
-properly; right now, all cases raise an exception.
-*)
-
-signature AM_SML =
-sig
- include ABSTRACT_MACHINE
- val save_result : (string * term) -> unit
- val set_compiled_rewriter : (term -> term) -> unit
- val list_nth : 'a list * int -> 'a
- val dump_output : (string option) Unsynchronized.ref
-end
-
-structure AM_SML : AM_SML = struct
-
-open AbstractMachine;
-
-val dump_output = Unsynchronized.ref (NONE: string option)
-
-type program = term Inttab.table * (term -> term)
-
-val saved_result = Unsynchronized.ref (NONE:(string*term)option)
-
-fun save_result r = (saved_result := SOME r)
-
-val list_nth = List.nth
-
-val compiled_rewriter = Unsynchronized.ref (NONE:(term -> term)Option.option)
-
-fun set_compiled_rewriter r = (compiled_rewriter := SOME r)
-
-fun count_patternvars PVar = 1
- | count_patternvars (PConst (_, ps)) =
- List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
-
-fun update_arity arity code a =
- (case Inttab.lookup arity code of
- NONE => Inttab.update_new (code, a) arity
- | SOME (a': int) => if a > a' then Inttab.update (code, a) arity else arity)
-
-(* We have to find out the maximal arity of each constant *)
-fun collect_pattern_arity PVar arity = arity
- | collect_pattern_arity (PConst (c, args)) arity = fold collect_pattern_arity args (update_arity arity c (length args))
-
-(* We also need to find out the maximal toplevel arity of each function constant *)
-fun collect_pattern_toplevel_arity PVar arity = raise Compile "internal error: collect_pattern_toplevel_arity"
- | collect_pattern_toplevel_arity (PConst (c, args)) arity = update_arity arity c (length args)
-
-local
-fun collect applevel (Var _) arity = arity
- | collect applevel (Const c) arity = update_arity arity c applevel
- | collect applevel (Abs m) arity = collect 0 m arity
- | collect applevel (App (a,b)) arity = collect 0 b (collect (applevel + 1) a arity)
-in
-fun collect_term_arity t arity = collect 0 t arity
-end
-
-fun collect_guard_arity (Guard (a,b)) arity = collect_term_arity b (collect_term_arity a arity)
-
-
-fun rep n x = if n < 0 then raise Compile "internal error: rep" else if n = 0 then [] else x::(rep (n-1) x)
-
-fun beta (Const c) = Const c
- | beta (Var i) = Var i
- | beta (App (Abs m, b)) = beta (unlift 0 (subst 0 m (lift 0 b)))
- | beta (App (a, b)) =
- (case beta a of
- Abs m => beta (App (Abs m, b))
- | a => App (a, beta b))
- | beta (Abs m) = Abs (beta m)
- | beta (Computed t) = Computed t
-and subst x (Const c) t = Const c
- | subst x (Var i) t = if i = x then t else Var i
- | subst x (App (a,b)) t = App (subst x a t, subst x b t)
- | subst x (Abs m) t = Abs (subst (x+1) m (lift 0 t))
-and lift level (Const c) = Const c
- | lift level (App (a,b)) = App (lift level a, lift level b)
- | lift level (Var i) = if i < level then Var i else Var (i+1)
- | lift level (Abs m) = Abs (lift (level + 1) m)
-and unlift level (Const c) = Const c
- | unlift level (App (a, b)) = App (unlift level a, unlift level b)
- | unlift level (Abs m) = Abs (unlift (level+1) m)
- | unlift level (Var i) = if i < level then Var i else Var (i-1)
-
-fun nlift level n (Var m) = if m < level then Var m else Var (m+n)
- | nlift level n (Const c) = Const c
- | nlift level n (App (a,b)) = App (nlift level n a, nlift level n b)
- | nlift level n (Abs b) = Abs (nlift (level+1) n b)
-
-fun subst_const (c, t) (Const c') = if c = c' then t else Const c'
- | subst_const _ (Var i) = Var i
- | subst_const ct (App (a, b)) = App (subst_const ct a, subst_const ct b)
- | subst_const ct (Abs m) = Abs (subst_const ct m)
-
-(* Remove all rules that are just parameterless rewrites. This is necessary because SML does not allow functions with no parameters. *)
-fun inline_rules rules =
- let
- fun term_contains_const c (App (a, b)) = term_contains_const c a orelse term_contains_const c b
- | term_contains_const c (Abs m) = term_contains_const c m
- | term_contains_const c (Var _) = false
- | term_contains_const c (Const c') = (c = c')
- fun find_rewrite [] = NONE
- | find_rewrite ((prems, PConst (c, []), r) :: _) =
- if check_freevars 0 r then
- if term_contains_const c r then
- raise Compile "parameterless rewrite is caught in cycle"
- else if not (null prems) then
- raise Compile "parameterless rewrite may not be guarded"
- else
- SOME (c, r)
- else raise Compile "unbound variable on right hand side or guards of rule"
- | find_rewrite (_ :: rules) = find_rewrite rules
- fun remove_rewrite _ [] = []
- | remove_rewrite (cr as (c, r)) ((rule as (prems', PConst (c', args), r')) :: rules) =
- if c = c' then
- if null args andalso r = r' andalso null prems' then remove_rewrite cr rules
- else raise Compile "incompatible parameterless rewrites found"
- else
- rule :: remove_rewrite cr rules
- | remove_rewrite cr (r :: rs) = r :: remove_rewrite cr rs
- fun pattern_contains_const c (PConst (c', args)) = c = c' orelse exists (pattern_contains_const c) args
- | pattern_contains_const c (PVar) = false
- fun inline_rewrite (ct as (c, _)) (prems, p, r) =
- if pattern_contains_const c p then
- raise Compile "parameterless rewrite cannot be used in pattern"
- else (map (fn (Guard (a, b)) => Guard (subst_const ct a, subst_const ct b)) prems, p, subst_const ct r)
- fun inline inlined rules =
- case find_rewrite rules of
- NONE => (Inttab.make inlined, rules)
- | SOME ct =>
- let
- val rules = map (inline_rewrite ct) (remove_rewrite ct rules)
- val inlined = ct :: (map o apsnd) (subst_const ct) inlined
- in inline inlined rules end
- in
- inline [] rules
- end
-
-
-(*
- Calculate the arity, the toplevel_arity, and adjust rules so that all toplevel pattern constants have maximal arity.
- Also beta reduce the adjusted right hand side of a rule.
-*)
-fun adjust_rules rules =
- let
- val arity = fold (fn (prems, p, t) => fn arity => fold collect_guard_arity prems (collect_term_arity t (collect_pattern_arity p arity))) rules Inttab.empty
- val toplevel_arity = fold (fn (_, p, _) => fn arity => collect_pattern_toplevel_arity p arity) rules Inttab.empty
- fun arity_of c = the (Inttab.lookup arity c)
- fun test_pattern PVar = ()
- | test_pattern (PConst (c, args)) = if (length args <> arity_of c) then raise Compile ("Constant inside pattern must have maximal arity") else (map test_pattern args; ())
- fun adjust_rule (_, PVar, _) = raise Compile ("pattern may not be a variable")
- | adjust_rule (_, PConst (_, []), _) = raise Compile ("cannot deal with rewrites that take no parameters")
- | adjust_rule (rule as (prems, p as PConst (c, args),t)) =
- let
- val patternvars_counted = count_patternvars p
- fun check_fv t = check_freevars patternvars_counted t
- val _ = if not (check_fv t) then raise Compile ("unbound variables on right hand side of rule") else ()
- val _ = if not (forall (fn (Guard (a,b)) => check_fv a andalso check_fv b) prems) then raise Compile ("unbound variables in guards") else ()
- val _ = map test_pattern args
- val len = length args
- val arity = arity_of c
- val lift = nlift 0
- fun addapps_tm n t = if n=0 then t else addapps_tm (n-1) (App (t, Var (n-1)))
- fun adjust_term n t = addapps_tm n (lift n t)
- fun adjust_guard n (Guard (a,b)) = Guard (lift n a, lift n b)
- in
- if len = arity then
- rule
- else if arity >= len then
- (map (adjust_guard (arity-len)) prems, PConst (c, args @ (rep (arity-len) PVar)), adjust_term (arity-len) t)
- else (raise Compile "internal error in adjust_rule")
- end
- fun beta_rule (prems, p, t) = ((prems, p, beta t) handle Match => raise Compile "beta_rule")
- in
- (arity, toplevel_arity, map (beta_rule o adjust_rule) rules)
- end
-
-fun print_term module arity_of toplevel_arity_of pattern_var_count pattern_lazy_var_count =
-let
- fun str x = string_of_int x
- fun protect_blank s = if exists_string Symbol.is_ascii_blank s then "(" ^ s ^")" else s
- val module_prefix = (case module of NONE => "" | SOME s => s^".")
- fun print_apps d f [] = f
- | print_apps d f (a::args) = print_apps d (module_prefix^"app "^(protect_blank f)^" "^(protect_blank (print_term d a))) args
- and print_call d (App (a, b)) args = print_call d a (b::args)
- | print_call d (Const c) args =
- (case arity_of c of
- NONE => print_apps d (module_prefix^"Const "^(str c)) args
- | SOME 0 => module_prefix^"C"^(str c)
- | SOME a =>
- let
- val len = length args
- in
- if a <= len then
- let
- val strict_a = (case toplevel_arity_of c of SOME sa => sa | NONE => a)
- val _ = if strict_a > a then raise Compile "strict" else ()
- val s = module_prefix^"c"^(str c)^(implode (map (fn t => " "^(protect_blank (print_term d t))) (List.take (args, strict_a))))
- val s = s^(implode (map (fn t => " (fn () => "^print_term d t^")") (List.drop (List.take (args, a), strict_a))))
- in
- print_apps d s (List.drop (args, a))
- end
- else
- let
- fun mk_apps n t = if n = 0 then t else mk_apps (n-1) (App (t, Var (n - 1)))
- fun mk_lambdas n t = if n = 0 then t else mk_lambdas (n-1) (Abs t)
- fun append_args [] t = t
- | append_args (c::cs) t = append_args cs (App (t, c))
- in
- print_term d (mk_lambdas (a-len) (mk_apps (a-len) (nlift 0 (a-len) (append_args args (Const c)))))
- end
- end)
- | print_call d t args = print_apps d (print_term d t) args
- and print_term d (Var x) =
- if x < d then
- "b"^(str (d-x-1))
- else
- let
- val n = pattern_var_count - (x-d) - 1
- val x = "x"^(str n)
- in
- if n < pattern_var_count - pattern_lazy_var_count then
- x
- else
- "("^x^" ())"
- end
- | print_term d (Abs c) = module_prefix^"Abs (fn b"^(str d)^" => "^(print_term (d + 1) c)^")"
- | print_term d t = print_call d t []
-in
- print_term 0
-end
-
-fun section n = if n = 0 then [] else (section (n-1))@[n-1]
-
-fun print_rule gnum arity_of toplevel_arity_of (guards, p, t) =
- let
- fun str x = string_of_int x
- fun print_pattern top n PVar = (n+1, "x"^(str n))
- | print_pattern top n (PConst (c, [])) = (n, (if top then "c" else "C")^(str c)^(if top andalso gnum > 0 then "_"^(str gnum) else ""))
- | print_pattern top n (PConst (c, args)) =
- let
- val f = (if top then "c" else "C")^(str c)^(if top andalso gnum > 0 then "_"^(str gnum) else "")
- val (n, s) = print_pattern_list 0 top (n, f) args
- in
- (n, s)
- end
- and print_pattern_list' counter top (n,p) [] = if top then (n,p) else (n,p^")")
- | print_pattern_list' counter top (n, p) (t::ts) =
- let
- val (n, t) = print_pattern false n t
- in
- print_pattern_list' (counter + 1) top (n, if top then p^" (a"^(str counter)^" as ("^t^"))" else p^", "^t) ts
- end
- and print_pattern_list counter top (n, p) (t::ts) =
- let
- val (n, t) = print_pattern false n t
- in
- print_pattern_list' (counter + 1) top (n, if top then p^" (a"^(str counter)^" as ("^t^"))" else p^" ("^t) ts
- end
- val c = (case p of PConst (c, _) => c | _ => raise Match)
- val (n, pattern) = print_pattern true 0 p
- val lazy_vars = the (arity_of c) - the (toplevel_arity_of c)
- fun print_tm tm = print_term NONE arity_of toplevel_arity_of n lazy_vars tm
- fun print_guard (Guard (a,b)) = "term_eq ("^(print_tm a)^") ("^(print_tm b)^")"
- val else_branch = "c"^(str c)^"_"^(str (gnum+1))^(implode (map (fn i => " a"^(str i)) (section (the (arity_of c)))))
- fun print_guards t [] = print_tm t
- | print_guards t (g::gs) = "if ("^(print_guard g)^")"^(implode (map (fn g => " andalso ("^(print_guard g)^")") gs))^" then ("^(print_tm t)^") else "^else_branch
- in
- (if null guards then gnum else gnum+1, pattern^" = "^(print_guards t guards))
- end
-
-fun group_rules rules =
- let
- fun add_rule (r as (_, PConst (c,_), _)) groups =
- let
- val rs = (case Inttab.lookup groups c of NONE => [] | SOME rs => rs)
- in
- Inttab.update (c, r::rs) groups
- end
- | add_rule _ _ = raise Compile "internal error group_rules"
- in
- fold_rev add_rule rules Inttab.empty
- end
-
-fun sml_prog name code rules =
- let
- val buffer = Unsynchronized.ref ""
- fun write s = (buffer := (!buffer)^s)
- fun writeln s = (write s; write "\n")
- fun writelist [] = ()
- | writelist (s::ss) = (writeln s; writelist ss)
- fun str i = string_of_int i
- val (inlinetab, rules) = inline_rules rules
- val (arity, toplevel_arity, rules) = adjust_rules rules
- val rules = group_rules rules
- val constants = Inttab.keys arity
- fun arity_of c = Inttab.lookup arity c
- fun toplevel_arity_of c = Inttab.lookup toplevel_arity c
- fun rep_str s n = implode (rep n s)
- fun indexed s n = s^(str n)
- fun string_of_tuple [] = ""
- | string_of_tuple (x::xs) = "("^x^(implode (map (fn s => ", "^s) xs))^")"
- fun string_of_args [] = ""
- | string_of_args (x::xs) = x^(implode (map (fn s => " "^s) xs))
- fun default_case gnum c =
- let
- val leftargs = implode (map (indexed " x") (section (the (arity_of c))))
- val rightargs = section (the (arity_of c))
- val strict_args = (case toplevel_arity_of c of NONE => the (arity_of c) | SOME sa => sa)
- val xs = map (fn n => if n < strict_args then "x"^(str n) else "x"^(str n)^"()") rightargs
- val right = (indexed "C" c)^" "^(string_of_tuple xs)
- val message = "(\"unresolved lazy call: " ^ string_of_int c ^ "\")"
- val right = if strict_args < the (arity_of c) then "raise AM_SML.Run "^message else right
- in
- (indexed "c" c)^(if gnum > 0 then "_"^(str gnum) else "")^leftargs^" = "^right
- end
-
- fun eval_rules c =
- let
- val arity = the (arity_of c)
- val strict_arity = (case toplevel_arity_of c of NONE => arity | SOME sa => sa)
- fun eval_rule n =
- let
- val sc = string_of_int c
- val left = fold (fn i => fn s => "AbstractMachine.App ("^s^(indexed ", x" i)^")") (section n) ("AbstractMachine.Const "^sc)
- fun arg i =
- let
- val x = indexed "x" i
- val x = if i < n then "(eval bounds "^x^")" else x
- val x = if i < strict_arity then x else "(fn () => "^x^")"
- in
- x
- end
- val right = "c"^sc^" "^(string_of_args (map arg (section arity)))
- val right = fold_rev (fn i => fn s => "Abs (fn "^(indexed "x" i)^" => "^s^")") (List.drop (section arity, n)) right
- val right = if arity > 0 then right else "C"^sc
- in
- " | eval bounds ("^left^") = "^right
- end
- in
- map eval_rule (rev (section (arity + 1)))
- end
-
- fun convert_computed_rules (c: int) : string list =
- let
- val arity = the (arity_of c)
- fun eval_rule () =
- let
- val sc = string_of_int c
- val left = fold (fn i => fn s => "AbstractMachine.App ("^s^(indexed ", x" i)^")") (section arity) ("AbstractMachine.Const "^sc)
- fun arg i = "(convert_computed "^(indexed "x" i)^")"
- val right = "C"^sc^" "^(string_of_tuple (map arg (section arity)))
- val right = if arity > 0 then right else "C"^sc
- in
- " | convert_computed ("^left^") = "^right
- end
- in
- [eval_rule ()]
- end
-
- fun mk_constr_type_args n = if n > 0 then " of Term "^(rep_str " * Term" (n-1)) else ""
- val _ = writelist [
- "structure "^name^" = struct",
- "",
- "datatype Term = Const of int | App of Term * Term | Abs of (Term -> Term)",
- " "^(implode (map (fn c => " | C"^(str c)^(mk_constr_type_args (the (arity_of c)))) constants)),
- ""]
- fun make_constr c argprefix = "(C"^(str c)^" "^(string_of_tuple (map (fn i => argprefix^(str i)) (section (the (arity_of c)))))^")"
- fun make_term_eq c = " | term_eq "^(make_constr c "a")^" "^(make_constr c "b")^" = "^
- (case the (arity_of c) of
- 0 => "true"
- | n =>
- let
- val eqs = map (fn i => "term_eq a"^(str i)^" b"^(str i)) (section n)
- val (eq, eqs) = (List.hd eqs, map (fn s => " andalso "^s) (List.tl eqs))
- in
- eq^(implode eqs)
- end)
- val _ = writelist [
- "fun term_eq (Const c1) (Const c2) = (c1 = c2)",
- " | term_eq (App (a1,a2)) (App (b1,b2)) = term_eq a1 b1 andalso term_eq a2 b2"]
- val _ = writelist (map make_term_eq constants)
- val _ = writelist [
- " | term_eq _ _ = false",
- ""
- ]
- val _ = writelist [
- "fun app (Abs a) b = a b",
- " | app a b = App (a, b)",
- ""]
- fun defcase gnum c = (case arity_of c of NONE => [] | SOME a => if a > 0 then [default_case gnum c] else [])
- fun writefundecl [] = ()
- | writefundecl (x::xs) = writelist ((("and "^x)::(map (fn s => " | "^s) xs)))
- fun list_group c = (case Inttab.lookup rules c of
- NONE => [defcase 0 c]
- | SOME rs =>
- let
- val rs =
- fold
- (fn r =>
- fn rs =>
- let
- val (gnum, l, rs) =
- (case rs of
- [] => (0, [], [])
- | (gnum, l)::rs => (gnum, l, rs))
- val (gnum', r) = print_rule gnum arity_of toplevel_arity_of r
- in
- if gnum' = gnum then
- (gnum, r::l)::rs
- else
- let
- val args = implode (map (fn i => " a"^(str i)) (section (the (arity_of c))))
- fun gnumc g = if g > 0 then "c"^(str c)^"_"^(str g)^args else "c"^(str c)^args
- val s = gnumc (gnum) ^ " = " ^ gnumc (gnum')
- in
- (gnum', [])::(gnum, s::r::l)::rs
- end
- end)
- rs []
- val rs = (case rs of [] => [(0,defcase 0 c)] | (gnum,l)::rs => (gnum, (defcase gnum c)@l)::rs)
- in
- rev (map (fn z => rev (snd z)) rs)
- end)
- val _ = map (fn z => (map writefundecl z; writeln "")) (map list_group constants)
- val _ = writelist [
- "fun convert (Const i) = AM_SML.Const i",
- " | convert (App (a, b)) = AM_SML.App (convert a, convert b)",
- " | convert (Abs _) = raise AM_SML.Run \"no abstraction in result allowed\""]
- fun make_convert c =
- let
- val args = map (indexed "a") (section (the (arity_of c)))
- val leftargs =
- case args of
- [] => ""
- | (x::xs) => "("^x^(implode (map (fn s => ", "^s) xs))^")"
- val args = map (indexed "convert a") (section (the (arity_of c)))
- val right = fold (fn x => fn s => "AM_SML.App ("^s^", "^x^")") args ("AM_SML.Const "^(str c))
- in
- " | convert (C"^(str c)^" "^leftargs^") = "^right
- end
- val _ = writelist (map make_convert constants)
- val _ = writelist [
- "",
- "fun convert_computed (AbstractMachine.Abs b) = raise AM_SML.Run \"no abstraction in convert_computed allowed\"",
- " | convert_computed (AbstractMachine.Var i) = raise AM_SML.Run \"no bound variables in convert_computed allowed\""]
- val _ = map (writelist o convert_computed_rules) constants
- val _ = writelist [
- " | convert_computed (AbstractMachine.Const c) = Const c",
- " | convert_computed (AbstractMachine.App (a, b)) = App (convert_computed a, convert_computed b)",
- " | convert_computed (AbstractMachine.Computed a) = raise AM_SML.Run \"no nesting in convert_computed allowed\""]
- val _ = writelist [
- "",
- "fun eval bounds (AbstractMachine.Abs m) = Abs (fn b => eval (b::bounds) m)",
- " | eval bounds (AbstractMachine.Var i) = AM_SML.list_nth (bounds, i)"]
- val _ = map (writelist o eval_rules) constants
- val _ = writelist [
- " | eval bounds (AbstractMachine.App (a, b)) = app (eval bounds a) (eval bounds b)",
- " | eval bounds (AbstractMachine.Const c) = Const c",
- " | eval bounds (AbstractMachine.Computed t) = convert_computed t"]
- val _ = writelist [
- "",
- "fun export term = AM_SML.save_result (\""^code^"\", convert term)",
- "",
- "val _ = AM_SML.set_compiled_rewriter (fn t => (convert (eval [] t)))",
- "",
- "end"]
- in
- (inlinetab, !buffer)
- end
-
-val guid_counter = Unsynchronized.ref 0
-fun get_guid () =
- let
- val c = !guid_counter
- val _ = guid_counter := !guid_counter + 1
- in
- string_of_int (Time.toMicroseconds (Time.now ())) ^ string_of_int c
- end
-
-
-fun writeTextFile name s = File.write (Path.explode name) s
-
-fun use_source src = use_text ML_Env.local_context (1, "") false src
-
-fun compile rules =
- let
- val guid = get_guid ()
- val code = Real.toString (random ())
- val name = "AMSML_"^guid
- val (inlinetab, source) = sml_prog name code rules
- val _ = case !dump_output of NONE => () | SOME p => writeTextFile p source
- val _ = compiled_rewriter := NONE
- val _ = use_source source
- in
- case !compiled_rewriter of
- NONE => raise Compile "broken link to compiled function"
- | SOME compiled_fun => (inlinetab, compiled_fun)
- end
-
-fun run (inlinetab, compiled_fun) t =
- let
- val _ = if check_freevars 0 t then () else raise Run ("can only compute closed terms")
- fun inline (Const c) = (case Inttab.lookup inlinetab c of NONE => Const c | SOME t => t)
- | inline (Var i) = Var i
- | inline (App (a, b)) = App (inline a, inline b)
- | inline (Abs m) = Abs (inline m)
- | inline (Computed t) = Computed t
- in
- compiled_fun (beta (inline t))
- end
-
-end
--- a/src/HOL/Matrix/Compute_Oracle/compute.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,653 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/compute.ML
- Author: Steven Obua
-*)
-
-signature COMPUTE = sig
-
- type computer
- type theorem
- type naming = int -> string
-
- datatype machine = BARRAS | BARRAS_COMPILED | HASKELL | SML
-
- (* Functions designated with a ! in front of them actually update the computer parameter *)
-
- exception Make of string
- val make : machine -> theory -> thm list -> computer
- val make_with_cache : machine -> theory -> term list -> thm list -> computer
- val theory_of : computer -> theory
- val hyps_of : computer -> term list
- val shyps_of : computer -> sort list
- (* ! *) val update : computer -> thm list -> unit
- (* ! *) val update_with_cache : computer -> term list -> thm list -> unit
-
- (* ! *) val set_naming : computer -> naming -> unit
- val naming_of : computer -> naming
-
- exception Compute of string
- val simplify : computer -> theorem -> thm
- val rewrite : computer -> cterm -> thm
-
- val make_theorem : computer -> thm -> string list -> theorem
- (* ! *) val instantiate : computer -> (string * cterm) list -> theorem -> theorem
- (* ! *) val evaluate_prem : computer -> int -> theorem -> theorem
- (* ! *) val modus_ponens : computer -> int -> thm -> theorem -> theorem
-
-end
-
-structure Compute :> COMPUTE = struct
-
-open Report;
-
-datatype machine = BARRAS | BARRAS_COMPILED | HASKELL | SML
-
-(* Terms are mapped to integer codes *)
-structure Encode :>
-sig
- type encoding
- val empty : encoding
- val insert : term -> encoding -> int * encoding
- val lookup_code : term -> encoding -> int option
- val lookup_term : int -> encoding -> term option
- val remove_code : int -> encoding -> encoding
- val remove_term : term -> encoding -> encoding
-end
-=
-struct
-
-type encoding = int * (int Termtab.table) * (term Inttab.table)
-
-val empty = (0, Termtab.empty, Inttab.empty)
-
-fun insert t (e as (count, term2int, int2term)) =
- (case Termtab.lookup term2int t of
- NONE => (count, (count+1, Termtab.update_new (t, count) term2int, Inttab.update_new (count, t) int2term))
- | SOME code => (code, e))
-
-fun lookup_code t (_, term2int, _) = Termtab.lookup term2int t
-
-fun lookup_term c (_, _, int2term) = Inttab.lookup int2term c
-
-fun remove_code c (e as (count, term2int, int2term)) =
- (case lookup_term c e of NONE => e | SOME t => (count, Termtab.delete t term2int, Inttab.delete c int2term))
-
-fun remove_term t (e as (count, term2int, int2term)) =
- (case lookup_code t e of NONE => e | SOME c => (count, Termtab.delete t term2int, Inttab.delete c int2term))
-
-end
-
-exception Make of string;
-exception Compute of string;
-
-local
- fun make_constant t encoding =
- let
- val (code, encoding) = Encode.insert t encoding
- in
- (encoding, AbstractMachine.Const code)
- end
-in
-
-fun remove_types encoding t =
- case t of
- Var _ => make_constant t encoding
- | Free _ => make_constant t encoding
- | Const _ => make_constant t encoding
- | Abs (_, _, t') =>
- let val (encoding, t'') = remove_types encoding t' in
- (encoding, AbstractMachine.Abs t'')
- end
- | a $ b =>
- let
- val (encoding, a) = remove_types encoding a
- val (encoding, b) = remove_types encoding b
- in
- (encoding, AbstractMachine.App (a,b))
- end
- | Bound b => (encoding, AbstractMachine.Var b)
-end
-
-local
- fun type_of (Free (_, ty)) = ty
- | type_of (Const (_, ty)) = ty
- | type_of (Var (_, ty)) = ty
- | type_of _ = raise Fail "infer_types: type_of error"
-in
-fun infer_types naming encoding =
- let
- fun infer_types _ bounds _ (AbstractMachine.Var v) = (Bound v, nth bounds v)
- | infer_types _ bounds _ (AbstractMachine.Const code) =
- let
- val c = the (Encode.lookup_term code encoding)
- in
- (c, type_of c)
- end
- | infer_types level bounds _ (AbstractMachine.App (a, b)) =
- let
- val (a, aty) = infer_types level bounds NONE a
- val (adom, arange) =
- case aty of
- Type ("fun", [dom, range]) => (dom, range)
- | _ => raise Fail "infer_types: function type expected"
- val (b, _) = infer_types level bounds (SOME adom) b
- in
- (a $ b, arange)
- end
- | infer_types level bounds (SOME (ty as Type ("fun", [dom, range]))) (AbstractMachine.Abs m) =
- let
- val (m, _) = infer_types (level+1) (dom::bounds) (SOME range) m
- in
- (Abs (naming level, dom, m), ty)
- end
- | infer_types _ _ NONE (AbstractMachine.Abs _) =
- raise Fail "infer_types: cannot infer type of abstraction"
-
- fun infer ty term =
- let
- val (term', _) = infer_types 0 [] (SOME ty) term
- in
- term'
- end
- in
- infer
- end
-end
-
-datatype prog =
- ProgBarras of AM_Interpreter.program
- | ProgBarrasC of AM_Compiler.program
- | ProgHaskell of AM_GHC.program
- | ProgSML of AM_SML.program
-
-fun machine_of_prog (ProgBarras _) = BARRAS
- | machine_of_prog (ProgBarrasC _) = BARRAS_COMPILED
- | machine_of_prog (ProgHaskell _) = HASKELL
- | machine_of_prog (ProgSML _) = SML
-
-type naming = int -> string
-
-fun default_naming i = "v_" ^ string_of_int i
-
-datatype computer = Computer of
- (theory_ref * Encode.encoding * term list * unit Sorttab.table * prog * unit Unsynchronized.ref * naming)
- option Unsynchronized.ref
-
-fun theory_of (Computer (Unsynchronized.ref (SOME (rthy,_,_,_,_,_,_)))) = Theory.deref rthy
-fun hyps_of (Computer (Unsynchronized.ref (SOME (_,_,hyps,_,_,_,_)))) = hyps
-fun shyps_of (Computer (Unsynchronized.ref (SOME (_,_,_,shyptable,_,_,_)))) = Sorttab.keys (shyptable)
-fun shyptab_of (Computer (Unsynchronized.ref (SOME (_,_,_,shyptable,_,_,_)))) = shyptable
-fun stamp_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,_,stamp,_)))) = stamp
-fun prog_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,prog,_,_)))) = prog
-fun encoding_of (Computer (Unsynchronized.ref (SOME (_,encoding,_,_,_,_,_)))) = encoding
-fun set_encoding (Computer (r as Unsynchronized.ref (SOME (p1,_,p2,p3,p4,p5,p6)))) encoding' =
- (r := SOME (p1,encoding',p2,p3,p4,p5,p6))
-fun naming_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,_,_,n)))) = n
-fun set_naming (Computer (r as Unsynchronized.ref (SOME (p1,p2,p3,p4,p5,p6,_)))) naming'=
- (r := SOME (p1,p2,p3,p4,p5,p6,naming'))
-
-fun ref_of (Computer r) = r
-
-datatype cthm = ComputeThm of term list * sort list * term
-
-fun thm2cthm th =
- let
- val {hyps, prop, tpairs, shyps, ...} = Thm.rep_thm th
- val _ = if not (null tpairs) then raise Make "theorems may not contain tpairs" else ()
- in
- ComputeThm (hyps, shyps, prop)
- end
-
-fun make_internal machine thy stamp encoding cache_pattern_terms raw_ths =
- let
- fun transfer (x:thm) = Thm.transfer thy x
- val ths = map (thm2cthm o Thm.strip_shyps o transfer) raw_ths
-
- fun make_pattern encoding n vars (AbstractMachine.Abs _) =
- raise (Make "no lambda abstractions allowed in pattern")
- | make_pattern encoding n vars (AbstractMachine.Var _) =
- raise (Make "no bound variables allowed in pattern")
- | make_pattern encoding n vars (AbstractMachine.Const code) =
- (case the (Encode.lookup_term code encoding) of
- Var _ => ((n+1, Inttab.update_new (code, n) vars, AbstractMachine.PVar)
- handle Inttab.DUP _ => raise (Make "no duplicate variable in pattern allowed"))
- | _ => (n, vars, AbstractMachine.PConst (code, [])))
- | make_pattern encoding n vars (AbstractMachine.App (a, b)) =
- let
- val (n, vars, pa) = make_pattern encoding n vars a
- val (n, vars, pb) = make_pattern encoding n vars b
- in
- case pa of
- AbstractMachine.PVar =>
- raise (Make "patterns may not start with a variable")
- | AbstractMachine.PConst (c, args) =>
- (n, vars, AbstractMachine.PConst (c, args@[pb]))
- end
-
- fun thm2rule (encoding, hyptable, shyptable) th =
- let
- val (ComputeThm (hyps, shyps, prop)) = th
- val hyptable = fold (fn h => Termtab.update (h, ())) hyps hyptable
- val shyptable = fold (fn sh => Sorttab.update (sh, ())) shyps shyptable
- val (prems, prop) = (Logic.strip_imp_prems prop, Logic.strip_imp_concl prop)
- val (a, b) = Logic.dest_equals prop
- handle TERM _ => raise (Make "theorems must be meta-level equations (with optional guards)")
- val a = Envir.eta_contract a
- val b = Envir.eta_contract b
- val prems = map Envir.eta_contract prems
-
- val (encoding, left) = remove_types encoding a
- val (encoding, right) = remove_types encoding b
- fun remove_types_of_guard encoding g =
- (let
- val (t1, t2) = Logic.dest_equals g
- val (encoding, t1) = remove_types encoding t1
- val (encoding, t2) = remove_types encoding t2
- in
- (encoding, AbstractMachine.Guard (t1, t2))
- end handle TERM _ => raise (Make "guards must be meta-level equations"))
- val (encoding, prems) = fold_rev (fn p => fn (encoding, ps) => let val (e, p) = remove_types_of_guard encoding p in (e, p::ps) end) prems (encoding, [])
-
- (* Principally, a check should be made here to see if the (meta-) hyps contain any of the variables of the rule.
- As it is, all variables of the rule are schematic, and there are no schematic variables in meta-hyps, therefore
- this check can be left out. *)
-
- val (vcount, vars, pattern) = make_pattern encoding 0 Inttab.empty left
- val _ = (case pattern of
- AbstractMachine.PVar =>
- raise (Make "patterns may not start with a variable")
- | _ => ())
-
- (* finally, provide a function for renaming the
- pattern bound variables on the right hand side *)
-
- fun rename level vars (var as AbstractMachine.Var _) = var
- | rename level vars (c as AbstractMachine.Const code) =
- (case Inttab.lookup vars code of
- NONE => c
- | SOME n => AbstractMachine.Var (vcount-n-1+level))
- | rename level vars (AbstractMachine.App (a, b)) =
- AbstractMachine.App (rename level vars a, rename level vars b)
- | rename level vars (AbstractMachine.Abs m) =
- AbstractMachine.Abs (rename (level+1) vars m)
-
- fun rename_guard (AbstractMachine.Guard (a,b)) =
- AbstractMachine.Guard (rename 0 vars a, rename 0 vars b)
- in
- ((encoding, hyptable, shyptable), (map rename_guard prems, pattern, rename 0 vars right))
- end
-
- val ((encoding, hyptable, shyptable), rules) =
- fold_rev (fn th => fn (encoding_hyptable, rules) =>
- let
- val (encoding_hyptable, rule) = thm2rule encoding_hyptable th
- in (encoding_hyptable, rule::rules) end)
- ths ((encoding, Termtab.empty, Sorttab.empty), [])
-
- fun make_cache_pattern t (encoding, cache_patterns) =
- let
- val (encoding, a) = remove_types encoding t
- val (_,_,p) = make_pattern encoding 0 Inttab.empty a
- in
- (encoding, p::cache_patterns)
- end
-
- val (encoding, _) = fold_rev make_cache_pattern cache_pattern_terms (encoding, [])
-
- val prog =
- case machine of
- BARRAS => ProgBarras (AM_Interpreter.compile rules)
- | BARRAS_COMPILED => ProgBarrasC (AM_Compiler.compile rules)
- | HASKELL => ProgHaskell (AM_GHC.compile rules)
- | SML => ProgSML (AM_SML.compile rules)
-
- fun has_witness s = not (null (Sign.witness_sorts thy [] [s]))
-
- val shyptable = fold Sorttab.delete (filter has_witness (Sorttab.keys (shyptable))) shyptable
-
- in (Theory.check_thy thy, encoding, Termtab.keys hyptable, shyptable, prog, stamp, default_naming) end
-
-fun make_with_cache machine thy cache_patterns raw_thms =
- Computer (Unsynchronized.ref (SOME (make_internal machine thy (Unsynchronized.ref ()) Encode.empty cache_patterns raw_thms)))
-
-fun make machine thy raw_thms = make_with_cache machine thy [] raw_thms
-
-fun update_with_cache computer cache_patterns raw_thms =
- let
- val c = make_internal (machine_of_prog (prog_of computer)) (theory_of computer) (stamp_of computer)
- (encoding_of computer) cache_patterns raw_thms
- val _ = (ref_of computer) := SOME c
- in
- ()
- end
-
-fun update computer raw_thms = update_with_cache computer [] raw_thms
-
-fun runprog (ProgBarras p) = AM_Interpreter.run p
- | runprog (ProgBarrasC p) = AM_Compiler.run p
- | runprog (ProgHaskell p) = AM_GHC.run p
- | runprog (ProgSML p) = AM_SML.run p
-
-(* ------------------------------------------------------------------------------------- *)
-(* An oracle for exporting theorems; must only be accessible from inside this structure! *)
-(* ------------------------------------------------------------------------------------- *)
-
-fun merge_hyps hyps1 hyps2 =
-let
- fun add hyps tab = fold (fn h => fn tab => Termtab.update (h, ()) tab) hyps tab
-in
- Termtab.keys (add hyps2 (add hyps1 Termtab.empty))
-end
-
-fun add_shyps shyps tab = fold (fn h => fn tab => Sorttab.update (h, ()) tab) shyps tab
-
-fun merge_shyps shyps1 shyps2 = Sorttab.keys (add_shyps shyps2 (add_shyps shyps1 Sorttab.empty))
-
-val (_, export_oracle) = Context.>>> (Context.map_theory_result
- (Thm.add_oracle (@{binding compute}, fn (thy, hyps, shyps, prop) =>
- let
- val shyptab = add_shyps shyps Sorttab.empty
- fun delete s shyptab = Sorttab.delete s shyptab handle Sorttab.UNDEF _ => shyptab
- fun delete_term t shyptab = fold delete (Sorts.insert_term t []) shyptab
- fun has_witness s = not (null (Sign.witness_sorts thy [] [s]))
- val shyptab = fold Sorttab.delete (filter has_witness (Sorttab.keys (shyptab))) shyptab
- val shyps = if Sorttab.is_empty shyptab then [] else Sorttab.keys (fold delete_term (prop::hyps) shyptab)
- val _ =
- if not (null shyps) then
- raise Compute ("dangling sort hypotheses: " ^
- commas (map (Syntax.string_of_sort_global thy) shyps))
- else ()
- in
- Thm.cterm_of thy (fold_rev (fn hyp => fn p => Logic.mk_implies (hyp, p)) hyps prop)
- end)));
-
-fun export_thm thy hyps shyps prop =
- let
- val th = export_oracle (thy, hyps, shyps, prop)
- val hyps = map (fn h => Thm.assume (cterm_of thy h)) hyps
- in
- fold (fn h => fn p => Thm.implies_elim p h) hyps th
- end
-
-(* --------- Rewrite ----------- *)
-
-fun rewrite computer ct =
- let
- val thy = Thm.theory_of_cterm ct
- val {t=t',T=ty,...} = rep_cterm ct
- val _ = Theory.assert_super (theory_of computer) thy
- val naming = naming_of computer
- val (encoding, t) = remove_types (encoding_of computer) t'
- val t = runprog (prog_of computer) t
- val t = infer_types naming encoding ty t
- val eq = Logic.mk_equals (t', t)
- in
- export_thm thy (hyps_of computer) (Sorttab.keys (shyptab_of computer)) eq
- end
-
-(* --------- Simplify ------------ *)
-
-datatype prem = EqPrem of AbstractMachine.term * AbstractMachine.term * Term.typ * int
- | Prem of AbstractMachine.term
-datatype theorem = Theorem of theory_ref * unit Unsynchronized.ref * (int * typ) Symtab.table * (AbstractMachine.term option) Inttab.table
- * prem list * AbstractMachine.term * term list * sort list
-
-
-exception ParamSimplify of computer * theorem
-
-fun make_theorem computer th vars =
-let
- val _ = Theory.assert_super (theory_of computer) (theory_of_thm th)
-
- val (ComputeThm (hyps, shyps, prop)) = thm2cthm th
-
- val encoding = encoding_of computer
-
- (* variables in the theorem are identified upfront *)
- fun collect_vars (Abs (_, _, t)) tab = collect_vars t tab
- | collect_vars (a $ b) tab = collect_vars b (collect_vars a tab)
- | collect_vars (Const _) tab = tab
- | collect_vars (Free _) tab = tab
- | collect_vars (Var ((s, i), ty)) tab =
- if List.find (fn x => x=s) vars = NONE then
- tab
- else
- (case Symtab.lookup tab s of
- SOME ((s',i'),ty') =>
- if s' <> s orelse i' <> i orelse ty <> ty' then
- raise Compute ("make_theorem: variable name '"^s^"' is not unique")
- else
- tab
- | NONE => Symtab.update (s, ((s, i), ty)) tab)
- val vartab = collect_vars prop Symtab.empty
- fun encodevar (s, t as (_, ty)) (encoding, tab) =
- let
- val (x, encoding) = Encode.insert (Var t) encoding
- in
- (encoding, Symtab.update (s, (x, ty)) tab)
- end
- val (encoding, vartab) = Symtab.fold encodevar vartab (encoding, Symtab.empty)
- val varsubst = Inttab.make (map (fn (_, (x, _)) => (x, NONE)) (Symtab.dest vartab))
-
- (* make the premises and the conclusion *)
- fun mk_prem encoding t =
- (let
- val (a, b) = Logic.dest_equals t
- val ty = type_of a
- val (encoding, a) = remove_types encoding a
- val (encoding, b) = remove_types encoding b
- val (eq, encoding) = Encode.insert (Const ("==", ty --> ty --> @{typ "prop"})) encoding
- in
- (encoding, EqPrem (a, b, ty, eq))
- end handle TERM _ => let val (encoding, t) = remove_types encoding t in (encoding, Prem t) end)
- val (encoding, prems) =
- (fold_rev (fn t => fn (encoding, l) =>
- case mk_prem encoding t of
- (encoding, t) => (encoding, t::l)) (Logic.strip_imp_prems prop) (encoding, []))
- val (encoding, concl) = remove_types encoding (Logic.strip_imp_concl prop)
- val _ = set_encoding computer encoding
-in
- Theorem (Theory.check_thy (theory_of_thm th), stamp_of computer, vartab, varsubst,
- prems, concl, hyps, shyps)
-end
-
-fun theory_of_theorem (Theorem (rthy,_,_,_,_,_,_,_)) = Theory.deref rthy
-fun update_theory thy (Theorem (_,p0,p1,p2,p3,p4,p5,p6)) =
- Theorem (Theory.check_thy thy,p0,p1,p2,p3,p4,p5,p6)
-fun stamp_of_theorem (Theorem (_,s, _, _, _, _, _, _)) = s
-fun vartab_of_theorem (Theorem (_,_,vt,_,_,_,_,_)) = vt
-fun varsubst_of_theorem (Theorem (_,_,_,vs,_,_,_,_)) = vs
-fun update_varsubst vs (Theorem (p0,p1,p2,_,p3,p4,p5,p6)) = Theorem (p0,p1,p2,vs,p3,p4,p5,p6)
-fun prems_of_theorem (Theorem (_,_,_,_,prems,_,_,_)) = prems
-fun update_prems prems (Theorem (p0,p1,p2,p3,_,p4,p5,p6)) = Theorem (p0,p1,p2,p3,prems,p4,p5,p6)
-fun concl_of_theorem (Theorem (_,_,_,_,_,concl,_,_)) = concl
-fun hyps_of_theorem (Theorem (_,_,_,_,_,_,hyps,_)) = hyps
-fun update_hyps hyps (Theorem (p0,p1,p2,p3,p4,p5,_,p6)) = Theorem (p0,p1,p2,p3,p4,p5,hyps,p6)
-fun shyps_of_theorem (Theorem (_,_,_,_,_,_,_,shyps)) = shyps
-fun update_shyps shyps (Theorem (p0,p1,p2,p3,p4,p5,p6,_)) = Theorem (p0,p1,p2,p3,p4,p5,p6,shyps)
-
-fun check_compatible computer th s =
- if stamp_of computer <> stamp_of_theorem th then
- raise Compute (s^": computer and theorem are incompatible")
- else ()
-
-fun instantiate computer insts th =
-let
- val _ = check_compatible computer th
-
- val thy = theory_of computer
-
- val vartab = vartab_of_theorem th
-
- fun rewrite computer t =
- let
- val (encoding, t) = remove_types (encoding_of computer) t
- val t = runprog (prog_of computer) t
- val _ = set_encoding computer encoding
- in
- t
- end
-
- fun assert_varfree vs t =
- if AbstractMachine.forall_consts (fn x => Inttab.lookup vs x = NONE) t then
- ()
- else
- raise Compute "instantiate: assert_varfree failed"
-
- fun assert_closed t =
- if AbstractMachine.closed t then
- ()
- else
- raise Compute "instantiate: not a closed term"
-
- fun compute_inst (s, ct) vs =
- let
- val _ = Theory.assert_super (theory_of_cterm ct) thy
- val ty = typ_of (ctyp_of_term ct)
- in
- (case Symtab.lookup vartab s of
- NONE => raise Compute ("instantiate: variable '"^s^"' not found in theorem")
- | SOME (x, ty') =>
- (case Inttab.lookup vs x of
- SOME (SOME _) => raise Compute ("instantiate: variable '"^s^"' has already been instantiated")
- | SOME NONE =>
- if ty <> ty' then
- raise Compute ("instantiate: wrong type for variable '"^s^"'")
- else
- let
- val t = rewrite computer (term_of ct)
- val _ = assert_varfree vs t
- val _ = assert_closed t
- in
- Inttab.update (x, SOME t) vs
- end
- | NONE => raise Compute "instantiate: internal error"))
- end
-
- val vs = fold compute_inst insts (varsubst_of_theorem th)
-in
- update_varsubst vs th
-end
-
-fun match_aterms subst =
- let
- exception no_match
- open AbstractMachine
- fun match subst (b as (Const c)) a =
- if a = b then subst
- else
- (case Inttab.lookup subst c of
- SOME (SOME a') => if a=a' then subst else raise no_match
- | SOME NONE => if AbstractMachine.closed a then
- Inttab.update (c, SOME a) subst
- else raise no_match
- | NONE => raise no_match)
- | match subst (b as (Var _)) a = if a=b then subst else raise no_match
- | match subst (App (u, v)) (App (u', v')) = match (match subst u u') v v'
- | match subst (Abs u) (Abs u') = match subst u u'
- | match subst _ _ = raise no_match
- in
- fn b => fn a => (SOME (match subst b a) handle no_match => NONE)
- end
-
-fun apply_subst vars_allowed subst =
- let
- open AbstractMachine
- fun app (t as (Const c)) =
- (case Inttab.lookup subst c of
- NONE => t
- | SOME (SOME t) => Computed t
- | SOME NONE => if vars_allowed then t else raise Compute "apply_subst: no vars allowed")
- | app (t as (Var _)) = t
- | app (App (u, v)) = App (app u, app v)
- | app (Abs m) = Abs (app m)
- in
- app
- end
-
-fun splicein n l L = List.take (L, n) @ l @ List.drop (L, n+1)
-
-fun evaluate_prem computer prem_no th =
-let
- val _ = check_compatible computer th
- val prems = prems_of_theorem th
- val varsubst = varsubst_of_theorem th
- fun run vars_allowed t =
- runprog (prog_of computer) (apply_subst vars_allowed varsubst t)
-in
- case nth prems prem_no of
- Prem _ => raise Compute "evaluate_prem: no equality premise"
- | EqPrem (a, b, ty, _) =>
- let
- val a' = run false a
- val b' = run true b
- in
- case match_aterms varsubst b' a' of
- NONE =>
- let
- fun mk s = Syntax.string_of_term_global Pure.thy
- (infer_types (naming_of computer) (encoding_of computer) ty s)
- val left = "computed left side: "^(mk a')
- val right = "computed right side: "^(mk b')
- in
- raise Compute ("evaluate_prem: cannot assign computed left to right hand side\n"^left^"\n"^right^"\n")
- end
- | SOME varsubst =>
- update_prems (splicein prem_no [] prems) (update_varsubst varsubst th)
- end
-end
-
-fun prem2term (Prem t) = t
- | prem2term (EqPrem (a,b,_,eq)) =
- AbstractMachine.App (AbstractMachine.App (AbstractMachine.Const eq, a), b)
-
-fun modus_ponens computer prem_no th' th =
-let
- val _ = check_compatible computer th
- val thy =
- let
- val thy1 = theory_of_theorem th
- val thy2 = theory_of_thm th'
- in
- if Theory.subthy (thy1, thy2) then thy2
- else if Theory.subthy (thy2, thy1) then thy1 else
- raise Compute "modus_ponens: theorems are not compatible with each other"
- end
- val th' = make_theorem computer th' []
- val varsubst = varsubst_of_theorem th
- fun run vars_allowed t =
- runprog (prog_of computer) (apply_subst vars_allowed varsubst t)
- val prems = prems_of_theorem th
- val prem = run true (prem2term (nth prems prem_no))
- val concl = run false (concl_of_theorem th')
-in
- case match_aterms varsubst prem concl of
- NONE => raise Compute "modus_ponens: conclusion does not match premise"
- | SOME varsubst =>
- let
- val th = update_varsubst varsubst th
- val th = update_prems (splicein prem_no (prems_of_theorem th') prems) th
- val th = update_hyps (merge_hyps (hyps_of_theorem th) (hyps_of_theorem th')) th
- val th = update_shyps (merge_shyps (shyps_of_theorem th) (shyps_of_theorem th')) th
- in
- update_theory thy th
- end
-end
-
-fun simplify computer th =
-let
- val _ = check_compatible computer th
- val varsubst = varsubst_of_theorem th
- val encoding = encoding_of computer
- val naming = naming_of computer
- fun infer t = infer_types naming encoding @{typ "prop"} t
- fun run t = infer (runprog (prog_of computer) (apply_subst true varsubst t))
- fun runprem p = run (prem2term p)
- val prop = Logic.list_implies (map runprem (prems_of_theorem th), run (concl_of_theorem th))
- val hyps = merge_hyps (hyps_of computer) (hyps_of_theorem th)
- val shyps = merge_shyps (shyps_of_theorem th) (Sorttab.keys (shyptab_of computer))
-in
- export_thm (theory_of_theorem th) hyps shyps prop
-end
-
-end
-
--- a/src/HOL/Matrix/Compute_Oracle/linker.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,470 +0,0 @@
-(* Title: HOL/Matrix/Compute_Oracle/linker.ML
- Author: Steven Obua
-
-This module solves the problem that the computing oracle does not
-instantiate polymorphic rules. By going through the PCompute
-interface, all possible instantiations are resolved by compiling new
-programs, if necessary. The obvious disadvantage of this approach is
-that in the worst case for each new term to be rewritten, a new
-program may be compiled.
-*)
-
-(*
- Given constants/frees c_1::t_1, c_2::t_2, ...., c_n::t_n,
- and constants/frees d_1::d_1, d_2::s_2, ..., d_m::s_m
-
- Find all substitutions S such that
- a) the domain of S is tvars (t_1, ..., t_n)
- b) there are indices i_1, ..., i_k, and j_1, ..., j_k with
- 1. S (c_i_1::t_i_1) = d_j_1::s_j_1, ..., S (c_i_k::t_i_k) = d_j_k::s_j_k
- 2. tvars (t_i_1, ..., t_i_k) = tvars (t_1, ..., t_n)
-*)
-signature LINKER =
-sig
- exception Link of string
-
- datatype constant = Constant of bool * string * typ
- val constant_of : term -> constant
-
- type instances
- type subst = Type.tyenv
-
- val empty : constant list -> instances
- val typ_of_constant : constant -> typ
- val add_instances : theory -> instances -> constant list -> subst list * instances
- val substs_of : instances -> subst list
- val is_polymorphic : constant -> bool
- val distinct_constants : constant list -> constant list
- val collect_consts : term list -> constant list
-end
-
-structure Linker : LINKER = struct
-
-exception Link of string;
-
-type subst = Type.tyenv
-
-datatype constant = Constant of bool * string * typ
-fun constant_of (Const (name, ty)) = Constant (false, name, ty)
- | constant_of (Free (name, ty)) = Constant (true, name, ty)
- | constant_of _ = raise Link "constant_of"
-
-fun bool_ord (x,y) = if x then (if y then EQUAL else GREATER) else (if y then LESS else EQUAL)
-fun constant_ord (Constant (x1,x2,x3), Constant (y1,y2,y3)) = (prod_ord (prod_ord bool_ord fast_string_ord) Term_Ord.typ_ord) (((x1,x2),x3), ((y1,y2),y3))
-fun constant_modty_ord (Constant (x1,x2,_), Constant (y1,y2,_)) = (prod_ord bool_ord fast_string_ord) ((x1,x2), (y1,y2))
-
-
-structure Consttab = Table(type key = constant val ord = constant_ord);
-structure ConsttabModTy = Table(type key = constant val ord = constant_modty_ord);
-
-fun typ_of_constant (Constant (_, _, ty)) = ty
-
-val empty_subst = (Vartab.empty : Type.tyenv)
-
-fun merge_subst (A:Type.tyenv) (B:Type.tyenv) =
- SOME (Vartab.fold (fn (v, t) =>
- fn tab =>
- (case Vartab.lookup tab v of
- NONE => Vartab.update (v, t) tab
- | SOME t' => if t = t' then tab else raise Type.TYPE_MATCH)) A B)
- handle Type.TYPE_MATCH => NONE
-
-fun subst_ord (A:Type.tyenv, B:Type.tyenv) =
- (list_ord (prod_ord Term_Ord.fast_indexname_ord (prod_ord Term_Ord.sort_ord Term_Ord.typ_ord))) (Vartab.dest A, Vartab.dest B)
-
-structure Substtab = Table(type key = Type.tyenv val ord = subst_ord);
-
-fun substtab_union c = Substtab.fold Substtab.update c
-fun substtab_unions [] = Substtab.empty
- | substtab_unions [c] = c
- | substtab_unions (c::cs) = substtab_union c (substtab_unions cs)
-
-datatype instances = Instances of unit ConsttabModTy.table * Type.tyenv Consttab.table Consttab.table * constant list list * unit Substtab.table
-
-fun is_polymorphic (Constant (_, _, ty)) = not (null (Term.add_tvarsT ty []))
-
-fun distinct_constants cs =
- Consttab.keys (fold (fn c => Consttab.update (c, ())) cs Consttab.empty)
-
-fun empty cs =
- let
- val cs = distinct_constants (filter is_polymorphic cs)
- val old_cs = cs
-(* fun collect_tvars ty tab = fold (fn v => fn tab => Typtab.update (TVar v, ()) tab) (Misc_Legacy.typ_tvars ty) tab
- val tvars_count = length (Typtab.keys (fold (fn c => fn tab => collect_tvars (typ_of_constant c) tab) cs Typtab.empty))
- fun tvars_of ty = collect_tvars ty Typtab.empty
- val cs = map (fn c => (c, tvars_of (typ_of_constant c))) cs
-
- fun tyunion A B =
- Typtab.fold
- (fn (v,()) => fn tab => Typtab.update (v, case Typtab.lookup tab v of NONE => 1 | SOME n => n+1) tab)
- A B
-
- fun is_essential A B =
- Typtab.fold
- (fn (v, ()) => fn essential => essential orelse (case Typtab.lookup B v of NONE => raise Link "is_essential" | SOME n => n=1))
- A false
-
- fun add_minimal (c', tvs') (tvs, cs) =
- let
- val tvs = tyunion tvs' tvs
- val cs = (c', tvs')::cs
- in
- if forall (fn (c',tvs') => is_essential tvs' tvs) cs then
- SOME (tvs, cs)
- else
- NONE
- end
-
- fun is_spanning (tvs, _) = (length (Typtab.keys tvs) = tvars_count)
-
- fun generate_minimal_subsets subsets [] = subsets
- | generate_minimal_subsets subsets (c::cs) =
- let
- val subsets' = map_filter (add_minimal c) subsets
- in
- generate_minimal_subsets (subsets@subsets') cs
- end*)
-
- val minimal_subsets = [old_cs] (*map (fn (tvs, cs) => map fst cs) (filter is_spanning (generate_minimal_subsets [(Typtab.empty, [])] cs))*)
-
- val constants = Consttab.keys (fold (fold (fn c => Consttab.update (c, ()))) minimal_subsets Consttab.empty)
-
- in
- Instances (
- fold (fn c => fn tab => ConsttabModTy.update (c, ()) tab) constants ConsttabModTy.empty,
- Consttab.make (map (fn c => (c, Consttab.empty : Type.tyenv Consttab.table)) constants),
- minimal_subsets, Substtab.empty)
- end
-
-local
-fun calc ctab substtab [] = substtab
- | calc ctab substtab (c::cs) =
- let
- val csubsts = map snd (Consttab.dest (the (Consttab.lookup ctab c)))
- fun merge_substs substtab subst =
- Substtab.fold (fn (s,_) =>
- fn tab =>
- (case merge_subst subst s of NONE => tab | SOME s => Substtab.update (s, ()) tab))
- substtab Substtab.empty
- val substtab = substtab_unions (map (merge_substs substtab) csubsts)
- in
- calc ctab substtab cs
- end
-in
-fun calc_substs ctab (cs:constant list) = calc ctab (Substtab.update (empty_subst, ()) Substtab.empty) cs
-end
-
-fun add_instances thy (Instances (cfilter, ctab,minsets,substs)) cs =
- let
-(* val _ = writeln (makestring ("add_instances: ", length_cs, length cs, length (Consttab.keys ctab)))*)
- fun calc_instantiations (constant as Constant (free, name, ty)) instantiations =
- Consttab.fold (fn (constant' as Constant (free', name', ty'), insttab) =>
- fn instantiations =>
- if free <> free' orelse name <> name' then
- instantiations
- else case Consttab.lookup insttab constant of
- SOME _ => instantiations
- | NONE => ((constant', (constant, Sign.typ_match thy (ty', ty) empty_subst))::instantiations
- handle Type.TYPE_MATCH => instantiations))
- ctab instantiations
- val instantiations = fold calc_instantiations cs []
- (*val _ = writeln ("instantiations = "^(makestring (length instantiations)))*)
- fun update_ctab (constant', entry) ctab =
- (case Consttab.lookup ctab constant' of
- NONE => raise Link "internal error: update_ctab"
- | SOME tab => Consttab.update (constant', Consttab.update entry tab) ctab)
- val ctab = fold update_ctab instantiations ctab
- val new_substs = fold (fn minset => fn substs => substtab_union (calc_substs ctab minset) substs)
- minsets Substtab.empty
- val (added_substs, substs) =
- Substtab.fold (fn (ns, _) =>
- fn (added, substtab) =>
- (case Substtab.lookup substs ns of
- NONE => (ns::added, Substtab.update (ns, ()) substtab)
- | SOME () => (added, substtab)))
- new_substs ([], substs)
- in
- (added_substs, Instances (cfilter, ctab, minsets, substs))
- end
-
-fun substs_of (Instances (_,_,_,substs)) = Substtab.keys substs
-
-
-local
-
-fun collect (Var _) tab = tab
- | collect (Bound _) tab = tab
- | collect (a $ b) tab = collect b (collect a tab)
- | collect (Abs (_, _, body)) tab = collect body tab
- | collect t tab = Consttab.update (constant_of t, ()) tab
-
-in
- fun collect_consts tms = Consttab.keys (fold collect tms Consttab.empty)
-end
-
-end
-
-signature PCOMPUTE =
-sig
- type pcomputer
-
- val make : Compute.machine -> theory -> thm list -> Linker.constant list -> pcomputer
- val make_with_cache : Compute.machine -> theory -> term list -> thm list -> Linker.constant list -> pcomputer
-
- val add_instances : pcomputer -> Linker.constant list -> bool
- val add_instances' : pcomputer -> term list -> bool
-
- val rewrite : pcomputer -> cterm list -> thm list
- val simplify : pcomputer -> Compute.theorem -> thm
-
- val make_theorem : pcomputer -> thm -> string list -> Compute.theorem
- val instantiate : pcomputer -> (string * cterm) list -> Compute.theorem -> Compute.theorem
- val evaluate_prem : pcomputer -> int -> Compute.theorem -> Compute.theorem
- val modus_ponens : pcomputer -> int -> thm -> Compute.theorem -> Compute.theorem
-
-end
-
-structure PCompute : PCOMPUTE = struct
-
-exception PCompute of string
-
-datatype theorem = MonoThm of thm | PolyThm of thm * Linker.instances * thm list
-datatype pattern = MonoPattern of term | PolyPattern of term * Linker.instances * term list
-
-datatype pcomputer =
- PComputer of theory_ref * Compute.computer * theorem list Unsynchronized.ref *
- pattern list Unsynchronized.ref
-
-(*fun collect_consts (Var x) = []
- | collect_consts (Bound _) = []
- | collect_consts (a $ b) = (collect_consts a)@(collect_consts b)
- | collect_consts (Abs (_, _, body)) = collect_consts body
- | collect_consts t = [Linker.constant_of t]*)
-
-fun computer_of (PComputer (_,computer,_,_)) = computer
-
-fun collect_consts_of_thm th =
- let
- val th = prop_of th
- val (prems, th) = (Logic.strip_imp_prems th, Logic.strip_imp_concl th)
- val (left, right) = Logic.dest_equals th
- in
- (Linker.collect_consts [left], Linker.collect_consts (right::prems))
- end
-
-fun create_theorem th =
-let
- val (left, right) = collect_consts_of_thm th
- val polycs = filter Linker.is_polymorphic left
- val tytab = fold (fn p => fn tab => fold (fn n => fn tab => Typtab.update (TVar n, ()) tab) (Misc_Legacy.typ_tvars (Linker.typ_of_constant p)) tab) polycs Typtab.empty
- fun check_const (c::cs) cs' =
- let
- val tvars = Misc_Legacy.typ_tvars (Linker.typ_of_constant c)
- val wrong = fold (fn n => fn wrong => wrong orelse is_none (Typtab.lookup tytab (TVar n))) tvars false
- in
- if wrong then raise PCompute "right hand side of theorem contains type variables which do not occur on the left hand side"
- else
- if null (tvars) then
- check_const cs (c::cs')
- else
- check_const cs cs'
- end
- | check_const [] cs' = cs'
- val monocs = check_const right []
-in
- if null (polycs) then
- (monocs, MonoThm th)
- else
- (monocs, PolyThm (th, Linker.empty polycs, []))
-end
-
-fun create_pattern pat =
-let
- val cs = Linker.collect_consts [pat]
- val polycs = filter Linker.is_polymorphic cs
-in
- if null (polycs) then
- MonoPattern pat
- else
- PolyPattern (pat, Linker.empty polycs, [])
-end
-
-fun create_computer machine thy pats ths =
- let
- fun add (MonoThm th) ths = th::ths
- | add (PolyThm (_, _, ths')) ths = ths'@ths
- fun addpat (MonoPattern p) pats = p::pats
- | addpat (PolyPattern (_, _, ps)) pats = ps@pats
- val ths = fold_rev add ths []
- val pats = fold_rev addpat pats []
- in
- Compute.make_with_cache machine thy pats ths
- end
-
-fun update_computer computer pats ths =
- let
- fun add (MonoThm th) ths = th::ths
- | add (PolyThm (_, _, ths')) ths = ths'@ths
- fun addpat (MonoPattern p) pats = p::pats
- | addpat (PolyPattern (_, _, ps)) pats = ps@pats
- val ths = fold_rev add ths []
- val pats = fold_rev addpat pats []
- in
- Compute.update_with_cache computer pats ths
- end
-
-fun conv_subst thy (subst : Type.tyenv) =
- map (fn (iname, (sort, ty)) => (ctyp_of thy (TVar (iname, sort)), ctyp_of thy ty)) (Vartab.dest subst)
-
-fun add_monos thy monocs pats ths =
- let
- val changed = Unsynchronized.ref false
- fun add monocs (th as (MonoThm _)) = ([], th)
- | add monocs (PolyThm (th, instances, instanceths)) =
- let
- val (newsubsts, instances) = Linker.add_instances thy instances monocs
- val _ = if not (null newsubsts) then changed := true else ()
- val newths = map (fn subst => Thm.instantiate (conv_subst thy subst, []) th) newsubsts
-(* val _ = if not (null newths) then (print ("added new theorems: ", newths); ()) else ()*)
- val newmonos = fold (fn th => fn monos => (snd (collect_consts_of_thm th))@monos) newths []
- in
- (newmonos, PolyThm (th, instances, instanceths@newths))
- end
- fun addpats monocs (pat as (MonoPattern _)) = pat
- | addpats monocs (PolyPattern (p, instances, instancepats)) =
- let
- val (newsubsts, instances) = Linker.add_instances thy instances monocs
- val _ = if not (null newsubsts) then changed := true else ()
- val newpats = map (fn subst => Envir.subst_term_types subst p) newsubsts
- in
- PolyPattern (p, instances, instancepats@newpats)
- end
- fun step monocs ths =
- fold_rev (fn th =>
- fn (newmonos, ths) =>
- let
- val (newmonos', th') = add monocs th
- in
- (newmonos'@newmonos, th'::ths)
- end)
- ths ([], [])
- fun loop monocs pats ths =
- let
- val (monocs', ths') = step monocs ths
- val pats' = map (addpats monocs) pats
- in
- if null (monocs') then
- (pats', ths')
- else
- loop monocs' pats' ths'
- end
- val result = loop monocs pats ths
- in
- (!changed, result)
- end
-
-datatype cthm = ComputeThm of term list * sort list * term
-
-fun thm2cthm th =
- let
- val {hyps, prop, shyps, ...} = Thm.rep_thm th
- in
- ComputeThm (hyps, shyps, prop)
- end
-
-val cthm_ord' = prod_ord (prod_ord (list_ord Term_Ord.term_ord) (list_ord Term_Ord.sort_ord)) Term_Ord.term_ord
-
-fun cthm_ord (ComputeThm (h1, sh1, p1), ComputeThm (h2, sh2, p2)) = cthm_ord' (((h1,sh1), p1), ((h2, sh2), p2))
-
-structure CThmtab = Table(type key = cthm val ord = cthm_ord)
-
-fun remove_duplicates ths =
- let
- val counter = Unsynchronized.ref 0
- val tab = Unsynchronized.ref (CThmtab.empty : unit CThmtab.table)
- val thstab = Unsynchronized.ref (Inttab.empty : thm Inttab.table)
- fun update th =
- let
- val key = thm2cthm th
- in
- case CThmtab.lookup (!tab) key of
- NONE => ((tab := CThmtab.update_new (key, ()) (!tab)); thstab := Inttab.update_new (!counter, th) (!thstab); counter := !counter + 1)
- | _ => ()
- end
- val _ = map update ths
- in
- map snd (Inttab.dest (!thstab))
- end
-
-fun make_with_cache machine thy pats ths cs =
- let
- val ths = remove_duplicates ths
- val (monocs, ths) = fold_rev (fn th =>
- fn (monocs, ths) =>
- let val (m, t) = create_theorem th in
- (m@monocs, t::ths)
- end)
- ths (cs, [])
- val pats = map create_pattern pats
- val (_, (pats, ths)) = add_monos thy monocs pats ths
- val computer = create_computer machine thy pats ths
- in
- PComputer (Theory.check_thy thy, computer, Unsynchronized.ref ths, Unsynchronized.ref pats)
- end
-
-fun make machine thy ths cs = make_with_cache machine thy [] ths cs
-
-fun add_instances (PComputer (thyref, computer, rths, rpats)) cs =
- let
- val thy = Theory.deref thyref
- val (changed, (pats, ths)) = add_monos thy cs (!rpats) (!rths)
- in
- if changed then
- (update_computer computer pats ths;
- rths := ths;
- rpats := pats;
- true)
- else
- false
-
- end
-
-fun add_instances' pc ts = add_instances pc (Linker.collect_consts ts)
-
-fun rewrite pc cts =
- let
- val _ = add_instances' pc (map term_of cts)
- val computer = (computer_of pc)
- in
- map (fn ct => Compute.rewrite computer ct) cts
- end
-
-fun simplify pc th = Compute.simplify (computer_of pc) th
-
-fun make_theorem pc th vars =
- let
- val _ = add_instances' pc [prop_of th]
-
- in
- Compute.make_theorem (computer_of pc) th vars
- end
-
-fun instantiate pc insts th =
- let
- val _ = add_instances' pc (map (term_of o snd) insts)
- in
- Compute.instantiate (computer_of pc) insts th
- end
-
-fun evaluate_prem pc prem_no th = Compute.evaluate_prem (computer_of pc) prem_no th
-
-fun modus_ponens pc prem_no th' th =
- let
- val _ = add_instances' pc [prop_of th']
- in
- Compute.modus_ponens (computer_of pc) prem_no th' th
- end
-
-
-end
--- a/src/HOL/Matrix/Compute_Oracle/report.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,33 +0,0 @@
-structure Report =
-struct
-
-local
-
- val report_depth = Unsynchronized.ref 0
- fun space n = if n <= 0 then "" else (space (n-1))^" "
- fun report_space () = space (!report_depth)
-
-in
-
-fun timeit f =
- let
- val t1 = Timing.start ()
- val x = f ()
- val t2 = Timing.message (Timing.result t1)
- val _ = writeln ((report_space ()) ^ "--> "^t2)
- in
- x
- end
-
-fun report s f =
-let
- val _ = writeln ((report_space ())^s)
- val _ = report_depth := !report_depth + 1
- val x = timeit f
- val _ = report_depth := !report_depth - 1
-in
- x
-end
-
-end
-end
\ No newline at end of file
--- a/src/HOL/Matrix/Cplex.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,67 +0,0 @@
-(* Title: HOL/Matrix/Cplex.thy
- Author: Steven Obua
-*)
-
-theory Cplex
-imports SparseMatrix LP ComputeFloat ComputeNumeral
-uses "Cplex_tools.ML" "CplexMatrixConverter.ML" "FloatSparseMatrixBuilder.ML"
- "fspmlp.ML" ("matrixlp.ML")
-begin
-
-lemma spm_mult_le_dual_prts:
- assumes
- "sorted_sparse_matrix A1"
- "sorted_sparse_matrix A2"
- "sorted_sparse_matrix c1"
- "sorted_sparse_matrix c2"
- "sorted_sparse_matrix y"
- "sorted_sparse_matrix r1"
- "sorted_sparse_matrix r2"
- "sorted_spvec b"
- "le_spmat [] y"
- "sparse_row_matrix A1 \<le> A"
- "A \<le> sparse_row_matrix A2"
- "sparse_row_matrix c1 \<le> c"
- "c \<le> sparse_row_matrix c2"
- "sparse_row_matrix r1 \<le> x"
- "x \<le> sparse_row_matrix r2"
- "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
- shows
- "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
- (let s1 = diff_spmat c1 (mult_spmat y A2); s2 = diff_spmat c2 (mult_spmat y A1) in
- add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
- (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))))"
- apply (simp add: Let_def)
- apply (insert assms)
- apply (simp add: sparse_row_matrix_op_simps algebra_simps)
- apply (rule mult_le_dual_prts[where A=A, simplified Let_def algebra_simps])
- apply (auto)
- done
-
-lemma spm_mult_le_dual_prts_no_let:
- assumes
- "sorted_sparse_matrix A1"
- "sorted_sparse_matrix A2"
- "sorted_sparse_matrix c1"
- "sorted_sparse_matrix c2"
- "sorted_sparse_matrix y"
- "sorted_sparse_matrix r1"
- "sorted_sparse_matrix r2"
- "sorted_spvec b"
- "le_spmat [] y"
- "sparse_row_matrix A1 \<le> A"
- "A \<le> sparse_row_matrix A2"
- "sparse_row_matrix c1 \<le> c"
- "c \<le> sparse_row_matrix c2"
- "sparse_row_matrix r1 \<le> x"
- "x \<le> sparse_row_matrix r2"
- "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
- shows
- "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
- (mult_est_spmat r1 r2 (diff_spmat c1 (mult_spmat y A2)) (diff_spmat c2 (mult_spmat y A1))))"
- by (simp add: assms mult_est_spmat_def spm_mult_le_dual_prts[where A=A, simplified Let_def])
-
-use "matrixlp.ML"
-
-end
-
--- a/src/HOL/Matrix/CplexMatrixConverter.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,128 +0,0 @@
-(* Title: HOL/Matrix/CplexMatrixConverter.ML
- Author: Steven Obua
-*)
-
-signature MATRIX_BUILDER =
-sig
- type vector
- type matrix
-
- val empty_vector : vector
- val empty_matrix : matrix
-
- exception Nat_expected of int
- val set_elem : vector -> int -> string -> vector
- val set_vector : matrix -> int -> vector -> matrix
-end;
-
-signature CPLEX_MATRIX_CONVERTER =
-sig
- structure cplex : CPLEX
- structure matrix_builder : MATRIX_BUILDER
- type vector = matrix_builder.vector
- type matrix = matrix_builder.matrix
- type naming = int * (int -> string) * (string -> int)
-
- exception Converter of string
-
- (* program must fulfill is_normed_cplexProg and must be an element of the image of elim_nonfree_bounds *)
- (* convert_prog maximize c A b naming *)
- val convert_prog : cplex.cplexProg -> bool * vector * matrix * vector * naming
-
- (* results must be optimal, converts_results returns the optimal value as string and the solution as vector *)
- (* convert_results results name2index *)
- val convert_results : cplex.cplexResult -> (string -> int) -> string * vector
-end;
-
-functor MAKE_CPLEX_MATRIX_CONVERTER (structure cplex: CPLEX and matrix_builder: MATRIX_BUILDER) : CPLEX_MATRIX_CONVERTER =
-struct
-
-structure cplex = cplex
-structure matrix_builder = matrix_builder
-type matrix = matrix_builder.matrix
-type vector = matrix_builder.vector
-type naming = int * (int -> string) * (string -> int)
-
-open matrix_builder
-open cplex
-
-exception Converter of string;
-
-fun neg_term (cplexNeg t) = t
- | neg_term (cplexSum ts) = cplexSum (map neg_term ts)
- | neg_term t = cplexNeg t
-
-fun convert_prog (cplexProg (_, goal, constrs, bounds)) =
- let
- fun build_naming index i2s s2i [] = (index, i2s, s2i)
- | build_naming index i2s s2i (cplexBounds (cplexNeg cplexInf, cplexLeq, cplexVar v, cplexLeq, cplexInf)::bounds)
- = build_naming (index+1) (Inttab.update (index, v) i2s) (Symtab.update_new (v, index) s2i) bounds
- | build_naming _ _ _ _ = raise (Converter "nonfree bound")
-
- val (varcount, i2s_tab, s2i_tab) = build_naming 0 Inttab.empty Symtab.empty bounds
-
- fun i2s i = case Inttab.lookup i2s_tab i of NONE => raise (Converter "index not found")
- | SOME n => n
- fun s2i s = case Symtab.lookup s2i_tab s of NONE => raise (Converter ("name not found: "^s))
- | SOME i => i
- fun num2str positive (cplexNeg t) = num2str (not positive) t
- | num2str positive (cplexNum num) = if positive then num else "-"^num
- | num2str _ _ = raise (Converter "term is not a (possibly signed) number")
-
- fun setprod vec positive (cplexNeg t) = setprod vec (not positive) t
- | setprod vec positive (cplexVar v) = set_elem vec (s2i v) (if positive then "1" else "-1")
- | setprod vec positive (cplexProd (cplexNum num, cplexVar v)) =
- set_elem vec (s2i v) (if positive then num else "-"^num)
- | setprod _ _ _ = raise (Converter "term is not a normed product")
-
- fun sum2vec (cplexSum ts) = fold (fn t => fn vec => setprod vec true t) ts empty_vector
- | sum2vec t = setprod empty_vector true t
-
- fun constrs2Ab j A b [] = (A, b)
- | constrs2Ab j A b ((_, cplexConstr (cplexLeq, (t1,t2)))::cs) =
- constrs2Ab (j+1) (set_vector A j (sum2vec t1)) (set_elem b j (num2str true t2)) cs
- | constrs2Ab j A b ((_, cplexConstr (cplexGeq, (t1,t2)))::cs) =
- constrs2Ab (j+1) (set_vector A j (sum2vec (neg_term t1))) (set_elem b j (num2str true (neg_term t2))) cs
- | constrs2Ab j A b ((_, cplexConstr (cplexEq, (t1,t2)))::cs) =
- constrs2Ab j A b ((NONE, cplexConstr (cplexLeq, (t1,t2)))::
- (NONE, cplexConstr (cplexGeq, (t1, t2)))::cs)
- | constrs2Ab _ _ _ _ = raise (Converter "no strict constraints allowed")
-
- val (A, b) = constrs2Ab 0 empty_matrix empty_vector constrs
-
- val (goal_maximize, goal_term) =
- case goal of
- (cplexMaximize t) => (true, t)
- | (cplexMinimize t) => (false, t)
- in
- (goal_maximize, sum2vec goal_term, A, b, (varcount, i2s, s2i))
- end
-
-fun convert_results (cplex.Optimal (opt, entries)) name2index =
- let
- fun setv (name, value) v = matrix_builder.set_elem v (name2index name) value
- in
- (opt, fold setv entries (matrix_builder.empty_vector))
- end
- | convert_results _ _ = raise (Converter "No optimal result")
-
-end;
-
-structure SimpleMatrixBuilder : MATRIX_BUILDER =
-struct
-type vector = (int * string) list
-type matrix = (int * vector) list
-
-val empty_matrix = []
-val empty_vector = []
-
-exception Nat_expected of int;
-
-fun set_elem v i s = v @ [(i, s)]
-
-fun set_vector m i v = m @ [(i, v)]
-
-end;
-
-structure SimpleCplexMatrixConverter =
- MAKE_CPLEX_MATRIX_CONVERTER(structure cplex = Cplex and matrix_builder = SimpleMatrixBuilder);
--- a/src/HOL/Matrix/Cplex_tools.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1192 +0,0 @@
-(* Title: HOL/Matrix/Cplex_tools.ML
- Author: Steven Obua
-*)
-
-signature CPLEX =
-sig
-
- datatype cplexTerm = cplexVar of string | cplexNum of string | cplexInf
- | cplexNeg of cplexTerm
- | cplexProd of cplexTerm * cplexTerm
- | cplexSum of (cplexTerm list)
-
- datatype cplexComp = cplexLe | cplexLeq | cplexEq | cplexGe | cplexGeq
-
- datatype cplexGoal = cplexMinimize of cplexTerm
- | cplexMaximize of cplexTerm
-
- datatype cplexConstr = cplexConstr of cplexComp *
- (cplexTerm * cplexTerm)
-
- datatype cplexBounds = cplexBounds of cplexTerm * cplexComp * cplexTerm
- * cplexComp * cplexTerm
- | cplexBound of cplexTerm * cplexComp * cplexTerm
-
- datatype cplexProg = cplexProg of string
- * cplexGoal
- * ((string option * cplexConstr)
- list)
- * cplexBounds list
-
- datatype cplexResult = Unbounded
- | Infeasible
- | Undefined
- | Optimal of string *
- (((* name *) string *
- (* value *) string) list)
-
- datatype cplexSolver = SOLVER_DEFAULT | SOLVER_CPLEX | SOLVER_GLPK
-
- exception Load_cplexFile of string
- exception Load_cplexResult of string
- exception Save_cplexFile of string
- exception Execute of string
-
- val load_cplexFile : string -> cplexProg
-
- val save_cplexFile : string -> cplexProg -> unit
-
- val elim_nonfree_bounds : cplexProg -> cplexProg
-
- val relax_strict_ineqs : cplexProg -> cplexProg
-
- val is_normed_cplexProg : cplexProg -> bool
-
- val get_solver : unit -> cplexSolver
- val set_solver : cplexSolver -> unit
- val solve : cplexProg -> cplexResult
-end;
-
-structure Cplex : CPLEX =
-struct
-
-datatype cplexSolver = SOLVER_DEFAULT | SOLVER_CPLEX | SOLVER_GLPK
-
-val cplexsolver = Unsynchronized.ref SOLVER_DEFAULT;
-fun get_solver () = !cplexsolver;
-fun set_solver s = (cplexsolver := s);
-
-exception Load_cplexFile of string;
-exception Load_cplexResult of string;
-exception Save_cplexFile of string;
-
-datatype cplexTerm = cplexVar of string
- | cplexNum of string
- | cplexInf
- | cplexNeg of cplexTerm
- | cplexProd of cplexTerm * cplexTerm
- | cplexSum of (cplexTerm list)
-datatype cplexComp = cplexLe | cplexLeq | cplexEq | cplexGe | cplexGeq
-datatype cplexGoal = cplexMinimize of cplexTerm | cplexMaximize of cplexTerm
-datatype cplexConstr = cplexConstr of cplexComp * (cplexTerm * cplexTerm)
-datatype cplexBounds = cplexBounds of cplexTerm * cplexComp * cplexTerm
- * cplexComp * cplexTerm
- | cplexBound of cplexTerm * cplexComp * cplexTerm
-datatype cplexProg = cplexProg of string
- * cplexGoal
- * ((string option * cplexConstr) list)
- * cplexBounds list
-
-fun rev_cmp cplexLe = cplexGe
- | rev_cmp cplexLeq = cplexGeq
- | rev_cmp cplexGe = cplexLe
- | rev_cmp cplexGeq = cplexLeq
- | rev_cmp cplexEq = cplexEq
-
-fun the NONE = raise (Load_cplexFile "SOME expected")
- | the (SOME x) = x;
-
-fun modulo_signed is_something (cplexNeg u) = is_something u
- | modulo_signed is_something u = is_something u
-
-fun is_Num (cplexNum _) = true
- | is_Num _ = false
-
-fun is_Inf cplexInf = true
- | is_Inf _ = false
-
-fun is_Var (cplexVar _) = true
- | is_Var _ = false
-
-fun is_Neg (cplexNeg _) = true
- | is_Neg _ = false
-
-fun is_normed_Prod (cplexProd (t1, t2)) =
- (is_Num t1) andalso (is_Var t2)
- | is_normed_Prod x = is_Var x
-
-fun is_normed_Sum (cplexSum ts) =
- (ts <> []) andalso forall (modulo_signed is_normed_Prod) ts
- | is_normed_Sum x = modulo_signed is_normed_Prod x
-
-fun is_normed_Constr (cplexConstr (_, (t1, t2))) =
- (is_normed_Sum t1) andalso (modulo_signed is_Num t2)
-
-fun is_Num_or_Inf x = is_Inf x orelse is_Num x
-
-fun is_normed_Bounds (cplexBounds (t1, c1, t2, c2, t3)) =
- (c1 = cplexLe orelse c1 = cplexLeq) andalso
- (c2 = cplexLe orelse c2 = cplexLeq) andalso
- is_Var t2 andalso
- modulo_signed is_Num_or_Inf t1 andalso
- modulo_signed is_Num_or_Inf t3
- | is_normed_Bounds (cplexBound (t1, c, t2)) =
- (is_Var t1 andalso (modulo_signed is_Num_or_Inf t2))
- orelse
- (c <> cplexEq andalso
- is_Var t2 andalso (modulo_signed is_Num_or_Inf t1))
-
-fun term_of_goal (cplexMinimize x) = x
- | term_of_goal (cplexMaximize x) = x
-
-fun is_normed_cplexProg (cplexProg (_, goal, constraints, bounds)) =
- is_normed_Sum (term_of_goal goal) andalso
- forall (fn (_,x) => is_normed_Constr x) constraints andalso
- forall is_normed_Bounds bounds
-
-fun is_NL s = s = "\n"
-
-fun is_blank s = forall (fn c => c <> #"\n" andalso Char.isSpace c) (String.explode s)
-
-fun is_num a =
- let
- val b = String.explode a
- fun num4 cs = forall Char.isDigit cs
- fun num3 [] = true
- | num3 (ds as (c::cs)) =
- if c = #"+" orelse c = #"-" then
- num4 cs
- else
- num4 ds
- fun num2 [] = true
- | num2 (c::cs) =
- if c = #"e" orelse c = #"E" then num3 cs
- else (Char.isDigit c) andalso num2 cs
- fun num1 [] = true
- | num1 (c::cs) =
- if c = #"." then num2 cs
- else if c = #"e" orelse c = #"E" then num3 cs
- else (Char.isDigit c) andalso num1 cs
- fun num [] = true
- | num (c::cs) =
- if c = #"." then num2 cs
- else (Char.isDigit c) andalso num1 cs
- in
- num b
- end
-
-fun is_delimiter s = s = "+" orelse s = "-" orelse s = ":"
-
-fun is_cmp s = s = "<" orelse s = ">" orelse s = "<="
- orelse s = ">=" orelse s = "="
-
-fun is_symbol a =
- let
- val symbol_char = String.explode "!\"#$%&()/,.;?@_`'{}|~"
- fun is_symbol_char c = Char.isAlphaNum c orelse
- exists (fn d => d=c) symbol_char
- fun is_symbol_start c = is_symbol_char c andalso
- not (Char.isDigit c) andalso
- not (c= #".")
- val b = String.explode a
- in
- b <> [] andalso is_symbol_start (hd b) andalso
- forall is_symbol_char b
- end
-
-fun to_upper s = String.implode (map Char.toUpper (String.explode s))
-
-fun keyword x =
- let
- val a = to_upper x
- in
- if a = "BOUNDS" orelse a = "BOUND" then
- SOME "BOUNDS"
- else if a = "MINIMIZE" orelse a = "MINIMUM" orelse a = "MIN" then
- SOME "MINIMIZE"
- else if a = "MAXIMIZE" orelse a = "MAXIMUM" orelse a = "MAX" then
- SOME "MAXIMIZE"
- else if a = "ST" orelse a = "S.T." orelse a = "ST." then
- SOME "ST"
- else if a = "FREE" orelse a = "END" then
- SOME a
- else if a = "GENERAL" orelse a = "GENERALS" orelse a = "GEN" then
- SOME "GENERAL"
- else if a = "INTEGER" orelse a = "INTEGERS" orelse a = "INT" then
- SOME "INTEGER"
- else if a = "BINARY" orelse a = "BINARIES" orelse a = "BIN" then
- SOME "BINARY"
- else if a = "INF" orelse a = "INFINITY" then
- SOME "INF"
- else
- NONE
- end
-
-val TOKEN_ERROR = ~1
-val TOKEN_BLANK = 0
-val TOKEN_NUM = 1
-val TOKEN_DELIMITER = 2
-val TOKEN_SYMBOL = 3
-val TOKEN_LABEL = 4
-val TOKEN_CMP = 5
-val TOKEN_KEYWORD = 6
-val TOKEN_NL = 7
-
-(* tokenize takes a list of chars as argument and returns a list of
- int * string pairs, each string representing a "cplex token",
- and each int being one of TOKEN_NUM, TOKEN_DELIMITER, TOKEN_CMP
- or TOKEN_SYMBOL *)
-fun tokenize s =
- let
- val flist = [(is_NL, TOKEN_NL),
- (is_blank, TOKEN_BLANK),
- (is_num, TOKEN_NUM),
- (is_delimiter, TOKEN_DELIMITER),
- (is_cmp, TOKEN_CMP),
- (is_symbol, TOKEN_SYMBOL)]
- fun match_helper [] s = (fn _ => false, TOKEN_ERROR)
- | match_helper (f::fs) s =
- if ((fst f) s) then f else match_helper fs s
- fun match s = match_helper flist s
- fun tok s =
- if s = "" then [] else
- let
- val h = String.substring (s,0,1)
- val (f, j) = match h
- fun len i =
- if size s = i then i
- else if f (String.substring (s,0,i+1)) then
- len (i+1)
- else i
- in
- if j < 0 then
- (if h = "\\" then []
- else raise (Load_cplexFile ("token expected, found: "
- ^s)))
- else
- let
- val l = len 1
- val u = String.substring (s,0,l)
- val v = String.extract (s,l,NONE)
- in
- if j = 0 then tok v else (j, u) :: tok v
- end
- end
- in
- tok s
- end
-
-exception Tokenize of string;
-
-fun tokenize_general flist s =
- let
- fun match_helper [] s = raise (Tokenize s)
- | match_helper (f::fs) s =
- if ((fst f) s) then f else match_helper fs s
- fun match s = match_helper flist s
- fun tok s =
- if s = "" then [] else
- let
- val h = String.substring (s,0,1)
- val (f, j) = match h
- fun len i =
- if size s = i then i
- else if f (String.substring (s,0,i+1)) then
- len (i+1)
- else i
- val l = len 1
- in
- (j, String.substring (s,0,l)) :: tok (String.extract (s,l,NONE))
- end
- in
- tok s
- end
-
-fun load_cplexFile name =
- let
- val f = TextIO.openIn name
- val ignore_NL = Unsynchronized.ref true
- val rest = Unsynchronized.ref []
-
- fun is_symbol s c = (fst c) = TOKEN_SYMBOL andalso (to_upper (snd c)) = s
-
- fun readToken_helper () =
- if length (!rest) > 0 then
- let val u = hd (!rest) in
- (
- rest := tl (!rest);
- SOME u
- )
- end
- else
- (case TextIO.inputLine f of
- NONE => NONE
- | SOME s =>
- let val t = tokenize s in
- if (length t >= 2 andalso
- snd(hd (tl t)) = ":")
- then
- rest := (TOKEN_LABEL, snd (hd t)) :: (tl (tl t))
- else if (length t >= 2) andalso is_symbol "SUBJECT" (hd (t))
- andalso is_symbol "TO" (hd (tl t))
- then
- rest := (TOKEN_SYMBOL, "ST") :: (tl (tl t))
- else
- rest := t;
- readToken_helper ()
- end)
-
- fun readToken_helper2 () =
- let val c = readToken_helper () in
- if c = NONE then NONE
- else if !ignore_NL andalso fst (the c) = TOKEN_NL then
- readToken_helper2 ()
- else if fst (the c) = TOKEN_SYMBOL
- andalso keyword (snd (the c)) <> NONE
- then SOME (TOKEN_KEYWORD, the (keyword (snd (the c))))
- else c
- end
-
- fun readToken () = readToken_helper2 ()
-
- fun pushToken a = rest := (a::(!rest))
-
- fun is_value token =
- fst token = TOKEN_NUM orelse (fst token = TOKEN_KEYWORD
- andalso snd token = "INF")
-
- fun get_value token =
- if fst token = TOKEN_NUM then
- cplexNum (snd token)
- else if fst token = TOKEN_KEYWORD andalso snd token = "INF"
- then
- cplexInf
- else
- raise (Load_cplexFile "num expected")
-
- fun readTerm_Product only_num =
- let val c = readToken () in
- if c = NONE then NONE
- else if fst (the c) = TOKEN_SYMBOL
- then (
- if only_num then (pushToken (the c); NONE)
- else SOME (cplexVar (snd (the c)))
- )
- else if only_num andalso is_value (the c) then
- SOME (get_value (the c))
- else if is_value (the c) then
- let val t1 = get_value (the c)
- val d = readToken ()
- in
- if d = NONE then SOME t1
- else if fst (the d) = TOKEN_SYMBOL then
- SOME (cplexProd (t1, cplexVar (snd (the d))))
- else
- (pushToken (the d); SOME t1)
- end
- else (pushToken (the c); NONE)
- end
-
- fun readTerm_Signed only_signed only_num =
- let
- val c = readToken ()
- in
- if c = NONE then NONE
- else
- let val d = the c in
- if d = (TOKEN_DELIMITER, "+") then
- readTerm_Product only_num
- else if d = (TOKEN_DELIMITER, "-") then
- SOME (cplexNeg (the (readTerm_Product
- only_num)))
- else (pushToken d;
- if only_signed then NONE
- else readTerm_Product only_num)
- end
- end
-
- fun readTerm_Sum first_signed =
- let val c = readTerm_Signed first_signed false in
- if c = NONE then [] else (the c)::(readTerm_Sum true)
- end
-
- fun readTerm () =
- let val c = readTerm_Sum false in
- if c = [] then NONE
- else if tl c = [] then SOME (hd c)
- else SOME (cplexSum c)
- end
-
- fun readLabeledTerm () =
- let val c = readToken () in
- if c = NONE then (NONE, NONE)
- else if fst (the c) = TOKEN_LABEL then
- let val t = readTerm () in
- if t = NONE then
- raise (Load_cplexFile ("term after label "^
- (snd (the c))^
- " expected"))
- else (SOME (snd (the c)), t)
- end
- else (pushToken (the c); (NONE, readTerm ()))
- end
-
- fun readGoal () =
- let
- val g = readToken ()
- in
- if g = SOME (TOKEN_KEYWORD, "MAXIMIZE") then
- cplexMaximize (the (snd (readLabeledTerm ())))
- else if g = SOME (TOKEN_KEYWORD, "MINIMIZE") then
- cplexMinimize (the (snd (readLabeledTerm ())))
- else raise (Load_cplexFile "MAXIMIZE or MINIMIZE expected")
- end
-
- fun str2cmp b =
- (case b of
- "<" => cplexLe
- | "<=" => cplexLeq
- | ">" => cplexGe
- | ">=" => cplexGeq
- | "=" => cplexEq
- | _ => raise (Load_cplexFile (b^" is no TOKEN_CMP")))
-
- fun readConstraint () =
- let
- val t = readLabeledTerm ()
- fun make_constraint b t1 t2 =
- cplexConstr
- (str2cmp b,
- (t1, t2))
- in
- if snd t = NONE then NONE
- else
- let val c = readToken () in
- if c = NONE orelse fst (the c) <> TOKEN_CMP
- then raise (Load_cplexFile "TOKEN_CMP expected")
- else
- let val n = readTerm_Signed false true in
- if n = NONE then
- raise (Load_cplexFile "num expected")
- else
- SOME (fst t,
- make_constraint (snd (the c))
- (the (snd t))
- (the n))
- end
- end
- end
-
- fun readST () =
- let
- fun readbody () =
- let val t = readConstraint () in
- if t = NONE then []
- else if (is_normed_Constr (snd (the t))) then
- (the t)::(readbody ())
- else if (fst (the t) <> NONE) then
- raise (Load_cplexFile
- ("constraint '"^(the (fst (the t)))^
- "'is not normed"))
- else
- raise (Load_cplexFile
- "constraint is not normed")
- end
- in
- if readToken () = SOME (TOKEN_KEYWORD, "ST")
- then
- readbody ()
- else
- raise (Load_cplexFile "ST expected")
- end
-
- fun readCmp () =
- let val c = readToken () in
- if c = NONE then NONE
- else if fst (the c) = TOKEN_CMP then
- SOME (str2cmp (snd (the c)))
- else (pushToken (the c); NONE)
- end
-
- fun skip_NL () =
- let val c = readToken () in
- if c <> NONE andalso fst (the c) = TOKEN_NL then
- skip_NL ()
- else
- (pushToken (the c); ())
- end
-
- fun make_bounds c t1 t2 =
- cplexBound (t1, c, t2)
-
- fun readBound () =
- let
- val _ = skip_NL ()
- val t1 = readTerm ()
- in
- if t1 = NONE then NONE
- else
- let
- val c1 = readCmp ()
- in
- if c1 = NONE then
- let
- val c = readToken ()
- in
- if c = SOME (TOKEN_KEYWORD, "FREE") then
- SOME (
- cplexBounds (cplexNeg cplexInf,
- cplexLeq,
- the t1,
- cplexLeq,
- cplexInf))
- else
- raise (Load_cplexFile "FREE expected")
- end
- else
- let
- val t2 = readTerm ()
- in
- if t2 = NONE then
- raise (Load_cplexFile "term expected")
- else
- let val c2 = readCmp () in
- if c2 = NONE then
- SOME (make_bounds (the c1)
- (the t1)
- (the t2))
- else
- SOME (
- cplexBounds (the t1,
- the c1,
- the t2,
- the c2,
- the (readTerm())))
- end
- end
- end
- end
-
- fun readBounds () =
- let
- fun makestring _ = "?"
- fun readbody () =
- let
- val b = readBound ()
- in
- if b = NONE then []
- else if (is_normed_Bounds (the b)) then
- (the b)::(readbody())
- else (
- raise (Load_cplexFile
- ("bounds are not normed in: "^
- (makestring (the b)))))
- end
- in
- if readToken () = SOME (TOKEN_KEYWORD, "BOUNDS") then
- readbody ()
- else raise (Load_cplexFile "BOUNDS expected")
- end
-
- fun readEnd () =
- if readToken () = SOME (TOKEN_KEYWORD, "END") then ()
- else raise (Load_cplexFile "END expected")
-
- val result_Goal = readGoal ()
- val result_ST = readST ()
- val _ = ignore_NL := false
- val result_Bounds = readBounds ()
- val _ = ignore_NL := true
- val _ = readEnd ()
- val _ = TextIO.closeIn f
- in
- cplexProg (name, result_Goal, result_ST, result_Bounds)
- end
-
-fun save_cplexFile filename (cplexProg (_, goal, constraints, bounds)) =
- let
- val f = TextIO.openOut filename
-
- fun basic_write s = TextIO.output(f, s)
-
- val linebuf = Unsynchronized.ref ""
- fun buf_flushline s =
- (basic_write (!linebuf);
- basic_write "\n";
- linebuf := s)
- fun buf_add s = linebuf := (!linebuf) ^ s
-
- fun write s =
- if (String.size s) + (String.size (!linebuf)) >= 250 then
- buf_flushline (" "^s)
- else
- buf_add s
-
- fun writeln s = (buf_add s; buf_flushline "")
-
- fun write_term (cplexVar x) = write x
- | write_term (cplexNum x) = write x
- | write_term cplexInf = write "inf"
- | write_term (cplexProd (cplexNum "1", b)) = write_term b
- | write_term (cplexProd (a, b)) =
- (write_term a; write " "; write_term b)
- | write_term (cplexNeg x) = (write " - "; write_term x)
- | write_term (cplexSum ts) = write_terms ts
- and write_terms [] = ()
- | write_terms (t::ts) =
- (if (not (is_Neg t)) then write " + " else ();
- write_term t; write_terms ts)
-
- fun write_goal (cplexMaximize term) =
- (writeln "MAXIMIZE"; write_term term; writeln "")
- | write_goal (cplexMinimize term) =
- (writeln "MINIMIZE"; write_term term; writeln "")
-
- fun write_cmp cplexLe = write "<"
- | write_cmp cplexLeq = write "<="
- | write_cmp cplexEq = write "="
- | write_cmp cplexGe = write ">"
- | write_cmp cplexGeq = write ">="
-
- fun write_constr (cplexConstr (cmp, (a,b))) =
- (write_term a;
- write " ";
- write_cmp cmp;
- write " ";
- write_term b)
-
- fun write_constraints [] = ()
- | write_constraints (c::cs) =
- (if (fst c <> NONE)
- then
- (write (the (fst c)); write ": ")
- else
- ();
- write_constr (snd c);
- writeln "";
- write_constraints cs)
-
- fun write_bounds [] = ()
- | write_bounds ((cplexBounds (t1,c1,t2,c2,t3))::bs) =
- ((if t1 = cplexNeg cplexInf andalso t3 = cplexInf
- andalso (c1 = cplexLeq orelse c1 = cplexLe)
- andalso (c2 = cplexLeq orelse c2 = cplexLe)
- then
- (write_term t2; write " free")
- else
- (write_term t1; write " "; write_cmp c1; write " ";
- write_term t2; write " "; write_cmp c2; write " ";
- write_term t3)
- ); writeln ""; write_bounds bs)
- | write_bounds ((cplexBound (t1, c, t2)) :: bs) =
- (write_term t1; write " ";
- write_cmp c; write " ";
- write_term t2; writeln ""; write_bounds bs)
-
- val _ = write_goal goal
- val _ = (writeln ""; writeln "ST")
- val _ = write_constraints constraints
- val _ = (writeln ""; writeln "BOUNDS")
- val _ = write_bounds bounds
- val _ = (writeln ""; writeln "END")
- val _ = TextIO.closeOut f
- in
- ()
- end
-
-fun norm_Constr (constr as cplexConstr (c, (t1, t2))) =
- if not (modulo_signed is_Num t2) andalso
- modulo_signed is_Num t1
- then
- [cplexConstr (rev_cmp c, (t2, t1))]
- else if (c = cplexLe orelse c = cplexLeq) andalso
- (t1 = (cplexNeg cplexInf) orelse t2 = cplexInf)
- then
- []
- else if (c = cplexGe orelse c = cplexGeq) andalso
- (t1 = cplexInf orelse t2 = cplexNeg cplexInf)
- then
- []
- else
- [constr]
-
-fun bound2constr (cplexBounds (t1,c1,t2,c2,t3)) =
- (norm_Constr(cplexConstr (c1, (t1, t2))))
- @ (norm_Constr(cplexConstr (c2, (t2, t3))))
- | bound2constr (cplexBound (t1, cplexEq, t2)) =
- (norm_Constr(cplexConstr (cplexLeq, (t1, t2))))
- @ (norm_Constr(cplexConstr (cplexLeq, (t2, t1))))
- | bound2constr (cplexBound (t1, c1, t2)) =
- norm_Constr(cplexConstr (c1, (t1,t2)))
-
-val emptyset = Symtab.empty
-
-fun singleton v = Symtab.update (v, ()) emptyset
-
-fun merge a b = Symtab.merge (op =) (a, b)
-
-fun mergemap f ts = fold (fn x => fn table => merge table (f x)) ts Symtab.empty
-
-fun diff a b = Symtab.fold (Symtab.delete_safe o fst) b a
-
-fun collect_vars (cplexVar v) = singleton v
- | collect_vars (cplexNeg t) = collect_vars t
- | collect_vars (cplexProd (t1, t2)) =
- merge (collect_vars t1) (collect_vars t2)
- | collect_vars (cplexSum ts) = mergemap collect_vars ts
- | collect_vars _ = emptyset
-
-(* Eliminates all nonfree bounds from the linear program and produces an
- equivalent program with only free bounds
- IF for the input program P holds: is_normed_cplexProg P *)
-fun elim_nonfree_bounds (cplexProg (name, goal, constraints, bounds)) =
- let
- fun collect_constr_vars (_, cplexConstr (_, (t1,_))) =
- (collect_vars t1)
-
- val cvars = merge (collect_vars (term_of_goal goal))
- (mergemap collect_constr_vars constraints)
-
- fun collect_lower_bounded_vars
- (cplexBounds (_, _, cplexVar v, _, _)) =
- singleton v
- | collect_lower_bounded_vars
- (cplexBound (_, cplexLe, cplexVar v)) =
- singleton v
- | collect_lower_bounded_vars
- (cplexBound (_, cplexLeq, cplexVar v)) =
- singleton v
- | collect_lower_bounded_vars
- (cplexBound (cplexVar v, cplexGe,_)) =
- singleton v
- | collect_lower_bounded_vars
- (cplexBound (cplexVar v, cplexGeq, _)) =
- singleton v
- | collect_lower_bounded_vars
- (cplexBound (cplexVar v, cplexEq, _)) =
- singleton v
- | collect_lower_bounded_vars _ = emptyset
-
- val lvars = mergemap collect_lower_bounded_vars bounds
- val positive_vars = diff cvars lvars
- val zero = cplexNum "0"
-
- fun make_pos_constr v =
- (NONE, cplexConstr (cplexGeq, ((cplexVar v), zero)))
-
- fun make_free_bound v =
- cplexBounds (cplexNeg cplexInf, cplexLeq,
- cplexVar v, cplexLeq,
- cplexInf)
-
- val pos_constrs = rev (Symtab.fold
- (fn (k, _) => cons (make_pos_constr k))
- positive_vars [])
- val bound_constrs = map (pair NONE)
- (maps bound2constr bounds)
- val constraints' = constraints @ pos_constrs @ bound_constrs
- val bounds' = rev (Symtab.fold (fn (v, _) => cons (make_free_bound v)) cvars []);
- in
- cplexProg (name, goal, constraints', bounds')
- end
-
-fun relax_strict_ineqs (cplexProg (name, goals, constrs, bounds)) =
- let
- fun relax cplexLe = cplexLeq
- | relax cplexGe = cplexGeq
- | relax x = x
-
- fun relax_constr (n, cplexConstr(c, (t1, t2))) =
- (n, cplexConstr(relax c, (t1, t2)))
-
- fun relax_bounds (cplexBounds (t1, c1, t2, c2, t3)) =
- cplexBounds (t1, relax c1, t2, relax c2, t3)
- | relax_bounds (cplexBound (t1, c, t2)) =
- cplexBound (t1, relax c, t2)
- in
- cplexProg (name,
- goals,
- map relax_constr constrs,
- map relax_bounds bounds)
- end
-
-datatype cplexResult = Unbounded
- | Infeasible
- | Undefined
- | Optimal of string * ((string * string) list)
-
-fun is_separator x = forall (fn c => c = #"-") (String.explode x)
-
-fun is_sign x = (x = "+" orelse x = "-")
-
-fun is_colon x = (x = ":")
-
-fun is_resultsymbol a =
- let
- val symbol_char = String.explode "!\"#$%&()/,.;?@_`'{}|~-"
- fun is_symbol_char c = Char.isAlphaNum c orelse
- exists (fn d => d=c) symbol_char
- fun is_symbol_start c = is_symbol_char c andalso
- not (Char.isDigit c) andalso
- not (c= #".") andalso
- not (c= #"-")
- val b = String.explode a
- in
- b <> [] andalso is_symbol_start (hd b) andalso
- forall is_symbol_char b
- end
-
-val TOKEN_SIGN = 100
-val TOKEN_COLON = 101
-val TOKEN_SEPARATOR = 102
-
-fun load_glpkResult name =
- let
- val flist = [(is_NL, TOKEN_NL),
- (is_blank, TOKEN_BLANK),
- (is_num, TOKEN_NUM),
- (is_sign, TOKEN_SIGN),
- (is_colon, TOKEN_COLON),
- (is_cmp, TOKEN_CMP),
- (is_resultsymbol, TOKEN_SYMBOL),
- (is_separator, TOKEN_SEPARATOR)]
-
- val tokenize = tokenize_general flist
-
- val f = TextIO.openIn name
-
- val rest = Unsynchronized.ref []
-
- fun readToken_helper () =
- if length (!rest) > 0 then
- let val u = hd (!rest) in
- (
- rest := tl (!rest);
- SOME u
- )
- end
- else
- (case TextIO.inputLine f of
- NONE => NONE
- | SOME s => (rest := tokenize s; readToken_helper()))
-
- fun is_tt tok ty = (tok <> NONE andalso (fst (the tok)) = ty)
-
- fun pushToken a = if a = NONE then () else (rest := ((the a)::(!rest)))
-
- fun readToken () =
- let val t = readToken_helper () in
- if is_tt t TOKEN_BLANK then
- readToken ()
- else if is_tt t TOKEN_NL then
- let val t2 = readToken_helper () in
- if is_tt t2 TOKEN_SIGN then
- (pushToken (SOME (TOKEN_SEPARATOR, "-")); t)
- else
- (pushToken t2; t)
- end
- else if is_tt t TOKEN_SIGN then
- let val t2 = readToken_helper () in
- if is_tt t2 TOKEN_NUM then
- (SOME (TOKEN_NUM, (snd (the t))^(snd (the t2))))
- else
- (pushToken t2; t)
- end
- else
- t
- end
-
- fun readRestOfLine P =
- let
- val t = readToken ()
- in
- if is_tt t TOKEN_NL orelse t = NONE
- then P
- else readRestOfLine P
- end
-
- fun readHeader () =
- let
- fun readStatus () = readRestOfLine ("STATUS", snd (the (readToken ())))
- fun readObjective () = readRestOfLine ("OBJECTIVE", snd (the (readToken (); readToken (); readToken ())))
- val t1 = readToken ()
- val t2 = readToken ()
- in
- if is_tt t1 TOKEN_SYMBOL andalso is_tt t2 TOKEN_COLON
- then
- case to_upper (snd (the t1)) of
- "STATUS" => (readStatus ())::(readHeader ())
- | "OBJECTIVE" => (readObjective())::(readHeader ())
- | _ => (readRestOfLine (); readHeader ())
- else
- (pushToken t2; pushToken t1; [])
- end
-
- fun skip_until_sep () =
- let val x = readToken () in
- if is_tt x TOKEN_SEPARATOR then
- readRestOfLine ()
- else
- skip_until_sep ()
- end
-
- fun load_value () =
- let
- val t1 = readToken ()
- val t2 = readToken ()
- in
- if is_tt t1 TOKEN_NUM andalso is_tt t2 TOKEN_SYMBOL then
- let
- val t = readToken ()
- val state = if is_tt t TOKEN_NL then readToken () else t
- val _ = if is_tt state TOKEN_SYMBOL then () else raise (Load_cplexResult "state expected")
- val k = readToken ()
- in
- if is_tt k TOKEN_NUM then
- readRestOfLine (SOME (snd (the t2), snd (the k)))
- else
- raise (Load_cplexResult "number expected")
- end
- else
- (pushToken t2; pushToken t1; NONE)
- end
-
- fun load_values () =
- let val v = load_value () in
- if v = NONE then [] else (the v)::(load_values ())
- end
-
- val header = readHeader ()
-
- val result =
- case AList.lookup (op =) header "STATUS" of
- SOME "INFEASIBLE" => Infeasible
- | SOME "UNBOUNDED" => Unbounded
- | SOME "OPTIMAL" => Optimal (the (AList.lookup (op =) header "OBJECTIVE"),
- (skip_until_sep ();
- skip_until_sep ();
- load_values ()))
- | _ => Undefined
-
- val _ = TextIO.closeIn f
- in
- result
- end
- handle (Tokenize s) => raise (Load_cplexResult ("Tokenize: "^s))
- | Option => raise (Load_cplexResult "Option")
-
-fun load_cplexResult name =
- let
- val flist = [(is_NL, TOKEN_NL),
- (is_blank, TOKEN_BLANK),
- (is_num, TOKEN_NUM),
- (is_sign, TOKEN_SIGN),
- (is_colon, TOKEN_COLON),
- (is_cmp, TOKEN_CMP),
- (is_resultsymbol, TOKEN_SYMBOL)]
-
- val tokenize = tokenize_general flist
-
- val f = TextIO.openIn name
-
- val rest = Unsynchronized.ref []
-
- fun readToken_helper () =
- if length (!rest) > 0 then
- let val u = hd (!rest) in
- (
- rest := tl (!rest);
- SOME u
- )
- end
- else
- (case TextIO.inputLine f of
- NONE => NONE
- | SOME s => (rest := tokenize s; readToken_helper()))
-
- fun is_tt tok ty = (tok <> NONE andalso (fst (the tok)) = ty)
-
- fun pushToken a = if a = NONE then () else (rest := ((the a)::(!rest)))
-
- fun readToken () =
- let val t = readToken_helper () in
- if is_tt t TOKEN_BLANK then
- readToken ()
- else if is_tt t TOKEN_SIGN then
- let val t2 = readToken_helper () in
- if is_tt t2 TOKEN_NUM then
- (SOME (TOKEN_NUM, (snd (the t))^(snd (the t2))))
- else
- (pushToken t2; t)
- end
- else
- t
- end
-
- fun readRestOfLine P =
- let
- val t = readToken ()
- in
- if is_tt t TOKEN_NL orelse t = NONE
- then P
- else readRestOfLine P
- end
-
- fun readHeader () =
- let
- fun readStatus () = readRestOfLine ("STATUS", snd (the (readToken ())))
- fun readObjective () =
- let
- val t = readToken ()
- in
- if is_tt t TOKEN_SYMBOL andalso to_upper (snd (the t)) = "VALUE" then
- readRestOfLine ("OBJECTIVE", snd (the (readToken())))
- else
- readRestOfLine ("OBJECTIVE_NAME", snd (the t))
- end
-
- val t = readToken ()
- in
- if is_tt t TOKEN_SYMBOL then
- case to_upper (snd (the t)) of
- "STATUS" => (readStatus ())::(readHeader ())
- | "OBJECTIVE" => (readObjective ())::(readHeader ())
- | "SECTION" => (pushToken t; [])
- | _ => (readRestOfLine (); readHeader ())
- else
- (readRestOfLine (); readHeader ())
- end
-
- fun skip_nls () =
- let val x = readToken () in
- if is_tt x TOKEN_NL then
- skip_nls ()
- else
- (pushToken x; ())
- end
-
- fun skip_paragraph () =
- if is_tt (readToken ()) TOKEN_NL then
- (if is_tt (readToken ()) TOKEN_NL then
- skip_nls ()
- else
- skip_paragraph ())
- else
- skip_paragraph ()
-
- fun load_value () =
- let
- val t1 = readToken ()
- val t1 = if is_tt t1 TOKEN_SYMBOL andalso snd (the t1) = "A" then readToken () else t1
- in
- if is_tt t1 TOKEN_NUM then
- let
- val name = readToken ()
- val status = readToken ()
- val value = readToken ()
- in
- if is_tt name TOKEN_SYMBOL andalso
- is_tt status TOKEN_SYMBOL andalso
- is_tt value TOKEN_NUM
- then
- readRestOfLine (SOME (snd (the name), snd (the value)))
- else
- raise (Load_cplexResult "column line expected")
- end
- else
- (pushToken t1; NONE)
- end
-
- fun load_values () =
- let val v = load_value () in
- if v = NONE then [] else (the v)::(load_values ())
- end
-
- val header = readHeader ()
-
- val result =
- case AList.lookup (op =) header "STATUS" of
- SOME "INFEASIBLE" => Infeasible
- | SOME "NONOPTIMAL" => Unbounded
- | SOME "OPTIMAL" => Optimal (the (AList.lookup (op =) header "OBJECTIVE"),
- (skip_paragraph ();
- skip_paragraph ();
- skip_paragraph ();
- skip_paragraph ();
- skip_paragraph ();
- load_values ()))
- | _ => Undefined
-
- val _ = TextIO.closeIn f
- in
- result
- end
- handle (Tokenize s) => raise (Load_cplexResult ("Tokenize: "^s))
- | Option => raise (Load_cplexResult "Option")
-
-exception Execute of string;
-
-fun tmp_file s = Path.implode (Path.expand (File.tmp_path (Path.basic s)));
-fun wrap s = "\""^s^"\"";
-
-fun solve_glpk prog =
- let
- val name = string_of_int (Time.toMicroseconds (Time.now ()))
- val lpname = tmp_file (name^".lp")
- val resultname = tmp_file (name^".txt")
- val _ = save_cplexFile lpname prog
- val cplex_path = getenv "GLPK_PATH"
- val cplex = if cplex_path = "" then "glpsol" else cplex_path
- val command = (wrap cplex)^" --lpt "^(wrap lpname)^" --output "^(wrap resultname)
- val answer = #1 (Isabelle_System.bash_output command)
- in
- let
- val result = load_glpkResult resultname
- val _ = OS.FileSys.remove lpname
- val _ = OS.FileSys.remove resultname
- in
- result
- end
- handle (Load_cplexResult s) => raise (Execute ("Load_cplexResult: "^s^"\nExecute: "^answer))
- | _ => raise (Execute answer) (* FIXME avoid handle _ *)
- end
-
-fun solve_cplex prog =
- let
- fun write_script s lp r =
- let
- val f = TextIO.openOut s
- val _ = TextIO.output (f, "read\n"^lp^"\noptimize\nwrite\n"^r^"\nquit")
- val _ = TextIO.closeOut f
- in
- ()
- end
-
- val name = string_of_int (Time.toMicroseconds (Time.now ()))
- val lpname = tmp_file (name^".lp")
- val resultname = tmp_file (name^".txt")
- val scriptname = tmp_file (name^".script")
- val _ = save_cplexFile lpname prog
- val _ = write_script scriptname lpname resultname
- in
- let
- val result = load_cplexResult resultname
- val _ = OS.FileSys.remove lpname
- val _ = OS.FileSys.remove resultname
- val _ = OS.FileSys.remove scriptname
- in
- result
- end
- end
-
-fun solve prog =
- case get_solver () of
- SOLVER_DEFAULT =>
- (case getenv "LP_SOLVER" of
- "CPLEX" => solve_cplex prog
- | "GLPK" => solve_glpk prog
- | _ => raise (Execute ("LP_SOLVER must be set to CPLEX or to GLPK")))
- | SOLVER_CPLEX => solve_cplex prog
- | SOLVER_GLPK => solve_glpk prog
-
-end;
--- a/src/HOL/Matrix/FloatSparseMatrixBuilder.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,284 +0,0 @@
-(* Title: HOL/Matrix/FloatSparseMatrixBuilder.ML
- Author: Steven Obua
-*)
-
-signature FLOAT_SPARSE_MATRIX_BUILDER =
-sig
- include MATRIX_BUILDER
-
- structure cplex : CPLEX
-
- type float = Float.float
- val approx_value : int -> (float -> float) -> string -> term * term
- val approx_vector : int -> (float -> float) -> vector -> term * term
- val approx_matrix : int -> (float -> float) -> matrix -> term * term
-
- val mk_spvec_entry : int -> float -> term
- val mk_spvec_entry' : int -> term -> term
- val mk_spmat_entry : int -> term -> term
- val spvecT: typ
- val spmatT: typ
-
- val v_elem_at : vector -> int -> string option
- val m_elem_at : matrix -> int -> vector option
- val v_only_elem : vector -> int option
- val v_fold : (int * string -> 'a -> 'a) -> vector -> 'a -> 'a
- val m_fold : (int * vector -> 'a -> 'a) -> matrix -> 'a -> 'a
-
- val transpose_matrix : matrix -> matrix
-
- val cut_vector : int -> vector -> vector
- val cut_matrix : vector -> int option -> matrix -> matrix
-
- val delete_matrix : int list -> matrix -> matrix
- val cut_matrix' : int list -> matrix -> matrix
- val delete_vector : int list -> vector -> vector
- val cut_vector' : int list -> vector -> vector
-
- val indices_of_matrix : matrix -> int list
- val indices_of_vector : vector -> int list
-
- (* cplexProg c A b *)
- val cplexProg : vector -> matrix -> vector -> cplex.cplexProg * (string -> int)
- (* dual_cplexProg c A b *)
- val dual_cplexProg : vector -> matrix -> vector -> cplex.cplexProg * (string -> int)
-end;
-
-structure FloatSparseMatrixBuilder : FLOAT_SPARSE_MATRIX_BUILDER =
-struct
-
-type float = Float.float
-structure Inttab = Table(type key = int val ord = rev_order o int_ord);
-
-type vector = string Inttab.table
-type matrix = vector Inttab.table
-
-val spvec_elemT = HOLogic.mk_prodT (HOLogic.natT, HOLogic.realT);
-val spvecT = HOLogic.listT spvec_elemT;
-val spmat_elemT = HOLogic.mk_prodT (HOLogic.natT, spvecT);
-val spmatT = HOLogic.listT spmat_elemT;
-
-fun approx_value prec f =
- FloatArith.approx_float prec (fn (x, y) => (f x, f y));
-
-fun mk_spvec_entry i f =
- HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, FloatArith.mk_float f);
-
-fun mk_spvec_entry' i x =
- HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, x);
-
-fun mk_spmat_entry i e =
- HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, e);
-
-fun approx_vector prec pprt vector =
- let
- fun app (index, s) (lower, upper) =
- let
- val (flower, fupper) = approx_value prec pprt s
- val index = HOLogic.mk_number HOLogic.natT index
- val elower = HOLogic.mk_prod (index, flower)
- val eupper = HOLogic.mk_prod (index, fupper)
- in (elower :: lower, eupper :: upper) end;
- in
- pairself (HOLogic.mk_list spvec_elemT) (Inttab.fold app vector ([], []))
- end;
-
-fun approx_matrix prec pprt vector =
- let
- fun app (index, v) (lower, upper) =
- let
- val (flower, fupper) = approx_vector prec pprt v
- val index = HOLogic.mk_number HOLogic.natT index
- val elower = HOLogic.mk_prod (index, flower)
- val eupper = HOLogic.mk_prod (index, fupper)
- in (elower :: lower, eupper :: upper) end;
- in
- pairself (HOLogic.mk_list spmat_elemT) (Inttab.fold app vector ([], []))
- end;
-
-exception Nat_expected of int;
-
-val zero_interval = approx_value 1 I "0"
-
-fun set_elem vector index str =
- if index < 0 then
- raise (Nat_expected index)
- else if (approx_value 1 I str) = zero_interval then
- vector
- else
- Inttab.update (index, str) vector
-
-fun set_vector matrix index vector =
- if index < 0 then
- raise (Nat_expected index)
- else if Inttab.is_empty vector then
- matrix
- else
- Inttab.update (index, vector) matrix
-
-val empty_matrix = Inttab.empty
-val empty_vector = Inttab.empty
-
-(* dual stuff *)
-
-structure cplex = Cplex
-
-fun transpose_matrix matrix =
- let
- fun upd j (i, s) =
- Inttab.map_default (i, Inttab.empty) (Inttab.update (j, s));
- fun updm (j, v) = Inttab.fold (upd j) v;
- in Inttab.fold updm matrix empty_matrix end;
-
-exception No_name of string;
-
-exception Superfluous_constr_right_hand_sides
-
-fun cplexProg c A b =
- let
- val ytable = Unsynchronized.ref Inttab.empty
- fun indexof s =
- if String.size s = 0 then raise (No_name s)
- else case Int.fromString (String.extract(s, 1, NONE)) of
- SOME i => i | NONE => raise (No_name s)
-
- fun nameof i =
- let
- val s = "x" ^ string_of_int i
- val _ = Unsynchronized.change ytable (Inttab.update (i, s))
- in
- s
- end
-
- fun split_numstr s =
- if String.isPrefix "-" s then (false,String.extract(s, 1, NONE))
- else if String.isPrefix "+" s then (true, String.extract(s, 1, NONE))
- else (true, s)
-
- fun mk_term index s =
- let
- val (p, s) = split_numstr s
- val prod = cplex.cplexProd (cplex.cplexNum s, cplex.cplexVar (nameof index))
- in
- if p then prod else cplex.cplexNeg prod
- end
-
- fun vec2sum vector =
- cplex.cplexSum (Inttab.fold (fn (index, s) => fn list => (mk_term index s) :: list) vector [])
-
- fun mk_constr index vector c =
- let
- val s = case Inttab.lookup c index of SOME s => s | NONE => "0"
- val (p, s) = split_numstr s
- val num = if p then cplex.cplexNum s else cplex.cplexNeg (cplex.cplexNum s)
- in
- (NONE, cplex.cplexConstr (cplex.cplexLeq, (vec2sum vector, num)))
- end
-
- fun delete index c = Inttab.delete index c handle Inttab.UNDEF _ => c
-
- val (list, b) = Inttab.fold
- (fn (index, v) => fn (list, c) => ((mk_constr index v c)::list, delete index c))
- A ([], b)
- val _ = if Inttab.is_empty b then () else raise Superfluous_constr_right_hand_sides
-
- fun mk_free y = cplex.cplexBounds (cplex.cplexNeg cplex.cplexInf, cplex.cplexLeq,
- cplex.cplexVar y, cplex.cplexLeq,
- cplex.cplexInf)
-
- val yvars = Inttab.fold (fn (_, y) => fn l => (mk_free y)::l) (!ytable) []
-
- val prog = cplex.cplexProg ("original", cplex.cplexMaximize (vec2sum c), list, yvars)
- in
- (prog, indexof)
- end
-
-
-fun dual_cplexProg c A b =
- let
- fun indexof s =
- if String.size s = 0 then raise (No_name s)
- else case Int.fromString (String.extract(s, 1, NONE)) of
- SOME i => i | NONE => raise (No_name s)
-
- fun nameof i = "y" ^ string_of_int i
-
- fun split_numstr s =
- if String.isPrefix "-" s then (false,String.extract(s, 1, NONE))
- else if String.isPrefix "+" s then (true, String.extract(s, 1, NONE))
- else (true, s)
-
- fun mk_term index s =
- let
- val (p, s) = split_numstr s
- val prod = cplex.cplexProd (cplex.cplexNum s, cplex.cplexVar (nameof index))
- in
- if p then prod else cplex.cplexNeg prod
- end
-
- fun vec2sum vector =
- cplex.cplexSum (Inttab.fold (fn (index, s) => fn list => (mk_term index s)::list) vector [])
-
- fun mk_constr index vector c =
- let
- val s = case Inttab.lookup c index of SOME s => s | NONE => "0"
- val (p, s) = split_numstr s
- val num = if p then cplex.cplexNum s else cplex.cplexNeg (cplex.cplexNum s)
- in
- (NONE, cplex.cplexConstr (cplex.cplexEq, (vec2sum vector, num)))
- end
-
- fun delete index c = Inttab.delete index c handle Inttab.UNDEF _ => c
-
- val (list, c) = Inttab.fold
- (fn (index, v) => fn (list, c) => ((mk_constr index v c)::list, delete index c))
- (transpose_matrix A) ([], c)
- val _ = if Inttab.is_empty c then () else raise Superfluous_constr_right_hand_sides
-
- val prog = cplex.cplexProg ("dual", cplex.cplexMinimize (vec2sum b), list, [])
- in
- (prog, indexof)
- end
-
-fun cut_vector size v =
- let
- val count = Unsynchronized.ref 0;
- fun app (i, s) = if (!count < size) then
- (count := !count +1 ; Inttab.update (i, s))
- else I
- in
- Inttab.fold app v empty_vector
- end
-
-fun cut_matrix vfilter vsize m =
- let
- fun app (i, v) =
- if is_none (Inttab.lookup vfilter i) then I
- else case vsize
- of NONE => Inttab.update (i, v)
- | SOME s => Inttab.update (i, cut_vector s v)
- in Inttab.fold app m empty_matrix end
-
-fun v_elem_at v i = Inttab.lookup v i
-fun m_elem_at m i = Inttab.lookup m i
-
-fun v_only_elem v =
- case Inttab.min_key v of
- NONE => NONE
- | SOME vmin => (case Inttab.max_key v of
- NONE => SOME vmin
- | SOME vmax => if vmin = vmax then SOME vmin else NONE)
-
-fun v_fold f = Inttab.fold f;
-fun m_fold f = Inttab.fold f;
-
-fun indices_of_vector v = Inttab.keys v
-fun indices_of_matrix m = Inttab.keys m
-fun delete_vector indices v = fold Inttab.delete indices v
-fun delete_matrix indices m = fold Inttab.delete indices m
-fun cut_matrix' indices _ = fold (fn i => fn m => (case Inttab.lookup m i of NONE => m | SOME v => Inttab.update (i, v) m)) indices Inttab.empty
-fun cut_vector' indices _ = fold (fn i => fn v => (case Inttab.lookup v i of NONE => v | SOME x => Inttab.update (i, x) v)) indices Inttab.empty
-
-
-
-end;
--- a/src/HOL/Matrix/LP.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,164 +0,0 @@
-(* Title: HOL/Matrix/LP.thy
- Author: Steven Obua
-*)
-
-theory LP
-imports Main "~~/src/HOL/Library/Lattice_Algebras"
-begin
-
-lemma le_add_right_mono:
- assumes
- "a <= b + (c::'a::ordered_ab_group_add)"
- "c <= d"
- shows "a <= b + d"
- apply (rule_tac order_trans[where y = "b+c"])
- apply (simp_all add: assms)
- done
-
-lemma linprog_dual_estimate:
- assumes
- "A * x \<le> (b::'a::lattice_ring)"
- "0 \<le> y"
- "abs (A - A') \<le> \<delta>A"
- "b \<le> b'"
- "abs (c - c') \<le> \<delta>c"
- "abs x \<le> r"
- shows
- "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
-proof -
- from assms have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
- from assms have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
- have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: algebra_simps)
- from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
- have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
- by (simp only: 4 estimate_by_abs)
- have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
- by (simp add: abs_le_mult)
- have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
- by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
- have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <= (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
- by (simp add: abs_triangle_ineq mult_right_mono)
- have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
- by (simp add: abs_le_mult mult_right_mono)
- have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
- have 11: "abs (c'-c) = abs (c-c')"
- by (subst 10, subst abs_minus_cancel, simp)
- have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
- by (simp add: 11 assms mult_right_mono)
- have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
- by (simp add: assms mult_right_mono mult_left_mono)
- have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
- apply (rule mult_left_mono)
- apply (simp add: assms)
- apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
- apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
- apply (simp_all)
- apply (rule order_trans[where y="abs (A-A')"], simp_all add: assms)
- apply (rule order_trans[where y="abs (c-c')"], simp_all add: assms)
- done
- from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
- by (simp)
- show ?thesis
- apply (rule le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
- apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified assms]])
- done
-qed
-
-lemma le_ge_imp_abs_diff_1:
- assumes
- "A1 <= (A::'a::lattice_ring)"
- "A <= A2"
- shows "abs (A-A1) <= A2-A1"
-proof -
- have "0 <= A - A1"
- proof -
- have 1: "A - A1 = A + (- A1)" by simp
- show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified assms])
- qed
- then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
- with assms show "abs (A-A1) <= (A2-A1)" by simp
-qed
-
-lemma mult_le_prts:
- assumes
- "a1 <= (a::'a::lattice_ring)"
- "a <= a2"
- "b1 <= b"
- "b <= b2"
- shows
- "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
-proof -
- have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
- apply (subst prts[symmetric])+
- apply simp
- done
- then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
- by (simp add: algebra_simps)
- moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
- by (simp_all add: assms mult_mono)
- moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
- proof -
- have "pprt a * nprt b <= pprt a * nprt b2"
- by (simp add: mult_left_mono assms)
- moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
- by (simp add: mult_right_mono_neg assms)
- ultimately show ?thesis
- by simp
- qed
- moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
- proof -
- have "nprt a * pprt b <= nprt a2 * pprt b"
- by (simp add: mult_right_mono assms)
- moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
- by (simp add: mult_left_mono_neg assms)
- ultimately show ?thesis
- by simp
- qed
- moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
- proof -
- have "nprt a * nprt b <= nprt a * nprt b1"
- by (simp add: mult_left_mono_neg assms)
- moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
- by (simp add: mult_right_mono_neg assms)
- ultimately show ?thesis
- by simp
- qed
- ultimately show ?thesis
- by - (rule add_mono | simp)+
-qed
-
-lemma mult_le_dual_prts:
- assumes
- "A * x \<le> (b::'a::lattice_ring)"
- "0 \<le> y"
- "A1 \<le> A"
- "A \<le> A2"
- "c1 \<le> c"
- "c \<le> c2"
- "r1 \<le> x"
- "x \<le> r2"
- shows
- "c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
- (is "_ <= _ + ?C")
-proof -
- from assms have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
- moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: algebra_simps)
- ultimately have "c * x + (y * A - c) * x <= y * b" by simp
- then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
- then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: algebra_simps)
- have s2: "c - y * A <= c2 - y * A1"
- by (simp add: diff_minus assms add_mono mult_left_mono)
- have s1: "c1 - y * A2 <= c - y * A"
- by (simp add: diff_minus assms add_mono mult_left_mono)
- have prts: "(c - y * A) * x <= ?C"
- apply (simp add: Let_def)
- apply (rule mult_le_prts)
- apply (simp_all add: assms s1 s2)
- done
- then have "y * b + (c - y * A) * x <= y * b + ?C"
- by simp
- with cx show ?thesis
- by(simp only:)
-qed
-
-end
\ No newline at end of file
--- a/src/HOL/Matrix/Matrix.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1836 +0,0 @@
-(* Title: HOL/Matrix/Matrix.thy
- Author: Steven Obua
-*)
-
-theory Matrix
-imports Main "~~/src/HOL/Library/Lattice_Algebras"
-begin
-
-type_synonym 'a infmatrix = "nat \<Rightarrow> nat \<Rightarrow> 'a"
-
-definition nonzero_positions :: "(nat \<Rightarrow> nat \<Rightarrow> 'a::zero) \<Rightarrow> (nat \<times> nat) set" where
- "nonzero_positions A = {pos. A (fst pos) (snd pos) ~= 0}"
-
-definition "matrix = {(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
-
-typedef (open) 'a matrix = "matrix :: (nat \<Rightarrow> nat \<Rightarrow> 'a::zero) set"
- unfolding matrix_def
-proof
- show "(\<lambda>j i. 0) \<in> {(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
- by (simp add: nonzero_positions_def)
-qed
-
-declare Rep_matrix_inverse[simp]
-
-lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
- by (induct A) (simp add: Abs_matrix_inverse matrix_def)
-
-definition nrows :: "('a::zero) matrix \<Rightarrow> nat" where
- "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
-
-definition ncols :: "('a::zero) matrix \<Rightarrow> nat" where
- "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))"
-
-lemma nrows:
- assumes hyp: "nrows A \<le> m"
- shows "(Rep_matrix A m n) = 0"
-proof cases
- assume "nonzero_positions(Rep_matrix A) = {}"
- then show "(Rep_matrix A m n) = 0" by (simp add: nonzero_positions_def)
-next
- assume a: "nonzero_positions(Rep_matrix A) \<noteq> {}"
- let ?S = "fst`(nonzero_positions(Rep_matrix A))"
- have c: "finite (?S)" by (simp add: finite_nonzero_positions)
- from hyp have d: "Max (?S) < m" by (simp add: a nrows_def)
- have "m \<notin> ?S"
- proof -
- have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max_ge [OF c])
- moreover from d have "~(m <= Max ?S)" by (simp)
- ultimately show "m \<notin> ?S" by (auto)
- qed
- thus "Rep_matrix A m n = 0" by (simp add: nonzero_positions_def image_Collect)
-qed
-
-definition transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix" where
- "transpose_infmatrix A j i == A i j"
-
-definition transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix" where
- "transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"
-
-declare transpose_infmatrix_def[simp]
-
-lemma transpose_infmatrix_twice[simp]: "transpose_infmatrix (transpose_infmatrix A) = A"
-by ((rule ext)+, simp)
-
-lemma transpose_infmatrix: "transpose_infmatrix (% j i. P j i) = (% j i. P i j)"
- apply (rule ext)+
- by simp
-
-lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
-apply (rule Abs_matrix_inverse)
-apply (simp add: matrix_def nonzero_positions_def image_def)
-proof -
- let ?A = "{pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}"
- let ?swap = "% pos. (snd pos, fst pos)"
- let ?B = "{pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}"
- have swap_image: "?swap`?A = ?B"
- apply (simp add: image_def)
- apply (rule set_eqI)
- apply (simp)
- proof
- fix y
- assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)"
- thus "Rep_matrix x (fst y) (snd y) \<noteq> 0"
- proof -
- from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
- then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
- qed
- next
- fix y
- assume hyp: "Rep_matrix x (fst y) (snd y) \<noteq> 0"
- show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))"
- by (rule exI[of _ "snd y"], rule exI[of _ "fst y"]) (simp add: hyp)
- qed
- then have "finite (?swap`?A)"
- proof -
- have "finite (nonzero_positions (Rep_matrix x))" by (simp add: finite_nonzero_positions)
- then have "finite ?B" by (simp add: nonzero_positions_def)
- with swap_image show "finite (?swap`?A)" by (simp)
- qed
- moreover
- have "inj_on ?swap ?A" by (simp add: inj_on_def)
- ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A])
-qed
-
-lemma infmatrixforward: "(x::'a infmatrix) = y \<Longrightarrow> \<forall> a b. x a b = y a b" by auto
-
-lemma transpose_infmatrix_inject: "(transpose_infmatrix A = transpose_infmatrix B) = (A = B)"
-apply (auto)
-apply (rule ext)+
-apply (simp add: transpose_infmatrix)
-apply (drule infmatrixforward)
-apply (simp)
-done
-
-lemma transpose_matrix_inject: "(transpose_matrix A = transpose_matrix B) = (A = B)"
-apply (simp add: transpose_matrix_def)
-apply (subst Rep_matrix_inject[THEN sym])+
-apply (simp only: transpose_infmatrix_closed transpose_infmatrix_inject)
-done
-
-lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
-by (simp add: transpose_matrix_def)
-
-lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
-by (simp add: transpose_matrix_def)
-
-lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A"
-by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
-
-lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A"
-by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
-
-lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0"
-proof -
- assume "ncols A <= n"
- then have "nrows (transpose_matrix A) <= n" by (simp)
- then have "Rep_matrix (transpose_matrix A) n m = 0" by (rule nrows)
- thus "Rep_matrix A m n = 0" by (simp add: transpose_matrix_def)
-qed
-
-lemma ncols_le: "(ncols A <= n) = (! j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
-apply (auto)
-apply (simp add: ncols)
-proof (simp add: ncols_def, auto)
- let ?P = "nonzero_positions (Rep_matrix A)"
- let ?p = "snd`?P"
- have a:"finite ?p" by (simp add: finite_nonzero_positions)
- let ?m = "Max ?p"
- assume "~(Suc (?m) <= n)"
- then have b:"n <= ?m" by (simp)
- fix a b
- assume "(a,b) \<in> ?P"
- then have "?p \<noteq> {}" by (auto)
- with a have "?m \<in> ?p" by (simp)
- moreover have "!x. (x \<in> ?p \<longrightarrow> (? y. (Rep_matrix A y x) \<noteq> 0))" by (simp add: nonzero_positions_def image_def)
- ultimately have "? y. (Rep_matrix A y ?m) \<noteq> 0" by (simp)
- moreover assume ?st
- ultimately show "False" using b by (simp)
-qed
-
-lemma less_ncols: "(n < ncols A) = (? j i. n <= i & (Rep_matrix A j i) \<noteq> 0)"
-proof -
- have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
- show ?thesis by (simp add: a ncols_le)
-qed
-
-lemma le_ncols: "(n <= ncols A) = (\<forall> m. (\<forall> j i. m <= i \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
-apply (auto)
-apply (subgoal_tac "ncols A <= m")
-apply (simp)
-apply (simp add: ncols_le)
-apply (drule_tac x="ncols A" in spec)
-by (simp add: ncols)
-
-lemma nrows_le: "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" (is ?s)
-proof -
- have "(nrows A <= n) = (ncols (transpose_matrix A) <= n)" by (simp)
- also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
- also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
- finally show "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
-qed
-
-lemma less_nrows: "(m < nrows A) = (? j i. m <= j & (Rep_matrix A j i) \<noteq> 0)"
-proof -
- have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
- show ?thesis by (simp add: a nrows_le)
-qed
-
-lemma le_nrows: "(n <= nrows A) = (\<forall> m. (\<forall> j i. m <= j \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
-apply (auto)
-apply (subgoal_tac "nrows A <= m")
-apply (simp)
-apply (simp add: nrows_le)
-apply (drule_tac x="nrows A" in spec)
-by (simp add: nrows)
-
-lemma nrows_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> m < nrows A"
-apply (case_tac "nrows A <= m")
-apply (simp_all add: nrows)
-done
-
-lemma ncols_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> n < ncols A"
-apply (case_tac "ncols A <= n")
-apply (simp_all add: ncols)
-done
-
-lemma finite_natarray1: "finite {x. x < (n::nat)}"
-apply (induct n)
-apply (simp)
-proof -
- fix n
- have "{x. x < Suc n} = insert n {x. x < n}" by (rule set_eqI, simp, arith)
- moreover assume "finite {x. x < n}"
- ultimately show "finite {x. x < Suc n}" by (simp)
-qed
-
-lemma finite_natarray2: "finite {pos. (fst pos) < (m::nat) & (snd pos) < (n::nat)}"
- apply (induct m)
- apply (simp+)
- proof -
- fix m::nat
- let ?s0 = "{pos. fst pos < m & snd pos < n}"
- let ?s1 = "{pos. fst pos < (Suc m) & snd pos < n}"
- let ?sd = "{pos. fst pos = m & snd pos < n}"
- assume f0: "finite ?s0"
- have f1: "finite ?sd"
- proof -
- let ?f = "% x. (m, x)"
- have "{pos. fst pos = m & snd pos < n} = ?f ` {x. x < n}" by (rule set_eqI, simp add: image_def, auto)
- moreover have "finite {x. x < n}" by (simp add: finite_natarray1)
- ultimately show "finite {pos. fst pos = m & snd pos < n}" by (simp)
- qed
- have su: "?s0 \<union> ?sd = ?s1" by (rule set_eqI, simp, arith)
- from f0 f1 have "finite (?s0 \<union> ?sd)" by (rule finite_UnI)
- with su show "finite ?s1" by (simp)
-qed
-
-lemma RepAbs_matrix:
- assumes aem: "? m. ! j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"? n. ! j i. (n <= i \<longrightarrow> x j i = 0)" (is ?en)
- shows "(Rep_matrix (Abs_matrix x)) = x"
-apply (rule Abs_matrix_inverse)
-apply (simp add: matrix_def nonzero_positions_def)
-proof -
- from aem obtain m where a: "! j i. m <= j \<longrightarrow> x j i = 0" by (blast)
- from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)
- let ?u = "{pos. x (fst pos) (snd pos) \<noteq> 0}"
- let ?v = "{pos. fst pos < m & snd pos < n}"
- have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
- from a b have "(?u \<inter> (-?v)) = {}"
- apply (simp)
- apply (rule set_eqI)
- apply (simp)
- apply auto
- by (rule c, auto)+
- then have d: "?u \<subseteq> ?v" by blast
- moreover have "finite ?v" by (simp add: finite_natarray2)
- ultimately show "finite ?u" by (rule finite_subset)
-qed
-
-definition apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix" where
- "apply_infmatrix f == % A. (% j i. f (A j i))"
-
-definition apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix" where
- "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
-
-definition combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix" where
- "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
-
-definition combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix" where
- "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
-
-lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
-by (simp add: apply_infmatrix_def)
-
-lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
-by (simp add: combine_infmatrix_def)
-
-definition commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool" where
-"commutative f == ! x y. f x y = f y x"
-
-definition associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
-"associative f == ! x y z. f (f x y) z = f x (f y z)"
-
-text{*
-To reason about associativity and commutativity of operations on matrices,
-let's take a step back and look at the general situtation: Assume that we have
-sets $A$ and $B$ with $B \subset A$ and an abstraction $u: A \rightarrow B$. This abstraction has to fulfill $u(b) = b$ for all $b \in B$, but is arbitrary otherwise.
-Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
-It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
-*}
-
-lemma combine_infmatrix_commute:
- "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
-by (simp add: commutative_def combine_infmatrix_def)
-
-lemma combine_matrix_commute:
-"commutative f \<Longrightarrow> commutative (combine_matrix f)"
-by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
-
-text{*
-On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
-as $f$ we take addition on $\ganz$, which is clearly associative. The abstraction is given by $u(a) = 0$ for $a \notin B$. Then we have
-\[ f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1, \]
-but on the other hand we have
-\[ f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.\]
-A way out of this problem is to assume that $f(A\times A)\subset A$ holds, and this is what we are going to do:
-*}
-
-lemma nonzero_positions_combine_infmatrix[simp]: "f 0 0 = 0 \<Longrightarrow> nonzero_positions (combine_infmatrix f A B) \<subseteq> (nonzero_positions A) \<union> (nonzero_positions B)"
-by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
-
-lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
-by (insert Rep_matrix [of A], simp add: matrix_def)
-
-lemma combine_infmatrix_closed [simp]:
- "f 0 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) = combine_infmatrix f (Rep_matrix A) (Rep_matrix B)"
-apply (rule Abs_matrix_inverse)
-apply (simp add: matrix_def)
-apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
-by (simp_all)
-
-text {* We need the next two lemmas only later, but it is analog to the above one, so we prove them now: *}
-lemma nonzero_positions_apply_infmatrix[simp]: "f 0 = 0 \<Longrightarrow> nonzero_positions (apply_infmatrix f A) \<subseteq> nonzero_positions A"
-by (rule subsetI, simp add: nonzero_positions_def apply_infmatrix_def, auto)
-
-lemma apply_infmatrix_closed [simp]:
- "f 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
-apply (rule Abs_matrix_inverse)
-apply (simp add: matrix_def)
-apply (rule finite_subset[of _ "nonzero_positions (Rep_matrix A)"])
-by (simp_all)
-
-lemma combine_infmatrix_assoc[simp]: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_infmatrix f)"
-by (simp add: associative_def combine_infmatrix_def)
-
-lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
-by (auto)
-
-lemma combine_matrix_assoc: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_matrix f)"
-apply (simp(no_asm) add: associative_def combine_matrix_def, auto)
-apply (rule comb [of Abs_matrix Abs_matrix])
-by (auto, insert combine_infmatrix_assoc[of f], simp add: associative_def)
-
-lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
-by (simp add: apply_matrix_def)
-
-lemma Rep_combine_matrix[simp]: "f 0 0 = 0 \<Longrightarrow> Rep_matrix (combine_matrix f A B) j i = f (Rep_matrix A j i) (Rep_matrix B j i)"
- by(simp add: combine_matrix_def)
-
-lemma combine_nrows_max: "f 0 0 = 0 \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
-by (simp add: nrows_le)
-
-lemma combine_ncols_max: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"
-by (simp add: ncols_le)
-
-lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A <= q \<Longrightarrow> nrows B <= q \<Longrightarrow> nrows(combine_matrix f A B) <= q"
- by (simp add: nrows_le)
-
-lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A <= q \<Longrightarrow> ncols B <= q \<Longrightarrow> ncols(combine_matrix f A B) <= q"
- by (simp add: ncols_le)
-
-definition zero_r_neutral :: "('a \<Rightarrow> 'b::zero \<Rightarrow> 'a) \<Rightarrow> bool" where
- "zero_r_neutral f == ! a. f a 0 = a"
-
-definition zero_l_neutral :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
- "zero_l_neutral f == ! a. f 0 a = a"
-
-definition zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool" where
- "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
-
-primrec foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
-where
- "foldseq f s 0 = s 0"
-| "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
-
-primrec foldseq_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
-where
- "foldseq_transposed f s 0 = s 0"
-| "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
-
-lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
-proof -
- assume a:"associative f"
- then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
- proof -
- fix n
- show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
- proof (induct n)
- show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
- next
- fix n
- assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
- have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
- show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
- proof (auto)
- fix N t
- assume Nsuc: "N <= Suc n"
- show "foldseq f t N = foldseq_transposed f t N"
- proof cases
- assume "N <= n"
- then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
- next
- assume "~(N <= n)"
- with Nsuc have Nsuceq: "N = Suc n" by simp
- have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
- have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
- show "foldseq f t N = foldseq_transposed f t N"
- apply (simp add: Nsuceq)
- apply (subst c)
- apply (simp)
- apply (case_tac "n = 0")
- apply (simp)
- apply (drule neqz)
- apply (erule exE)
- apply (simp)
- apply (subst assocf)
- proof -
- fix m
- assume "n = Suc m & Suc m <= n"
- then have mless: "Suc m <= n" by arith
- then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
- apply (subst c)
- by simp+
- have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
- have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
- apply (subst c)
- by (simp add: mless)+
- have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
- from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
- qed
- qed
- qed
- qed
- qed
- show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
- qed
-
-lemma foldseq_distr: "\<lbrakk>associative f; commutative f\<rbrakk> \<Longrightarrow> foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)"
-proof -
- assume assoc: "associative f"
- assume comm: "commutative f"
- from assoc have a:"!! x y z. f (f x y) z = f x (f y z)" by (simp add: associative_def)
- from comm have b: "!! x y. f x y = f y x" by (simp add: commutative_def)
- from assoc comm have c: "!! x y z. f x (f y z) = f y (f x z)" by (simp add: commutative_def associative_def)
- have "!! n. (! u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
- apply (induct_tac n)
- apply (simp+, auto)
- by (simp add: a b c)
- then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
-qed
-
-theorem "\<lbrakk>associative f; associative g; \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); ? x y. (f x) \<noteq> (f y); ? x y. (g x) \<noteq> (g y); f x x = x; g x x = x\<rbrakk> \<Longrightarrow> f=g | (! y. f y x = y) | (! y. g y x = y)"
-oops
-(* Model found
-
-Trying to find a model that refutes: \<lbrakk>associative f; associative g;
- \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); \<exists>x y. f x \<noteq> f y;
- \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
-\<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)
-Searching for a model of size 1, translating term... invoking SAT solver... no model found.
-Searching for a model of size 2, translating term... invoking SAT solver... no model found.
-Searching for a model of size 3, translating term... invoking SAT solver...
-Model found:
-Size of types: 'a: 3
-x: a1
-g: (a0\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1), a1\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a1, a2\<mapsto>a0), a2\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1))
-f: (a0\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0), a1\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a1, a2\<mapsto>a1), a2\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0))
-*)
-
-lemma foldseq_zero:
-assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
-shows "foldseq f s n = 0"
-proof -
- have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
- apply (induct_tac n)
- apply (simp)
- by (simp add: fz)
- then show "foldseq f s n = 0" by (simp add: sz)
-qed
-
-lemma foldseq_significant_positions:
- assumes p: "! i. i <= N \<longrightarrow> S i = T i"
- shows "foldseq f S N = foldseq f T N"
-proof -
- have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
- apply (induct_tac m)
- apply (simp)
- apply (simp)
- apply (auto)
- proof -
- fix n
- fix s::"nat\<Rightarrow>'a"
- fix t::"nat\<Rightarrow>'a"
- assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
- assume b: "\<forall>i\<le>Suc n. s i = t i"
- have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
- have d:"!! s t. (\<forall>i\<le>n. s i = t i) \<Longrightarrow> foldseq f s n = foldseq f t n" by (simp add: a)
- show "f (t 0) (foldseq f (\<lambda>k. s (Suc k)) n) = f (t 0) (foldseq f (\<lambda>k. t (Suc k)) n)" by (rule c, simp add: d b)
- qed
- with p show ?thesis by simp
-qed
-
-lemma foldseq_tail:
- assumes "M <= N"
- shows "foldseq f S N = foldseq f (% k. (if k < M then (S k) else (foldseq f (% k. S(k+M)) (N-M)))) M"
-proof -
- have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
- have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
- have "!! n. ! m s. m <= n \<longrightarrow> foldseq f s n = foldseq f (% k. (if k < m then (s k) else (foldseq f (% k. s(k+m)) (n-m)))) m"
- apply (induct_tac n)
- apply (simp)
- apply (simp)
- apply (auto)
- apply (case_tac "m = Suc na")
- apply (simp)
- apply (rule a)
- apply (rule foldseq_significant_positions)
- apply (auto)
- apply (drule suc, simp+)
- proof -
- fix na m s
- assume suba:"\<forall>m\<le>na. \<forall>s. foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m"
- assume subb:"m <= na"
- from suba have subc:"!! m s. m <= na \<Longrightarrow>foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m" by simp
- have subd: "foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m =
- foldseq f (% k. s(Suc k)) na"
- by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
- from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
- show "f (s 0) (foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m) =
- foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
- apply (simp add: subd)
- apply (cases "m = 0")
- apply (simp)
- apply (drule sube)
- apply (auto)
- apply (rule a)
- by (simp add: subc cong del: if_cong)
- qed
- then show ?thesis using assms by simp
-qed
-
-lemma foldseq_zerotail:
- assumes
- fz: "f 0 0 = 0"
- and sz: "! i. n <= i \<longrightarrow> s i = 0"
- and nm: "n <= m"
- shows
- "foldseq f s n = foldseq f s m"
-proof -
- show "foldseq f s n = foldseq f s m"
- apply (simp add: foldseq_tail[OF nm, of f s])
- apply (rule foldseq_significant_positions)
- apply (auto)
- apply (subst foldseq_zero)
- by (simp add: fz sz)+
-qed
-
-lemma foldseq_zerotail2:
- assumes "! x. f x 0 = x"
- and "! i. n < i \<longrightarrow> s i = 0"
- and nm: "n <= m"
- shows "foldseq f s n = foldseq f s m"
-proof -
- have "f 0 0 = 0" by (simp add: assms)
- have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
- have c: "0 <= m" by simp
- have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
- show ?thesis
- apply (subst foldseq_tail[OF nm])
- apply (rule foldseq_significant_positions)
- apply (auto)
- apply (case_tac "m=n")
- apply (simp+)
- apply (drule b[OF nm])
- apply (auto)
- apply (case_tac "k=0")
- apply (simp add: assms)
- apply (drule d)
- apply (auto)
- apply (simp add: assms foldseq_zero)
- done
-qed
-
-lemma foldseq_zerostart:
- "! x. f 0 (f 0 x) = f 0 x \<Longrightarrow> ! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
-proof -
- assume f00x: "! x. f 0 (f 0 x) = f 0 x"
- have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
- apply (induct n)
- apply (simp)
- apply (rule allI, rule impI)
- proof -
- fix n
- fix s
- have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
- assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
- from b have c:"!! s. (\<forall>i\<le>n. s i = 0) \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
- assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
- show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
- apply (subst a)
- apply (subst c)
- by (simp add: d f00x)+
- qed
- then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
-qed
-
-lemma foldseq_zerostart2:
- "! x. f 0 x = x \<Longrightarrow> ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
-proof -
- assume a:"! i. i<n \<longrightarrow> s i = 0"
- assume x:"! x. f 0 x = x"
- from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
- have b: "!! i l. i < Suc l = (i <= l)" by arith
- have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
- show "foldseq f s n = s n"
- apply (case_tac "n=0")
- apply (simp)
- apply (insert a)
- apply (drule d)
- apply (auto)
- apply (simp add: b)
- apply (insert f00x)
- apply (drule foldseq_zerostart)
- by (simp add: x)+
-qed
-
-lemma foldseq_almostzero:
- assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
- shows "foldseq f s n = (if (j <= n) then (s j) else 0)"
-proof -
- from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
- from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
- show ?thesis
- apply auto
- apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
- apply simp
- apply (subst foldseq_zerostart2)
- apply (simp add: f0x a)+
- apply (subst foldseq_zero)
- by (simp add: s0 f0x)+
-qed
-
-lemma foldseq_distr_unary:
- assumes "!! a b. g (f a b) = f (g a) (g b)"
- shows "g(foldseq f s n) = foldseq f (% x. g(s x)) n"
-proof -
- have "! s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
- apply (induct_tac n)
- apply (simp)
- apply (simp)
- apply (auto)
- apply (drule_tac x="% k. s (Suc k)" in spec)
- by (simp add: assms)
- then show ?thesis by simp
-qed
-
-definition mult_matrix_n :: "nat \<Rightarrow> (('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix" where
- "mult_matrix_n n fmul fadd A B == Abs_matrix(% j i. foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
-
-definition mult_matrix :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix" where
- "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
-
-lemma mult_matrix_n:
- assumes "ncols A \<le> n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
- shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B"
-proof -
- show ?thesis using assms
- apply (simp add: mult_matrix_def mult_matrix_n_def)
- apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
- apply (rule foldseq_zerotail, simp_all add: nrows_le ncols_le assms)
- done
-qed
-
-lemma mult_matrix_nm:
- assumes "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
- shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
-proof -
- from assms have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B"
- by (simp add: mult_matrix_n)
- also from assms have "\<dots> = mult_matrix_n m fmul fadd A B"
- by (simp add: mult_matrix_n[THEN sym])
- finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
-qed
-
-definition r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
- "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
-
-definition l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
- "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
-
-definition distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
- "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd"
-
-lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
-lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
-
-lemma r_distributive_matrix:
- assumes
- "r_distributive fmul fadd"
- "associative fadd"
- "commutative fadd"
- "fadd 0 0 = 0"
- "! a. fmul a 0 = 0"
- "! a. fmul 0 a = 0"
- shows "r_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
-proof -
- from assms show ?thesis
- apply (simp add: r_distributive_def mult_matrix_def, auto)
- proof -
- fix a::"'a matrix"
- fix u::"'b matrix"
- fix v::"'b matrix"
- let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
- from assms show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
- combine_matrix fadd (mult_matrix_n (max (ncols a) (nrows u)) fmul fadd a u) (mult_matrix_n (max (ncols a) (nrows v)) fmul fadd a v)"
- apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (subst mult_matrix_nm[of _ _ v ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (subst mult_matrix_nm[of _ _ u ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
- apply (simp add: combine_matrix_def combine_infmatrix_def)
- apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
- apply (simplesubst RepAbs_matrix)
- apply (simp, auto)
- apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
- apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
- apply (subst RepAbs_matrix)
- apply (simp, auto)
- apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
- apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
- done
- qed
-qed
-
-lemma l_distributive_matrix:
- assumes
- "l_distributive fmul fadd"
- "associative fadd"
- "commutative fadd"
- "fadd 0 0 = 0"
- "! a. fmul a 0 = 0"
- "! a. fmul 0 a = 0"
- shows "l_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
-proof -
- from assms show ?thesis
- apply (simp add: l_distributive_def mult_matrix_def, auto)
- proof -
- fix a::"'b matrix"
- fix u::"'a matrix"
- fix v::"'a matrix"
- let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
- from assms show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
- combine_matrix fadd (mult_matrix_n (max (ncols u) (nrows a)) fmul fadd u a) (mult_matrix_n (max (ncols v) (nrows a)) fmul fadd v a)"
- apply (subst mult_matrix_nm[of v _ _ ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (subst mult_matrix_nm[of u _ _ ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
- apply (simp add: max1 max2 combine_nrows combine_ncols)+
- apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
- apply (simp add: combine_matrix_def combine_infmatrix_def)
- apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
- apply (simplesubst RepAbs_matrix)
- apply (simp, auto)
- apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
- apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
- apply (subst RepAbs_matrix)
- apply (simp, auto)
- apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
- apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
- done
- qed
-qed
-
-instantiation matrix :: (zero) zero
-begin
-
-definition zero_matrix_def: "0 = Abs_matrix (\<lambda>j i. 0)"
-
-instance ..
-
-end
-
-lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
- apply (simp add: zero_matrix_def)
- apply (subst RepAbs_matrix)
- by (auto)
-
-lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
-proof -
- have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
- show "nrows 0 = 0" by (rule a, subst nrows_le, simp)
-qed
-
-lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
-proof -
- have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
- show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
-qed
-
-lemma combine_matrix_zero_l_neutral: "zero_l_neutral f \<Longrightarrow> zero_l_neutral (combine_matrix f)"
- by (simp add: zero_l_neutral_def combine_matrix_def combine_infmatrix_def)
-
-lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
- by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
-
-lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
- apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)
- apply (auto)
- by (subst foldseq_zero, (simp add: zero_matrix_def)+)+
-
-lemma mult_matrix_n_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd A 0 = 0"
- apply (simp add: mult_matrix_n_def)
- apply (subst foldseq_zero)
- by (simp_all add: zero_matrix_def)
-
-lemma mult_matrix_n_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd 0 A = 0"
- apply (simp add: mult_matrix_n_def)
- apply (subst foldseq_zero)
- by (simp_all add: zero_matrix_def)
-
-lemma mult_matrix_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd 0 A = 0"
-by (simp add: mult_matrix_def)
-
-lemma mult_matrix_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd A 0 = 0"
-by (simp add: mult_matrix_def)
-
-lemma apply_matrix_zero[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f 0 = 0"
- apply (simp add: apply_matrix_def apply_infmatrix_def)
- by (simp add: zero_matrix_def)
-
-lemma combine_matrix_zero: "f 0 0 = 0 \<Longrightarrow> combine_matrix f 0 0 = 0"
- apply (simp add: combine_matrix_def combine_infmatrix_def)
- by (simp add: zero_matrix_def)
-
-lemma transpose_matrix_zero[simp]: "transpose_matrix 0 = 0"
-apply (simp add: transpose_matrix_def zero_matrix_def RepAbs_matrix)
-apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
-apply (simp add: RepAbs_matrix)
-done
-
-lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
- apply (simp add: apply_matrix_def apply_infmatrix_def)
- by (simp add: zero_matrix_def)
-
-definition singleton_matrix :: "nat \<Rightarrow> nat \<Rightarrow> ('a::zero) \<Rightarrow> 'a matrix" where
- "singleton_matrix j i a == Abs_matrix(% m n. if j = m & i = n then a else 0)"
-
-definition move_matrix :: "('a::zero) matrix \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a matrix" where
- "move_matrix A y x == Abs_matrix(% j i. if (((int j)-y) < 0) | (((int i)-x) < 0) then 0 else Rep_matrix A (nat ((int j)-y)) (nat ((int i)-x)))"
-
-definition take_rows :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
- "take_rows A r == Abs_matrix(% j i. if (j < r) then (Rep_matrix A j i) else 0)"
-
-definition take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
- "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
-
-definition column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
- "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
-
-definition row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
- "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
-
-lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
-apply (simp add: singleton_matrix_def)
-apply (auto)
-apply (subst RepAbs_matrix)
-apply (rule exI[of _ "Suc m"], simp)
-apply (rule exI[of _ "Suc n"], simp+)
-by (subst RepAbs_matrix, rule exI[of _ "Suc j"], simp, rule exI[of _ "Suc i"], simp+)+
-
-lemma apply_singleton_matrix[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f (singleton_matrix j i x) = (singleton_matrix j i (f x))"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (simp)
-done
-
-lemma singleton_matrix_zero[simp]: "singleton_matrix j i 0 = 0"
- by (simp add: singleton_matrix_def zero_matrix_def)
-
-lemma nrows_singleton[simp]: "nrows(singleton_matrix j i e) = (if e = 0 then 0 else Suc j)"
-proof-
-have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
-from th show ?thesis
-apply (auto)
-apply (rule le_antisym)
-apply (subst nrows_le)
-apply (simp add: singleton_matrix_def, auto)
-apply (subst RepAbs_matrix)
-apply auto
-apply (simp add: Suc_le_eq)
-apply (rule not_leE)
-apply (subst nrows_le)
-by simp
-qed
-
-lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
-proof-
-have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
-from th show ?thesis
-apply (auto)
-apply (rule le_antisym)
-apply (subst ncols_le)
-apply (simp add: singleton_matrix_def, auto)
-apply (subst RepAbs_matrix)
-apply auto
-apply (simp add: Suc_le_eq)
-apply (rule not_leE)
-apply (subst ncols_le)
-by simp
-qed
-
-lemma combine_singleton: "f 0 0 = 0 \<Longrightarrow> combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
-apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
-apply (subst RepAbs_matrix)
-apply (rule exI[of _ "Suc j"], simp)
-apply (rule exI[of _ "Suc i"], simp)
-apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
-apply (subst RepAbs_matrix)
-apply (rule exI[of _ "Suc j"], simp)
-apply (rule exI[of _ "Suc i"], simp)
-by simp
-
-lemma transpose_singleton[simp]: "transpose_matrix (singleton_matrix j i a) = singleton_matrix i j a"
-apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
-apply (simp)
-done
-
-lemma Rep_move_matrix[simp]:
- "Rep_matrix (move_matrix A y x) j i =
- (if (((int j)-y) < 0) | (((int i)-x) < 0) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
-apply (simp add: move_matrix_def)
-apply (auto)
-by (subst RepAbs_matrix,
- rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
- rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
-
-lemma move_matrix_0_0[simp]: "move_matrix A 0 0 = A"
-by (simp add: move_matrix_def)
-
-lemma move_matrix_ortho: "move_matrix A j i = move_matrix (move_matrix A j 0) 0 i"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (simp)
-done
-
-lemma transpose_move_matrix[simp]:
- "transpose_matrix (move_matrix A x y) = move_matrix (transpose_matrix A) y x"
-apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
-apply (simp)
-done
-
-lemma move_matrix_singleton[simp]: "move_matrix (singleton_matrix u v x) j i =
- (if (j + int u < 0) | (i + int v < 0) then 0 else (singleton_matrix (nat (j + int u)) (nat (i + int v)) x))"
- apply (subst Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply (case_tac "j + int u < 0")
- apply (simp, arith)
- apply (case_tac "i + int v < 0")
- apply (simp, arith)
- apply simp
- apply arith
- done
-
-lemma Rep_take_columns[simp]:
- "Rep_matrix (take_columns A c) j i =
- (if i < c then (Rep_matrix A j i) else 0)"
-apply (simp add: take_columns_def)
-apply (simplesubst RepAbs_matrix)
-apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
-apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
-done
-
-lemma Rep_take_rows[simp]:
- "Rep_matrix (take_rows A r) j i =
- (if j < r then (Rep_matrix A j i) else 0)"
-apply (simp add: take_rows_def)
-apply (simplesubst RepAbs_matrix)
-apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
-apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
-done
-
-lemma Rep_column_of_matrix[simp]:
- "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then (Rep_matrix A j c) else 0)"
- by (simp add: column_of_matrix_def)
-
-lemma Rep_row_of_matrix[simp]:
- "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then (Rep_matrix A r i) else 0)"
- by (simp add: row_of_matrix_def)
-
-lemma column_of_matrix: "ncols A <= n \<Longrightarrow> column_of_matrix A n = 0"
-apply (subst Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by (simp add: ncols)
-
-lemma row_of_matrix: "nrows A <= n \<Longrightarrow> row_of_matrix A n = 0"
-apply (subst Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by (simp add: nrows)
-
-lemma mult_matrix_singleton_right[simp]:
- assumes
- "! x. fmul x 0 = 0"
- "! x. fmul 0 x = 0"
- "! x. fadd 0 x = x"
- "! x. fadd x 0 = x"
- shows "(mult_matrix fmul fadd A (singleton_matrix j i e)) = apply_matrix (% x. fmul x e) (move_matrix (column_of_matrix A j) 0 (int i))"
- apply (simp add: mult_matrix_def)
- apply (subst mult_matrix_nm[of _ _ _ "max (ncols A) (Suc j)"])
- apply (auto)
- apply (simp add: assms)+
- apply (simp add: mult_matrix_n_def apply_matrix_def apply_infmatrix_def)
- apply (rule comb[of "Abs_matrix" "Abs_matrix"], auto, (rule ext)+)
- apply (subst foldseq_almostzero[of _ j])
- apply (simp add: assms)+
- apply (auto)
- done
-
-lemma mult_matrix_ext:
- assumes
- eprem:
- "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
- and fprems:
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "! a. fadd a 0 = a"
- "! a. fadd 0 a = a"
- and contraprems:
- "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
- shows
- "A = B"
-proof(rule contrapos_np[of "False"], simp)
- assume a: "A \<noteq> B"
- have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
- have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
- apply (rule contrapos_np[of "False"], simp+)
- apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
- by (simp add: Rep_matrix_inject a)
- then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
- from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
- let ?S = "singleton_matrix I 0 e"
- let ?comp = "mult_matrix fmul fadd"
- have d: "!!x f g. f = g \<Longrightarrow> f x = g x" by blast
- have e: "(% x. fmul x e) 0 = 0" by (simp add: assms)
- have "~(?comp A ?S = ?comp B ?S)"
- apply (rule notI)
- apply (simp add: fprems eprops)
- apply (simp add: Rep_matrix_inject[THEN sym])
- apply (drule d[of _ _ "J"], drule d[of _ _ "0"])
- by (simp add: e c eprops)
- with contraprems show "False" by simp
-qed
-
-definition foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
- "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
-
-definition foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
- "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
-
-lemma foldmatrix_transpose:
- assumes
- "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
- shows
- "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
-proof -
- have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
- have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
- apply (induct n)
- apply (simp add: foldmatrix_def foldmatrix_transposed_def assms)+
- apply (auto)
- by (drule_tac x="(% j i. A j (Suc i))" in forall, simp)
- show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
- apply (simp add: foldmatrix_def foldmatrix_transposed_def)
- apply (induct m, simp)
- apply (simp)
- apply (insert tworows)
- apply (drule_tac x="% j i. (if j = 0 then (foldseq_transposed g (\<lambda>u. A u i) m) else (A (Suc m) i))" in spec)
- by (simp add: foldmatrix_def foldmatrix_transposed_def)
-qed
-
-lemma foldseq_foldseq:
-assumes
- "associative f"
- "associative g"
- "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
-shows
- "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
- apply (insert foldmatrix_transpose[of g f A m n])
- by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] assms)
-
-lemma mult_n_nrows:
-assumes
-"! a. fmul 0 a = 0"
-"! a. fmul a 0 = 0"
-"fadd 0 0 = 0"
-shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
-apply (subst nrows_le)
-apply (simp add: mult_matrix_n_def)
-apply (subst RepAbs_matrix)
-apply (rule_tac x="nrows A" in exI)
-apply (simp add: nrows assms foldseq_zero)
-apply (rule_tac x="ncols B" in exI)
-apply (simp add: ncols assms foldseq_zero)
-apply (simp add: nrows assms foldseq_zero)
-done
-
-lemma mult_n_ncols:
-assumes
-"! a. fmul 0 a = 0"
-"! a. fmul a 0 = 0"
-"fadd 0 0 = 0"
-shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
-apply (subst ncols_le)
-apply (simp add: mult_matrix_n_def)
-apply (subst RepAbs_matrix)
-apply (rule_tac x="nrows A" in exI)
-apply (simp add: nrows assms foldseq_zero)
-apply (rule_tac x="ncols B" in exI)
-apply (simp add: ncols assms foldseq_zero)
-apply (simp add: ncols assms foldseq_zero)
-done
-
-lemma mult_nrows:
-assumes
-"! a. fmul 0 a = 0"
-"! a. fmul a 0 = 0"
-"fadd 0 0 = 0"
-shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
-by (simp add: mult_matrix_def mult_n_nrows assms)
-
-lemma mult_ncols:
-assumes
-"! a. fmul 0 a = 0"
-"! a. fmul a 0 = 0"
-"fadd 0 0 = 0"
-shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
-by (simp add: mult_matrix_def mult_n_ncols assms)
-
-lemma nrows_move_matrix_le: "nrows (move_matrix A j i) <= nat((int (nrows A)) + j)"
- apply (auto simp add: nrows_le)
- apply (rule nrows)
- apply (arith)
- done
-
-lemma ncols_move_matrix_le: "ncols (move_matrix A j i) <= nat((int (ncols A)) + i)"
- apply (auto simp add: ncols_le)
- apply (rule ncols)
- apply (arith)
- done
-
-lemma mult_matrix_assoc:
- assumes
- "! a. fmul1 0 a = 0"
- "! a. fmul1 a 0 = 0"
- "! a. fmul2 0 a = 0"
- "! a. fmul2 a 0 = 0"
- "fadd1 0 0 = 0"
- "fadd2 0 0 = 0"
- "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
- "associative fadd1"
- "associative fadd2"
- "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
- "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
- "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
- shows "mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B) C = mult_matrix fmul1 fadd1 A (mult_matrix fmul2 fadd2 B C)"
-proof -
- have comb_left: "!! A B x y. A = B \<Longrightarrow> (Rep_matrix (Abs_matrix A)) x y = (Rep_matrix(Abs_matrix B)) x y" by blast
- have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n) x = foldseq fadd1 (% k. fmul2 (s k) x) n"
- by (rule_tac g1 = "% y. fmul2 y x" in ssubst [OF foldseq_distr_unary], insert assms, simp_all)
- have fmul1fadd2fold: "!! x s n. fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (% k. fmul1 x (s k)) n"
- using assms by (rule_tac g1 = "% y. fmul1 x y" in ssubst [OF foldseq_distr_unary], simp_all)
- let ?N = "max (ncols A) (max (ncols B) (max (nrows B) (nrows C)))"
- show ?thesis
- apply (simp add: Rep_matrix_inject[THEN sym])
- apply (rule ext)+
- apply (simp add: mult_matrix_def)
- apply (simplesubst mult_matrix_nm[of _ "max (ncols (mult_matrix_n (max (ncols A) (nrows B)) fmul1 fadd1 A B)) (nrows C)" _ "max (ncols B) (nrows C)"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simplesubst mult_matrix_nm[of _ "max (ncols A) (nrows (mult_matrix_n (max (ncols B) (nrows C)) fmul2 fadd2 B C))" _ "max (ncols A) (nrows B)"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
- apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
- apply (simp add: mult_matrix_n_def)
- apply (rule comb_left)
- apply ((rule ext)+, simp)
- apply (simplesubst RepAbs_matrix)
- apply (rule exI[of _ "nrows B"])
- apply (simp add: nrows assms foldseq_zero)
- apply (rule exI[of _ "ncols C"])
- apply (simp add: assms ncols foldseq_zero)
- apply (subst RepAbs_matrix)
- apply (rule exI[of _ "nrows A"])
- apply (simp add: nrows assms foldseq_zero)
- apply (rule exI[of _ "ncols B"])
- apply (simp add: assms ncols foldseq_zero)
- apply (simp add: fmul2fadd1fold fmul1fadd2fold assms)
- apply (subst foldseq_foldseq)
- apply (simp add: assms)+
- apply (simp add: transpose_infmatrix)
- done
-qed
-
-lemma
- assumes
- "! a. fmul1 0 a = 0"
- "! a. fmul1 a 0 = 0"
- "! a. fmul2 0 a = 0"
- "! a. fmul2 a 0 = 0"
- "fadd1 0 0 = 0"
- "fadd2 0 0 = 0"
- "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
- "associative fadd1"
- "associative fadd2"
- "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
- "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
- "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
- shows
- "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)"
-apply (rule ext)+
-apply (simp add: comp_def )
-apply (simp add: mult_matrix_assoc assms)
-done
-
-lemma mult_matrix_assoc_simple:
- assumes
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "fadd 0 0 = 0"
- "associative fadd"
- "commutative fadd"
- "associative fmul"
- "distributive fmul fadd"
- shows "mult_matrix fmul fadd (mult_matrix fmul fadd A B) C = mult_matrix fmul fadd A (mult_matrix fmul fadd B C)"
-proof -
- have "!! a b c d. fadd (fadd a b) (fadd c d) = fadd (fadd a c) (fadd b d)"
- using assms by (simp add: associative_def commutative_def)
- then show ?thesis
- apply (subst mult_matrix_assoc)
- using assms
- apply simp_all
- apply (simp_all add: associative_def distributive_def l_distributive_def r_distributive_def)
- done
-qed
-
-lemma transpose_apply_matrix: "f 0 = 0 \<Longrightarrow> transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
-apply (simp add: Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by simp
-
-lemma transpose_combine_matrix: "f 0 0 = 0 \<Longrightarrow> transpose_matrix (combine_matrix f A B) = combine_matrix f (transpose_matrix A) (transpose_matrix B)"
-apply (simp add: Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by simp
-
-lemma Rep_mult_matrix:
- assumes
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "fadd 0 0 = 0"
- shows
- "Rep_matrix(mult_matrix fmul fadd A B) j i =
- foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
-apply (simp add: mult_matrix_def mult_matrix_n_def)
-apply (subst RepAbs_matrix)
-apply (rule exI[of _ "nrows A"], insert assms, simp add: nrows foldseq_zero)
-apply (rule exI[of _ "ncols B"], insert assms, simp add: ncols foldseq_zero)
-apply simp
-done
-
-lemma transpose_mult_matrix:
- assumes
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "fadd 0 0 = 0"
- "! x y. fmul y x = fmul x y"
- shows
- "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
- apply (simp add: Rep_matrix_inject[THEN sym])
- apply (rule ext)+
- using assms
- apply (simp add: Rep_mult_matrix max_ac)
- done
-
-lemma column_transpose_matrix: "column_of_matrix (transpose_matrix A) n = transpose_matrix (row_of_matrix A n)"
-apply (simp add: Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by simp
-
-lemma take_columns_transpose_matrix: "take_columns (transpose_matrix A) n = transpose_matrix (take_rows A n)"
-apply (simp add: Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-by simp
-
-instantiation matrix :: ("{zero, ord}") ord
-begin
-
-definition
- le_matrix_def: "A \<le> B \<longleftrightarrow> (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
-
-definition
- less_def: "A < (B\<Colon>'a matrix) \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
-
-instance ..
-
-end
-
-instance matrix :: ("{zero, order}") order
-apply intro_classes
-apply (simp_all add: le_matrix_def less_def)
-apply (auto)
-apply (drule_tac x=j in spec, drule_tac x=j in spec)
-apply (drule_tac x=i in spec, drule_tac x=i in spec)
-apply (simp)
-apply (simp add: Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
-apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
-apply simp
-done
-
-lemma le_apply_matrix:
- assumes
- "f 0 = 0"
- "! x y. x <= y \<longrightarrow> f x <= f y"
- "(a::('a::{ord, zero}) matrix) <= b"
- shows
- "apply_matrix f a <= apply_matrix f b"
- using assms by (simp add: le_matrix_def)
-
-lemma le_combine_matrix:
- assumes
- "f 0 0 = 0"
- "! a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
- "A <= B"
- "C <= D"
- shows
- "combine_matrix f A C <= combine_matrix f B D"
- using assms by (simp add: le_matrix_def)
-
-lemma le_left_combine_matrix:
- assumes
- "f 0 0 = 0"
- "! a b c. a <= b \<longrightarrow> f c a <= f c b"
- "A <= B"
- shows
- "combine_matrix f C A <= combine_matrix f C B"
- using assms by (simp add: le_matrix_def)
-
-lemma le_right_combine_matrix:
- assumes
- "f 0 0 = 0"
- "! a b c. a <= b \<longrightarrow> f a c <= f b c"
- "A <= B"
- shows
- "combine_matrix f A C <= combine_matrix f B C"
- using assms by (simp add: le_matrix_def)
-
-lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
- by (simp add: le_matrix_def, auto)
-
-lemma le_foldseq:
- assumes
- "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
- "! i. i <= n \<longrightarrow> s i <= t i"
- shows
- "foldseq f s n <= foldseq f t n"
-proof -
- have "! s t. (! i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n"
- by (induct n) (simp_all add: assms)
- then show "foldseq f s n <= foldseq f t n" using assms by simp
-qed
-
-lemma le_left_mult:
- assumes
- "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
- "! c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "fadd 0 0 = 0"
- "0 <= C"
- "A <= B"
- shows
- "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
- using assms
- apply (simp add: le_matrix_def Rep_mult_matrix)
- apply (auto)
- apply (simplesubst foldseq_zerotail[of _ _ _ "max (ncols C) (max (nrows A) (nrows B))"], simp_all add: nrows ncols max1 max2)+
- apply (rule le_foldseq)
- apply (auto)
- done
-
-lemma le_right_mult:
- assumes
- "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
- "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
- "! a. fmul 0 a = 0"
- "! a. fmul a 0 = 0"
- "fadd 0 0 = 0"
- "0 <= C"
- "A <= B"
- shows
- "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
- using assms
- apply (simp add: le_matrix_def Rep_mult_matrix)
- apply (auto)
- apply (simplesubst foldseq_zerotail[of _ _ _ "max (nrows C) (max (ncols A) (ncols B))"], simp_all add: nrows ncols max1 max2)+
- apply (rule le_foldseq)
- apply (auto)
- done
-
-lemma spec2: "! j i. P j i \<Longrightarrow> P j i" by blast
-lemma neg_imp: "(\<not> Q \<longrightarrow> \<not> P) \<Longrightarrow> P \<longrightarrow> Q" by blast
-
-lemma singleton_matrix_le[simp]: "(singleton_matrix j i a <= singleton_matrix j i b) = (a <= (b::_::order))"
- by (auto simp add: le_matrix_def)
-
-lemma singleton_le_zero[simp]: "(singleton_matrix j i x <= 0) = (x <= (0::'a::{order,zero}))"
- apply (auto)
- apply (simp add: le_matrix_def)
- apply (drule_tac j=j and i=i in spec2)
- apply (simp)
- apply (simp add: le_matrix_def)
- done
-
-lemma singleton_ge_zero[simp]: "(0 <= singleton_matrix j i x) = ((0::'a::{order,zero}) <= x)"
- apply (auto)
- apply (simp add: le_matrix_def)
- apply (drule_tac j=j and i=i in spec2)
- apply (simp)
- apply (simp add: le_matrix_def)
- done
-
-lemma move_matrix_le_zero[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= 0) = (A <= (0::('a::{order,zero}) matrix))"
- apply (auto simp add: le_matrix_def)
- apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
- apply (auto)
- done
-
-lemma move_matrix_zero_le[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (0 <= move_matrix A j i) = ((0::('a::{order,zero}) matrix) <= A)"
- apply (auto simp add: le_matrix_def)
- apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
- apply (auto)
- done
-
-lemma move_matrix_le_move_matrix_iff[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= move_matrix B j i) = (A <= (B::('a::{order,zero}) matrix))"
- apply (auto simp add: le_matrix_def)
- apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
- apply (auto)
- done
-
-instantiation matrix :: ("{lattice, zero}") lattice
-begin
-
-definition "inf = combine_matrix inf"
-
-definition "sup = combine_matrix sup"
-
-instance
- by default (auto simp add: le_infI le_matrix_def inf_matrix_def sup_matrix_def)
-
-end
-
-instantiation matrix :: ("{plus, zero}") plus
-begin
-
-definition
- plus_matrix_def: "A + B = combine_matrix (op +) A B"
-
-instance ..
-
-end
-
-instantiation matrix :: ("{uminus, zero}") uminus
-begin
-
-definition
- minus_matrix_def: "- A = apply_matrix uminus A"
-
-instance ..
-
-end
-
-instantiation matrix :: ("{minus, zero}") minus
-begin
-
-definition
- diff_matrix_def: "A - B = combine_matrix (op -) A B"
-
-instance ..
-
-end
-
-instantiation matrix :: ("{plus, times, zero}") times
-begin
-
-definition
- times_matrix_def: "A * B = mult_matrix (op *) (op +) A B"
-
-instance ..
-
-end
-
-instantiation matrix :: ("{lattice, uminus, zero}") abs
-begin
-
-definition
- abs_matrix_def: "abs (A \<Colon> 'a matrix) = sup A (- A)"
-
-instance ..
-
-end
-
-instance matrix :: (monoid_add) monoid_add
-proof
- fix A B C :: "'a matrix"
- show "A + B + C = A + (B + C)"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
- apply (simp_all add: add_assoc)
- done
- show "0 + A = A"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
- apply (simp)
- done
- show "A + 0 = A"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_zero_r_neutral [simplified zero_r_neutral_def, THEN spec])
- apply (simp)
- done
-qed
-
-instance matrix :: (comm_monoid_add) comm_monoid_add
-proof
- fix A B :: "'a matrix"
- show "A + B = B + A"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
- apply (simp_all add: add_commute)
- done
- show "0 + A = A"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
- apply (simp)
- done
-qed
-
-instance matrix :: (group_add) group_add
-proof
- fix A B :: "'a matrix"
- show "- A + A = 0"
- by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
- show "A - B = A + - B"
- by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject [symmetric] ext diff_minus)
-qed
-
-instance matrix :: (ab_group_add) ab_group_add
-proof
- fix A B :: "'a matrix"
- show "- A + A = 0"
- by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
- show "A - B = A + - B"
- by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
-qed
-
-instance matrix :: (ordered_ab_group_add) ordered_ab_group_add
-proof
- fix A B C :: "'a matrix"
- assume "A <= B"
- then show "C + A <= C + B"
- apply (simp add: plus_matrix_def)
- apply (rule le_left_combine_matrix)
- apply (simp_all)
- done
-qed
-
-instance matrix :: (lattice_ab_group_add) semilattice_inf_ab_group_add ..
-instance matrix :: (lattice_ab_group_add) semilattice_sup_ab_group_add ..
-
-instance matrix :: (semiring_0) semiring_0
-proof
- fix A B C :: "'a matrix"
- show "A * B * C = A * (B * C)"
- apply (simp add: times_matrix_def)
- apply (rule mult_matrix_assoc)
- apply (simp_all add: associative_def algebra_simps)
- done
- show "(A + B) * C = A * C + B * C"
- apply (simp add: times_matrix_def plus_matrix_def)
- apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
- apply (simp_all add: associative_def commutative_def algebra_simps)
- done
- show "A * (B + C) = A * B + A * C"
- apply (simp add: times_matrix_def plus_matrix_def)
- apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
- apply (simp_all add: associative_def commutative_def algebra_simps)
- done
- show "0 * A = 0" by (simp add: times_matrix_def)
- show "A * 0 = 0" by (simp add: times_matrix_def)
-qed
-
-instance matrix :: (ring) ring ..
-
-instance matrix :: (ordered_ring) ordered_ring
-proof
- fix A B C :: "'a matrix"
- assume a: "A \<le> B"
- assume b: "0 \<le> C"
- from a b show "C * A \<le> C * B"
- apply (simp add: times_matrix_def)
- apply (rule le_left_mult)
- apply (simp_all add: add_mono mult_left_mono)
- done
- from a b show "A * C \<le> B * C"
- apply (simp add: times_matrix_def)
- apply (rule le_right_mult)
- apply (simp_all add: add_mono mult_right_mono)
- done
-qed
-
-instance matrix :: (lattice_ring) lattice_ring
-proof
- fix A B C :: "('a :: lattice_ring) matrix"
- show "abs A = sup A (-A)"
- by (simp add: abs_matrix_def)
-qed
-
-lemma Rep_matrix_add[simp]:
- "Rep_matrix ((a::('a::monoid_add)matrix)+b) j i = (Rep_matrix a j i) + (Rep_matrix b j i)"
- by (simp add: plus_matrix_def)
-
-lemma Rep_matrix_mult: "Rep_matrix ((a::('a::semiring_0) matrix) * b) j i =
- foldseq (op +) (% k. (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
-apply (simp add: times_matrix_def)
-apply (simp add: Rep_mult_matrix)
-done
-
-lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a)
- \<Longrightarrow> apply_matrix f ((a::('a::monoid_add) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (simp)
-done
-
-lemma singleton_matrix_add: "singleton_matrix j i ((a::_::monoid_add)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (simp)
-done
-
-lemma nrows_mult: "nrows ((A::('a::semiring_0) matrix) * B) <= nrows A"
-by (simp add: times_matrix_def mult_nrows)
-
-lemma ncols_mult: "ncols ((A::('a::semiring_0) matrix) * B) <= ncols B"
-by (simp add: times_matrix_def mult_ncols)
-
-definition
- one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix" where
- "one_matrix n = Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
-
-lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
-apply (simp add: one_matrix_def)
-apply (simplesubst RepAbs_matrix)
-apply (rule exI[of _ n], simp add: split_if)+
-by (simp add: split_if)
-
-lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
-proof -
- have "?r <= n" by (simp add: nrows_le)
- moreover have "n <= ?r" by (simp add:le_nrows, arith)
- ultimately show "?r = n" by simp
-qed
-
-lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
-proof -
- have "?r <= n" by (simp add: ncols_le)
- moreover have "n <= ?r" by (simp add: le_ncols, arith)
- ultimately show "?r = n" by simp
-qed
-
-lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{semiring_1}) matrix) * (one_matrix n) = A"
-apply (subst Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-apply (simp add: times_matrix_def Rep_mult_matrix)
-apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
-apply (simp_all)
-by (simp add: ncols)
-
-lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::semiring_1) matrix)"
-apply (subst Rep_matrix_inject[THEN sym])
-apply (rule ext)+
-apply (simp add: times_matrix_def Rep_mult_matrix)
-apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
-apply (simp_all)
-by (simp add: nrows)
-
-lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
-apply (simp add: times_matrix_def)
-apply (subst transpose_mult_matrix)
-apply (simp_all add: mult_commute)
-done
-
-lemma transpose_matrix_add: "transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B"
-by (simp add: plus_matrix_def transpose_combine_matrix)
-
-lemma transpose_matrix_diff: "transpose_matrix ((A::('a::group_add) matrix)-B) = transpose_matrix A - transpose_matrix B"
-by (simp add: diff_matrix_def transpose_combine_matrix)
-
-lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::group_add) matrix)) = - transpose_matrix (A::'a matrix)"
-by (simp add: minus_matrix_def transpose_apply_matrix)
-
-definition right_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
- "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A"
-
-definition left_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
- "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A"
-
-definition inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
- "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
-
-lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
-apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
-by (simp add: right_inverse_matrix_def)
-
-lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
-apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A])
-by (simp add: left_inverse_matrix_def)
-
-lemma left_right_inverse_matrix_unique:
- assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
- shows "X = Y"
-proof -
- have "Y = Y * one_matrix (nrows A)"
- apply (subst one_matrix_mult_right)
- using assms
- apply (simp_all add: left_inverse_matrix_def)
- done
- also have "\<dots> = Y * (A * X)"
- apply (insert assms)
- apply (frule right_inverse_matrix_dim)
- by (simp add: right_inverse_matrix_def)
- also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
- also have "\<dots> = X"
- apply (insert assms)
- apply (frule left_inverse_matrix_dim)
- apply (simp_all add: left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
- done
- ultimately show "X = Y" by (simp)
-qed
-
-lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
- by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
-
-lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
- by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
-
-lemma zero_imp_mult_zero: "(a::'a::semiring_0) = 0 | b = 0 \<Longrightarrow> a * b = 0"
-by auto
-
-lemma Rep_matrix_zero_imp_mult_zero:
- "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0 \<Longrightarrow> A * B = (0::('a::lattice_ring) matrix)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
-done
-
-lemma add_nrows: "nrows (A::('a::monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
-apply (simp add: plus_matrix_def)
-apply (rule combine_nrows)
-apply (simp_all)
-done
-
-lemma move_matrix_row_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) j 0 = (move_matrix A j 0) * B"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (auto simp add: Rep_matrix_mult foldseq_zero)
-apply (rule_tac foldseq_zerotail[symmetric])
-apply (auto simp add: nrows zero_imp_mult_zero max2)
-apply (rule order_trans)
-apply (rule ncols_move_matrix_le)
-apply (simp add: max1)
-done
-
-lemma move_matrix_col_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) 0 i = A * (move_matrix B 0 i)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (auto simp add: Rep_matrix_mult foldseq_zero)
-apply (rule_tac foldseq_zerotail[symmetric])
-apply (auto simp add: ncols zero_imp_mult_zero max1)
-apply (rule order_trans)
-apply (rule nrows_move_matrix_le)
-apply (simp add: max2)
-done
-
-lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::monoid_add) matrix)) = (move_matrix A j i) + (move_matrix B j i)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (simp)
-done
-
-lemma move_matrix_mult: "move_matrix ((A::('a::semiring_0) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
-by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
-
-definition scalar_mult :: "('a::ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix" where
- "scalar_mult a m == apply_matrix (op * a) m"
-
-lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0"
-by (simp add: scalar_mult_def)
-
-lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
-by (simp add: scalar_mult_def apply_matrix_add algebra_simps)
-
-lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)"
-by (simp add: scalar_mult_def)
-
-lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
-apply (subst Rep_matrix_inject[symmetric])
-apply (rule ext)+
-apply (auto)
-done
-
-lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
-by (simp add: minus_matrix_def)
-
-lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lattice_ab_group_add)) x y = abs (Rep_matrix A x y)"
-by (simp add: abs_lattice sup_matrix_def)
-
-end
--- a/src/HOL/Matrix/ROOT.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2 +0,0 @@
-
-use_thy "Cplex";
--- a/src/HOL/Matrix/SparseMatrix.thy Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1070 +0,0 @@
-(* Title: HOL/Matrix/SparseMatrix.thy
- Author: Steven Obua
-*)
-
-theory SparseMatrix
-imports Matrix
-begin
-
-type_synonym 'a spvec = "(nat * 'a) list"
-type_synonym 'a spmat = "'a spvec spvec"
-
-definition sparse_row_vector :: "('a::ab_group_add) spvec \<Rightarrow> 'a matrix"
- where "sparse_row_vector arr = foldl (% m x. m + (singleton_matrix 0 (fst x) (snd x))) 0 arr"
-
-definition sparse_row_matrix :: "('a::ab_group_add) spmat \<Rightarrow> 'a matrix"
- where "sparse_row_matrix arr = foldl (% m r. m + (move_matrix (sparse_row_vector (snd r)) (int (fst r)) 0)) 0 arr"
-
-code_datatype sparse_row_vector sparse_row_matrix
-
-lemma sparse_row_vector_empty [simp]: "sparse_row_vector [] = 0"
- by (simp add: sparse_row_vector_def)
-
-lemma sparse_row_matrix_empty [simp]: "sparse_row_matrix [] = 0"
- by (simp add: sparse_row_matrix_def)
-
-lemmas [code] = sparse_row_vector_empty [symmetric]
-
-lemma foldl_distrstart: "! a x y. (f (g x y) a = g x (f y a)) \<Longrightarrow> (foldl f (g x y) l = g x (foldl f y l))"
- by (induct l arbitrary: x y, auto)
-
-lemma sparse_row_vector_cons[simp]:
- "sparse_row_vector (a # arr) = (singleton_matrix 0 (fst a) (snd a)) + (sparse_row_vector arr)"
- apply (induct arr)
- apply (auto simp add: sparse_row_vector_def)
- apply (simp add: foldl_distrstart [of "\<lambda>m x. m + singleton_matrix 0 (fst x) (snd x)" "\<lambda>x m. singleton_matrix 0 (fst x) (snd x) + m"])
- done
-
-lemma sparse_row_vector_append[simp]:
- "sparse_row_vector (a @ b) = (sparse_row_vector a) + (sparse_row_vector b)"
- by (induct a) auto
-
-lemma nrows_spvec[simp]: "nrows (sparse_row_vector x) <= (Suc 0)"
- apply (induct x)
- apply (simp_all add: add_nrows)
- done
-
-lemma sparse_row_matrix_cons: "sparse_row_matrix (a#arr) = ((move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0)) + sparse_row_matrix arr"
- apply (induct arr)
- apply (auto simp add: sparse_row_matrix_def)
- apply (simp add: foldl_distrstart[of "\<lambda>m x. m + (move_matrix (sparse_row_vector (snd x)) (int (fst x)) 0)"
- "% a m. (move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0) + m"])
- done
-
-lemma sparse_row_matrix_append: "sparse_row_matrix (arr@brr) = (sparse_row_matrix arr) + (sparse_row_matrix brr)"
- apply (induct arr)
- apply (auto simp add: sparse_row_matrix_cons)
- done
-
-primrec sorted_spvec :: "'a spvec \<Rightarrow> bool"
-where
- "sorted_spvec [] = True"
-| sorted_spvec_step: "sorted_spvec (a#as) = (case as of [] \<Rightarrow> True | b#bs \<Rightarrow> ((fst a < fst b) & (sorted_spvec as)))"
-
-primrec sorted_spmat :: "'a spmat \<Rightarrow> bool"
-where
- "sorted_spmat [] = True"
-| "sorted_spmat (a#as) = ((sorted_spvec (snd a)) & (sorted_spmat as))"
-
-declare sorted_spvec.simps [simp del]
-
-lemma sorted_spvec_empty[simp]: "sorted_spvec [] = True"
-by (simp add: sorted_spvec.simps)
-
-lemma sorted_spvec_cons1: "sorted_spvec (a#as) \<Longrightarrow> sorted_spvec as"
-apply (induct as)
-apply (auto simp add: sorted_spvec.simps)
-done
-
-lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) \<Longrightarrow> sorted_spvec (a#t)"
-apply (induct t)
-apply (auto simp add: sorted_spvec.simps)
-done
-
-lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) \<Longrightarrow> fst a < fst b"
-apply (auto simp add: sorted_spvec.simps)
-done
-
-lemma sorted_sparse_row_vector_zero[rule_format]: "m <= n \<Longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_vector arr) j m = 0"
-apply (induct arr)
-apply (auto)
-apply (frule sorted_spvec_cons2,simp)+
-apply (frule sorted_spvec_cons3, simp)
-done
-
-lemma sorted_sparse_row_matrix_zero[rule_format]: "m <= n \<Longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_matrix arr) m j = 0"
- apply (induct arr)
- apply (auto)
- apply (frule sorted_spvec_cons2, simp)
- apply (frule sorted_spvec_cons3, simp)
- apply (simp add: sparse_row_matrix_cons)
- done
-
-primrec minus_spvec :: "('a::ab_group_add) spvec \<Rightarrow> 'a spvec"
-where
- "minus_spvec [] = []"
-| "minus_spvec (a#as) = (fst a, -(snd a))#(minus_spvec as)"
-
-primrec abs_spvec :: "('a::lattice_ab_group_add_abs) spvec \<Rightarrow> 'a spvec"
-where
- "abs_spvec [] = []"
-| "abs_spvec (a#as) = (fst a, abs (snd a))#(abs_spvec as)"
-
-lemma sparse_row_vector_minus:
- "sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"
- apply (induct v)
- apply (simp_all add: sparse_row_vector_cons)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- done
-
-instance matrix :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
-apply default
-unfolding abs_matrix_def .. (*FIXME move*)
-
-lemma sparse_row_vector_abs:
- "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"
- apply (induct v)
- apply simp_all
- apply (frule_tac sorted_spvec_cons1, simp)
- apply (simp only: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply auto
- apply (subgoal_tac "Rep_matrix (sparse_row_vector v) 0 a = 0")
- apply (simp)
- apply (rule sorted_sparse_row_vector_zero)
- apply auto
- done
-
-lemma sorted_spvec_minus_spvec:
- "sorted_spvec v \<Longrightarrow> sorted_spvec (minus_spvec v)"
- apply (induct v)
- apply (simp)
- apply (frule sorted_spvec_cons1, simp)
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spvec_abs_spvec:
- "sorted_spvec v \<Longrightarrow> sorted_spvec (abs_spvec v)"
- apply (induct v)
- apply (simp)
- apply (frule sorted_spvec_cons1, simp)
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-definition "smult_spvec y = map (% a. (fst a, y * snd a))"
-
-lemma smult_spvec_empty[simp]: "smult_spvec y [] = []"
- by (simp add: smult_spvec_def)
-
-lemma smult_spvec_cons: "smult_spvec y (a#arr) = (fst a, y * (snd a)) # (smult_spvec y arr)"
- by (simp add: smult_spvec_def)
-
-fun addmult_spvec :: "('a::ring) \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
-where
- "addmult_spvec y arr [] = arr"
-| "addmult_spvec y [] brr = smult_spvec y brr"
-| "addmult_spvec y ((i,a)#arr) ((j,b)#brr) = (
- if i < j then ((i,a)#(addmult_spvec y arr ((j,b)#brr)))
- else (if (j < i) then ((j, y * b)#(addmult_spvec y ((i,a)#arr) brr))
- else ((i, a + y*b)#(addmult_spvec y arr brr))))"
-(* Steven used termination "measure (% (y, a, b). length a + (length b))" *)
-
-lemma addmult_spvec_empty1[simp]: "addmult_spvec y [] a = smult_spvec y a"
- by (induct a) auto
-
-lemma addmult_spvec_empty2[simp]: "addmult_spvec y a [] = a"
- by (induct a) auto
-
-lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lattice_ring)) 0 = 0 \<Longrightarrow>
- sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
- apply (induct a)
- apply (simp_all add: apply_matrix_add)
- done
-
-lemma sparse_row_vector_smult: "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"
- apply (induct a)
- apply (simp_all add: smult_spvec_cons scalar_mult_add)
- done
-
-lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lattice_ring) a b) =
- (sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
- apply (induct y a b rule: addmult_spvec.induct)
- apply (simp add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)+
- done
-
-lemma sorted_smult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec (smult_spvec y a)"
- apply (auto simp add: smult_spvec_def)
- apply (induct a)
- apply (auto simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spvec_addmult_spvec_helper: "\<lbrakk>sorted_spvec (addmult_spvec y ((a, b) # arr) brr); aa < a; sorted_spvec ((a, b) # arr);
- sorted_spvec ((aa, ba) # brr)\<rbrakk> \<Longrightarrow> sorted_spvec ((aa, y * ba) # addmult_spvec y ((a, b) # arr) brr)"
- apply (induct brr)
- apply (auto simp add: sorted_spvec.simps)
- done
-
-lemma sorted_spvec_addmult_spvec_helper2:
- "\<lbrakk>sorted_spvec (addmult_spvec y arr ((aa, ba) # brr)); a < aa; sorted_spvec ((a, b) # arr); sorted_spvec ((aa, ba) # brr)\<rbrakk>
- \<Longrightarrow> sorted_spvec ((a, b) # addmult_spvec y arr ((aa, ba) # brr))"
- apply (induct arr)
- apply (auto simp add: smult_spvec_def sorted_spvec.simps)
- done
-
-lemma sorted_spvec_addmult_spvec_helper3[rule_format]:
- "sorted_spvec (addmult_spvec y arr brr) \<longrightarrow> sorted_spvec ((aa, b) # arr) \<longrightarrow> sorted_spvec ((aa, ba) # brr)
- \<longrightarrow> sorted_spvec ((aa, b + y * ba) # (addmult_spvec y arr brr))"
- apply (induct y arr brr rule: addmult_spvec.induct)
- apply (simp_all add: sorted_spvec.simps smult_spvec_def split:list.split)
- done
-
-lemma sorted_addmult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> sorted_spvec (addmult_spvec y a b)"
- apply (induct y a b rule: addmult_spvec.induct)
- apply (simp_all add: sorted_smult_spvec)
- apply (rule conjI, intro strip)
- apply (case_tac "~(i < j)")
- apply (simp_all)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp add: sorted_spvec_addmult_spvec_helper)
- apply (intro strip | rule conjI)+
- apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (simp add: sorted_spvec_addmult_spvec_helper2)
- apply (intro strip)
- apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp)
- apply (simp_all add: sorted_spvec_addmult_spvec_helper3)
- done
-
-fun mult_spvec_spmat :: "('a::lattice_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec"
-where
-(* recdef mult_spvec_spmat "measure (% (c, arr, brr). (length arr) + (length brr))" *)
- "mult_spvec_spmat c [] brr = c"
-| "mult_spvec_spmat c arr [] = c"
-| "mult_spvec_spmat c ((i,a)#arr) ((j,b)#brr) = (
- if (i < j) then mult_spvec_spmat c arr ((j,b)#brr)
- else if (j < i) then mult_spvec_spmat c ((i,a)#arr) brr
- else mult_spvec_spmat (addmult_spvec a c b) arr brr)"
-
-lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lattice_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow>
- sparse_row_vector (mult_spvec_spmat c a B) = (sparse_row_vector c) + (sparse_row_vector a) * (sparse_row_matrix B)"
-proof -
- have comp_1: "!! a b. a < b \<Longrightarrow> Suc 0 <= nat ((int b)-(int a))" by arith
- have not_iff: "!! a b. a = b \<Longrightarrow> (~ a) = (~ b)" by simp
- have max_helper: "!! a b. ~ (a <= max (Suc a) b) \<Longrightarrow> False"
- by arith
- {
- fix a
- fix v
- assume a:"a < nrows(sparse_row_vector v)"
- have b:"nrows(sparse_row_vector v) <= 1" by simp
- note dummy = less_le_trans[of a "nrows (sparse_row_vector v)" 1, OF a b]
- then have "a = 0" by simp
- }
- note nrows_helper = this
- show ?thesis
- apply (induct c a B rule: mult_spvec_spmat.induct)
- apply simp+
- apply (rule conjI)
- apply (intro strip)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp add: algebra_simps sparse_row_matrix_cons)
- apply (simplesubst Rep_matrix_zero_imp_mult_zero)
- apply (simp)
- apply (rule disjI2)
- apply (intro strip)
- apply (subst nrows)
- apply (rule order_trans[of _ 1])
- apply (simp add: comp_1)+
- apply (subst Rep_matrix_zero_imp_mult_zero)
- apply (intro strip)
- apply (case_tac "k <= j")
- apply (rule_tac m1 = k and n1 = i and a1 = a in ssubst[OF sorted_sparse_row_vector_zero])
- apply (simp_all)
- apply (rule disjI2)
- apply (rule nrows)
- apply (rule order_trans[of _ 1])
- apply (simp_all add: comp_1)
-
- apply (intro strip | rule conjI)+
- apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (simp add: algebra_simps)
- apply (subst Rep_matrix_zero_imp_mult_zero)
- apply (simp)
- apply (rule disjI2)
- apply (intro strip)
- apply (simp add: sparse_row_matrix_cons)
- apply (case_tac "i <= j")
- apply (erule sorted_sparse_row_matrix_zero)
- apply (simp_all)
- apply (intro strip)
- apply (case_tac "i=j")
- apply (simp_all)
- apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp add: sparse_row_matrix_cons algebra_simps sparse_row_vector_addmult_spvec)
- apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
- apply (auto)
- apply (rule sorted_sparse_row_matrix_zero)
- apply (simp_all)
- apply (rule_tac A1 = "sparse_row_vector arr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
- apply (auto)
- apply (rule_tac m=k and n = j and a = a and arr=arr in sorted_sparse_row_vector_zero)
- apply (simp_all)
- apply (drule nrows_notzero)
- apply (drule nrows_helper)
- apply (arith)
-
- apply (subst Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply (simp)
- apply (subst Rep_matrix_mult)
- apply (rule_tac j1=j in ssubst[OF foldseq_almostzero])
- apply (simp_all)
- apply (intro strip, rule conjI)
- apply (intro strip)
- apply (drule_tac max_helper)
- apply (simp)
- apply (auto)
- apply (rule zero_imp_mult_zero)
- apply (rule disjI2)
- apply (rule nrows)
- apply (rule order_trans[of _ 1])
- apply (simp)
- apply (simp)
- done
-qed
-
-lemma sorted_mult_spvec_spmat[rule_format]:
- "sorted_spvec (c::('a::lattice_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
- apply (induct c a B rule: mult_spvec_spmat.induct)
- apply (simp_all add: sorted_addmult_spvec)
- done
-
-primrec mult_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
-where
- "mult_spmat [] A = []"
-| "mult_spmat (a#as) A = (fst a, mult_spvec_spmat [] (snd a) A)#(mult_spmat as A)"
-
-lemma sparse_row_mult_spmat:
- "sorted_spmat A \<Longrightarrow> sorted_spvec B \<Longrightarrow>
- sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
- apply (induct A)
- apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
- done
-
-lemma sorted_spvec_mult_spmat[rule_format]:
- "sorted_spvec (A::('a::lattice_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"
- apply (induct A)
- apply (auto)
- apply (drule sorted_spvec_cons1, simp)
- apply (case_tac A)
- apply (auto simp add: sorted_spvec.simps)
- done
-
-lemma sorted_spmat_mult_spmat:
- "sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
- apply (induct A)
- apply (auto simp add: sorted_mult_spvec_spmat)
- done
-
-
-fun add_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
-where
-(* "measure (% (a, b). length a + (length b))" *)
- "add_spvec arr [] = arr"
-| "add_spvec [] brr = brr"
-| "add_spvec ((i,a)#arr) ((j,b)#brr) = (
- if i < j then (i,a)#(add_spvec arr ((j,b)#brr))
- else if (j < i) then (j,b) # add_spvec ((i,a)#arr) brr
- else (i, a+b) # add_spvec arr brr)"
-
-lemma add_spvec_empty1[simp]: "add_spvec [] a = a"
-by (cases a, auto)
-
-lemma sparse_row_vector_add: "sparse_row_vector (add_spvec a b) = (sparse_row_vector a) + (sparse_row_vector b)"
- apply (induct a b rule: add_spvec.induct)
- apply (simp_all add: singleton_matrix_add)
- done
-
-fun add_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
-where
-(* "measure (% (A,B). (length A)+(length B))" *)
- "add_spmat [] bs = bs"
-| "add_spmat as [] = as"
-| "add_spmat ((i,a)#as) ((j,b)#bs) = (
- if i < j then
- (i,a) # add_spmat as ((j,b)#bs)
- else if j < i then
- (j,b) # add_spmat ((i,a)#as) bs
- else
- (i, add_spvec a b) # add_spmat as bs)"
-
-lemma add_spmat_Nil2[simp]: "add_spmat as [] = as"
-by(cases as) auto
-
-lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat A B) = (sparse_row_matrix A) + (sparse_row_matrix B)"
- apply (induct A B rule: add_spmat.induct)
- apply (auto simp add: sparse_row_matrix_cons sparse_row_vector_add move_matrix_add)
- done
-
-lemmas [code] = sparse_row_add_spmat [symmetric]
-lemmas [code] = sparse_row_vector_add [symmetric]
-
-lemma sorted_add_spvec_helper1[rule_format]: "add_spvec ((a,b)#arr) brr = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"
- proof -
- have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spvec x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
- by (induct brr rule: add_spvec.induct) (auto split:if_splits)
- then show ?thesis
- by (case_tac brr, auto)
- qed
-
-lemma sorted_add_spmat_helper1[rule_format]: "add_spmat ((a,b)#arr) brr = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"
- proof -
- have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spmat x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
- by (rule add_spmat.induct) (auto split:if_splits)
- then show ?thesis
- by (case_tac brr, auto)
- qed
-
-lemma sorted_add_spvec_helper: "add_spvec arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
- apply (induct arr brr rule: add_spvec.induct)
- apply (auto split:if_splits)
- done
-
-lemma sorted_add_spmat_helper: "add_spmat arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
- apply (induct arr brr rule: add_spmat.induct)
- apply (auto split:if_splits)
- done
-
-lemma add_spvec_commute: "add_spvec a b = add_spvec b a"
-by (induct a b rule: add_spvec.induct) auto
-
-lemma add_spmat_commute: "add_spmat a b = add_spmat b a"
- apply (induct a b rule: add_spmat.induct)
- apply (simp_all add: add_spvec_commute)
- done
-
-lemma sorted_add_spvec_helper2: "add_spvec ((a,b)#arr) brr = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"
- apply (drule sorted_add_spvec_helper1)
- apply (auto)
- apply (case_tac brr)
- apply (simp_all)
- apply (drule_tac sorted_spvec_cons3)
- apply (simp)
- done
-
-lemma sorted_add_spmat_helper2: "add_spmat ((a,b)#arr) brr = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"
- apply (drule sorted_add_spmat_helper1)
- apply (auto)
- apply (case_tac brr)
- apply (simp_all)
- apply (drule_tac sorted_spvec_cons3)
- apply (simp)
- done
-
-lemma sorted_spvec_add_spvec[rule_format]: "sorted_spvec a \<longrightarrow> sorted_spvec b \<longrightarrow> sorted_spvec (add_spvec a b)"
- apply (induct a b rule: add_spvec.induct)
- apply (simp_all)
- apply (rule conjI)
- apply (clarsimp)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp)
- apply (subst sorted_spvec_step)
- apply (clarsimp simp: sorted_add_spvec_helper2 split: list.split)
- apply (clarify)
- apply (rule conjI)
- apply (clarify)
- apply (frule_tac as=arr in sorted_spvec_cons1, simp)
- apply (subst sorted_spvec_step)
- apply (clarsimp simp: sorted_add_spvec_helper2 add_spvec_commute split: list.split)
- apply (clarify)
- apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp)
- apply (subst sorted_spvec_step)
- apply (simp split: list.split)
- apply (clarsimp)
- apply (drule_tac sorted_add_spvec_helper)
- apply (auto simp: neq_Nil_conv)
- apply (drule sorted_spvec_cons3)
- apply (simp)
- apply (drule sorted_spvec_cons3)
- apply (simp)
- done
-
-lemma sorted_spvec_add_spmat[rule_format]: "sorted_spvec A \<longrightarrow> sorted_spvec B \<longrightarrow> sorted_spvec (add_spmat A B)"
- apply (induct A B rule: add_spmat.induct)
- apply (simp_all)
- apply (rule conjI)
- apply (intro strip)
- apply (simp)
- apply (frule_tac as=bs in sorted_spvec_cons1)
- apply (simp)
- apply (subst sorted_spvec_step)
- apply (simp split: list.split)
- apply (clarify, simp)
- apply (simp add: sorted_add_spmat_helper2)
- apply (clarify)
- apply (rule conjI)
- apply (clarify)
- apply (frule_tac as=as in sorted_spvec_cons1, simp)
- apply (subst sorted_spvec_step)
- apply (clarsimp simp: sorted_add_spmat_helper2 add_spmat_commute split: list.split)
- apply (clarsimp)
- apply (frule_tac as=as in sorted_spvec_cons1)
- apply (frule_tac as=bs in sorted_spvec_cons1)
- apply (simp)
- apply (subst sorted_spvec_step)
- apply (simp split: list.split)
- apply (clarify, simp)
- apply (drule_tac sorted_add_spmat_helper)
- apply (auto simp:neq_Nil_conv)
- apply (drule sorted_spvec_cons3)
- apply (simp)
- apply (drule sorted_spvec_cons3)
- apply (simp)
- done
-
-lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (add_spmat A B)"
- apply (induct A B rule: add_spmat.induct)
- apply (simp_all add: sorted_spvec_add_spvec)
- done
-
-fun le_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool"
-where
-(* "measure (% (a,b). (length a) + (length b))" *)
- "le_spvec [] [] = True"
-| "le_spvec ((_,a)#as) [] = (a <= 0 & le_spvec as [])"
-| "le_spvec [] ((_,b)#bs) = (0 <= b & le_spvec [] bs)"
-| "le_spvec ((i,a)#as) ((j,b)#bs) = (
- if (i < j) then a <= 0 & le_spvec as ((j,b)#bs)
- else if (j < i) then 0 <= b & le_spvec ((i,a)#as) bs
- else a <= b & le_spvec as bs)"
-
-fun le_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool"
-where
-(* "measure (% (a,b). (length a) + (length b))" *)
- "le_spmat [] [] = True"
-| "le_spmat ((i,a)#as) [] = (le_spvec a [] & le_spmat as [])"
-| "le_spmat [] ((j,b)#bs) = (le_spvec [] b & le_spmat [] bs)"
-| "le_spmat ((i,a)#as) ((j,b)#bs) = (
- if i < j then (le_spvec a [] & le_spmat as ((j,b)#bs))
- else if j < i then (le_spvec [] b & le_spmat ((i,a)#as) bs)
- else (le_spvec a b & le_spmat as bs))"
-
-definition disj_matrices :: "('a::zero) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
- "disj_matrices A B \<longleftrightarrow>
- (! j i. (Rep_matrix A j i \<noteq> 0) \<longrightarrow> (Rep_matrix B j i = 0)) & (! j i. (Rep_matrix B j i \<noteq> 0) \<longrightarrow> (Rep_matrix A j i = 0))"
-
-declare [[simp_depth_limit = 6]]
-
-lemma disj_matrices_contr1: "disj_matrices A B \<Longrightarrow> Rep_matrix A j i \<noteq> 0 \<Longrightarrow> Rep_matrix B j i = 0"
- by (simp add: disj_matrices_def)
-
-lemma disj_matrices_contr2: "disj_matrices A B \<Longrightarrow> Rep_matrix B j i \<noteq> 0 \<Longrightarrow> Rep_matrix A j i = 0"
- by (simp add: disj_matrices_def)
-
-
-lemma disj_matrices_add: "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (A + B <= C + D) = (A <= C & B <= (D::('a::lattice_ab_group_add) matrix))"
- apply (auto)
- apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
- apply (intro strip)
- apply (erule conjE)+
- apply (drule_tac j=j and i=i in spec2)+
- apply (case_tac "Rep_matrix B j i = 0")
- apply (case_tac "Rep_matrix D j i = 0")
- apply (simp_all)
- apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
- apply (intro strip)
- apply (erule conjE)+
- apply (drule_tac j=j and i=i in spec2)+
- apply (case_tac "Rep_matrix A j i = 0")
- apply (case_tac "Rep_matrix C j i = 0")
- apply (simp_all)
- apply (erule add_mono)
- apply (assumption)
- done
-
-lemma disj_matrices_zero1[simp]: "disj_matrices 0 B"
-by (simp add: disj_matrices_def)
-
-lemma disj_matrices_zero2[simp]: "disj_matrices A 0"
-by (simp add: disj_matrices_def)
-
-lemma disj_matrices_commute: "disj_matrices A B = disj_matrices B A"
-by (auto simp add: disj_matrices_def)
-
-lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>
- (A + B <= 0) = (A <= 0 & (B::('a::lattice_ab_group_add) matrix) <= 0)"
-by (rule disj_matrices_add[of A B 0 0, simplified])
-
-lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>
- (0 <= A + B) = (0 <= A & 0 <= (B::('a::lattice_ab_group_add) matrix))"
-by (rule disj_matrices_add[of 0 0 A B, simplified])
-
-lemma disj_matrices_add_x_le: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (A <= B + C) = (A <= C & 0 <= (B::('a::lattice_ab_group_add) matrix))"
-by (auto simp add: disj_matrices_add[of 0 A B C, simplified])
-
-lemma disj_matrices_add_le_x: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
- (B + A <= C) = (A <= C & (B::('a::lattice_ab_group_add) matrix) <= 0)"
-by (auto simp add: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)
-
-lemma disj_sparse_row_singleton: "i <= j \<Longrightarrow> sorted_spvec((j,y)#v) \<Longrightarrow> disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"
- apply (simp add: disj_matrices_def)
- apply (rule conjI)
- apply (rule neg_imp)
- apply (simp)
- apply (intro strip)
- apply (rule sorted_sparse_row_vector_zero)
- apply (simp_all)
- apply (intro strip)
- apply (rule sorted_sparse_row_vector_zero)
- apply (simp_all)
- done
-
-lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lattice_ab_group_add) matrix) (B+C)"
- apply (simp add: disj_matrices_def)
- apply (auto)
- apply (drule_tac j=j and i=i in spec2)+
- apply (case_tac "Rep_matrix B j i = 0")
- apply (case_tac "Rep_matrix C j i = 0")
- apply (simp_all)
- done
-
-lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lattice_ab_group_add) matrix)"
- by (simp add: disj_matrices_x_add disj_matrices_commute)
-
-lemma disj_singleton_matrices[simp]: "disj_matrices (singleton_matrix j i x) (singleton_matrix u v y) = (j \<noteq> u | i \<noteq> v | x = 0 | y = 0)"
- by (auto simp add: disj_matrices_def)
-
-lemma disj_move_sparse_vec_mat[simplified disj_matrices_commute]:
- "j <= a \<Longrightarrow> sorted_spvec((a,c)#as) \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector b) (int j) i) (sparse_row_matrix as)"
- apply (auto simp add: disj_matrices_def)
- apply (drule nrows_notzero)
- apply (drule less_le_trans[OF _ nrows_spvec])
- apply (subgoal_tac "ja = j")
- apply (simp add: sorted_sparse_row_matrix_zero)
- apply (arith)
- apply (rule nrows)
- apply (rule order_trans[of _ 1 _])
- apply (simp)
- apply (case_tac "nat (int ja - int j) = 0")
- apply (case_tac "ja = j")
- apply (simp add: sorted_sparse_row_matrix_zero)
- apply arith+
- done
-
-lemma disj_move_sparse_row_vector_twice:
- "j \<noteq> u \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"
- apply (auto simp add: disj_matrices_def)
- apply (rule nrows, rule order_trans[of _ 1], simp, drule nrows_notzero, drule less_le_trans[OF _ nrows_spvec], arith)+
- done
-
-lemma le_spvec_iff_sparse_row_le[rule_format]: "(sorted_spvec a) \<longrightarrow> (sorted_spvec b) \<longrightarrow> (le_spvec a b) = (sparse_row_vector a <= sparse_row_vector b)"
- apply (induct a b rule: le_spvec.induct)
- apply (simp_all add: sorted_spvec_cons1 disj_matrices_add_le_zero disj_matrices_add_zero_le
- disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
- apply (rule conjI, intro strip)
- apply (simp add: sorted_spvec_cons1)
- apply (subst disj_matrices_add_x_le)
- apply (simp add: disj_sparse_row_singleton[OF less_imp_le] disj_matrices_x_add disj_matrices_commute)
- apply (simp add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
- apply (simp, blast)
- apply (intro strip, rule conjI, intro strip)
- apply (simp add: sorted_spvec_cons1)
- apply (subst disj_matrices_add_le_x)
- apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_sparse_row_singleton[OF less_imp_le] disj_matrices_commute disj_matrices_x_add)
- apply (blast)
- apply (intro strip)
- apply (simp add: sorted_spvec_cons1)
- apply (case_tac "a=b", simp_all)
- apply (subst disj_matrices_add)
- apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
- done
-
-lemma le_spvec_empty2_sparse_row[rule_format]: "sorted_spvec b \<longrightarrow> le_spvec b [] = (sparse_row_vector b <= 0)"
- apply (induct b)
- apply (simp_all add: sorted_spvec_cons1)
- apply (intro strip)
- apply (subst disj_matrices_add_le_zero)
- apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
- done
-
-lemma le_spvec_empty1_sparse_row[rule_format]: "(sorted_spvec b) \<longrightarrow> (le_spvec [] b = (0 <= sparse_row_vector b))"
- apply (induct b)
- apply (simp_all add: sorted_spvec_cons1)
- apply (intro strip)
- apply (subst disj_matrices_add_zero_le)
- apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
- done
-
-lemma le_spmat_iff_sparse_row_le[rule_format]: "(sorted_spvec A) \<longrightarrow> (sorted_spmat A) \<longrightarrow> (sorted_spvec B) \<longrightarrow> (sorted_spmat B) \<longrightarrow>
- le_spmat A B = (sparse_row_matrix A <= sparse_row_matrix B)"
- apply (induct A B rule: le_spmat.induct)
- apply (simp add: sparse_row_matrix_cons disj_matrices_add_le_zero disj_matrices_add_zero_le disj_move_sparse_vec_mat[OF order_refl]
- disj_matrices_commute sorted_spvec_cons1 le_spvec_empty2_sparse_row le_spvec_empty1_sparse_row)+
- apply (rule conjI, intro strip)
- apply (simp add: sorted_spvec_cons1)
- apply (subst disj_matrices_add_x_le)
- apply (rule disj_matrices_add_x)
- apply (simp add: disj_move_sparse_row_vector_twice)
- apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
- apply (simp add: disj_move_sparse_vec_mat[OF order_refl] disj_matrices_commute)
- apply (simp, blast)
- apply (intro strip, rule conjI, intro strip)
- apply (simp add: sorted_spvec_cons1)
- apply (subst disj_matrices_add_le_x)
- apply (simp add: disj_move_sparse_vec_mat[OF order_refl])
- apply (rule disj_matrices_x_add)
- apply (simp add: disj_move_sparse_row_vector_twice)
- apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
- apply (simp, blast)
- apply (intro strip)
- apply (case_tac "i=j")
- apply (simp_all)
- apply (subst disj_matrices_add)
- apply (simp_all add: disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl])
- apply (simp add: sorted_spvec_cons1 le_spvec_iff_sparse_row_le)
- done
-
-declare [[simp_depth_limit = 999]]
-
-primrec abs_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
-where
- "abs_spmat [] = []"
-| "abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"
-
-primrec minus_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
-where
- "minus_spmat [] = []"
-| "minus_spmat (a#as) = (fst a, minus_spvec (snd a))#(minus_spmat as)"
-
-lemma sparse_row_matrix_minus:
- "sparse_row_matrix (minus_spmat A) = - (sparse_row_matrix A)"
- apply (induct A)
- apply (simp_all add: sparse_row_vector_minus sparse_row_matrix_cons)
- apply (subst Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- done
-
-lemma Rep_sparse_row_vector_zero: "x \<noteq> 0 \<Longrightarrow> Rep_matrix (sparse_row_vector v) x y = 0"
-proof -
- assume x:"x \<noteq> 0"
- have r:"nrows (sparse_row_vector v) <= Suc 0" by (rule nrows_spvec)
- show ?thesis
- apply (rule nrows)
- apply (subgoal_tac "Suc 0 <= x")
- apply (insert r)
- apply (simp only:)
- apply (insert x)
- apply arith
- done
-qed
-
-lemma sparse_row_matrix_abs:
- "sorted_spvec A \<Longrightarrow> sorted_spmat A \<Longrightarrow> sparse_row_matrix (abs_spmat A) = abs (sparse_row_matrix A)"
- apply (induct A)
- apply (simp_all add: sparse_row_vector_abs sparse_row_matrix_cons)
- apply (frule_tac sorted_spvec_cons1, simp)
- apply (simplesubst Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply auto
- apply (case_tac "x=a")
- apply (simp)
- apply (simplesubst sorted_sparse_row_matrix_zero)
- apply auto
- apply (simplesubst Rep_sparse_row_vector_zero)
- apply simp_all
- done
-
-lemma sorted_spvec_minus_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (minus_spmat A)"
- apply (induct A)
- apply (simp)
- apply (frule sorted_spvec_cons1, simp)
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spvec_abs_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (abs_spmat A)"
- apply (induct A)
- apply (simp)
- apply (frule sorted_spvec_cons1, simp)
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"
- apply (induct A)
- apply (simp_all add: sorted_spvec_minus_spvec)
- done
-
-lemma sorted_spmat_abs_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (abs_spmat A)"
- apply (induct A)
- apply (simp_all add: sorted_spvec_abs_spvec)
- done
-
-definition diff_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
- where "diff_spmat A B = add_spmat A (minus_spmat B)"
-
-lemma sorted_spmat_diff_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (diff_spmat A B)"
- by (simp add: diff_spmat_def sorted_spmat_minus_spmat sorted_spmat_add_spmat)
-
-lemma sorted_spvec_diff_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec B \<Longrightarrow> sorted_spvec (diff_spmat A B)"
- by (simp add: diff_spmat_def sorted_spvec_minus_spmat sorted_spvec_add_spmat)
-
-lemma sparse_row_diff_spmat: "sparse_row_matrix (diff_spmat A B ) = (sparse_row_matrix A) - (sparse_row_matrix B)"
- by (simp add: diff_spmat_def sparse_row_add_spmat sparse_row_matrix_minus)
-
-definition sorted_sparse_matrix :: "'a spmat \<Rightarrow> bool"
- where "sorted_sparse_matrix A \<longleftrightarrow> sorted_spvec A & sorted_spmat A"
-
-lemma sorted_sparse_matrix_imp_spvec: "sorted_sparse_matrix A \<Longrightarrow> sorted_spvec A"
- by (simp add: sorted_sparse_matrix_def)
-
-lemma sorted_sparse_matrix_imp_spmat: "sorted_sparse_matrix A \<Longrightarrow> sorted_spmat A"
- by (simp add: sorted_sparse_matrix_def)
-
-lemmas sorted_sp_simps =
- sorted_spvec.simps
- sorted_spmat.simps
- sorted_sparse_matrix_def
-
-lemma bool1: "(\<not> True) = False" by blast
-lemma bool2: "(\<not> False) = True" by blast
-lemma bool3: "((P\<Colon>bool) \<and> True) = P" by blast
-lemma bool4: "(True \<and> (P\<Colon>bool)) = P" by blast
-lemma bool5: "((P\<Colon>bool) \<and> False) = False" by blast
-lemma bool6: "(False \<and> (P\<Colon>bool)) = False" by blast
-lemma bool7: "((P\<Colon>bool) \<or> True) = True" by blast
-lemma bool8: "(True \<or> (P\<Colon>bool)) = True" by blast
-lemma bool9: "((P\<Colon>bool) \<or> False) = P" by blast
-lemma bool10: "(False \<or> (P\<Colon>bool)) = P" by blast
-lemmas boolarith = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10
-
-lemma if_case_eq: "(if b then x else y) = (case b of True => x | False => y)" by simp
-
-primrec pprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
-where
- "pprt_spvec [] = []"
-| "pprt_spvec (a#as) = (fst a, pprt (snd a)) # (pprt_spvec as)"
-
-primrec nprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
-where
- "nprt_spvec [] = []"
-| "nprt_spvec (a#as) = (fst a, nprt (snd a)) # (nprt_spvec as)"
-
-primrec pprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
-where
- "pprt_spmat [] = []"
-| "pprt_spmat (a#as) = (fst a, pprt_spvec (snd a))#(pprt_spmat as)"
-
-primrec nprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
-where
- "nprt_spmat [] = []"
-| "nprt_spmat (a#as) = (fst a, nprt_spvec (snd a))#(nprt_spmat as)"
-
-
-lemma pprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
- apply (simp add: pprt_def sup_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- apply (case_tac "Rep_matrix A x xa \<noteq> 0")
- apply (simp_all add: disj_matrices_contr1)
- done
-
-lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
- apply (simp add: nprt_def inf_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- apply (case_tac "Rep_matrix A x xa \<noteq> 0")
- apply (simp_all add: disj_matrices_contr1)
- done
-
-lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (pprt x)"
- apply (simp add: pprt_def sup_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- done
-
-lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (nprt x)"
- apply (simp add: nprt_def inf_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply simp
- done
-
-lemma less_imp_le: "a < b \<Longrightarrow> a <= (b::_::order)" by (simp add: less_def)
-
-lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
- apply (induct v)
- apply (simp_all)
- apply (frule sorted_spvec_cons1, auto)
- apply (subst pprt_add)
- apply (subst disj_matrices_commute)
- apply (rule disj_sparse_row_singleton)
- apply auto
- done
-
-lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
- apply (induct v)
- apply (simp_all)
- apply (frule sorted_spvec_cons1, auto)
- apply (subst nprt_add)
- apply (subst disj_matrices_commute)
- apply (rule disj_sparse_row_singleton)
- apply auto
- done
-
-
-lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
- apply (simp add: pprt_def)
- apply (simp add: sup_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply (simp)
- done
-
-lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
- apply (simp add: nprt_def)
- apply (simp add: inf_matrix_def)
- apply (simp add: Rep_matrix_inject[symmetric])
- apply (rule ext)+
- apply (simp)
- done
-
-lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
- apply (induct m)
- apply simp
- apply simp
- apply (frule sorted_spvec_cons1)
- apply (simp add: sparse_row_matrix_cons sparse_row_vector_pprt)
- apply (subst pprt_add)
- apply (subst disj_matrices_commute)
- apply (rule disj_move_sparse_vec_mat)
- apply auto
- apply (simp add: sorted_spvec.simps)
- apply (simp split: list.split)
- apply auto
- apply (simp add: pprt_move_matrix)
- done
-
-lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
- apply (induct m)
- apply simp
- apply simp
- apply (frule sorted_spvec_cons1)
- apply (simp add: sparse_row_matrix_cons sparse_row_vector_nprt)
- apply (subst nprt_add)
- apply (subst disj_matrices_commute)
- apply (rule disj_move_sparse_vec_mat)
- apply auto
- apply (simp add: sorted_spvec.simps)
- apply (simp split: list.split)
- apply auto
- apply (simp add: nprt_move_matrix)
- done
-
-lemma sorted_pprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (pprt_spvec v)"
- apply (induct v)
- apply (simp)
- apply (frule sorted_spvec_cons1)
- apply simp
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_nprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (nprt_spvec v)"
- apply (induct v)
- apply (simp)
- apply (frule sorted_spvec_cons1)
- apply simp
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spvec_pprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (pprt_spmat m)"
- apply (induct m)
- apply (simp)
- apply (frule sorted_spvec_cons1)
- apply simp
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spvec_nprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (nprt_spmat m)"
- apply (induct m)
- apply (simp)
- apply (frule sorted_spvec_cons1)
- apply simp
- apply (simp add: sorted_spvec.simps split:list.split_asm)
- done
-
-lemma sorted_spmat_pprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (pprt_spmat m)"
- apply (induct m)
- apply (simp_all add: sorted_pprt_spvec)
- done
-
-lemma sorted_spmat_nprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (nprt_spmat m)"
- apply (induct m)
- apply (simp_all add: sorted_nprt_spvec)
- done
-
-definition mult_est_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
- "mult_est_spmat r1 r2 s1 s2 =
- add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
- (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"
-
-lemmas sparse_row_matrix_op_simps =
- sorted_sparse_matrix_imp_spmat sorted_sparse_matrix_imp_spvec
- sparse_row_add_spmat sorted_spvec_add_spmat sorted_spmat_add_spmat
- sparse_row_diff_spmat sorted_spvec_diff_spmat sorted_spmat_diff_spmat
- sparse_row_matrix_minus sorted_spvec_minus_spmat sorted_spmat_minus_spmat
- sparse_row_mult_spmat sorted_spvec_mult_spmat sorted_spmat_mult_spmat
- sparse_row_matrix_abs sorted_spvec_abs_spmat sorted_spmat_abs_spmat
- le_spmat_iff_sparse_row_le
- sparse_row_matrix_pprt sorted_spvec_pprt_spmat sorted_spmat_pprt_spmat
- sparse_row_matrix_nprt sorted_spvec_nprt_spmat sorted_spmat_nprt_spmat
-
-lemma zero_eq_Numeral0: "(0::_::number_ring) = Numeral0" by simp
-
-lemmas sparse_row_matrix_arith_simps[simplified zero_eq_Numeral0] =
- mult_spmat.simps mult_spvec_spmat.simps
- addmult_spvec.simps
- smult_spvec_empty smult_spvec_cons
- add_spmat.simps add_spvec.simps
- minus_spmat.simps minus_spvec.simps
- abs_spmat.simps abs_spvec.simps
- diff_spmat_def
- le_spmat.simps le_spvec.simps
- pprt_spmat.simps pprt_spvec.simps
- nprt_spmat.simps nprt_spvec.simps
- mult_est_spmat_def
-
-
-(*lemma spm_linprog_dual_estimate_1:
- assumes
- "sorted_sparse_matrix A1"
- "sorted_sparse_matrix A2"
- "sorted_sparse_matrix c1"
- "sorted_sparse_matrix c2"
- "sorted_sparse_matrix y"
- "sorted_spvec b"
- "sorted_spvec r"
- "le_spmat ([], y)"
- "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
- "sparse_row_matrix A1 <= A"
- "A <= sparse_row_matrix A2"
- "sparse_row_matrix c1 <= c"
- "c <= sparse_row_matrix c2"
- "abs x \<le> sparse_row_matrix r"
- shows
- "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b, mult_spmat (add_spmat (add_spmat (mult_spmat y (diff_spmat A2 A1),
- abs_spmat (diff_spmat (mult_spmat y A1) c1)), diff_spmat c2 c1)) r))"
- by (insert prems, simp add: sparse_row_matrix_op_simps linprog_dual_estimate_1[where A=A])
-*)
-
-end
--- a/src/HOL/Matrix/document/root.tex Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,25 +0,0 @@
-\documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym}
-
-% this should be the last package used
-\usepackage{pdfsetup}
-
-% urls in roman style, theory text in math-similar italics
-\urlstyle{rm}
-\isabellestyle{it}
-
-\newcommand{\ganz}{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}
-
-\begin{document}
-
-\title{Matrix}
-\author{Steven Obua}
-\maketitle
-
-%\tableofcontents
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-\end{document}
--- a/src/HOL/Matrix/fspmlp.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,313 +0,0 @@
-(* Title: HOL/Matrix/fspmlp.ML
- Author: Steven Obua
-*)
-
-signature FSPMLP =
-sig
- type linprog
- type vector = FloatSparseMatrixBuilder.vector
- type matrix = FloatSparseMatrixBuilder.matrix
-
- val y : linprog -> term
- val A : linprog -> term * term
- val b : linprog -> term
- val c : linprog -> term * term
- val r12 : linprog -> term * term
-
- exception Load of string
-
- val load : string -> int -> bool -> linprog
-end
-
-structure Fspmlp : FSPMLP =
-struct
-
-type vector = FloatSparseMatrixBuilder.vector
-type matrix = FloatSparseMatrixBuilder.matrix
-
-type linprog = term * (term * term) * term * (term * term) * (term * term)
-
-fun y (c1, _, _, _, _) = c1
-fun A (_, c2, _, _, _) = c2
-fun b (_, _, c3, _, _) = c3
-fun c (_, _, _, c4, _) = c4
-fun r12 (_, _, _, _, c6) = c6
-
-structure CplexFloatSparseMatrixConverter =
-MAKE_CPLEX_MATRIX_CONVERTER(structure cplex = Cplex and matrix_builder = FloatSparseMatrixBuilder);
-
-datatype bound_type = LOWER | UPPER
-
-fun intbound_ord ((i1: int, b1),(i2,b2)) =
- if i1 < i2 then LESS
- else if i1 = i2 then
- (if b1 = b2 then EQUAL else if b1=LOWER then LESS else GREATER)
- else GREATER
-
-structure Inttab = Table(type key = int val ord = (rev_order o int_ord));
-
-structure VarGraph = Table(type key = int*bound_type val ord = intbound_ord);
-(* key -> (float option) * (int -> (float * (((float * float) * key) list)))) *)
-(* dest_key -> (sure_bound * (row_index -> (row_bound * (((coeff_lower * coeff_upper) * src_key) list)))) *)
-
-exception Internal of string;
-
-fun add_row_bound g dest_key row_index row_bound =
- let
- val x =
- case VarGraph.lookup g dest_key of
- NONE => (NONE, Inttab.update (row_index, (row_bound, [])) Inttab.empty)
- | SOME (sure_bound, f) =>
- (sure_bound,
- case Inttab.lookup f row_index of
- NONE => Inttab.update (row_index, (row_bound, [])) f
- | SOME _ => raise (Internal "add_row_bound"))
- in
- VarGraph.update (dest_key, x) g
- end
-
-fun update_sure_bound g (key as (_, btype)) bound =
- let
- val x =
- case VarGraph.lookup g key of
- NONE => (SOME bound, Inttab.empty)
- | SOME (NONE, f) => (SOME bound, f)
- | SOME (SOME old_bound, f) =>
- (SOME ((case btype of
- UPPER => Float.min
- | LOWER => Float.max)
- old_bound bound), f)
- in
- VarGraph.update (key, x) g
- end
-
-fun get_sure_bound g key =
- case VarGraph.lookup g key of
- NONE => NONE
- | SOME (sure_bound, _) => sure_bound
-
-(*fun get_row_bound g key row_index =
- case VarGraph.lookup g key of
- NONE => NONE
- | SOME (sure_bound, f) =>
- (case Inttab.lookup f row_index of
- NONE => NONE
- | SOME (row_bound, _) => (sure_bound, row_bound))*)
-
-fun add_edge g src_key dest_key row_index coeff =
- case VarGraph.lookup g dest_key of
- NONE => raise (Internal "add_edge: dest_key not found")
- | SOME (sure_bound, f) =>
- (case Inttab.lookup f row_index of
- NONE => raise (Internal "add_edge: row_index not found")
- | SOME (row_bound, sources) =>
- VarGraph.update (dest_key, (sure_bound, Inttab.update (row_index, (row_bound, (coeff, src_key) :: sources)) f)) g)
-
-fun split_graph g =
- let
- fun split (key, (sure_bound, _)) (r1, r2) = case sure_bound
- of NONE => (r1, r2)
- | SOME bound => (case key
- of (u, UPPER) => (r1, Inttab.update (u, bound) r2)
- | (u, LOWER) => (Inttab.update (u, bound) r1, r2))
- in VarGraph.fold split g (Inttab.empty, Inttab.empty) end
-
-(* If safe is true, termination is guaranteed, but the sure bounds may be not optimal (relative to the algorithm).
- If safe is false, termination is not guaranteed, but on termination the sure bounds are optimal (relative to the algorithm) *)
-fun propagate_sure_bounds safe names g =
- let
- (* returns NONE if no new sure bound could be calculated, otherwise the new sure bound is returned *)
- fun calc_sure_bound_from_sources g (key as (_, btype)) =
- let
- fun mult_upper x (lower, upper) =
- if Float.sign x = LESS then
- Float.mult x lower
- else
- Float.mult x upper
-
- fun mult_lower x (lower, upper) =
- if Float.sign x = LESS then
- Float.mult x upper
- else
- Float.mult x lower
-
- val mult_btype = case btype of UPPER => mult_upper | LOWER => mult_lower
-
- fun calc_sure_bound (_, (row_bound, sources)) sure_bound =
- let
- fun add_src_bound (coeff, src_key) sum =
- case sum of
- NONE => NONE
- | SOME x =>
- (case get_sure_bound g src_key of
- NONE => NONE
- | SOME src_sure_bound => SOME (Float.add x (mult_btype src_sure_bound coeff)))
- in
- case fold add_src_bound sources (SOME row_bound) of
- NONE => sure_bound
- | new_sure_bound as (SOME new_bound) =>
- (case sure_bound of
- NONE => new_sure_bound
- | SOME old_bound =>
- SOME (case btype of
- UPPER => Float.min old_bound new_bound
- | LOWER => Float.max old_bound new_bound))
- end
- in
- case VarGraph.lookup g key of
- NONE => NONE
- | SOME (sure_bound, f) =>
- let
- val x = Inttab.fold calc_sure_bound f sure_bound
- in
- if x = sure_bound then NONE else x
- end
- end
-
- fun propagate (key, _) (g, b) =
- case calc_sure_bound_from_sources g key of
- NONE => (g,b)
- | SOME bound => (update_sure_bound g key bound,
- if safe then
- case get_sure_bound g key of
- NONE => true
- | _ => b
- else
- true)
-
- val (g, b) = VarGraph.fold propagate g (g, false)
- in
- if b then propagate_sure_bounds safe names g else g
- end
-
-exception Load of string;
-
-val empty_spvec = @{term "Nil :: real spvec"};
-fun cons_spvec x xs = @{term "Cons :: nat * real => real spvec => real spvec"} $ x $ xs;
-val empty_spmat = @{term "Nil :: real spmat"};
-fun cons_spmat x xs = @{term "Cons :: nat * real spvec => real spmat => real spmat"} $ x $ xs;
-
-fun calcr safe_propagation xlen names prec A b =
- let
- fun test_1 (lower, upper) =
- if lower = upper then
- (if Float.eq (lower, (~1, 0)) then ~1
- else if Float.eq (lower, (1, 0)) then 1
- else 0)
- else 0
-
- fun calcr (row_index, a) g =
- let
- val b = FloatSparseMatrixBuilder.v_elem_at b row_index
- val (_, b2) = FloatArith.approx_decstr_by_bin prec (case b of NONE => "0" | SOME b => b)
- val approx_a = FloatSparseMatrixBuilder.v_fold (fn (i, s) => fn l =>
- (i, FloatArith.approx_decstr_by_bin prec s)::l) a []
-
- fun fold_dest_nodes (dest_index, dest_value) g =
- let
- val dest_test = test_1 dest_value
- in
- if dest_test = 0 then
- g
- else let
- val (dest_key as (_, dest_btype), row_bound) =
- if dest_test = ~1 then
- ((dest_index, LOWER), Float.neg b2)
- else
- ((dest_index, UPPER), b2)
-
- fun fold_src_nodes (src_index, src_value as (src_lower, src_upper)) g =
- if src_index = dest_index then g
- else
- let
- val coeff = case dest_btype of
- UPPER => (Float.neg src_upper, Float.neg src_lower)
- | LOWER => src_value
- in
- if Float.sign src_lower = LESS then
- add_edge g (src_index, UPPER) dest_key row_index coeff
- else
- add_edge g (src_index, LOWER) dest_key row_index coeff
- end
- in
- fold fold_src_nodes approx_a (add_row_bound g dest_key row_index row_bound)
- end
- end
- in
- case approx_a of
- [] => g
- | [(u, a)] =>
- let
- val atest = test_1 a
- in
- if atest = ~1 then
- update_sure_bound g (u, LOWER) (Float.neg b2)
- else if atest = 1 then
- update_sure_bound g (u, UPPER) b2
- else
- g
- end
- | _ => fold fold_dest_nodes approx_a g
- end
-
- val g = FloatSparseMatrixBuilder.m_fold calcr A VarGraph.empty
-
- val g = propagate_sure_bounds safe_propagation names g
-
- val (r1, r2) = split_graph g
-
- fun add_row_entry m index f vname value =
- let
- val v = (case value of
- SOME value => FloatSparseMatrixBuilder.mk_spvec_entry 0 value
- | NONE => FloatSparseMatrixBuilder.mk_spvec_entry' 0 (f $ (Var ((vname,0), HOLogic.realT))))
- val vec = cons_spvec v empty_spvec
- in
- cons_spmat (FloatSparseMatrixBuilder.mk_spmat_entry index vec) m
- end
-
- fun abs_estimate i r1 r2 =
- if i = 0 then
- let val e = empty_spmat in (e, e) end
- else
- let
- val index = xlen-i
- val (r12_1, r12_2) = abs_estimate (i-1) r1 r2
- val b1 = Inttab.lookup r1 index
- val b2 = Inttab.lookup r2 index
- in
- (add_row_entry r12_1 index @{term "lbound :: real => real"} ((names index)^"l") b1,
- add_row_entry r12_2 index @{term "ubound :: real => real"} ((names index)^"u") b2)
- end
-
- val (r1, r2) = abs_estimate xlen r1 r2
-
- in
- (r1, r2)
- end
-
-fun load filename prec safe_propagation =
- let
- val prog = Cplex.load_cplexFile filename
- val prog = Cplex.elim_nonfree_bounds prog
- val prog = Cplex.relax_strict_ineqs prog
- val (maximize, c, A, b, (xlen, names, _)) = CplexFloatSparseMatrixConverter.convert_prog prog
- val (r1, r2) = calcr safe_propagation xlen names prec A b
- val _ = if maximize then () else raise Load "sorry, cannot handle minimization problems"
- val (dualprog, indexof) = FloatSparseMatrixBuilder.dual_cplexProg c A b
- val results = Cplex.solve dualprog
- val (_, v) = CplexFloatSparseMatrixConverter.convert_results results indexof
- (*val A = FloatSparseMatrixBuilder.cut_matrix v NONE A*)
- fun id x = x
- val v = FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 v
- val b = FloatSparseMatrixBuilder.transpose_matrix (FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 b)
- val c = FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 c
- val (y1, _) = FloatSparseMatrixBuilder.approx_matrix prec Float.positive_part v
- val A = FloatSparseMatrixBuilder.approx_matrix prec id A
- val (_,b2) = FloatSparseMatrixBuilder.approx_matrix prec id b
- val c = FloatSparseMatrixBuilder.approx_matrix prec id c
- in
- (y1, A, b2, c, (r1, r2))
- end handle CplexFloatSparseMatrixConverter.Converter s => (raise (Load ("Converter: "^s)))
-
-end
--- a/src/HOL/Matrix/matrixlp.ML Sat Mar 17 12:26:19 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,59 +0,0 @@
-(* Title: HOL/Matrix/matrixlp.ML
- Author: Steven Obua
-*)
-
-signature MATRIX_LP =
-sig
- val matrix_compute : cterm -> thm
- val matrix_simplify : thm -> thm
- val prove_bound : string -> int -> thm
- val float2real : string * string -> Real.real
-end
-
-structure MatrixLP : MATRIX_LP =
-struct
-
-val compute_thms = ComputeHOL.prep_thms @{thms "ComputeHOL.compute_list_case" "ComputeHOL.compute_let"
- "ComputeHOL.compute_if" "ComputeFloat.arith" "SparseMatrix.sparse_row_matrix_arith_simps"
- "ComputeHOL.compute_bool" "ComputeHOL.compute_pair"
- "SparseMatrix.sorted_sp_simps"
- "ComputeNumeral.natnorm"}; (*"ComputeNumeral.number_norm"*)
-
-val spm_mult_le_dual_prts_no_let_real = @{thm "spm_mult_le_dual_prts_no_let" [where ?'a = real]}
-
-fun lp_dual_estimate_prt lptfile prec =
- let
- val cert = cterm_of @{theory}
- fun var s x = (cert (Var ((s, 0), FloatSparseMatrixBuilder.spmatT)), x)
- val l = Fspmlp.load lptfile prec false
- val (y, (A1, A2), (c1, c2), b, (r1, r2)) =
- let
- open Fspmlp
- in
- (y l |> cert, A l |> pairself cert, c l |> pairself cert, b l |> cert, r12 l |> pairself cert)
- end
- in
- Thm.instantiate ([],
- [var "A1" A1, var "A2" A2, var "y" y, var "c1" c1, var "c2" c2, var "r1" r1, var "r2" r2, var "b" b])
- spm_mult_le_dual_prts_no_let_real
- end
-
-val computer = PCompute.make Compute.SML @{theory} compute_thms []
-
-fun matrix_compute c = hd (PCompute.rewrite computer [c])
-
-fun matrix_simplify th =
- let
- val simp_th = matrix_compute (cprop_of th)
- val th = Thm.strip_shyps (Thm.equal_elim simp_th th)
- fun removeTrue th = removeTrue (Thm.implies_elim th TrueI) handle THM _ => th
- in
- removeTrue th
- end
-
-val prove_bound = matrix_simplify oo lp_dual_estimate_prt;
-
-val realFromStr = the o Real.fromString;
-fun float2real (x, y) = realFromStr x * Math.pow (2.0, realFromStr y);
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/ComputeFloat.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,309 @@
+(* Title: HOL/Matrix/ComputeFloat.thy
+ Author: Steven Obua
+*)
+
+header {* Floating Point Representation of the Reals *}
+
+theory ComputeFloat
+imports Complex_Main "~~/src/HOL/Library/Lattice_Algebras"
+uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML")
+begin
+
+definition int_of_real :: "real \<Rightarrow> int"
+ where "int_of_real x = (SOME y. real y = x)"
+
+definition real_is_int :: "real \<Rightarrow> bool"
+ where "real_is_int x = (EX (u::int). x = real u)"
+
+lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
+ by (auto simp add: real_is_int_def int_of_real_def)
+
+lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
+by (auto simp add: real_is_int_def int_of_real_def)
+
+lemma int_of_real_real[simp]: "int_of_real (real x) = x"
+by (simp add: int_of_real_def)
+
+lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
+by (auto simp add: int_of_real_def real_is_int_def)
+
+lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"
+by (auto simp add: int_of_real_def real_is_int_def)
+
+lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
+apply (subst real_is_int_def2)
+apply (simp add: real_is_int_add_int_of_real real_int_of_real)
+done
+
+lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)"
+by (auto simp add: int_of_real_def real_is_int_def)
+
+lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
+apply (subst real_is_int_def2)
+apply (simp add: int_of_real_sub real_int_of_real)
+done
+
+lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
+by (auto simp add: real_is_int_def)
+
+lemma int_of_real_mult:
+ assumes "real_is_int a" "real_is_int b"
+ shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
+ using assms
+ by (auto simp add: real_is_int_def real_of_int_mult[symmetric]
+ simp del: real_of_int_mult)
+
+lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
+apply (subst real_is_int_def2)
+apply (simp add: int_of_real_mult)
+done
+
+lemma real_is_int_0[simp]: "real_is_int (0::real)"
+by (simp add: real_is_int_def int_of_real_def)
+
+lemma real_is_int_1[simp]: "real_is_int (1::real)"
+proof -
+ have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
+ also have "\<dots> = True" by (simp only: real_is_int_real)
+ ultimately show ?thesis by auto
+qed
+
+lemma real_is_int_n1: "real_is_int (-1::real)"
+proof -
+ have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
+ also have "\<dots> = True" by (simp only: real_is_int_real)
+ ultimately show ?thesis by auto
+qed
+
+lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
+ by (auto simp: real_is_int_def intro!: exI[of _ "number_of x"])
+
+lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
+by (simp add: int_of_real_def)
+
+lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
+proof -
+ have 1: "(1::real) = real (1::int)" by auto
+ show ?thesis by (simp only: 1 int_of_real_real)
+qed
+
+lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
+ unfolding int_of_real_def
+ by (intro some_equality)
+ (auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
+
+lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
+by (rule zdiv_int)
+
+lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
+by (rule zmod_int)
+
+lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
+by arith
+
+lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
+ by auto
+
+lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
+ by simp
+
+lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
+ by simp
+
+lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
+ by simp
+
+lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
+ by simp
+
+lemma int_pow_0: "(a::int)^(Numeral0) = 1"
+ by simp
+
+lemma int_pow_1: "(a::int)^(Numeral1) = a"
+ by simp
+
+lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
+ by simp
+
+lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
+ by simp
+
+lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
+ by simp
+
+lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
+ by simp
+
+lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
+ by simp
+
+lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
+proof -
+ have 1:"((-1)::nat) = 0"
+ by simp
+ show ?thesis by (simp add: 1)
+qed
+
+lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
+ by simp
+
+lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
+ by simp
+
+lemma lift_bool: "x \<Longrightarrow> x=True"
+ by simp
+
+lemma nlift_bool: "~x \<Longrightarrow> x=False"
+ by simp
+
+lemma not_false_eq_true: "(~ False) = True" by simp
+
+lemma not_true_eq_false: "(~ True) = False" by simp
+
+lemmas binarith =
+ normalize_bin_simps
+ pred_bin_simps succ_bin_simps
+ add_bin_simps minus_bin_simps mult_bin_simps
+
+lemma int_eq_number_of_eq:
+ "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"
+ by (rule eq_number_of_eq)
+
+lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
+ by (simp only: iszero_number_of_Pls)
+
+lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
+ by simp
+
+lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)"
+ by simp
+
+lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)"
+ by simp
+
+lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
+ unfolding neg_def number_of_is_id by simp
+
+lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
+ by simp
+
+lemma int_neg_number_of_Min: "neg (-1::int)"
+ by simp
+
+lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)"
+ by simp
+
+lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)"
+ by simp
+
+lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
+ unfolding neg_def number_of_is_id by (simp add: not_less)
+
+lemmas intarithrel =
+ int_eq_number_of_eq
+ lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0
+ lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
+ int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq
+
+lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
+ by simp
+
+lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
+ by simp
+
+lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"
+ by simp
+
+lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
+ by simp
+
+lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
+
+lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
+
+lemmas powerarith = nat_number_of zpower_number_of_even
+ zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
+ zpower_Pls zpower_Min
+
+definition float :: "(int \<times> int) \<Rightarrow> real" where
+ "float = (\<lambda>(a, b). real a * 2 powr real b)"
+
+lemma float_add_l0: "float (0, e) + x = x"
+ by (simp add: float_def)
+
+lemma float_add_r0: "x + float (0, e) = x"
+ by (simp add: float_def)
+
+lemma float_add:
+ "float (a1, e1) + float (a2, e2) =
+ (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) else float (a1*2^(nat (e1-e2))+a2, e2))"
+ by (simp add: float_def algebra_simps powr_realpow[symmetric] powr_divide2[symmetric])
+
+lemma float_mult_l0: "float (0, e) * x = float (0, 0)"
+ by (simp add: float_def)
+
+lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
+ by (simp add: float_def)
+
+lemma float_mult:
+ "float (a1, e1) * float (a2, e2) = (float (a1 * a2, e1 + e2))"
+ by (simp add: float_def powr_add)
+
+lemma float_minus:
+ "- (float (a,b)) = float (-a, b)"
+ by (simp add: float_def)
+
+lemma zero_le_float:
+ "(0 <= float (a,b)) = (0 <= a)"
+ using powr_gt_zero[of 2 "real b", arith]
+ by (simp add: float_def zero_le_mult_iff)
+
+lemma float_le_zero:
+ "(float (a,b) <= 0) = (a <= 0)"
+ using powr_gt_zero[of 2 "real b", arith]
+ by (simp add: float_def mult_le_0_iff)
+
+lemma float_abs:
+ "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
+ using powr_gt_zero[of 2 "real b", arith]
+ by (simp add: float_def abs_if mult_less_0_iff)
+
+lemma float_zero:
+ "float (0, b) = 0"
+ by (simp add: float_def)
+
+lemma float_pprt:
+ "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
+ by (auto simp add: zero_le_float float_le_zero float_zero)
+
+lemma float_nprt:
+ "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
+ by (auto simp add: zero_le_float float_le_zero float_zero)
+
+definition lbound :: "real \<Rightarrow> real"
+ where "lbound x = min 0 x"
+
+definition ubound :: "real \<Rightarrow> real"
+ where "ubound x = max 0 x"
+
+lemma lbound: "lbound x \<le> x"
+ by (simp add: lbound_def)
+
+lemma ubound: "x \<le> ubound x"
+ by (simp add: ubound_def)
+
+lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"
+ by (auto simp: float_def lbound_def)
+
+lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"
+ by (auto simp: float_def ubound_def)
+
+lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0
+ float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound
+
+(* for use with the compute oracle *)
+lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
+
+use "~~/src/HOL/Tools/float_arith.ML"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/ComputeHOL.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,187 @@
+theory ComputeHOL
+imports Complex_Main "Compute_Oracle/Compute_Oracle"
+begin
+
+lemma Trueprop_eq_eq: "Trueprop X == (X == True)" by (simp add: atomize_eq)
+lemma meta_eq_trivial: "x == y \<Longrightarrow> x == y" by simp
+lemma meta_eq_imp_eq: "x == y \<Longrightarrow> x = y" by auto
+lemma eq_trivial: "x = y \<Longrightarrow> x = y" by auto
+lemma bool_to_true: "x :: bool \<Longrightarrow> x == True" by simp
+lemma transmeta_1: "x = y \<Longrightarrow> y == z \<Longrightarrow> x = z" by simp
+lemma transmeta_2: "x == y \<Longrightarrow> y = z \<Longrightarrow> x = z" by simp
+lemma transmeta_3: "x == y \<Longrightarrow> y == z \<Longrightarrow> x = z" by simp
+
+
+(**** compute_if ****)
+
+lemma If_True: "If True = (\<lambda> x y. x)" by ((rule ext)+,auto)
+lemma If_False: "If False = (\<lambda> x y. y)" by ((rule ext)+, auto)
+
+lemmas compute_if = If_True If_False
+
+(**** compute_bool ****)
+
+lemma bool1: "(\<not> True) = False" by blast
+lemma bool2: "(\<not> False) = True" by blast
+lemma bool3: "(P \<and> True) = P" by blast
+lemma bool4: "(True \<and> P) = P" by blast
+lemma bool5: "(P \<and> False) = False" by blast
+lemma bool6: "(False \<and> P) = False" by blast
+lemma bool7: "(P \<or> True) = True" by blast
+lemma bool8: "(True \<or> P) = True" by blast
+lemma bool9: "(P \<or> False) = P" by blast
+lemma bool10: "(False \<or> P) = P" by blast
+lemma bool11: "(True \<longrightarrow> P) = P" by blast
+lemma bool12: "(P \<longrightarrow> True) = True" by blast
+lemma bool13: "(True \<longrightarrow> P) = P" by blast
+lemma bool14: "(P \<longrightarrow> False) = (\<not> P)" by blast
+lemma bool15: "(False \<longrightarrow> P) = True" by blast
+lemma bool16: "(False = False) = True" by blast
+lemma bool17: "(True = True) = True" by blast
+lemma bool18: "(False = True) = False" by blast
+lemma bool19: "(True = False) = False" by blast
+
+lemmas compute_bool = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10 bool11 bool12 bool13 bool14 bool15 bool16 bool17 bool18 bool19
+
+
+(*** compute_pair ***)
+
+lemma compute_fst: "fst (x,y) = x" by simp
+lemma compute_snd: "snd (x,y) = y" by simp
+lemma compute_pair_eq: "((a, b) = (c, d)) = (a = c \<and> b = d)" by auto
+
+lemma prod_case_simp: "prod_case f (x,y) = f x y" by simp
+
+lemmas compute_pair = compute_fst compute_snd compute_pair_eq prod_case_simp
+
+(*** compute_option ***)
+
+lemma compute_the: "the (Some x) = x" by simp
+lemma compute_None_Some_eq: "(None = Some x) = False" by auto
+lemma compute_Some_None_eq: "(Some x = None) = False" by auto
+lemma compute_None_None_eq: "(None = None) = True" by auto
+lemma compute_Some_Some_eq: "(Some x = Some y) = (x = y)" by auto
+
+definition option_case_compute :: "'b option \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
+ where "option_case_compute opt a f = option_case a f opt"
+
+lemma option_case_compute: "option_case = (\<lambda> a f opt. option_case_compute opt a f)"
+ by (simp add: option_case_compute_def)
+
+lemma option_case_compute_None: "option_case_compute None = (\<lambda> a f. a)"
+ apply (rule ext)+
+ apply (simp add: option_case_compute_def)
+ done
+
+lemma option_case_compute_Some: "option_case_compute (Some x) = (\<lambda> a f. f x)"
+ apply (rule ext)+
+ apply (simp add: option_case_compute_def)
+ done
+
+lemmas compute_option_case = option_case_compute option_case_compute_None option_case_compute_Some
+
+lemmas compute_option = compute_the compute_None_Some_eq compute_Some_None_eq compute_None_None_eq compute_Some_Some_eq compute_option_case
+
+(**** compute_list_length ****)
+
+lemma length_cons:"length (x#xs) = 1 + (length xs)"
+ by simp
+
+lemma length_nil: "length [] = 0"
+ by simp
+
+lemmas compute_list_length = length_nil length_cons
+
+(*** compute_list_case ***)
+
+definition list_case_compute :: "'b list \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'a) \<Rightarrow> 'a"
+ where "list_case_compute l a f = list_case a f l"
+
+lemma list_case_compute: "list_case = (\<lambda> (a::'a) f (l::'b list). list_case_compute l a f)"
+ apply (rule ext)+
+ apply (simp add: list_case_compute_def)
+ done
+
+lemma list_case_compute_empty: "list_case_compute ([]::'b list) = (\<lambda> (a::'a) f. a)"
+ apply (rule ext)+
+ apply (simp add: list_case_compute_def)
+ done
+
+lemma list_case_compute_cons: "list_case_compute (u#v) = (\<lambda> (a::'a) f. (f (u::'b) v))"
+ apply (rule ext)+
+ apply (simp add: list_case_compute_def)
+ done
+
+lemmas compute_list_case = list_case_compute list_case_compute_empty list_case_compute_cons
+
+(*** compute_list_nth ***)
+(* Of course, you will need computation with nats for this to work \<dots> *)
+
+lemma compute_list_nth: "((x#xs) ! n) = (if n = 0 then x else (xs ! (n - 1)))"
+ by (cases n, auto)
+
+(*** compute_list ***)
+
+lemmas compute_list = compute_list_case compute_list_length compute_list_nth
+
+(*** compute_let ***)
+
+lemmas compute_let = Let_def
+
+(***********************)
+(* Everything together *)
+(***********************)
+
+lemmas compute_hol = compute_if compute_bool compute_pair compute_option compute_list compute_let
+
+ML {*
+signature ComputeHOL =
+sig
+ val prep_thms : thm list -> thm list
+ val to_meta_eq : thm -> thm
+ val to_hol_eq : thm -> thm
+ val symmetric : thm -> thm
+ val trans : thm -> thm -> thm
+end
+
+structure ComputeHOL : ComputeHOL =
+struct
+
+local
+fun lhs_of eq = fst (Thm.dest_equals (cprop_of eq));
+in
+fun rewrite_conv [] ct = raise CTERM ("rewrite_conv", [ct])
+ | rewrite_conv (eq :: eqs) ct =
+ Thm.instantiate (Thm.match (lhs_of eq, ct)) eq
+ handle Pattern.MATCH => rewrite_conv eqs ct;
+end
+
+val convert_conditions = Conv.fconv_rule (Conv.prems_conv ~1 (Conv.try_conv (rewrite_conv [@{thm "Trueprop_eq_eq"}])))
+
+val eq_th = @{thm "HOL.eq_reflection"}
+val meta_eq_trivial = @{thm "ComputeHOL.meta_eq_trivial"}
+val bool_to_true = @{thm "ComputeHOL.bool_to_true"}
+
+fun to_meta_eq th = eq_th OF [th] handle THM _ => meta_eq_trivial OF [th] handle THM _ => bool_to_true OF [th]
+
+fun to_hol_eq th = @{thm "meta_eq_imp_eq"} OF [th] handle THM _ => @{thm "eq_trivial"} OF [th]
+
+fun prep_thms ths = map (convert_conditions o to_meta_eq) ths
+
+fun symmetric th = @{thm "HOL.sym"} OF [th] handle THM _ => @{thm "Pure.symmetric"} OF [th]
+
+local
+ val trans_HOL = @{thm "HOL.trans"}
+ val trans_HOL_1 = @{thm "ComputeHOL.transmeta_1"}
+ val trans_HOL_2 = @{thm "ComputeHOL.transmeta_2"}
+ val trans_HOL_3 = @{thm "ComputeHOL.transmeta_3"}
+ fun tr [] th1 th2 = trans_HOL OF [th1, th2]
+ | tr (t::ts) th1 th2 = (t OF [th1, th2] handle THM _ => tr ts th1 th2)
+in
+ fun trans th1 th2 = tr [trans_HOL, trans_HOL_1, trans_HOL_2, trans_HOL_3] th1 th2
+end
+
+end
+*}
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/ComputeNumeral.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,189 @@
+theory ComputeNumeral
+imports ComputeHOL ComputeFloat
+begin
+
+(* normalization of bit strings *)
+lemmas bitnorm = normalize_bin_simps
+
+(* neg for bit strings *)
+lemma neg1: "neg Int.Pls = False" by (simp add: Int.Pls_def)
+lemma neg2: "neg Int.Min = True" apply (subst Int.Min_def) by auto
+lemma neg3: "neg (Int.Bit0 x) = neg x" apply (simp add: neg_def) apply (subst Bit0_def) by auto
+lemma neg4: "neg (Int.Bit1 x) = neg x" apply (simp add: neg_def) apply (subst Bit1_def) by auto
+lemmas bitneg = neg1 neg2 neg3 neg4
+
+(* iszero for bit strings *)
+lemma iszero1: "iszero Int.Pls = True" by (simp add: Int.Pls_def iszero_def)
+lemma iszero2: "iszero Int.Min = False" apply (subst Int.Min_def) apply (subst iszero_def) by simp
+lemma iszero3: "iszero (Int.Bit0 x) = iszero x" apply (subst Int.Bit0_def) apply (subst iszero_def)+ by auto
+lemma iszero4: "iszero (Int.Bit1 x) = False" apply (subst Int.Bit1_def) apply (subst iszero_def)+ apply simp by arith
+lemmas bitiszero = iszero1 iszero2 iszero3 iszero4
+
+(* lezero for bit strings *)
+definition "lezero x \<longleftrightarrow> x \<le> 0"
+lemma lezero1: "lezero Int.Pls = True" unfolding Int.Pls_def lezero_def by auto
+lemma lezero2: "lezero Int.Min = True" unfolding Int.Min_def lezero_def by auto
+lemma lezero3: "lezero (Int.Bit0 x) = lezero x" unfolding Int.Bit0_def lezero_def by auto
+lemma lezero4: "lezero (Int.Bit1 x) = neg x" unfolding Int.Bit1_def lezero_def neg_def by auto
+lemmas bitlezero = lezero1 lezero2 lezero3 lezero4
+
+(* equality for bit strings *)
+lemmas biteq = eq_bin_simps
+
+(* x < y for bit strings *)
+lemmas bitless = less_bin_simps
+
+(* x \<le> y for bit strings *)
+lemmas bitle = le_bin_simps
+
+(* succ for bit strings *)
+lemmas bitsucc = succ_bin_simps
+
+(* pred for bit strings *)
+lemmas bitpred = pred_bin_simps
+
+(* unary minus for bit strings *)
+lemmas bituminus = minus_bin_simps
+
+(* addition for bit strings *)
+lemmas bitadd = add_bin_simps
+
+(* multiplication for bit strings *)
+lemma mult_Pls_right: "x * Int.Pls = Int.Pls" by (simp add: Pls_def)
+lemma mult_Min_right: "x * Int.Min = - x" by (subst mult_commute) simp
+lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0)
+lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp
+lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)"
+ unfolding Bit0_def Bit1_def by (simp add: algebra_simps)
+lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1
+
+lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul
+
+definition "nat_norm_number_of (x::nat) = x"
+
+lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)"
+ apply (simp add: nat_norm_number_of_def)
+ unfolding lezero_def iszero_def neg_def
+ apply (simp add: numeral_simps)
+ done
+
+(* Normalization of nat literals *)
+lemma natnorm0: "(0::nat) = number_of (Int.Pls)" by auto
+lemma natnorm1: "(1 :: nat) = number_of (Int.Bit1 Int.Pls)" by auto
+lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of
+
+(* Suc *)
+lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Int.succ x))" by (auto simp add: number_of_is_id)
+
+(* Addition for nat *)
+lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by auto
+
+(* Subtraction for nat *)
+lemma natsub: "(number_of x) - ((number_of y)::nat) =
+ (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))"
+ unfolding nat_norm_number_of
+ by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def)
+
+(* Multiplication for nat *)
+lemma natmul: "(number_of x) * ((number_of y)::nat) =
+ (if neg x then 0 else (if neg y then 0 else number_of (x * y)))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by (simp add: nat_mult_distrib)
+
+lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \<and> lezero y) \<or> (x = y))"
+ by (auto simp add: iszero_def lezero_def neg_def number_of_is_id)
+
+lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \<and> (\<not> (lezero y)))"
+ by (simp add: lezero_def numeral_simps not_le)
+
+lemma natle: "(((number_of x)::nat) \<le> (number_of y)) = (y < x \<longrightarrow> lezero x)"
+ by (auto simp add: number_of_is_id lezero_def nat_number_of_def)
+
+fun natfac :: "nat \<Rightarrow> nat"
+ where "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))"
+
+lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps
+
+lemma number_eq: "(((number_of x)::'a::{number_ring, linordered_idom}) = (number_of y)) = (x = y)"
+ unfolding number_of_eq
+ apply simp
+ done
+
+lemma number_le: "(((number_of x)::'a::{number_ring, linordered_idom}) \<le> (number_of y)) = (x \<le> y)"
+ unfolding number_of_eq
+ apply simp
+ done
+
+lemma number_less: "(((number_of x)::'a::{number_ring, linordered_idom}) < (number_of y)) = (x < y)"
+ unfolding number_of_eq
+ apply simp
+ done
+
+lemma number_diff: "((number_of x)::'a::{number_ring, linordered_idom}) - number_of y = number_of (x + (- y))"
+ apply (subst diff_number_of_eq)
+ apply simp
+ done
+
+lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric]
+
+lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less
+
+lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)"
+ by (simp only: real_of_nat_number_of number_of_is_id)
+
+lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)"
+ by simp
+
+lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of
+
+lemmas zpowerarith = zpower_number_of_even
+ zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
+ zpower_Pls zpower_Min
+
+(* div, mod *)
+
+lemma adjust: "adjust b (q, r) = (if 0 \<le> r - b then (2 * q + 1, r - b) else (2 * q, r))"
+ by (auto simp only: adjust_def)
+
+lemma divmod: "divmod_int a b = (if 0\<le>a then
+ if 0\<le>b then posDivAlg a b
+ else if a=0 then (0, 0)
+ else apsnd uminus (negDivAlg (-a) (-b))
+ else
+ if 0<b then negDivAlg a b
+ else apsnd uminus (posDivAlg (-a) (-b)))"
+ by (auto simp only: divmod_int_def)
+
+lemmas compute_div_mod = div_int_def mod_int_def divmod adjust apsnd_def map_pair_def posDivAlg.simps negDivAlg.simps
+
+
+
+(* collecting all the theorems *)
+
+lemma even_Pls: "even (Int.Pls) = True"
+ apply (unfold Pls_def even_def)
+ by simp
+
+lemma even_Min: "even (Int.Min) = False"
+ apply (unfold Min_def even_def)
+ by simp
+
+lemma even_B0: "even (Int.Bit0 x) = True"
+ apply (unfold Bit0_def)
+ by simp
+
+lemma even_B1: "even (Int.Bit1 x) = False"
+ apply (unfold Bit1_def)
+ by simp
+
+lemma even_number_of: "even ((number_of w)::int) = even w"
+ by (simp only: number_of_is_id)
+
+lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of
+
+lemmas compute_numeral = compute_if compute_let compute_pair compute_bool
+ compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/Compute_Oracle.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,11 @@
+(* Title: HOL/Matrix/Compute_Oracle/Compute_Oracle.thy
+ Author: Steven Obua, TU Munich
+
+Steven Obua's evaluator.
+*)
+
+theory Compute_Oracle imports HOL
+uses "am.ML" "am_compiler.ML" "am_interpreter.ML" "am_ghc.ML" "am_sml.ML" "report.ML" "compute.ML" "linker.ML"
+begin
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/am.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,71 @@
+signature ABSTRACT_MACHINE =
+sig
+
+datatype term = Var of int | Const of int | App of term * term | Abs of term | Computed of term
+
+datatype pattern = PVar | PConst of int * (pattern list)
+
+datatype guard = Guard of term * term
+
+type program
+
+exception Compile of string;
+
+(* The de-Bruijn index 0 occurring on the right hand side refers to the LAST pattern variable, when traversing the pattern from left to right,
+ 1 to the second last, and so on. *)
+val compile : (guard list * pattern * term) list -> program
+
+exception Run of string;
+val run : program -> term -> term
+
+(* Utilities *)
+
+val check_freevars : int -> term -> bool
+val forall_consts : (int -> bool) -> term -> bool
+val closed : term -> bool
+val erase_Computed : term -> term
+
+end
+
+structure AbstractMachine : ABSTRACT_MACHINE =
+struct
+
+datatype term = Var of int | Const of int | App of term * term | Abs of term | Computed of term
+
+datatype pattern = PVar | PConst of int * (pattern list)
+
+datatype guard = Guard of term * term
+
+type program = unit
+
+exception Compile of string;
+
+fun erase_Computed (Computed t) = erase_Computed t
+ | erase_Computed (App (t1, t2)) = App (erase_Computed t1, erase_Computed t2)
+ | erase_Computed (Abs t) = Abs (erase_Computed t)
+ | erase_Computed t = t
+
+(*Returns true iff at most 0 .. (free-1) occur unbound. therefore
+ check_freevars 0 t iff t is closed*)
+fun check_freevars free (Var x) = x < free
+ | check_freevars free (Const _) = true
+ | check_freevars free (App (u, v)) = check_freevars free u andalso check_freevars free v
+ | check_freevars free (Abs m) = check_freevars (free+1) m
+ | check_freevars free (Computed t) = check_freevars free t
+
+fun forall_consts pred (Const c) = pred c
+ | forall_consts pred (Var _) = true
+ | forall_consts pred (App (u,v)) = forall_consts pred u
+ andalso forall_consts pred v
+ | forall_consts pred (Abs m) = forall_consts pred m
+ | forall_consts pred (Computed t) = forall_consts pred t
+
+fun closed t = check_freevars 0 t
+
+fun compile _ = raise Compile "abstract machine stub"
+
+exception Run of string;
+
+fun run _ _ = raise Run "abstract machine stub"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/am_compiler.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,208 @@
+(* Title: HOL/Matrix/Compute_Oracle/am_compiler.ML
+ Author: Steven Obua
+*)
+
+signature COMPILING_AM =
+sig
+ include ABSTRACT_MACHINE
+
+ val set_compiled_rewriter : (term -> term) -> unit
+ val list_nth : 'a list * int -> 'a
+ val list_map : ('a -> 'b) -> 'a list -> 'b list
+end
+
+structure AM_Compiler : COMPILING_AM = struct
+
+val list_nth = List.nth;
+val list_map = map;
+
+open AbstractMachine;
+
+val compiled_rewriter = Unsynchronized.ref (NONE:(term -> term)Option.option)
+
+fun set_compiled_rewriter r = (compiled_rewriter := SOME r)
+
+type program = (term -> term)
+
+fun count_patternvars PVar = 1
+ | count_patternvars (PConst (_, ps)) =
+ List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
+
+fun print_rule (p, t) =
+ let
+ fun str x = string_of_int x
+ fun print_pattern n PVar = (n+1, "x"^(str n))
+ | print_pattern n (PConst (c, [])) = (n, "c"^(str c))
+ | print_pattern n (PConst (c, args)) =
+ let
+ val h = print_pattern n (PConst (c,[]))
+ in
+ print_pattern_list h args
+ end
+ and print_pattern_list r [] = r
+ | print_pattern_list (n, p) (t::ts) =
+ let
+ val (n, t) = print_pattern n t
+ in
+ print_pattern_list (n, "App ("^p^", "^t^")") ts
+ end
+
+ val (n, pattern) = print_pattern 0 p
+ val pattern =
+ if exists_string Symbol.is_ascii_blank pattern then "(" ^ pattern ^")"
+ else pattern
+
+ fun print_term d (Var x) = "Var " ^ str x
+ | print_term d (Const c) = "c" ^ str c
+ | print_term d (App (a,b)) = "App (" ^ print_term d a ^ ", " ^ print_term d b ^ ")"
+ | print_term d (Abs c) = "Abs (" ^ print_term (d + 1) c ^ ")"
+ | print_term d (Computed c) = print_term d c
+
+ fun listvars n = if n = 0 then "x0" else "x"^(str n)^", "^(listvars (n-1))
+
+ val term = print_term 0 t
+ val term =
+ if n > 0 then "Closure (["^(listvars (n-1))^"], "^term^")"
+ else "Closure ([], "^term^")"
+
+ in
+ " | weak_reduce (false, stack, "^pattern^") = Continue (false, stack, "^term^")"
+ end
+
+fun constants_of PVar = []
+ | constants_of (PConst (c, ps)) = c :: maps constants_of ps
+
+fun constants_of_term (Var _) = []
+ | constants_of_term (Abs m) = constants_of_term m
+ | constants_of_term (App (a,b)) = (constants_of_term a)@(constants_of_term b)
+ | constants_of_term (Const c) = [c]
+ | constants_of_term (Computed c) = constants_of_term c
+
+fun load_rules sname name prog =
+ let
+ val buffer = Unsynchronized.ref ""
+ fun write s = (buffer := (!buffer)^s)
+ fun writeln s = (write s; write "\n")
+ fun writelist [] = ()
+ | writelist (s::ss) = (writeln s; writelist ss)
+ fun str i = string_of_int i
+ val _ = writelist [
+ "structure "^name^" = struct",
+ "",
+ "datatype term = Dummy | App of term * term | Abs of term | Var of int | Const of int | Closure of term list * term"]
+ val constants = distinct (op =) (maps (fn (p, r) => ((constants_of p)@(constants_of_term r))) prog)
+ val _ = map (fn x => write (" | c"^(str x))) constants
+ val _ = writelist [
+ "",
+ "datatype stack = SEmpty | SAppL of term * stack | SAppR of term * stack | SAbs of stack",
+ "",
+ "type state = bool * stack * term",
+ "",
+ "datatype loopstate = Continue of state | Stop of stack * term",
+ "",
+ "fun proj_C (Continue s) = s",
+ " | proj_C _ = raise Match",
+ "",
+ "fun proj_S (Stop s) = s",
+ " | proj_S _ = raise Match",
+ "",
+ "fun cont (Continue _) = true",
+ " | cont _ = false",
+ "",
+ "fun do_reduction reduce p =",
+ " let",
+ " val s = Unsynchronized.ref (Continue p)",
+ " val _ = while cont (!s) do (s := reduce (proj_C (!s)))",
+ " in",
+ " proj_S (!s)",
+ " end",
+ ""]
+
+ val _ = writelist [
+ "fun weak_reduce (false, stack, Closure (e, App (a, b))) = Continue (false, SAppL (Closure (e, b), stack), Closure (e, a))",
+ " | weak_reduce (false, SAppL (b, stack), Closure (e, Abs m)) = Continue (false, stack, Closure (b::e, m))",
+ " | weak_reduce (false, stack, c as Closure (e, Abs m)) = Continue (true, stack, c)",
+ " | weak_reduce (false, stack, Closure (e, Var n)) = Continue (false, stack, case "^sname^".list_nth (e, n) of Dummy => Var n | r => r)",
+ " | weak_reduce (false, stack, Closure (e, c)) = Continue (false, stack, c)"]
+ val _ = writelist (map print_rule prog)
+ val _ = writelist [
+ " | weak_reduce (false, stack, clos) = Continue (true, stack, clos)",
+ " | weak_reduce (true, SAppR (a, stack), b) = Continue (false, stack, App (a,b))",
+ " | weak_reduce (true, s as (SAppL (b, stack)), a) = Continue (false, SAppR (a, stack), b)",
+ " | weak_reduce (true, stack, c) = Stop (stack, c)",
+ "",
+ "fun strong_reduce (false, stack, Closure (e, Abs m)) =",
+ " let",
+ " val (stack', wnf) = do_reduction weak_reduce (false, SEmpty, Closure (Dummy::e, m))",
+ " in",
+ " case stack' of",
+ " SEmpty => Continue (false, SAbs stack, wnf)",
+ " | _ => raise ("^sname^".Run \"internal error in strong: weak failed\")",
+ " end",
+ " | strong_reduce (false, stack, clos as (App (u, v))) = Continue (false, SAppL (v, stack), u)",
+ " | strong_reduce (false, stack, clos) = Continue (true, stack, clos)",
+ " | strong_reduce (true, SAbs stack, m) = Continue (false, stack, Abs m)",
+ " | strong_reduce (true, SAppL (b, stack), a) = Continue (false, SAppR (a, stack), b)",
+ " | strong_reduce (true, SAppR (a, stack), b) = Continue (true, stack, App (a, b))",
+ " | strong_reduce (true, stack, clos) = Stop (stack, clos)",
+ ""]
+
+ val ic = "(case c of "^(implode (map (fn c => (str c)^" => c"^(str c)^" | ") constants))^" _ => Const c)"
+ val _ = writelist [
+ "fun importTerm ("^sname^".Var x) = Var x",
+ " | importTerm ("^sname^".Const c) = "^ic,
+ " | importTerm ("^sname^".App (a, b)) = App (importTerm a, importTerm b)",
+ " | importTerm ("^sname^".Abs m) = Abs (importTerm m)",
+ ""]
+
+ fun ec c = " | exportTerm c"^(str c)^" = "^sname^".Const "^(str c)
+ val _ = writelist [
+ "fun exportTerm (Var x) = "^sname^".Var x",
+ " | exportTerm (Const c) = "^sname^".Const c",
+ " | exportTerm (App (a,b)) = "^sname^".App (exportTerm a, exportTerm b)",
+ " | exportTerm (Abs m) = "^sname^".Abs (exportTerm m)",
+ " | exportTerm (Closure (closlist, clos)) = raise ("^sname^".Run \"internal error, cannot export Closure\")",
+ " | exportTerm Dummy = raise ("^sname^".Run \"internal error, cannot export Dummy\")"]
+ val _ = writelist (map ec constants)
+
+ val _ = writelist [
+ "",
+ "fun rewrite t = ",
+ " let",
+ " val (stack, wnf) = do_reduction weak_reduce (false, SEmpty, Closure ([], importTerm t))",
+ " in",
+ " case stack of ",
+ " SEmpty => (case do_reduction strong_reduce (false, SEmpty, wnf) of",
+ " (SEmpty, snf) => exportTerm snf",
+ " | _ => raise ("^sname^".Run \"internal error in rewrite: strong failed\"))",
+ " | _ => (raise ("^sname^".Run \"internal error in rewrite: weak failed\"))",
+ " end",
+ "",
+ "val _ = "^sname^".set_compiled_rewriter rewrite",
+ "",
+ "end;"]
+
+ in
+ compiled_rewriter := NONE;
+ use_text ML_Env.local_context (1, "") false (!buffer);
+ case !compiled_rewriter of
+ NONE => raise (Compile "cannot communicate with compiled function")
+ | SOME r => (compiled_rewriter := NONE; r)
+ end
+
+fun compile eqs =
+ let
+ val _ = if exists (fn (a,_,_) => not (null a)) eqs then raise Compile ("cannot deal with guards") else ()
+ val eqs = map (fn (_,b,c) => (b,c)) eqs
+ fun check (p, r) = if check_freevars (count_patternvars p) r then () else raise Compile ("unbound variables in rule")
+ val _ = map (fn (p, r) =>
+ (check (p, r);
+ case p of PVar => raise (Compile "pattern is just a variable") | _ => ())) eqs
+ in
+ load_rules "AM_Compiler" "AM_compiled_code" eqs
+ end
+
+fun run prog t = prog t
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/am_ghc.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,324 @@
+(* Title: HOL/Matrix/Compute_Oracle/am_ghc.ML
+ Author: Steven Obua
+*)
+
+structure AM_GHC : ABSTRACT_MACHINE =
+struct
+
+open AbstractMachine;
+
+type program = string * string * (int Inttab.table)
+
+fun count_patternvars PVar = 1
+ | count_patternvars (PConst (_, ps)) =
+ List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
+
+fun update_arity arity code a =
+ (case Inttab.lookup arity code of
+ NONE => Inttab.update_new (code, a) arity
+ | SOME (a': int) => if a > a' then Inttab.update (code, a) arity else arity)
+
+(* We have to find out the maximal arity of each constant *)
+fun collect_pattern_arity PVar arity = arity
+ | collect_pattern_arity (PConst (c, args)) arity = fold collect_pattern_arity args (update_arity arity c (length args))
+
+local
+fun collect applevel (Var _) arity = arity
+ | collect applevel (Const c) arity = update_arity arity c applevel
+ | collect applevel (Abs m) arity = collect 0 m arity
+ | collect applevel (App (a,b)) arity = collect 0 b (collect (applevel + 1) a arity)
+in
+fun collect_term_arity t arity = collect 0 t arity
+end
+
+fun nlift level n (Var m) = if m < level then Var m else Var (m+n)
+ | nlift level n (Const c) = Const c
+ | nlift level n (App (a,b)) = App (nlift level n a, nlift level n b)
+ | nlift level n (Abs b) = Abs (nlift (level+1) n b)
+
+fun rep n x = if n = 0 then [] else x::(rep (n-1) x)
+
+fun adjust_rules rules =
+ let
+ val arity = fold (fn (p, t) => fn arity => collect_term_arity t (collect_pattern_arity p arity)) rules Inttab.empty
+ fun arity_of c = the (Inttab.lookup arity c)
+ fun adjust_pattern PVar = PVar
+ | adjust_pattern (C as PConst (c, args)) = if (length args <> arity_of c) then raise Compile ("Constant inside pattern must have maximal arity") else C
+ fun adjust_rule (PVar, _) = raise Compile ("pattern may not be a variable")
+ | adjust_rule (rule as (p as PConst (c, args),t)) =
+ let
+ val _ = if not (check_freevars (count_patternvars p) t) then raise Compile ("unbound variables on right hand side") else ()
+ val args = map adjust_pattern args
+ val len = length args
+ val arity = arity_of c
+ fun lift level n (Var m) = if m < level then Var m else Var (m+n)
+ | lift level n (Const c) = Const c
+ | lift level n (App (a,b)) = App (lift level n a, lift level n b)
+ | lift level n (Abs b) = Abs (lift (level+1) n b)
+ val lift = lift 0
+ fun adjust_term n t = if n=0 then t else adjust_term (n-1) (App (t, Var (n-1)))
+ in
+ if len = arity then
+ rule
+ else if arity >= len then
+ (PConst (c, args @ (rep (arity-len) PVar)), adjust_term (arity-len) (lift (arity-len) t))
+ else (raise Compile "internal error in adjust_rule")
+ end
+ in
+ (arity, map adjust_rule rules)
+ end
+
+fun print_term arity_of n =
+let
+ fun str x = string_of_int x
+ fun protect_blank s = if exists_string Symbol.is_ascii_blank s then "(" ^ s ^")" else s
+
+ fun print_apps d f [] = f
+ | print_apps d f (a::args) = print_apps d ("app "^(protect_blank f)^" "^(protect_blank (print_term d a))) args
+ and print_call d (App (a, b)) args = print_call d a (b::args)
+ | print_call d (Const c) args =
+ (case arity_of c of
+ NONE => print_apps d ("Const "^(str c)) args
+ | SOME a =>
+ let
+ val len = length args
+ in
+ if a <= len then
+ let
+ val s = "c"^(str c)^(implode (map (fn t => " "^(protect_blank (print_term d t))) (List.take (args, a))))
+ in
+ print_apps d s (List.drop (args, a))
+ end
+ else
+ let
+ fun mk_apps n t = if n = 0 then t else mk_apps (n-1) (App (t, Var (n-1)))
+ fun mk_lambdas n t = if n = 0 then t else mk_lambdas (n-1) (Abs t)
+ fun append_args [] t = t
+ | append_args (c::cs) t = append_args cs (App (t, c))
+ in
+ print_term d (mk_lambdas (a-len) (mk_apps (a-len) (nlift 0 (a-len) (append_args args (Const c)))))
+ end
+ end)
+ | print_call d t args = print_apps d (print_term d t) args
+ and print_term d (Var x) = if x < d then "b"^(str (d-x-1)) else "x"^(str (n-(x-d)-1))
+ | print_term d (Abs c) = "Abs (\\b"^(str d)^" -> "^(print_term (d + 1) c)^")"
+ | print_term d t = print_call d t []
+in
+ print_term 0
+end
+
+fun print_rule arity_of (p, t) =
+ let
+ fun str x = string_of_int x
+ fun print_pattern top n PVar = (n+1, "x"^(str n))
+ | print_pattern top n (PConst (c, [])) = (n, (if top then "c" else "C")^(str c))
+ | print_pattern top n (PConst (c, args)) =
+ let
+ val (n,s) = print_pattern_list (n, (if top then "c" else "C")^(str c)) args
+ in
+ (n, if top then s else "("^s^")")
+ end
+ and print_pattern_list r [] = r
+ | print_pattern_list (n, p) (t::ts) =
+ let
+ val (n, t) = print_pattern false n t
+ in
+ print_pattern_list (n, p^" "^t) ts
+ end
+ val (n, pattern) = print_pattern true 0 p
+ in
+ pattern^" = "^(print_term arity_of n t)
+ end
+
+fun group_rules rules =
+ let
+ fun add_rule (r as (PConst (c,_), _)) groups =
+ let
+ val rs = (case Inttab.lookup groups c of NONE => [] | SOME rs => rs)
+ in
+ Inttab.update (c, r::rs) groups
+ end
+ | add_rule _ _ = raise Compile "internal error group_rules"
+ in
+ fold_rev add_rule rules Inttab.empty
+ end
+
+fun haskell_prog name rules =
+ let
+ val buffer = Unsynchronized.ref ""
+ fun write s = (buffer := (!buffer)^s)
+ fun writeln s = (write s; write "\n")
+ fun writelist [] = ()
+ | writelist (s::ss) = (writeln s; writelist ss)
+ fun str i = string_of_int i
+ val (arity, rules) = adjust_rules rules
+ val rules = group_rules rules
+ val constants = Inttab.keys arity
+ fun arity_of c = Inttab.lookup arity c
+ fun rep_str s n = implode (rep n s)
+ fun indexed s n = s^(str n)
+ fun section n = if n = 0 then [] else (section (n-1))@[n-1]
+ fun make_show c =
+ let
+ val args = section (the (arity_of c))
+ in
+ " show ("^(indexed "C" c)^(implode (map (indexed " a") args))^") = "
+ ^"\""^(indexed "C" c)^"\""^(implode (map (fn a => "++(show "^(indexed "a" a)^")") args))
+ end
+ fun default_case c =
+ let
+ val args = implode (map (indexed " x") (section (the (arity_of c))))
+ in
+ (indexed "c" c)^args^" = "^(indexed "C" c)^args
+ end
+ val _ = writelist [
+ "module "^name^" where",
+ "",
+ "data Term = Const Integer | App Term Term | Abs (Term -> Term)",
+ " "^(implode (map (fn c => " | C"^(str c)^(rep_str " Term" (the (arity_of c)))) constants)),
+ "",
+ "instance Show Term where"]
+ val _ = writelist (map make_show constants)
+ val _ = writelist [
+ " show (Const c) = \"c\"++(show c)",
+ " show (App a b) = \"A\"++(show a)++(show b)",
+ " show (Abs _) = \"L\"",
+ ""]
+ val _ = writelist [
+ "app (Abs a) b = a b",
+ "app a b = App a b",
+ "",
+ "calc s c = writeFile s (show c)",
+ ""]
+ fun list_group c = (writelist (case Inttab.lookup rules c of
+ NONE => [default_case c, ""]
+ | SOME (rs as ((PConst (_, []), _)::rs')) =>
+ if not (null rs') then raise Compile "multiple declaration of constant"
+ else (map (print_rule arity_of) rs) @ [""]
+ | SOME rs => (map (print_rule arity_of) rs) @ [default_case c, ""]))
+ val _ = map list_group constants
+ in
+ (arity, !buffer)
+ end
+
+val guid_counter = Unsynchronized.ref 0
+fun get_guid () =
+ let
+ val c = !guid_counter
+ val _ = guid_counter := !guid_counter + 1
+ in
+ string_of_int (Time.toMicroseconds (Time.now ())) ^ string_of_int c
+ end
+
+fun tmp_file s = Path.implode (Path.expand (File.tmp_path (Path.basic s)));
+
+fun writeTextFile name s = File.write (Path.explode name) s
+
+fun fileExists name = ((OS.FileSys.fileSize name; true) handle OS.SysErr _ => false)
+
+fun compile eqs =
+ let
+ val _ = if exists (fn (a,_,_) => not (null a)) eqs then raise Compile ("cannot deal with guards") else ()
+ val eqs = map (fn (_,b,c) => (b,c)) eqs
+ val guid = get_guid ()
+ val module = "AMGHC_Prog_"^guid
+ val (arity, source) = haskell_prog module eqs
+ val module_file = tmp_file (module^".hs")
+ val object_file = tmp_file (module^".o")
+ val _ = writeTextFile module_file source
+ val _ = Isabelle_System.bash ("exec \"$ISABELLE_GHC\" -c " ^ module_file)
+ val _ =
+ if not (fileExists object_file) then
+ raise Compile ("Failure compiling haskell code (ISABELLE_GHC='" ^ getenv "ISABELLE_GHC" ^ "')")
+ else ()
+ in
+ (guid, module_file, arity)
+ end
+
+fun readResultFile name = File.read (Path.explode name)
+
+fun parse_result arity_of result =
+ let
+ val result = String.explode result
+ fun shift NONE x = SOME x
+ | shift (SOME y) x = SOME (y*10 + x)
+ fun parse_int' x (#"0"::rest) = parse_int' (shift x 0) rest
+ | parse_int' x (#"1"::rest) = parse_int' (shift x 1) rest
+ | parse_int' x (#"2"::rest) = parse_int' (shift x 2) rest
+ | parse_int' x (#"3"::rest) = parse_int' (shift x 3) rest
+ | parse_int' x (#"4"::rest) = parse_int' (shift x 4) rest
+ | parse_int' x (#"5"::rest) = parse_int' (shift x 5) rest
+ | parse_int' x (#"6"::rest) = parse_int' (shift x 6) rest
+ | parse_int' x (#"7"::rest) = parse_int' (shift x 7) rest
+ | parse_int' x (#"8"::rest) = parse_int' (shift x 8) rest
+ | parse_int' x (#"9"::rest) = parse_int' (shift x 9) rest
+ | parse_int' x rest = (x, rest)
+ fun parse_int rest = parse_int' NONE rest
+
+ fun parse (#"C"::rest) =
+ (case parse_int rest of
+ (SOME c, rest) =>
+ let
+ val (args, rest) = parse_list (the (arity_of c)) rest
+ fun app_args [] t = t
+ | app_args (x::xs) t = app_args xs (App (t, x))
+ in
+ (app_args args (Const c), rest)
+ end
+ | (NONE, _) => raise Run "parse C")
+ | parse (#"c"::rest) =
+ (case parse_int rest of
+ (SOME c, rest) => (Const c, rest)
+ | _ => raise Run "parse c")
+ | parse (#"A"::rest) =
+ let
+ val (a, rest) = parse rest
+ val (b, rest) = parse rest
+ in
+ (App (a,b), rest)
+ end
+ | parse (#"L"::_) = raise Run "there may be no abstraction in the result"
+ | parse _ = raise Run "invalid result"
+ and parse_list n rest =
+ if n = 0 then
+ ([], rest)
+ else
+ let
+ val (x, rest) = parse rest
+ val (xs, rest) = parse_list (n-1) rest
+ in
+ (x::xs, rest)
+ end
+ val (parsed, rest) = parse result
+ fun is_blank (#" "::rest) = is_blank rest
+ | is_blank (#"\n"::rest) = is_blank rest
+ | is_blank [] = true
+ | is_blank _ = false
+ in
+ if is_blank rest then parsed else raise Run "non-blank suffix in result file"
+ end
+
+fun run (guid, module_file, arity) t =
+ let
+ val _ = if check_freevars 0 t then () else raise Run ("can only compute closed terms")
+ fun arity_of c = Inttab.lookup arity c
+ val callguid = get_guid()
+ val module = "AMGHC_Prog_"^guid
+ val call = module^"_Call_"^callguid
+ val result_file = tmp_file (module^"_Result_"^callguid^".txt")
+ val call_file = tmp_file (call^".hs")
+ val term = print_term arity_of 0 t
+ val call_source = "module "^call^" where\n\nimport "^module^"\n\ncall = "^module^".calc \""^result_file^"\" ("^term^")"
+ val _ = writeTextFile call_file call_source
+ val _ = Isabelle_System.bash ("exec \"$ISABELLE_GHC\" -e \""^call^".call\" "^module_file^" "^call_file)
+ val result = readResultFile result_file handle IO.Io _ =>
+ raise Run ("Failure running haskell compiler (ISABELLE_GHC='" ^ getenv "ISABELLE_GHC" ^ "')")
+ val t' = parse_result arity_of result
+ val _ = OS.FileSys.remove call_file
+ val _ = OS.FileSys.remove result_file
+ in
+ t'
+ end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/am_interpreter.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,211 @@
+(* Title: HOL/Matrix/Compute_Oracle/am_interpreter.ML
+ Author: Steven Obua
+*)
+
+signature AM_BARRAS =
+sig
+ include ABSTRACT_MACHINE
+ val max_reductions : int option Unsynchronized.ref
+end
+
+structure AM_Interpreter : AM_BARRAS = struct
+
+open AbstractMachine;
+
+datatype closure = CDummy | CVar of int | CConst of int
+ | CApp of closure * closure | CAbs of closure
+ | Closure of (closure list) * closure
+
+structure prog_struct = Table(type key = int*int val ord = prod_ord int_ord int_ord);
+
+datatype program = Program of ((pattern * closure * (closure*closure) list) list) prog_struct.table
+
+datatype stack = SEmpty | SAppL of closure * stack | SAppR of closure * stack | SAbs of stack
+
+fun clos_of_term (Var x) = CVar x
+ | clos_of_term (Const c) = CConst c
+ | clos_of_term (App (u, v)) = CApp (clos_of_term u, clos_of_term v)
+ | clos_of_term (Abs u) = CAbs (clos_of_term u)
+ | clos_of_term (Computed t) = clos_of_term t
+
+fun term_of_clos (CVar x) = Var x
+ | term_of_clos (CConst c) = Const c
+ | term_of_clos (CApp (u, v)) = App (term_of_clos u, term_of_clos v)
+ | term_of_clos (CAbs u) = Abs (term_of_clos u)
+ | term_of_clos (Closure _) = raise (Run "internal error: closure in normalized term found")
+ | term_of_clos CDummy = raise (Run "internal error: dummy in normalized term found")
+
+fun resolve_closure closures (CVar x) = (case nth closures x of CDummy => CVar x | r => r)
+ | resolve_closure closures (CConst c) = CConst c
+ | resolve_closure closures (CApp (u, v)) = CApp (resolve_closure closures u, resolve_closure closures v)
+ | resolve_closure closures (CAbs u) = CAbs (resolve_closure (CDummy::closures) u)
+ | resolve_closure closures (CDummy) = raise (Run "internal error: resolve_closure applied to CDummy")
+ | resolve_closure closures (Closure (e, u)) = resolve_closure e u
+
+fun resolve_closure' c = resolve_closure [] c
+
+fun resolve_stack tm SEmpty = tm
+ | resolve_stack tm (SAppL (c, s)) = resolve_stack (CApp (tm, resolve_closure' c)) s
+ | resolve_stack tm (SAppR (c, s)) = resolve_stack (CApp (resolve_closure' c, tm)) s
+ | resolve_stack tm (SAbs s) = resolve_stack (CAbs tm) s
+
+fun resolve (stack, closure) =
+ let
+ val _ = writeln "start resolving"
+ val t = resolve_stack (resolve_closure' closure) stack
+ val _ = writeln "finished resolving"
+ in
+ t
+ end
+
+fun strip_closure args (CApp (a,b)) = strip_closure (b::args) a
+ | strip_closure args x = (x, args)
+
+fun len_head_of_closure n (CApp (a, _)) = len_head_of_closure (n+1) a
+ | len_head_of_closure n x = (n, x)
+
+
+(* earlier occurrence of PVar corresponds to higher de Bruijn index *)
+fun pattern_match args PVar clos = SOME (clos::args)
+ | pattern_match args (PConst (c, patterns)) clos =
+ let
+ val (f, closargs) = strip_closure [] clos
+ in
+ case f of
+ CConst d =>
+ if c = d then
+ pattern_match_list args patterns closargs
+ else
+ NONE
+ | _ => NONE
+ end
+and pattern_match_list args [] [] = SOME args
+ | pattern_match_list args (p::ps) (c::cs) =
+ (case pattern_match args p c of
+ NONE => NONE
+ | SOME args => pattern_match_list args ps cs)
+ | pattern_match_list _ _ _ = NONE
+
+fun count_patternvars PVar = 1
+ | count_patternvars (PConst (_, ps)) = List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
+
+fun pattern_key (PConst (c, ps)) = (c, length ps)
+ | pattern_key _ = raise (Compile "pattern reduces to variable")
+
+(*Returns true iff at most 0 .. (free-1) occur unbound. therefore
+ check_freevars 0 t iff t is closed*)
+fun check_freevars free (Var x) = x < free
+ | check_freevars free (Const _) = true
+ | check_freevars free (App (u, v)) = check_freevars free u andalso check_freevars free v
+ | check_freevars free (Abs m) = check_freevars (free+1) m
+ | check_freevars free (Computed t) = check_freevars free t
+
+fun compile eqs =
+ let
+ fun check p r = if check_freevars p r then () else raise Compile ("unbound variables in rule")
+ fun check_guard p (Guard (a,b)) = (check p a; check p b)
+ fun clos_of_guard (Guard (a,b)) = (clos_of_term a, clos_of_term b)
+ val eqs = map (fn (guards, p, r) => let val pcount = count_patternvars p val _ = map (check_guard pcount) (guards) val _ = check pcount r in
+ (pattern_key p, (p, clos_of_term r, map clos_of_guard guards)) end) eqs
+ fun merge (k, a) table = prog_struct.update (k, case prog_struct.lookup table k of NONE => [a] | SOME l => a::l) table
+ val p = fold merge eqs prog_struct.empty
+ in
+ Program p
+ end
+
+
+type state = bool * program * stack * closure
+
+datatype loopstate = Continue of state | Stop of stack * closure
+
+fun proj_C (Continue s) = s
+ | proj_C _ = raise Match
+
+exception InterruptedExecution of stack * closure
+
+fun proj_S (Stop s) = s
+ | proj_S (Continue (_,_,s,c)) = (s,c)
+
+fun cont (Continue _) = true
+ | cont _ = false
+
+val max_reductions = Unsynchronized.ref (NONE : int option)
+
+fun do_reduction reduce p =
+ let
+ val s = Unsynchronized.ref (Continue p)
+ val counter = Unsynchronized.ref 0
+ val _ = case !max_reductions of
+ NONE => while cont (!s) do (s := reduce (proj_C (!s)))
+ | SOME m => while cont (!s) andalso (!counter < m) do (s := reduce (proj_C (!s)); counter := (!counter) + 1)
+ in
+ case !max_reductions of
+ SOME m => if !counter >= m then raise InterruptedExecution (proj_S (!s)) else proj_S (!s)
+ | NONE => proj_S (!s)
+ end
+
+fun match_rules prog n [] clos = NONE
+ | match_rules prog n ((p,eq,guards)::rs) clos =
+ case pattern_match [] p clos of
+ NONE => match_rules prog (n+1) rs clos
+ | SOME args => if forall (guard_checks prog args) guards then SOME (Closure (args, eq)) else match_rules prog (n+1) rs clos
+and guard_checks prog args (a,b) = (simp prog (Closure (args, a)) = simp prog (Closure (args, b)))
+and match_closure (p as (Program prog)) clos =
+ case len_head_of_closure 0 clos of
+ (len, CConst c) =>
+ (case prog_struct.lookup prog (c, len) of
+ NONE => NONE
+ | SOME rules => match_rules p 0 rules clos)
+ | _ => NONE
+
+and weak_reduce (false, prog, stack, Closure (e, CApp (a, b))) = Continue (false, prog, SAppL (Closure (e, b), stack), Closure (e, a))
+ | weak_reduce (false, prog, SAppL (b, stack), Closure (e, CAbs m)) = Continue (false, prog, stack, Closure (b::e, m))
+ | weak_reduce (false, prog, stack, Closure (e, CVar n)) = Continue (false, prog, stack, case nth e n of CDummy => CVar n | r => r)
+ | weak_reduce (false, prog, stack, Closure (_, c as CConst _)) = Continue (false, prog, stack, c)
+ | weak_reduce (false, prog, stack, clos) =
+ (case match_closure prog clos of
+ NONE => Continue (true, prog, stack, clos)
+ | SOME r => Continue (false, prog, stack, r))
+ | weak_reduce (true, prog, SAppR (a, stack), b) = Continue (false, prog, stack, CApp (a,b))
+ | weak_reduce (true, prog, SAppL (b, stack), a) = Continue (false, prog, SAppR (a, stack), b)
+ | weak_reduce (true, prog, stack, c) = Stop (stack, c)
+
+and strong_reduce (false, prog, stack, Closure (e, CAbs m)) =
+ (let
+ val (stack', wnf) = do_reduction weak_reduce (false, prog, SEmpty, Closure (CDummy::e, m))
+ in
+ case stack' of
+ SEmpty => Continue (false, prog, SAbs stack, wnf)
+ | _ => raise (Run "internal error in strong: weak failed")
+ end handle InterruptedExecution state => raise InterruptedExecution (stack, resolve state))
+ | strong_reduce (false, prog, stack, CApp (u, v)) = Continue (false, prog, SAppL (v, stack), u)
+ | strong_reduce (false, prog, stack, clos) = Continue (true, prog, stack, clos)
+ | strong_reduce (true, prog, SAbs stack, m) = Continue (false, prog, stack, CAbs m)
+ | strong_reduce (true, prog, SAppL (b, stack), a) = Continue (false, prog, SAppR (a, stack), b)
+ | strong_reduce (true, prog, SAppR (a, stack), b) = Continue (true, prog, stack, CApp (a, b))
+ | strong_reduce (true, prog, stack, clos) = Stop (stack, clos)
+
+and simp prog t =
+ (let
+ val (stack, wnf) = do_reduction weak_reduce (false, prog, SEmpty, t)
+ in
+ case stack of
+ SEmpty => (case do_reduction strong_reduce (false, prog, SEmpty, wnf) of
+ (SEmpty, snf) => snf
+ | _ => raise (Run "internal error in run: strong failed"))
+ | _ => raise (Run "internal error in run: weak failed")
+ end handle InterruptedExecution state => resolve state)
+
+
+fun run prog t =
+ (let
+ val (stack, wnf) = do_reduction weak_reduce (false, prog, SEmpty, Closure ([], clos_of_term t))
+ in
+ case stack of
+ SEmpty => (case do_reduction strong_reduce (false, prog, SEmpty, wnf) of
+ (SEmpty, snf) => term_of_clos snf
+ | _ => raise (Run "internal error in run: strong failed"))
+ | _ => raise (Run "internal error in run: weak failed")
+ end handle InterruptedExecution state => term_of_clos (resolve state))
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/am_sml.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,517 @@
+(* Title: HOL/Matrix/Compute_Oracle/am_sml.ML
+ Author: Steven Obua
+
+TODO: "parameterless rewrite cannot be used in pattern": In a lot of
+cases it CAN be used, and these cases should be handled
+properly; right now, all cases raise an exception.
+*)
+
+signature AM_SML =
+sig
+ include ABSTRACT_MACHINE
+ val save_result : (string * term) -> unit
+ val set_compiled_rewriter : (term -> term) -> unit
+ val list_nth : 'a list * int -> 'a
+ val dump_output : (string option) Unsynchronized.ref
+end
+
+structure AM_SML : AM_SML = struct
+
+open AbstractMachine;
+
+val dump_output = Unsynchronized.ref (NONE: string option)
+
+type program = term Inttab.table * (term -> term)
+
+val saved_result = Unsynchronized.ref (NONE:(string*term)option)
+
+fun save_result r = (saved_result := SOME r)
+
+val list_nth = List.nth
+
+val compiled_rewriter = Unsynchronized.ref (NONE:(term -> term)Option.option)
+
+fun set_compiled_rewriter r = (compiled_rewriter := SOME r)
+
+fun count_patternvars PVar = 1
+ | count_patternvars (PConst (_, ps)) =
+ List.foldl (fn (p, count) => (count_patternvars p)+count) 0 ps
+
+fun update_arity arity code a =
+ (case Inttab.lookup arity code of
+ NONE => Inttab.update_new (code, a) arity
+ | SOME (a': int) => if a > a' then Inttab.update (code, a) arity else arity)
+
+(* We have to find out the maximal arity of each constant *)
+fun collect_pattern_arity PVar arity = arity
+ | collect_pattern_arity (PConst (c, args)) arity = fold collect_pattern_arity args (update_arity arity c (length args))
+
+(* We also need to find out the maximal toplevel arity of each function constant *)
+fun collect_pattern_toplevel_arity PVar arity = raise Compile "internal error: collect_pattern_toplevel_arity"
+ | collect_pattern_toplevel_arity (PConst (c, args)) arity = update_arity arity c (length args)
+
+local
+fun collect applevel (Var _) arity = arity
+ | collect applevel (Const c) arity = update_arity arity c applevel
+ | collect applevel (Abs m) arity = collect 0 m arity
+ | collect applevel (App (a,b)) arity = collect 0 b (collect (applevel + 1) a arity)
+in
+fun collect_term_arity t arity = collect 0 t arity
+end
+
+fun collect_guard_arity (Guard (a,b)) arity = collect_term_arity b (collect_term_arity a arity)
+
+
+fun rep n x = if n < 0 then raise Compile "internal error: rep" else if n = 0 then [] else x::(rep (n-1) x)
+
+fun beta (Const c) = Const c
+ | beta (Var i) = Var i
+ | beta (App (Abs m, b)) = beta (unlift 0 (subst 0 m (lift 0 b)))
+ | beta (App (a, b)) =
+ (case beta a of
+ Abs m => beta (App (Abs m, b))
+ | a => App (a, beta b))
+ | beta (Abs m) = Abs (beta m)
+ | beta (Computed t) = Computed t
+and subst x (Const c) t = Const c
+ | subst x (Var i) t = if i = x then t else Var i
+ | subst x (App (a,b)) t = App (subst x a t, subst x b t)
+ | subst x (Abs m) t = Abs (subst (x+1) m (lift 0 t))
+and lift level (Const c) = Const c
+ | lift level (App (a,b)) = App (lift level a, lift level b)
+ | lift level (Var i) = if i < level then Var i else Var (i+1)
+ | lift level (Abs m) = Abs (lift (level + 1) m)
+and unlift level (Const c) = Const c
+ | unlift level (App (a, b)) = App (unlift level a, unlift level b)
+ | unlift level (Abs m) = Abs (unlift (level+1) m)
+ | unlift level (Var i) = if i < level then Var i else Var (i-1)
+
+fun nlift level n (Var m) = if m < level then Var m else Var (m+n)
+ | nlift level n (Const c) = Const c
+ | nlift level n (App (a,b)) = App (nlift level n a, nlift level n b)
+ | nlift level n (Abs b) = Abs (nlift (level+1) n b)
+
+fun subst_const (c, t) (Const c') = if c = c' then t else Const c'
+ | subst_const _ (Var i) = Var i
+ | subst_const ct (App (a, b)) = App (subst_const ct a, subst_const ct b)
+ | subst_const ct (Abs m) = Abs (subst_const ct m)
+
+(* Remove all rules that are just parameterless rewrites. This is necessary because SML does not allow functions with no parameters. *)
+fun inline_rules rules =
+ let
+ fun term_contains_const c (App (a, b)) = term_contains_const c a orelse term_contains_const c b
+ | term_contains_const c (Abs m) = term_contains_const c m
+ | term_contains_const c (Var _) = false
+ | term_contains_const c (Const c') = (c = c')
+ fun find_rewrite [] = NONE
+ | find_rewrite ((prems, PConst (c, []), r) :: _) =
+ if check_freevars 0 r then
+ if term_contains_const c r then
+ raise Compile "parameterless rewrite is caught in cycle"
+ else if not (null prems) then
+ raise Compile "parameterless rewrite may not be guarded"
+ else
+ SOME (c, r)
+ else raise Compile "unbound variable on right hand side or guards of rule"
+ | find_rewrite (_ :: rules) = find_rewrite rules
+ fun remove_rewrite _ [] = []
+ | remove_rewrite (cr as (c, r)) ((rule as (prems', PConst (c', args), r')) :: rules) =
+ if c = c' then
+ if null args andalso r = r' andalso null prems' then remove_rewrite cr rules
+ else raise Compile "incompatible parameterless rewrites found"
+ else
+ rule :: remove_rewrite cr rules
+ | remove_rewrite cr (r :: rs) = r :: remove_rewrite cr rs
+ fun pattern_contains_const c (PConst (c', args)) = c = c' orelse exists (pattern_contains_const c) args
+ | pattern_contains_const c (PVar) = false
+ fun inline_rewrite (ct as (c, _)) (prems, p, r) =
+ if pattern_contains_const c p then
+ raise Compile "parameterless rewrite cannot be used in pattern"
+ else (map (fn (Guard (a, b)) => Guard (subst_const ct a, subst_const ct b)) prems, p, subst_const ct r)
+ fun inline inlined rules =
+ case find_rewrite rules of
+ NONE => (Inttab.make inlined, rules)
+ | SOME ct =>
+ let
+ val rules = map (inline_rewrite ct) (remove_rewrite ct rules)
+ val inlined = ct :: (map o apsnd) (subst_const ct) inlined
+ in inline inlined rules end
+ in
+ inline [] rules
+ end
+
+
+(*
+ Calculate the arity, the toplevel_arity, and adjust rules so that all toplevel pattern constants have maximal arity.
+ Also beta reduce the adjusted right hand side of a rule.
+*)
+fun adjust_rules rules =
+ let
+ val arity = fold (fn (prems, p, t) => fn arity => fold collect_guard_arity prems (collect_term_arity t (collect_pattern_arity p arity))) rules Inttab.empty
+ val toplevel_arity = fold (fn (_, p, _) => fn arity => collect_pattern_toplevel_arity p arity) rules Inttab.empty
+ fun arity_of c = the (Inttab.lookup arity c)
+ fun test_pattern PVar = ()
+ | test_pattern (PConst (c, args)) = if (length args <> arity_of c) then raise Compile ("Constant inside pattern must have maximal arity") else (map test_pattern args; ())
+ fun adjust_rule (_, PVar, _) = raise Compile ("pattern may not be a variable")
+ | adjust_rule (_, PConst (_, []), _) = raise Compile ("cannot deal with rewrites that take no parameters")
+ | adjust_rule (rule as (prems, p as PConst (c, args),t)) =
+ let
+ val patternvars_counted = count_patternvars p
+ fun check_fv t = check_freevars patternvars_counted t
+ val _ = if not (check_fv t) then raise Compile ("unbound variables on right hand side of rule") else ()
+ val _ = if not (forall (fn (Guard (a,b)) => check_fv a andalso check_fv b) prems) then raise Compile ("unbound variables in guards") else ()
+ val _ = map test_pattern args
+ val len = length args
+ val arity = arity_of c
+ val lift = nlift 0
+ fun addapps_tm n t = if n=0 then t else addapps_tm (n-1) (App (t, Var (n-1)))
+ fun adjust_term n t = addapps_tm n (lift n t)
+ fun adjust_guard n (Guard (a,b)) = Guard (lift n a, lift n b)
+ in
+ if len = arity then
+ rule
+ else if arity >= len then
+ (map (adjust_guard (arity-len)) prems, PConst (c, args @ (rep (arity-len) PVar)), adjust_term (arity-len) t)
+ else (raise Compile "internal error in adjust_rule")
+ end
+ fun beta_rule (prems, p, t) = ((prems, p, beta t) handle Match => raise Compile "beta_rule")
+ in
+ (arity, toplevel_arity, map (beta_rule o adjust_rule) rules)
+ end
+
+fun print_term module arity_of toplevel_arity_of pattern_var_count pattern_lazy_var_count =
+let
+ fun str x = string_of_int x
+ fun protect_blank s = if exists_string Symbol.is_ascii_blank s then "(" ^ s ^")" else s
+ val module_prefix = (case module of NONE => "" | SOME s => s^".")
+ fun print_apps d f [] = f
+ | print_apps d f (a::args) = print_apps d (module_prefix^"app "^(protect_blank f)^" "^(protect_blank (print_term d a))) args
+ and print_call d (App (a, b)) args = print_call d a (b::args)
+ | print_call d (Const c) args =
+ (case arity_of c of
+ NONE => print_apps d (module_prefix^"Const "^(str c)) args
+ | SOME 0 => module_prefix^"C"^(str c)
+ | SOME a =>
+ let
+ val len = length args
+ in
+ if a <= len then
+ let
+ val strict_a = (case toplevel_arity_of c of SOME sa => sa | NONE => a)
+ val _ = if strict_a > a then raise Compile "strict" else ()
+ val s = module_prefix^"c"^(str c)^(implode (map (fn t => " "^(protect_blank (print_term d t))) (List.take (args, strict_a))))
+ val s = s^(implode (map (fn t => " (fn () => "^print_term d t^")") (List.drop (List.take (args, a), strict_a))))
+ in
+ print_apps d s (List.drop (args, a))
+ end
+ else
+ let
+ fun mk_apps n t = if n = 0 then t else mk_apps (n-1) (App (t, Var (n - 1)))
+ fun mk_lambdas n t = if n = 0 then t else mk_lambdas (n-1) (Abs t)
+ fun append_args [] t = t
+ | append_args (c::cs) t = append_args cs (App (t, c))
+ in
+ print_term d (mk_lambdas (a-len) (mk_apps (a-len) (nlift 0 (a-len) (append_args args (Const c)))))
+ end
+ end)
+ | print_call d t args = print_apps d (print_term d t) args
+ and print_term d (Var x) =
+ if x < d then
+ "b"^(str (d-x-1))
+ else
+ let
+ val n = pattern_var_count - (x-d) - 1
+ val x = "x"^(str n)
+ in
+ if n < pattern_var_count - pattern_lazy_var_count then
+ x
+ else
+ "("^x^" ())"
+ end
+ | print_term d (Abs c) = module_prefix^"Abs (fn b"^(str d)^" => "^(print_term (d + 1) c)^")"
+ | print_term d t = print_call d t []
+in
+ print_term 0
+end
+
+fun section n = if n = 0 then [] else (section (n-1))@[n-1]
+
+fun print_rule gnum arity_of toplevel_arity_of (guards, p, t) =
+ let
+ fun str x = string_of_int x
+ fun print_pattern top n PVar = (n+1, "x"^(str n))
+ | print_pattern top n (PConst (c, [])) = (n, (if top then "c" else "C")^(str c)^(if top andalso gnum > 0 then "_"^(str gnum) else ""))
+ | print_pattern top n (PConst (c, args)) =
+ let
+ val f = (if top then "c" else "C")^(str c)^(if top andalso gnum > 0 then "_"^(str gnum) else "")
+ val (n, s) = print_pattern_list 0 top (n, f) args
+ in
+ (n, s)
+ end
+ and print_pattern_list' counter top (n,p) [] = if top then (n,p) else (n,p^")")
+ | print_pattern_list' counter top (n, p) (t::ts) =
+ let
+ val (n, t) = print_pattern false n t
+ in
+ print_pattern_list' (counter + 1) top (n, if top then p^" (a"^(str counter)^" as ("^t^"))" else p^", "^t) ts
+ end
+ and print_pattern_list counter top (n, p) (t::ts) =
+ let
+ val (n, t) = print_pattern false n t
+ in
+ print_pattern_list' (counter + 1) top (n, if top then p^" (a"^(str counter)^" as ("^t^"))" else p^" ("^t) ts
+ end
+ val c = (case p of PConst (c, _) => c | _ => raise Match)
+ val (n, pattern) = print_pattern true 0 p
+ val lazy_vars = the (arity_of c) - the (toplevel_arity_of c)
+ fun print_tm tm = print_term NONE arity_of toplevel_arity_of n lazy_vars tm
+ fun print_guard (Guard (a,b)) = "term_eq ("^(print_tm a)^") ("^(print_tm b)^")"
+ val else_branch = "c"^(str c)^"_"^(str (gnum+1))^(implode (map (fn i => " a"^(str i)) (section (the (arity_of c)))))
+ fun print_guards t [] = print_tm t
+ | print_guards t (g::gs) = "if ("^(print_guard g)^")"^(implode (map (fn g => " andalso ("^(print_guard g)^")") gs))^" then ("^(print_tm t)^") else "^else_branch
+ in
+ (if null guards then gnum else gnum+1, pattern^" = "^(print_guards t guards))
+ end
+
+fun group_rules rules =
+ let
+ fun add_rule (r as (_, PConst (c,_), _)) groups =
+ let
+ val rs = (case Inttab.lookup groups c of NONE => [] | SOME rs => rs)
+ in
+ Inttab.update (c, r::rs) groups
+ end
+ | add_rule _ _ = raise Compile "internal error group_rules"
+ in
+ fold_rev add_rule rules Inttab.empty
+ end
+
+fun sml_prog name code rules =
+ let
+ val buffer = Unsynchronized.ref ""
+ fun write s = (buffer := (!buffer)^s)
+ fun writeln s = (write s; write "\n")
+ fun writelist [] = ()
+ | writelist (s::ss) = (writeln s; writelist ss)
+ fun str i = string_of_int i
+ val (inlinetab, rules) = inline_rules rules
+ val (arity, toplevel_arity, rules) = adjust_rules rules
+ val rules = group_rules rules
+ val constants = Inttab.keys arity
+ fun arity_of c = Inttab.lookup arity c
+ fun toplevel_arity_of c = Inttab.lookup toplevel_arity c
+ fun rep_str s n = implode (rep n s)
+ fun indexed s n = s^(str n)
+ fun string_of_tuple [] = ""
+ | string_of_tuple (x::xs) = "("^x^(implode (map (fn s => ", "^s) xs))^")"
+ fun string_of_args [] = ""
+ | string_of_args (x::xs) = x^(implode (map (fn s => " "^s) xs))
+ fun default_case gnum c =
+ let
+ val leftargs = implode (map (indexed " x") (section (the (arity_of c))))
+ val rightargs = section (the (arity_of c))
+ val strict_args = (case toplevel_arity_of c of NONE => the (arity_of c) | SOME sa => sa)
+ val xs = map (fn n => if n < strict_args then "x"^(str n) else "x"^(str n)^"()") rightargs
+ val right = (indexed "C" c)^" "^(string_of_tuple xs)
+ val message = "(\"unresolved lazy call: " ^ string_of_int c ^ "\")"
+ val right = if strict_args < the (arity_of c) then "raise AM_SML.Run "^message else right
+ in
+ (indexed "c" c)^(if gnum > 0 then "_"^(str gnum) else "")^leftargs^" = "^right
+ end
+
+ fun eval_rules c =
+ let
+ val arity = the (arity_of c)
+ val strict_arity = (case toplevel_arity_of c of NONE => arity | SOME sa => sa)
+ fun eval_rule n =
+ let
+ val sc = string_of_int c
+ val left = fold (fn i => fn s => "AbstractMachine.App ("^s^(indexed ", x" i)^")") (section n) ("AbstractMachine.Const "^sc)
+ fun arg i =
+ let
+ val x = indexed "x" i
+ val x = if i < n then "(eval bounds "^x^")" else x
+ val x = if i < strict_arity then x else "(fn () => "^x^")"
+ in
+ x
+ end
+ val right = "c"^sc^" "^(string_of_args (map arg (section arity)))
+ val right = fold_rev (fn i => fn s => "Abs (fn "^(indexed "x" i)^" => "^s^")") (List.drop (section arity, n)) right
+ val right = if arity > 0 then right else "C"^sc
+ in
+ " | eval bounds ("^left^") = "^right
+ end
+ in
+ map eval_rule (rev (section (arity + 1)))
+ end
+
+ fun convert_computed_rules (c: int) : string list =
+ let
+ val arity = the (arity_of c)
+ fun eval_rule () =
+ let
+ val sc = string_of_int c
+ val left = fold (fn i => fn s => "AbstractMachine.App ("^s^(indexed ", x" i)^")") (section arity) ("AbstractMachine.Const "^sc)
+ fun arg i = "(convert_computed "^(indexed "x" i)^")"
+ val right = "C"^sc^" "^(string_of_tuple (map arg (section arity)))
+ val right = if arity > 0 then right else "C"^sc
+ in
+ " | convert_computed ("^left^") = "^right
+ end
+ in
+ [eval_rule ()]
+ end
+
+ fun mk_constr_type_args n = if n > 0 then " of Term "^(rep_str " * Term" (n-1)) else ""
+ val _ = writelist [
+ "structure "^name^" = struct",
+ "",
+ "datatype Term = Const of int | App of Term * Term | Abs of (Term -> Term)",
+ " "^(implode (map (fn c => " | C"^(str c)^(mk_constr_type_args (the (arity_of c)))) constants)),
+ ""]
+ fun make_constr c argprefix = "(C"^(str c)^" "^(string_of_tuple (map (fn i => argprefix^(str i)) (section (the (arity_of c)))))^")"
+ fun make_term_eq c = " | term_eq "^(make_constr c "a")^" "^(make_constr c "b")^" = "^
+ (case the (arity_of c) of
+ 0 => "true"
+ | n =>
+ let
+ val eqs = map (fn i => "term_eq a"^(str i)^" b"^(str i)) (section n)
+ val (eq, eqs) = (List.hd eqs, map (fn s => " andalso "^s) (List.tl eqs))
+ in
+ eq^(implode eqs)
+ end)
+ val _ = writelist [
+ "fun term_eq (Const c1) (Const c2) = (c1 = c2)",
+ " | term_eq (App (a1,a2)) (App (b1,b2)) = term_eq a1 b1 andalso term_eq a2 b2"]
+ val _ = writelist (map make_term_eq constants)
+ val _ = writelist [
+ " | term_eq _ _ = false",
+ ""
+ ]
+ val _ = writelist [
+ "fun app (Abs a) b = a b",
+ " | app a b = App (a, b)",
+ ""]
+ fun defcase gnum c = (case arity_of c of NONE => [] | SOME a => if a > 0 then [default_case gnum c] else [])
+ fun writefundecl [] = ()
+ | writefundecl (x::xs) = writelist ((("and "^x)::(map (fn s => " | "^s) xs)))
+ fun list_group c = (case Inttab.lookup rules c of
+ NONE => [defcase 0 c]
+ | SOME rs =>
+ let
+ val rs =
+ fold
+ (fn r =>
+ fn rs =>
+ let
+ val (gnum, l, rs) =
+ (case rs of
+ [] => (0, [], [])
+ | (gnum, l)::rs => (gnum, l, rs))
+ val (gnum', r) = print_rule gnum arity_of toplevel_arity_of r
+ in
+ if gnum' = gnum then
+ (gnum, r::l)::rs
+ else
+ let
+ val args = implode (map (fn i => " a"^(str i)) (section (the (arity_of c))))
+ fun gnumc g = if g > 0 then "c"^(str c)^"_"^(str g)^args else "c"^(str c)^args
+ val s = gnumc (gnum) ^ " = " ^ gnumc (gnum')
+ in
+ (gnum', [])::(gnum, s::r::l)::rs
+ end
+ end)
+ rs []
+ val rs = (case rs of [] => [(0,defcase 0 c)] | (gnum,l)::rs => (gnum, (defcase gnum c)@l)::rs)
+ in
+ rev (map (fn z => rev (snd z)) rs)
+ end)
+ val _ = map (fn z => (map writefundecl z; writeln "")) (map list_group constants)
+ val _ = writelist [
+ "fun convert (Const i) = AM_SML.Const i",
+ " | convert (App (a, b)) = AM_SML.App (convert a, convert b)",
+ " | convert (Abs _) = raise AM_SML.Run \"no abstraction in result allowed\""]
+ fun make_convert c =
+ let
+ val args = map (indexed "a") (section (the (arity_of c)))
+ val leftargs =
+ case args of
+ [] => ""
+ | (x::xs) => "("^x^(implode (map (fn s => ", "^s) xs))^")"
+ val args = map (indexed "convert a") (section (the (arity_of c)))
+ val right = fold (fn x => fn s => "AM_SML.App ("^s^", "^x^")") args ("AM_SML.Const "^(str c))
+ in
+ " | convert (C"^(str c)^" "^leftargs^") = "^right
+ end
+ val _ = writelist (map make_convert constants)
+ val _ = writelist [
+ "",
+ "fun convert_computed (AbstractMachine.Abs b) = raise AM_SML.Run \"no abstraction in convert_computed allowed\"",
+ " | convert_computed (AbstractMachine.Var i) = raise AM_SML.Run \"no bound variables in convert_computed allowed\""]
+ val _ = map (writelist o convert_computed_rules) constants
+ val _ = writelist [
+ " | convert_computed (AbstractMachine.Const c) = Const c",
+ " | convert_computed (AbstractMachine.App (a, b)) = App (convert_computed a, convert_computed b)",
+ " | convert_computed (AbstractMachine.Computed a) = raise AM_SML.Run \"no nesting in convert_computed allowed\""]
+ val _ = writelist [
+ "",
+ "fun eval bounds (AbstractMachine.Abs m) = Abs (fn b => eval (b::bounds) m)",
+ " | eval bounds (AbstractMachine.Var i) = AM_SML.list_nth (bounds, i)"]
+ val _ = map (writelist o eval_rules) constants
+ val _ = writelist [
+ " | eval bounds (AbstractMachine.App (a, b)) = app (eval bounds a) (eval bounds b)",
+ " | eval bounds (AbstractMachine.Const c) = Const c",
+ " | eval bounds (AbstractMachine.Computed t) = convert_computed t"]
+ val _ = writelist [
+ "",
+ "fun export term = AM_SML.save_result (\""^code^"\", convert term)",
+ "",
+ "val _ = AM_SML.set_compiled_rewriter (fn t => (convert (eval [] t)))",
+ "",
+ "end"]
+ in
+ (inlinetab, !buffer)
+ end
+
+val guid_counter = Unsynchronized.ref 0
+fun get_guid () =
+ let
+ val c = !guid_counter
+ val _ = guid_counter := !guid_counter + 1
+ in
+ string_of_int (Time.toMicroseconds (Time.now ())) ^ string_of_int c
+ end
+
+
+fun writeTextFile name s = File.write (Path.explode name) s
+
+fun use_source src = use_text ML_Env.local_context (1, "") false src
+
+fun compile rules =
+ let
+ val guid = get_guid ()
+ val code = Real.toString (random ())
+ val name = "AMSML_"^guid
+ val (inlinetab, source) = sml_prog name code rules
+ val _ = case !dump_output of NONE => () | SOME p => writeTextFile p source
+ val _ = compiled_rewriter := NONE
+ val _ = use_source source
+ in
+ case !compiled_rewriter of
+ NONE => raise Compile "broken link to compiled function"
+ | SOME compiled_fun => (inlinetab, compiled_fun)
+ end
+
+fun run (inlinetab, compiled_fun) t =
+ let
+ val _ = if check_freevars 0 t then () else raise Run ("can only compute closed terms")
+ fun inline (Const c) = (case Inttab.lookup inlinetab c of NONE => Const c | SOME t => t)
+ | inline (Var i) = Var i
+ | inline (App (a, b)) = App (inline a, inline b)
+ | inline (Abs m) = Abs (inline m)
+ | inline (Computed t) = Computed t
+ in
+ compiled_fun (beta (inline t))
+ end
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/compute.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,653 @@
+(* Title: HOL/Matrix/Compute_Oracle/compute.ML
+ Author: Steven Obua
+*)
+
+signature COMPUTE = sig
+
+ type computer
+ type theorem
+ type naming = int -> string
+
+ datatype machine = BARRAS | BARRAS_COMPILED | HASKELL | SML
+
+ (* Functions designated with a ! in front of them actually update the computer parameter *)
+
+ exception Make of string
+ val make : machine -> theory -> thm list -> computer
+ val make_with_cache : machine -> theory -> term list -> thm list -> computer
+ val theory_of : computer -> theory
+ val hyps_of : computer -> term list
+ val shyps_of : computer -> sort list
+ (* ! *) val update : computer -> thm list -> unit
+ (* ! *) val update_with_cache : computer -> term list -> thm list -> unit
+
+ (* ! *) val set_naming : computer -> naming -> unit
+ val naming_of : computer -> naming
+
+ exception Compute of string
+ val simplify : computer -> theorem -> thm
+ val rewrite : computer -> cterm -> thm
+
+ val make_theorem : computer -> thm -> string list -> theorem
+ (* ! *) val instantiate : computer -> (string * cterm) list -> theorem -> theorem
+ (* ! *) val evaluate_prem : computer -> int -> theorem -> theorem
+ (* ! *) val modus_ponens : computer -> int -> thm -> theorem -> theorem
+
+end
+
+structure Compute :> COMPUTE = struct
+
+open Report;
+
+datatype machine = BARRAS | BARRAS_COMPILED | HASKELL | SML
+
+(* Terms are mapped to integer codes *)
+structure Encode :>
+sig
+ type encoding
+ val empty : encoding
+ val insert : term -> encoding -> int * encoding
+ val lookup_code : term -> encoding -> int option
+ val lookup_term : int -> encoding -> term option
+ val remove_code : int -> encoding -> encoding
+ val remove_term : term -> encoding -> encoding
+end
+=
+struct
+
+type encoding = int * (int Termtab.table) * (term Inttab.table)
+
+val empty = (0, Termtab.empty, Inttab.empty)
+
+fun insert t (e as (count, term2int, int2term)) =
+ (case Termtab.lookup term2int t of
+ NONE => (count, (count+1, Termtab.update_new (t, count) term2int, Inttab.update_new (count, t) int2term))
+ | SOME code => (code, e))
+
+fun lookup_code t (_, term2int, _) = Termtab.lookup term2int t
+
+fun lookup_term c (_, _, int2term) = Inttab.lookup int2term c
+
+fun remove_code c (e as (count, term2int, int2term)) =
+ (case lookup_term c e of NONE => e | SOME t => (count, Termtab.delete t term2int, Inttab.delete c int2term))
+
+fun remove_term t (e as (count, term2int, int2term)) =
+ (case lookup_code t e of NONE => e | SOME c => (count, Termtab.delete t term2int, Inttab.delete c int2term))
+
+end
+
+exception Make of string;
+exception Compute of string;
+
+local
+ fun make_constant t encoding =
+ let
+ val (code, encoding) = Encode.insert t encoding
+ in
+ (encoding, AbstractMachine.Const code)
+ end
+in
+
+fun remove_types encoding t =
+ case t of
+ Var _ => make_constant t encoding
+ | Free _ => make_constant t encoding
+ | Const _ => make_constant t encoding
+ | Abs (_, _, t') =>
+ let val (encoding, t'') = remove_types encoding t' in
+ (encoding, AbstractMachine.Abs t'')
+ end
+ | a $ b =>
+ let
+ val (encoding, a) = remove_types encoding a
+ val (encoding, b) = remove_types encoding b
+ in
+ (encoding, AbstractMachine.App (a,b))
+ end
+ | Bound b => (encoding, AbstractMachine.Var b)
+end
+
+local
+ fun type_of (Free (_, ty)) = ty
+ | type_of (Const (_, ty)) = ty
+ | type_of (Var (_, ty)) = ty
+ | type_of _ = raise Fail "infer_types: type_of error"
+in
+fun infer_types naming encoding =
+ let
+ fun infer_types _ bounds _ (AbstractMachine.Var v) = (Bound v, nth bounds v)
+ | infer_types _ bounds _ (AbstractMachine.Const code) =
+ let
+ val c = the (Encode.lookup_term code encoding)
+ in
+ (c, type_of c)
+ end
+ | infer_types level bounds _ (AbstractMachine.App (a, b)) =
+ let
+ val (a, aty) = infer_types level bounds NONE a
+ val (adom, arange) =
+ case aty of
+ Type ("fun", [dom, range]) => (dom, range)
+ | _ => raise Fail "infer_types: function type expected"
+ val (b, _) = infer_types level bounds (SOME adom) b
+ in
+ (a $ b, arange)
+ end
+ | infer_types level bounds (SOME (ty as Type ("fun", [dom, range]))) (AbstractMachine.Abs m) =
+ let
+ val (m, _) = infer_types (level+1) (dom::bounds) (SOME range) m
+ in
+ (Abs (naming level, dom, m), ty)
+ end
+ | infer_types _ _ NONE (AbstractMachine.Abs _) =
+ raise Fail "infer_types: cannot infer type of abstraction"
+
+ fun infer ty term =
+ let
+ val (term', _) = infer_types 0 [] (SOME ty) term
+ in
+ term'
+ end
+ in
+ infer
+ end
+end
+
+datatype prog =
+ ProgBarras of AM_Interpreter.program
+ | ProgBarrasC of AM_Compiler.program
+ | ProgHaskell of AM_GHC.program
+ | ProgSML of AM_SML.program
+
+fun machine_of_prog (ProgBarras _) = BARRAS
+ | machine_of_prog (ProgBarrasC _) = BARRAS_COMPILED
+ | machine_of_prog (ProgHaskell _) = HASKELL
+ | machine_of_prog (ProgSML _) = SML
+
+type naming = int -> string
+
+fun default_naming i = "v_" ^ string_of_int i
+
+datatype computer = Computer of
+ (theory_ref * Encode.encoding * term list * unit Sorttab.table * prog * unit Unsynchronized.ref * naming)
+ option Unsynchronized.ref
+
+fun theory_of (Computer (Unsynchronized.ref (SOME (rthy,_,_,_,_,_,_)))) = Theory.deref rthy
+fun hyps_of (Computer (Unsynchronized.ref (SOME (_,_,hyps,_,_,_,_)))) = hyps
+fun shyps_of (Computer (Unsynchronized.ref (SOME (_,_,_,shyptable,_,_,_)))) = Sorttab.keys (shyptable)
+fun shyptab_of (Computer (Unsynchronized.ref (SOME (_,_,_,shyptable,_,_,_)))) = shyptable
+fun stamp_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,_,stamp,_)))) = stamp
+fun prog_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,prog,_,_)))) = prog
+fun encoding_of (Computer (Unsynchronized.ref (SOME (_,encoding,_,_,_,_,_)))) = encoding
+fun set_encoding (Computer (r as Unsynchronized.ref (SOME (p1,_,p2,p3,p4,p5,p6)))) encoding' =
+ (r := SOME (p1,encoding',p2,p3,p4,p5,p6))
+fun naming_of (Computer (Unsynchronized.ref (SOME (_,_,_,_,_,_,n)))) = n
+fun set_naming (Computer (r as Unsynchronized.ref (SOME (p1,p2,p3,p4,p5,p6,_)))) naming'=
+ (r := SOME (p1,p2,p3,p4,p5,p6,naming'))
+
+fun ref_of (Computer r) = r
+
+datatype cthm = ComputeThm of term list * sort list * term
+
+fun thm2cthm th =
+ let
+ val {hyps, prop, tpairs, shyps, ...} = Thm.rep_thm th
+ val _ = if not (null tpairs) then raise Make "theorems may not contain tpairs" else ()
+ in
+ ComputeThm (hyps, shyps, prop)
+ end
+
+fun make_internal machine thy stamp encoding cache_pattern_terms raw_ths =
+ let
+ fun transfer (x:thm) = Thm.transfer thy x
+ val ths = map (thm2cthm o Thm.strip_shyps o transfer) raw_ths
+
+ fun make_pattern encoding n vars (AbstractMachine.Abs _) =
+ raise (Make "no lambda abstractions allowed in pattern")
+ | make_pattern encoding n vars (AbstractMachine.Var _) =
+ raise (Make "no bound variables allowed in pattern")
+ | make_pattern encoding n vars (AbstractMachine.Const code) =
+ (case the (Encode.lookup_term code encoding) of
+ Var _ => ((n+1, Inttab.update_new (code, n) vars, AbstractMachine.PVar)
+ handle Inttab.DUP _ => raise (Make "no duplicate variable in pattern allowed"))
+ | _ => (n, vars, AbstractMachine.PConst (code, [])))
+ | make_pattern encoding n vars (AbstractMachine.App (a, b)) =
+ let
+ val (n, vars, pa) = make_pattern encoding n vars a
+ val (n, vars, pb) = make_pattern encoding n vars b
+ in
+ case pa of
+ AbstractMachine.PVar =>
+ raise (Make "patterns may not start with a variable")
+ | AbstractMachine.PConst (c, args) =>
+ (n, vars, AbstractMachine.PConst (c, args@[pb]))
+ end
+
+ fun thm2rule (encoding, hyptable, shyptable) th =
+ let
+ val (ComputeThm (hyps, shyps, prop)) = th
+ val hyptable = fold (fn h => Termtab.update (h, ())) hyps hyptable
+ val shyptable = fold (fn sh => Sorttab.update (sh, ())) shyps shyptable
+ val (prems, prop) = (Logic.strip_imp_prems prop, Logic.strip_imp_concl prop)
+ val (a, b) = Logic.dest_equals prop
+ handle TERM _ => raise (Make "theorems must be meta-level equations (with optional guards)")
+ val a = Envir.eta_contract a
+ val b = Envir.eta_contract b
+ val prems = map Envir.eta_contract prems
+
+ val (encoding, left) = remove_types encoding a
+ val (encoding, right) = remove_types encoding b
+ fun remove_types_of_guard encoding g =
+ (let
+ val (t1, t2) = Logic.dest_equals g
+ val (encoding, t1) = remove_types encoding t1
+ val (encoding, t2) = remove_types encoding t2
+ in
+ (encoding, AbstractMachine.Guard (t1, t2))
+ end handle TERM _ => raise (Make "guards must be meta-level equations"))
+ val (encoding, prems) = fold_rev (fn p => fn (encoding, ps) => let val (e, p) = remove_types_of_guard encoding p in (e, p::ps) end) prems (encoding, [])
+
+ (* Principally, a check should be made here to see if the (meta-) hyps contain any of the variables of the rule.
+ As it is, all variables of the rule are schematic, and there are no schematic variables in meta-hyps, therefore
+ this check can be left out. *)
+
+ val (vcount, vars, pattern) = make_pattern encoding 0 Inttab.empty left
+ val _ = (case pattern of
+ AbstractMachine.PVar =>
+ raise (Make "patterns may not start with a variable")
+ | _ => ())
+
+ (* finally, provide a function for renaming the
+ pattern bound variables on the right hand side *)
+
+ fun rename level vars (var as AbstractMachine.Var _) = var
+ | rename level vars (c as AbstractMachine.Const code) =
+ (case Inttab.lookup vars code of
+ NONE => c
+ | SOME n => AbstractMachine.Var (vcount-n-1+level))
+ | rename level vars (AbstractMachine.App (a, b)) =
+ AbstractMachine.App (rename level vars a, rename level vars b)
+ | rename level vars (AbstractMachine.Abs m) =
+ AbstractMachine.Abs (rename (level+1) vars m)
+
+ fun rename_guard (AbstractMachine.Guard (a,b)) =
+ AbstractMachine.Guard (rename 0 vars a, rename 0 vars b)
+ in
+ ((encoding, hyptable, shyptable), (map rename_guard prems, pattern, rename 0 vars right))
+ end
+
+ val ((encoding, hyptable, shyptable), rules) =
+ fold_rev (fn th => fn (encoding_hyptable, rules) =>
+ let
+ val (encoding_hyptable, rule) = thm2rule encoding_hyptable th
+ in (encoding_hyptable, rule::rules) end)
+ ths ((encoding, Termtab.empty, Sorttab.empty), [])
+
+ fun make_cache_pattern t (encoding, cache_patterns) =
+ let
+ val (encoding, a) = remove_types encoding t
+ val (_,_,p) = make_pattern encoding 0 Inttab.empty a
+ in
+ (encoding, p::cache_patterns)
+ end
+
+ val (encoding, _) = fold_rev make_cache_pattern cache_pattern_terms (encoding, [])
+
+ val prog =
+ case machine of
+ BARRAS => ProgBarras (AM_Interpreter.compile rules)
+ | BARRAS_COMPILED => ProgBarrasC (AM_Compiler.compile rules)
+ | HASKELL => ProgHaskell (AM_GHC.compile rules)
+ | SML => ProgSML (AM_SML.compile rules)
+
+ fun has_witness s = not (null (Sign.witness_sorts thy [] [s]))
+
+ val shyptable = fold Sorttab.delete (filter has_witness (Sorttab.keys (shyptable))) shyptable
+
+ in (Theory.check_thy thy, encoding, Termtab.keys hyptable, shyptable, prog, stamp, default_naming) end
+
+fun make_with_cache machine thy cache_patterns raw_thms =
+ Computer (Unsynchronized.ref (SOME (make_internal machine thy (Unsynchronized.ref ()) Encode.empty cache_patterns raw_thms)))
+
+fun make machine thy raw_thms = make_with_cache machine thy [] raw_thms
+
+fun update_with_cache computer cache_patterns raw_thms =
+ let
+ val c = make_internal (machine_of_prog (prog_of computer)) (theory_of computer) (stamp_of computer)
+ (encoding_of computer) cache_patterns raw_thms
+ val _ = (ref_of computer) := SOME c
+ in
+ ()
+ end
+
+fun update computer raw_thms = update_with_cache computer [] raw_thms
+
+fun runprog (ProgBarras p) = AM_Interpreter.run p
+ | runprog (ProgBarrasC p) = AM_Compiler.run p
+ | runprog (ProgHaskell p) = AM_GHC.run p
+ | runprog (ProgSML p) = AM_SML.run p
+
+(* ------------------------------------------------------------------------------------- *)
+(* An oracle for exporting theorems; must only be accessible from inside this structure! *)
+(* ------------------------------------------------------------------------------------- *)
+
+fun merge_hyps hyps1 hyps2 =
+let
+ fun add hyps tab = fold (fn h => fn tab => Termtab.update (h, ()) tab) hyps tab
+in
+ Termtab.keys (add hyps2 (add hyps1 Termtab.empty))
+end
+
+fun add_shyps shyps tab = fold (fn h => fn tab => Sorttab.update (h, ()) tab) shyps tab
+
+fun merge_shyps shyps1 shyps2 = Sorttab.keys (add_shyps shyps2 (add_shyps shyps1 Sorttab.empty))
+
+val (_, export_oracle) = Context.>>> (Context.map_theory_result
+ (Thm.add_oracle (@{binding compute}, fn (thy, hyps, shyps, prop) =>
+ let
+ val shyptab = add_shyps shyps Sorttab.empty
+ fun delete s shyptab = Sorttab.delete s shyptab handle Sorttab.UNDEF _ => shyptab
+ fun delete_term t shyptab = fold delete (Sorts.insert_term t []) shyptab
+ fun has_witness s = not (null (Sign.witness_sorts thy [] [s]))
+ val shyptab = fold Sorttab.delete (filter has_witness (Sorttab.keys (shyptab))) shyptab
+ val shyps = if Sorttab.is_empty shyptab then [] else Sorttab.keys (fold delete_term (prop::hyps) shyptab)
+ val _ =
+ if not (null shyps) then
+ raise Compute ("dangling sort hypotheses: " ^
+ commas (map (Syntax.string_of_sort_global thy) shyps))
+ else ()
+ in
+ Thm.cterm_of thy (fold_rev (fn hyp => fn p => Logic.mk_implies (hyp, p)) hyps prop)
+ end)));
+
+fun export_thm thy hyps shyps prop =
+ let
+ val th = export_oracle (thy, hyps, shyps, prop)
+ val hyps = map (fn h => Thm.assume (cterm_of thy h)) hyps
+ in
+ fold (fn h => fn p => Thm.implies_elim p h) hyps th
+ end
+
+(* --------- Rewrite ----------- *)
+
+fun rewrite computer ct =
+ let
+ val thy = Thm.theory_of_cterm ct
+ val {t=t',T=ty,...} = rep_cterm ct
+ val _ = Theory.assert_super (theory_of computer) thy
+ val naming = naming_of computer
+ val (encoding, t) = remove_types (encoding_of computer) t'
+ val t = runprog (prog_of computer) t
+ val t = infer_types naming encoding ty t
+ val eq = Logic.mk_equals (t', t)
+ in
+ export_thm thy (hyps_of computer) (Sorttab.keys (shyptab_of computer)) eq
+ end
+
+(* --------- Simplify ------------ *)
+
+datatype prem = EqPrem of AbstractMachine.term * AbstractMachine.term * Term.typ * int
+ | Prem of AbstractMachine.term
+datatype theorem = Theorem of theory_ref * unit Unsynchronized.ref * (int * typ) Symtab.table * (AbstractMachine.term option) Inttab.table
+ * prem list * AbstractMachine.term * term list * sort list
+
+
+exception ParamSimplify of computer * theorem
+
+fun make_theorem computer th vars =
+let
+ val _ = Theory.assert_super (theory_of computer) (theory_of_thm th)
+
+ val (ComputeThm (hyps, shyps, prop)) = thm2cthm th
+
+ val encoding = encoding_of computer
+
+ (* variables in the theorem are identified upfront *)
+ fun collect_vars (Abs (_, _, t)) tab = collect_vars t tab
+ | collect_vars (a $ b) tab = collect_vars b (collect_vars a tab)
+ | collect_vars (Const _) tab = tab
+ | collect_vars (Free _) tab = tab
+ | collect_vars (Var ((s, i), ty)) tab =
+ if List.find (fn x => x=s) vars = NONE then
+ tab
+ else
+ (case Symtab.lookup tab s of
+ SOME ((s',i'),ty') =>
+ if s' <> s orelse i' <> i orelse ty <> ty' then
+ raise Compute ("make_theorem: variable name '"^s^"' is not unique")
+ else
+ tab
+ | NONE => Symtab.update (s, ((s, i), ty)) tab)
+ val vartab = collect_vars prop Symtab.empty
+ fun encodevar (s, t as (_, ty)) (encoding, tab) =
+ let
+ val (x, encoding) = Encode.insert (Var t) encoding
+ in
+ (encoding, Symtab.update (s, (x, ty)) tab)
+ end
+ val (encoding, vartab) = Symtab.fold encodevar vartab (encoding, Symtab.empty)
+ val varsubst = Inttab.make (map (fn (_, (x, _)) => (x, NONE)) (Symtab.dest vartab))
+
+ (* make the premises and the conclusion *)
+ fun mk_prem encoding t =
+ (let
+ val (a, b) = Logic.dest_equals t
+ val ty = type_of a
+ val (encoding, a) = remove_types encoding a
+ val (encoding, b) = remove_types encoding b
+ val (eq, encoding) = Encode.insert (Const ("==", ty --> ty --> @{typ "prop"})) encoding
+ in
+ (encoding, EqPrem (a, b, ty, eq))
+ end handle TERM _ => let val (encoding, t) = remove_types encoding t in (encoding, Prem t) end)
+ val (encoding, prems) =
+ (fold_rev (fn t => fn (encoding, l) =>
+ case mk_prem encoding t of
+ (encoding, t) => (encoding, t::l)) (Logic.strip_imp_prems prop) (encoding, []))
+ val (encoding, concl) = remove_types encoding (Logic.strip_imp_concl prop)
+ val _ = set_encoding computer encoding
+in
+ Theorem (Theory.check_thy (theory_of_thm th), stamp_of computer, vartab, varsubst,
+ prems, concl, hyps, shyps)
+end
+
+fun theory_of_theorem (Theorem (rthy,_,_,_,_,_,_,_)) = Theory.deref rthy
+fun update_theory thy (Theorem (_,p0,p1,p2,p3,p4,p5,p6)) =
+ Theorem (Theory.check_thy thy,p0,p1,p2,p3,p4,p5,p6)
+fun stamp_of_theorem (Theorem (_,s, _, _, _, _, _, _)) = s
+fun vartab_of_theorem (Theorem (_,_,vt,_,_,_,_,_)) = vt
+fun varsubst_of_theorem (Theorem (_,_,_,vs,_,_,_,_)) = vs
+fun update_varsubst vs (Theorem (p0,p1,p2,_,p3,p4,p5,p6)) = Theorem (p0,p1,p2,vs,p3,p4,p5,p6)
+fun prems_of_theorem (Theorem (_,_,_,_,prems,_,_,_)) = prems
+fun update_prems prems (Theorem (p0,p1,p2,p3,_,p4,p5,p6)) = Theorem (p0,p1,p2,p3,prems,p4,p5,p6)
+fun concl_of_theorem (Theorem (_,_,_,_,_,concl,_,_)) = concl
+fun hyps_of_theorem (Theorem (_,_,_,_,_,_,hyps,_)) = hyps
+fun update_hyps hyps (Theorem (p0,p1,p2,p3,p4,p5,_,p6)) = Theorem (p0,p1,p2,p3,p4,p5,hyps,p6)
+fun shyps_of_theorem (Theorem (_,_,_,_,_,_,_,shyps)) = shyps
+fun update_shyps shyps (Theorem (p0,p1,p2,p3,p4,p5,p6,_)) = Theorem (p0,p1,p2,p3,p4,p5,p6,shyps)
+
+fun check_compatible computer th s =
+ if stamp_of computer <> stamp_of_theorem th then
+ raise Compute (s^": computer and theorem are incompatible")
+ else ()
+
+fun instantiate computer insts th =
+let
+ val _ = check_compatible computer th
+
+ val thy = theory_of computer
+
+ val vartab = vartab_of_theorem th
+
+ fun rewrite computer t =
+ let
+ val (encoding, t) = remove_types (encoding_of computer) t
+ val t = runprog (prog_of computer) t
+ val _ = set_encoding computer encoding
+ in
+ t
+ end
+
+ fun assert_varfree vs t =
+ if AbstractMachine.forall_consts (fn x => Inttab.lookup vs x = NONE) t then
+ ()
+ else
+ raise Compute "instantiate: assert_varfree failed"
+
+ fun assert_closed t =
+ if AbstractMachine.closed t then
+ ()
+ else
+ raise Compute "instantiate: not a closed term"
+
+ fun compute_inst (s, ct) vs =
+ let
+ val _ = Theory.assert_super (theory_of_cterm ct) thy
+ val ty = typ_of (ctyp_of_term ct)
+ in
+ (case Symtab.lookup vartab s of
+ NONE => raise Compute ("instantiate: variable '"^s^"' not found in theorem")
+ | SOME (x, ty') =>
+ (case Inttab.lookup vs x of
+ SOME (SOME _) => raise Compute ("instantiate: variable '"^s^"' has already been instantiated")
+ | SOME NONE =>
+ if ty <> ty' then
+ raise Compute ("instantiate: wrong type for variable '"^s^"'")
+ else
+ let
+ val t = rewrite computer (term_of ct)
+ val _ = assert_varfree vs t
+ val _ = assert_closed t
+ in
+ Inttab.update (x, SOME t) vs
+ end
+ | NONE => raise Compute "instantiate: internal error"))
+ end
+
+ val vs = fold compute_inst insts (varsubst_of_theorem th)
+in
+ update_varsubst vs th
+end
+
+fun match_aterms subst =
+ let
+ exception no_match
+ open AbstractMachine
+ fun match subst (b as (Const c)) a =
+ if a = b then subst
+ else
+ (case Inttab.lookup subst c of
+ SOME (SOME a') => if a=a' then subst else raise no_match
+ | SOME NONE => if AbstractMachine.closed a then
+ Inttab.update (c, SOME a) subst
+ else raise no_match
+ | NONE => raise no_match)
+ | match subst (b as (Var _)) a = if a=b then subst else raise no_match
+ | match subst (App (u, v)) (App (u', v')) = match (match subst u u') v v'
+ | match subst (Abs u) (Abs u') = match subst u u'
+ | match subst _ _ = raise no_match
+ in
+ fn b => fn a => (SOME (match subst b a) handle no_match => NONE)
+ end
+
+fun apply_subst vars_allowed subst =
+ let
+ open AbstractMachine
+ fun app (t as (Const c)) =
+ (case Inttab.lookup subst c of
+ NONE => t
+ | SOME (SOME t) => Computed t
+ | SOME NONE => if vars_allowed then t else raise Compute "apply_subst: no vars allowed")
+ | app (t as (Var _)) = t
+ | app (App (u, v)) = App (app u, app v)
+ | app (Abs m) = Abs (app m)
+ in
+ app
+ end
+
+fun splicein n l L = List.take (L, n) @ l @ List.drop (L, n+1)
+
+fun evaluate_prem computer prem_no th =
+let
+ val _ = check_compatible computer th
+ val prems = prems_of_theorem th
+ val varsubst = varsubst_of_theorem th
+ fun run vars_allowed t =
+ runprog (prog_of computer) (apply_subst vars_allowed varsubst t)
+in
+ case nth prems prem_no of
+ Prem _ => raise Compute "evaluate_prem: no equality premise"
+ | EqPrem (a, b, ty, _) =>
+ let
+ val a' = run false a
+ val b' = run true b
+ in
+ case match_aterms varsubst b' a' of
+ NONE =>
+ let
+ fun mk s = Syntax.string_of_term_global Pure.thy
+ (infer_types (naming_of computer) (encoding_of computer) ty s)
+ val left = "computed left side: "^(mk a')
+ val right = "computed right side: "^(mk b')
+ in
+ raise Compute ("evaluate_prem: cannot assign computed left to right hand side\n"^left^"\n"^right^"\n")
+ end
+ | SOME varsubst =>
+ update_prems (splicein prem_no [] prems) (update_varsubst varsubst th)
+ end
+end
+
+fun prem2term (Prem t) = t
+ | prem2term (EqPrem (a,b,_,eq)) =
+ AbstractMachine.App (AbstractMachine.App (AbstractMachine.Const eq, a), b)
+
+fun modus_ponens computer prem_no th' th =
+let
+ val _ = check_compatible computer th
+ val thy =
+ let
+ val thy1 = theory_of_theorem th
+ val thy2 = theory_of_thm th'
+ in
+ if Theory.subthy (thy1, thy2) then thy2
+ else if Theory.subthy (thy2, thy1) then thy1 else
+ raise Compute "modus_ponens: theorems are not compatible with each other"
+ end
+ val th' = make_theorem computer th' []
+ val varsubst = varsubst_of_theorem th
+ fun run vars_allowed t =
+ runprog (prog_of computer) (apply_subst vars_allowed varsubst t)
+ val prems = prems_of_theorem th
+ val prem = run true (prem2term (nth prems prem_no))
+ val concl = run false (concl_of_theorem th')
+in
+ case match_aterms varsubst prem concl of
+ NONE => raise Compute "modus_ponens: conclusion does not match premise"
+ | SOME varsubst =>
+ let
+ val th = update_varsubst varsubst th
+ val th = update_prems (splicein prem_no (prems_of_theorem th') prems) th
+ val th = update_hyps (merge_hyps (hyps_of_theorem th) (hyps_of_theorem th')) th
+ val th = update_shyps (merge_shyps (shyps_of_theorem th) (shyps_of_theorem th')) th
+ in
+ update_theory thy th
+ end
+end
+
+fun simplify computer th =
+let
+ val _ = check_compatible computer th
+ val varsubst = varsubst_of_theorem th
+ val encoding = encoding_of computer
+ val naming = naming_of computer
+ fun infer t = infer_types naming encoding @{typ "prop"} t
+ fun run t = infer (runprog (prog_of computer) (apply_subst true varsubst t))
+ fun runprem p = run (prem2term p)
+ val prop = Logic.list_implies (map runprem (prems_of_theorem th), run (concl_of_theorem th))
+ val hyps = merge_hyps (hyps_of computer) (hyps_of_theorem th)
+ val shyps = merge_shyps (shyps_of_theorem th) (Sorttab.keys (shyptab_of computer))
+in
+ export_thm (theory_of_theorem th) hyps shyps prop
+end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/linker.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,470 @@
+(* Title: HOL/Matrix/Compute_Oracle/linker.ML
+ Author: Steven Obua
+
+This module solves the problem that the computing oracle does not
+instantiate polymorphic rules. By going through the PCompute
+interface, all possible instantiations are resolved by compiling new
+programs, if necessary. The obvious disadvantage of this approach is
+that in the worst case for each new term to be rewritten, a new
+program may be compiled.
+*)
+
+(*
+ Given constants/frees c_1::t_1, c_2::t_2, ...., c_n::t_n,
+ and constants/frees d_1::d_1, d_2::s_2, ..., d_m::s_m
+
+ Find all substitutions S such that
+ a) the domain of S is tvars (t_1, ..., t_n)
+ b) there are indices i_1, ..., i_k, and j_1, ..., j_k with
+ 1. S (c_i_1::t_i_1) = d_j_1::s_j_1, ..., S (c_i_k::t_i_k) = d_j_k::s_j_k
+ 2. tvars (t_i_1, ..., t_i_k) = tvars (t_1, ..., t_n)
+*)
+signature LINKER =
+sig
+ exception Link of string
+
+ datatype constant = Constant of bool * string * typ
+ val constant_of : term -> constant
+
+ type instances
+ type subst = Type.tyenv
+
+ val empty : constant list -> instances
+ val typ_of_constant : constant -> typ
+ val add_instances : theory -> instances -> constant list -> subst list * instances
+ val substs_of : instances -> subst list
+ val is_polymorphic : constant -> bool
+ val distinct_constants : constant list -> constant list
+ val collect_consts : term list -> constant list
+end
+
+structure Linker : LINKER = struct
+
+exception Link of string;
+
+type subst = Type.tyenv
+
+datatype constant = Constant of bool * string * typ
+fun constant_of (Const (name, ty)) = Constant (false, name, ty)
+ | constant_of (Free (name, ty)) = Constant (true, name, ty)
+ | constant_of _ = raise Link "constant_of"
+
+fun bool_ord (x,y) = if x then (if y then EQUAL else GREATER) else (if y then LESS else EQUAL)
+fun constant_ord (Constant (x1,x2,x3), Constant (y1,y2,y3)) = (prod_ord (prod_ord bool_ord fast_string_ord) Term_Ord.typ_ord) (((x1,x2),x3), ((y1,y2),y3))
+fun constant_modty_ord (Constant (x1,x2,_), Constant (y1,y2,_)) = (prod_ord bool_ord fast_string_ord) ((x1,x2), (y1,y2))
+
+
+structure Consttab = Table(type key = constant val ord = constant_ord);
+structure ConsttabModTy = Table(type key = constant val ord = constant_modty_ord);
+
+fun typ_of_constant (Constant (_, _, ty)) = ty
+
+val empty_subst = (Vartab.empty : Type.tyenv)
+
+fun merge_subst (A:Type.tyenv) (B:Type.tyenv) =
+ SOME (Vartab.fold (fn (v, t) =>
+ fn tab =>
+ (case Vartab.lookup tab v of
+ NONE => Vartab.update (v, t) tab
+ | SOME t' => if t = t' then tab else raise Type.TYPE_MATCH)) A B)
+ handle Type.TYPE_MATCH => NONE
+
+fun subst_ord (A:Type.tyenv, B:Type.tyenv) =
+ (list_ord (prod_ord Term_Ord.fast_indexname_ord (prod_ord Term_Ord.sort_ord Term_Ord.typ_ord))) (Vartab.dest A, Vartab.dest B)
+
+structure Substtab = Table(type key = Type.tyenv val ord = subst_ord);
+
+fun substtab_union c = Substtab.fold Substtab.update c
+fun substtab_unions [] = Substtab.empty
+ | substtab_unions [c] = c
+ | substtab_unions (c::cs) = substtab_union c (substtab_unions cs)
+
+datatype instances = Instances of unit ConsttabModTy.table * Type.tyenv Consttab.table Consttab.table * constant list list * unit Substtab.table
+
+fun is_polymorphic (Constant (_, _, ty)) = not (null (Term.add_tvarsT ty []))
+
+fun distinct_constants cs =
+ Consttab.keys (fold (fn c => Consttab.update (c, ())) cs Consttab.empty)
+
+fun empty cs =
+ let
+ val cs = distinct_constants (filter is_polymorphic cs)
+ val old_cs = cs
+(* fun collect_tvars ty tab = fold (fn v => fn tab => Typtab.update (TVar v, ()) tab) (Misc_Legacy.typ_tvars ty) tab
+ val tvars_count = length (Typtab.keys (fold (fn c => fn tab => collect_tvars (typ_of_constant c) tab) cs Typtab.empty))
+ fun tvars_of ty = collect_tvars ty Typtab.empty
+ val cs = map (fn c => (c, tvars_of (typ_of_constant c))) cs
+
+ fun tyunion A B =
+ Typtab.fold
+ (fn (v,()) => fn tab => Typtab.update (v, case Typtab.lookup tab v of NONE => 1 | SOME n => n+1) tab)
+ A B
+
+ fun is_essential A B =
+ Typtab.fold
+ (fn (v, ()) => fn essential => essential orelse (case Typtab.lookup B v of NONE => raise Link "is_essential" | SOME n => n=1))
+ A false
+
+ fun add_minimal (c', tvs') (tvs, cs) =
+ let
+ val tvs = tyunion tvs' tvs
+ val cs = (c', tvs')::cs
+ in
+ if forall (fn (c',tvs') => is_essential tvs' tvs) cs then
+ SOME (tvs, cs)
+ else
+ NONE
+ end
+
+ fun is_spanning (tvs, _) = (length (Typtab.keys tvs) = tvars_count)
+
+ fun generate_minimal_subsets subsets [] = subsets
+ | generate_minimal_subsets subsets (c::cs) =
+ let
+ val subsets' = map_filter (add_minimal c) subsets
+ in
+ generate_minimal_subsets (subsets@subsets') cs
+ end*)
+
+ val minimal_subsets = [old_cs] (*map (fn (tvs, cs) => map fst cs) (filter is_spanning (generate_minimal_subsets [(Typtab.empty, [])] cs))*)
+
+ val constants = Consttab.keys (fold (fold (fn c => Consttab.update (c, ()))) minimal_subsets Consttab.empty)
+
+ in
+ Instances (
+ fold (fn c => fn tab => ConsttabModTy.update (c, ()) tab) constants ConsttabModTy.empty,
+ Consttab.make (map (fn c => (c, Consttab.empty : Type.tyenv Consttab.table)) constants),
+ minimal_subsets, Substtab.empty)
+ end
+
+local
+fun calc ctab substtab [] = substtab
+ | calc ctab substtab (c::cs) =
+ let
+ val csubsts = map snd (Consttab.dest (the (Consttab.lookup ctab c)))
+ fun merge_substs substtab subst =
+ Substtab.fold (fn (s,_) =>
+ fn tab =>
+ (case merge_subst subst s of NONE => tab | SOME s => Substtab.update (s, ()) tab))
+ substtab Substtab.empty
+ val substtab = substtab_unions (map (merge_substs substtab) csubsts)
+ in
+ calc ctab substtab cs
+ end
+in
+fun calc_substs ctab (cs:constant list) = calc ctab (Substtab.update (empty_subst, ()) Substtab.empty) cs
+end
+
+fun add_instances thy (Instances (cfilter, ctab,minsets,substs)) cs =
+ let
+(* val _ = writeln (makestring ("add_instances: ", length_cs, length cs, length (Consttab.keys ctab)))*)
+ fun calc_instantiations (constant as Constant (free, name, ty)) instantiations =
+ Consttab.fold (fn (constant' as Constant (free', name', ty'), insttab) =>
+ fn instantiations =>
+ if free <> free' orelse name <> name' then
+ instantiations
+ else case Consttab.lookup insttab constant of
+ SOME _ => instantiations
+ | NONE => ((constant', (constant, Sign.typ_match thy (ty', ty) empty_subst))::instantiations
+ handle Type.TYPE_MATCH => instantiations))
+ ctab instantiations
+ val instantiations = fold calc_instantiations cs []
+ (*val _ = writeln ("instantiations = "^(makestring (length instantiations)))*)
+ fun update_ctab (constant', entry) ctab =
+ (case Consttab.lookup ctab constant' of
+ NONE => raise Link "internal error: update_ctab"
+ | SOME tab => Consttab.update (constant', Consttab.update entry tab) ctab)
+ val ctab = fold update_ctab instantiations ctab
+ val new_substs = fold (fn minset => fn substs => substtab_union (calc_substs ctab minset) substs)
+ minsets Substtab.empty
+ val (added_substs, substs) =
+ Substtab.fold (fn (ns, _) =>
+ fn (added, substtab) =>
+ (case Substtab.lookup substs ns of
+ NONE => (ns::added, Substtab.update (ns, ()) substtab)
+ | SOME () => (added, substtab)))
+ new_substs ([], substs)
+ in
+ (added_substs, Instances (cfilter, ctab, minsets, substs))
+ end
+
+fun substs_of (Instances (_,_,_,substs)) = Substtab.keys substs
+
+
+local
+
+fun collect (Var _) tab = tab
+ | collect (Bound _) tab = tab
+ | collect (a $ b) tab = collect b (collect a tab)
+ | collect (Abs (_, _, body)) tab = collect body tab
+ | collect t tab = Consttab.update (constant_of t, ()) tab
+
+in
+ fun collect_consts tms = Consttab.keys (fold collect tms Consttab.empty)
+end
+
+end
+
+signature PCOMPUTE =
+sig
+ type pcomputer
+
+ val make : Compute.machine -> theory -> thm list -> Linker.constant list -> pcomputer
+ val make_with_cache : Compute.machine -> theory -> term list -> thm list -> Linker.constant list -> pcomputer
+
+ val add_instances : pcomputer -> Linker.constant list -> bool
+ val add_instances' : pcomputer -> term list -> bool
+
+ val rewrite : pcomputer -> cterm list -> thm list
+ val simplify : pcomputer -> Compute.theorem -> thm
+
+ val make_theorem : pcomputer -> thm -> string list -> Compute.theorem
+ val instantiate : pcomputer -> (string * cterm) list -> Compute.theorem -> Compute.theorem
+ val evaluate_prem : pcomputer -> int -> Compute.theorem -> Compute.theorem
+ val modus_ponens : pcomputer -> int -> thm -> Compute.theorem -> Compute.theorem
+
+end
+
+structure PCompute : PCOMPUTE = struct
+
+exception PCompute of string
+
+datatype theorem = MonoThm of thm | PolyThm of thm * Linker.instances * thm list
+datatype pattern = MonoPattern of term | PolyPattern of term * Linker.instances * term list
+
+datatype pcomputer =
+ PComputer of theory_ref * Compute.computer * theorem list Unsynchronized.ref *
+ pattern list Unsynchronized.ref
+
+(*fun collect_consts (Var x) = []
+ | collect_consts (Bound _) = []
+ | collect_consts (a $ b) = (collect_consts a)@(collect_consts b)
+ | collect_consts (Abs (_, _, body)) = collect_consts body
+ | collect_consts t = [Linker.constant_of t]*)
+
+fun computer_of (PComputer (_,computer,_,_)) = computer
+
+fun collect_consts_of_thm th =
+ let
+ val th = prop_of th
+ val (prems, th) = (Logic.strip_imp_prems th, Logic.strip_imp_concl th)
+ val (left, right) = Logic.dest_equals th
+ in
+ (Linker.collect_consts [left], Linker.collect_consts (right::prems))
+ end
+
+fun create_theorem th =
+let
+ val (left, right) = collect_consts_of_thm th
+ val polycs = filter Linker.is_polymorphic left
+ val tytab = fold (fn p => fn tab => fold (fn n => fn tab => Typtab.update (TVar n, ()) tab) (Misc_Legacy.typ_tvars (Linker.typ_of_constant p)) tab) polycs Typtab.empty
+ fun check_const (c::cs) cs' =
+ let
+ val tvars = Misc_Legacy.typ_tvars (Linker.typ_of_constant c)
+ val wrong = fold (fn n => fn wrong => wrong orelse is_none (Typtab.lookup tytab (TVar n))) tvars false
+ in
+ if wrong then raise PCompute "right hand side of theorem contains type variables which do not occur on the left hand side"
+ else
+ if null (tvars) then
+ check_const cs (c::cs')
+ else
+ check_const cs cs'
+ end
+ | check_const [] cs' = cs'
+ val monocs = check_const right []
+in
+ if null (polycs) then
+ (monocs, MonoThm th)
+ else
+ (monocs, PolyThm (th, Linker.empty polycs, []))
+end
+
+fun create_pattern pat =
+let
+ val cs = Linker.collect_consts [pat]
+ val polycs = filter Linker.is_polymorphic cs
+in
+ if null (polycs) then
+ MonoPattern pat
+ else
+ PolyPattern (pat, Linker.empty polycs, [])
+end
+
+fun create_computer machine thy pats ths =
+ let
+ fun add (MonoThm th) ths = th::ths
+ | add (PolyThm (_, _, ths')) ths = ths'@ths
+ fun addpat (MonoPattern p) pats = p::pats
+ | addpat (PolyPattern (_, _, ps)) pats = ps@pats
+ val ths = fold_rev add ths []
+ val pats = fold_rev addpat pats []
+ in
+ Compute.make_with_cache machine thy pats ths
+ end
+
+fun update_computer computer pats ths =
+ let
+ fun add (MonoThm th) ths = th::ths
+ | add (PolyThm (_, _, ths')) ths = ths'@ths
+ fun addpat (MonoPattern p) pats = p::pats
+ | addpat (PolyPattern (_, _, ps)) pats = ps@pats
+ val ths = fold_rev add ths []
+ val pats = fold_rev addpat pats []
+ in
+ Compute.update_with_cache computer pats ths
+ end
+
+fun conv_subst thy (subst : Type.tyenv) =
+ map (fn (iname, (sort, ty)) => (ctyp_of thy (TVar (iname, sort)), ctyp_of thy ty)) (Vartab.dest subst)
+
+fun add_monos thy monocs pats ths =
+ let
+ val changed = Unsynchronized.ref false
+ fun add monocs (th as (MonoThm _)) = ([], th)
+ | add monocs (PolyThm (th, instances, instanceths)) =
+ let
+ val (newsubsts, instances) = Linker.add_instances thy instances monocs
+ val _ = if not (null newsubsts) then changed := true else ()
+ val newths = map (fn subst => Thm.instantiate (conv_subst thy subst, []) th) newsubsts
+(* val _ = if not (null newths) then (print ("added new theorems: ", newths); ()) else ()*)
+ val newmonos = fold (fn th => fn monos => (snd (collect_consts_of_thm th))@monos) newths []
+ in
+ (newmonos, PolyThm (th, instances, instanceths@newths))
+ end
+ fun addpats monocs (pat as (MonoPattern _)) = pat
+ | addpats monocs (PolyPattern (p, instances, instancepats)) =
+ let
+ val (newsubsts, instances) = Linker.add_instances thy instances monocs
+ val _ = if not (null newsubsts) then changed := true else ()
+ val newpats = map (fn subst => Envir.subst_term_types subst p) newsubsts
+ in
+ PolyPattern (p, instances, instancepats@newpats)
+ end
+ fun step monocs ths =
+ fold_rev (fn th =>
+ fn (newmonos, ths) =>
+ let
+ val (newmonos', th') = add monocs th
+ in
+ (newmonos'@newmonos, th'::ths)
+ end)
+ ths ([], [])
+ fun loop monocs pats ths =
+ let
+ val (monocs', ths') = step monocs ths
+ val pats' = map (addpats monocs) pats
+ in
+ if null (monocs') then
+ (pats', ths')
+ else
+ loop monocs' pats' ths'
+ end
+ val result = loop monocs pats ths
+ in
+ (!changed, result)
+ end
+
+datatype cthm = ComputeThm of term list * sort list * term
+
+fun thm2cthm th =
+ let
+ val {hyps, prop, shyps, ...} = Thm.rep_thm th
+ in
+ ComputeThm (hyps, shyps, prop)
+ end
+
+val cthm_ord' = prod_ord (prod_ord (list_ord Term_Ord.term_ord) (list_ord Term_Ord.sort_ord)) Term_Ord.term_ord
+
+fun cthm_ord (ComputeThm (h1, sh1, p1), ComputeThm (h2, sh2, p2)) = cthm_ord' (((h1,sh1), p1), ((h2, sh2), p2))
+
+structure CThmtab = Table(type key = cthm val ord = cthm_ord)
+
+fun remove_duplicates ths =
+ let
+ val counter = Unsynchronized.ref 0
+ val tab = Unsynchronized.ref (CThmtab.empty : unit CThmtab.table)
+ val thstab = Unsynchronized.ref (Inttab.empty : thm Inttab.table)
+ fun update th =
+ let
+ val key = thm2cthm th
+ in
+ case CThmtab.lookup (!tab) key of
+ NONE => ((tab := CThmtab.update_new (key, ()) (!tab)); thstab := Inttab.update_new (!counter, th) (!thstab); counter := !counter + 1)
+ | _ => ()
+ end
+ val _ = map update ths
+ in
+ map snd (Inttab.dest (!thstab))
+ end
+
+fun make_with_cache machine thy pats ths cs =
+ let
+ val ths = remove_duplicates ths
+ val (monocs, ths) = fold_rev (fn th =>
+ fn (monocs, ths) =>
+ let val (m, t) = create_theorem th in
+ (m@monocs, t::ths)
+ end)
+ ths (cs, [])
+ val pats = map create_pattern pats
+ val (_, (pats, ths)) = add_monos thy monocs pats ths
+ val computer = create_computer machine thy pats ths
+ in
+ PComputer (Theory.check_thy thy, computer, Unsynchronized.ref ths, Unsynchronized.ref pats)
+ end
+
+fun make machine thy ths cs = make_with_cache machine thy [] ths cs
+
+fun add_instances (PComputer (thyref, computer, rths, rpats)) cs =
+ let
+ val thy = Theory.deref thyref
+ val (changed, (pats, ths)) = add_monos thy cs (!rpats) (!rths)
+ in
+ if changed then
+ (update_computer computer pats ths;
+ rths := ths;
+ rpats := pats;
+ true)
+ else
+ false
+
+ end
+
+fun add_instances' pc ts = add_instances pc (Linker.collect_consts ts)
+
+fun rewrite pc cts =
+ let
+ val _ = add_instances' pc (map term_of cts)
+ val computer = (computer_of pc)
+ in
+ map (fn ct => Compute.rewrite computer ct) cts
+ end
+
+fun simplify pc th = Compute.simplify (computer_of pc) th
+
+fun make_theorem pc th vars =
+ let
+ val _ = add_instances' pc [prop_of th]
+
+ in
+ Compute.make_theorem (computer_of pc) th vars
+ end
+
+fun instantiate pc insts th =
+ let
+ val _ = add_instances' pc (map (term_of o snd) insts)
+ in
+ Compute.instantiate (computer_of pc) insts th
+ end
+
+fun evaluate_prem pc prem_no th = Compute.evaluate_prem (computer_of pc) prem_no th
+
+fun modus_ponens pc prem_no th' th =
+ let
+ val _ = add_instances' pc [prop_of th']
+ in
+ Compute.modus_ponens (computer_of pc) prem_no th' th
+ end
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Compute_Oracle/report.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,33 @@
+structure Report =
+struct
+
+local
+
+ val report_depth = Unsynchronized.ref 0
+ fun space n = if n <= 0 then "" else (space (n-1))^" "
+ fun report_space () = space (!report_depth)
+
+in
+
+fun timeit f =
+ let
+ val t1 = Timing.start ()
+ val x = f ()
+ val t2 = Timing.message (Timing.result t1)
+ val _ = writeln ((report_space ()) ^ "--> "^t2)
+ in
+ x
+ end
+
+fun report s f =
+let
+ val _ = writeln ((report_space ())^s)
+ val _ = report_depth := !report_depth + 1
+ val x = timeit f
+ val _ = report_depth := !report_depth - 1
+in
+ x
+end
+
+end
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Cplex.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,67 @@
+(* Title: HOL/Matrix/Cplex.thy
+ Author: Steven Obua
+*)
+
+theory Cplex
+imports SparseMatrix LP ComputeFloat ComputeNumeral
+uses "Cplex_tools.ML" "CplexMatrixConverter.ML" "FloatSparseMatrixBuilder.ML"
+ "fspmlp.ML" ("matrixlp.ML")
+begin
+
+lemma spm_mult_le_dual_prts:
+ assumes
+ "sorted_sparse_matrix A1"
+ "sorted_sparse_matrix A2"
+ "sorted_sparse_matrix c1"
+ "sorted_sparse_matrix c2"
+ "sorted_sparse_matrix y"
+ "sorted_sparse_matrix r1"
+ "sorted_sparse_matrix r2"
+ "sorted_spvec b"
+ "le_spmat [] y"
+ "sparse_row_matrix A1 \<le> A"
+ "A \<le> sparse_row_matrix A2"
+ "sparse_row_matrix c1 \<le> c"
+ "c \<le> sparse_row_matrix c2"
+ "sparse_row_matrix r1 \<le> x"
+ "x \<le> sparse_row_matrix r2"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
+ shows
+ "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
+ (let s1 = diff_spmat c1 (mult_spmat y A2); s2 = diff_spmat c2 (mult_spmat y A1) in
+ add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
+ (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))))"
+ apply (simp add: Let_def)
+ apply (insert assms)
+ apply (simp add: sparse_row_matrix_op_simps algebra_simps)
+ apply (rule mult_le_dual_prts[where A=A, simplified Let_def algebra_simps])
+ apply (auto)
+ done
+
+lemma spm_mult_le_dual_prts_no_let:
+ assumes
+ "sorted_sparse_matrix A1"
+ "sorted_sparse_matrix A2"
+ "sorted_sparse_matrix c1"
+ "sorted_sparse_matrix c2"
+ "sorted_sparse_matrix y"
+ "sorted_sparse_matrix r1"
+ "sorted_sparse_matrix r2"
+ "sorted_spvec b"
+ "le_spmat [] y"
+ "sparse_row_matrix A1 \<le> A"
+ "A \<le> sparse_row_matrix A2"
+ "sparse_row_matrix c1 \<le> c"
+ "c \<le> sparse_row_matrix c2"
+ "sparse_row_matrix r1 \<le> x"
+ "x \<le> sparse_row_matrix r2"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
+ shows
+ "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b)
+ (mult_est_spmat r1 r2 (diff_spmat c1 (mult_spmat y A2)) (diff_spmat c2 (mult_spmat y A1))))"
+ by (simp add: assms mult_est_spmat_def spm_mult_le_dual_prts[where A=A, simplified Let_def])
+
+use "matrixlp.ML"
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/CplexMatrixConverter.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,128 @@
+(* Title: HOL/Matrix/CplexMatrixConverter.ML
+ Author: Steven Obua
+*)
+
+signature MATRIX_BUILDER =
+sig
+ type vector
+ type matrix
+
+ val empty_vector : vector
+ val empty_matrix : matrix
+
+ exception Nat_expected of int
+ val set_elem : vector -> int -> string -> vector
+ val set_vector : matrix -> int -> vector -> matrix
+end;
+
+signature CPLEX_MATRIX_CONVERTER =
+sig
+ structure cplex : CPLEX
+ structure matrix_builder : MATRIX_BUILDER
+ type vector = matrix_builder.vector
+ type matrix = matrix_builder.matrix
+ type naming = int * (int -> string) * (string -> int)
+
+ exception Converter of string
+
+ (* program must fulfill is_normed_cplexProg and must be an element of the image of elim_nonfree_bounds *)
+ (* convert_prog maximize c A b naming *)
+ val convert_prog : cplex.cplexProg -> bool * vector * matrix * vector * naming
+
+ (* results must be optimal, converts_results returns the optimal value as string and the solution as vector *)
+ (* convert_results results name2index *)
+ val convert_results : cplex.cplexResult -> (string -> int) -> string * vector
+end;
+
+functor MAKE_CPLEX_MATRIX_CONVERTER (structure cplex: CPLEX and matrix_builder: MATRIX_BUILDER) : CPLEX_MATRIX_CONVERTER =
+struct
+
+structure cplex = cplex
+structure matrix_builder = matrix_builder
+type matrix = matrix_builder.matrix
+type vector = matrix_builder.vector
+type naming = int * (int -> string) * (string -> int)
+
+open matrix_builder
+open cplex
+
+exception Converter of string;
+
+fun neg_term (cplexNeg t) = t
+ | neg_term (cplexSum ts) = cplexSum (map neg_term ts)
+ | neg_term t = cplexNeg t
+
+fun convert_prog (cplexProg (_, goal, constrs, bounds)) =
+ let
+ fun build_naming index i2s s2i [] = (index, i2s, s2i)
+ | build_naming index i2s s2i (cplexBounds (cplexNeg cplexInf, cplexLeq, cplexVar v, cplexLeq, cplexInf)::bounds)
+ = build_naming (index+1) (Inttab.update (index, v) i2s) (Symtab.update_new (v, index) s2i) bounds
+ | build_naming _ _ _ _ = raise (Converter "nonfree bound")
+
+ val (varcount, i2s_tab, s2i_tab) = build_naming 0 Inttab.empty Symtab.empty bounds
+
+ fun i2s i = case Inttab.lookup i2s_tab i of NONE => raise (Converter "index not found")
+ | SOME n => n
+ fun s2i s = case Symtab.lookup s2i_tab s of NONE => raise (Converter ("name not found: "^s))
+ | SOME i => i
+ fun num2str positive (cplexNeg t) = num2str (not positive) t
+ | num2str positive (cplexNum num) = if positive then num else "-"^num
+ | num2str _ _ = raise (Converter "term is not a (possibly signed) number")
+
+ fun setprod vec positive (cplexNeg t) = setprod vec (not positive) t
+ | setprod vec positive (cplexVar v) = set_elem vec (s2i v) (if positive then "1" else "-1")
+ | setprod vec positive (cplexProd (cplexNum num, cplexVar v)) =
+ set_elem vec (s2i v) (if positive then num else "-"^num)
+ | setprod _ _ _ = raise (Converter "term is not a normed product")
+
+ fun sum2vec (cplexSum ts) = fold (fn t => fn vec => setprod vec true t) ts empty_vector
+ | sum2vec t = setprod empty_vector true t
+
+ fun constrs2Ab j A b [] = (A, b)
+ | constrs2Ab j A b ((_, cplexConstr (cplexLeq, (t1,t2)))::cs) =
+ constrs2Ab (j+1) (set_vector A j (sum2vec t1)) (set_elem b j (num2str true t2)) cs
+ | constrs2Ab j A b ((_, cplexConstr (cplexGeq, (t1,t2)))::cs) =
+ constrs2Ab (j+1) (set_vector A j (sum2vec (neg_term t1))) (set_elem b j (num2str true (neg_term t2))) cs
+ | constrs2Ab j A b ((_, cplexConstr (cplexEq, (t1,t2)))::cs) =
+ constrs2Ab j A b ((NONE, cplexConstr (cplexLeq, (t1,t2)))::
+ (NONE, cplexConstr (cplexGeq, (t1, t2)))::cs)
+ | constrs2Ab _ _ _ _ = raise (Converter "no strict constraints allowed")
+
+ val (A, b) = constrs2Ab 0 empty_matrix empty_vector constrs
+
+ val (goal_maximize, goal_term) =
+ case goal of
+ (cplexMaximize t) => (true, t)
+ | (cplexMinimize t) => (false, t)
+ in
+ (goal_maximize, sum2vec goal_term, A, b, (varcount, i2s, s2i))
+ end
+
+fun convert_results (cplex.Optimal (opt, entries)) name2index =
+ let
+ fun setv (name, value) v = matrix_builder.set_elem v (name2index name) value
+ in
+ (opt, fold setv entries (matrix_builder.empty_vector))
+ end
+ | convert_results _ _ = raise (Converter "No optimal result")
+
+end;
+
+structure SimpleMatrixBuilder : MATRIX_BUILDER =
+struct
+type vector = (int * string) list
+type matrix = (int * vector) list
+
+val empty_matrix = []
+val empty_vector = []
+
+exception Nat_expected of int;
+
+fun set_elem v i s = v @ [(i, s)]
+
+fun set_vector m i v = m @ [(i, v)]
+
+end;
+
+structure SimpleCplexMatrixConverter =
+ MAKE_CPLEX_MATRIX_CONVERTER(structure cplex = Cplex and matrix_builder = SimpleMatrixBuilder);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Cplex_tools.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,1192 @@
+(* Title: HOL/Matrix/Cplex_tools.ML
+ Author: Steven Obua
+*)
+
+signature CPLEX =
+sig
+
+ datatype cplexTerm = cplexVar of string | cplexNum of string | cplexInf
+ | cplexNeg of cplexTerm
+ | cplexProd of cplexTerm * cplexTerm
+ | cplexSum of (cplexTerm list)
+
+ datatype cplexComp = cplexLe | cplexLeq | cplexEq | cplexGe | cplexGeq
+
+ datatype cplexGoal = cplexMinimize of cplexTerm
+ | cplexMaximize of cplexTerm
+
+ datatype cplexConstr = cplexConstr of cplexComp *
+ (cplexTerm * cplexTerm)
+
+ datatype cplexBounds = cplexBounds of cplexTerm * cplexComp * cplexTerm
+ * cplexComp * cplexTerm
+ | cplexBound of cplexTerm * cplexComp * cplexTerm
+
+ datatype cplexProg = cplexProg of string
+ * cplexGoal
+ * ((string option * cplexConstr)
+ list)
+ * cplexBounds list
+
+ datatype cplexResult = Unbounded
+ | Infeasible
+ | Undefined
+ | Optimal of string *
+ (((* name *) string *
+ (* value *) string) list)
+
+ datatype cplexSolver = SOLVER_DEFAULT | SOLVER_CPLEX | SOLVER_GLPK
+
+ exception Load_cplexFile of string
+ exception Load_cplexResult of string
+ exception Save_cplexFile of string
+ exception Execute of string
+
+ val load_cplexFile : string -> cplexProg
+
+ val save_cplexFile : string -> cplexProg -> unit
+
+ val elim_nonfree_bounds : cplexProg -> cplexProg
+
+ val relax_strict_ineqs : cplexProg -> cplexProg
+
+ val is_normed_cplexProg : cplexProg -> bool
+
+ val get_solver : unit -> cplexSolver
+ val set_solver : cplexSolver -> unit
+ val solve : cplexProg -> cplexResult
+end;
+
+structure Cplex : CPLEX =
+struct
+
+datatype cplexSolver = SOLVER_DEFAULT | SOLVER_CPLEX | SOLVER_GLPK
+
+val cplexsolver = Unsynchronized.ref SOLVER_DEFAULT;
+fun get_solver () = !cplexsolver;
+fun set_solver s = (cplexsolver := s);
+
+exception Load_cplexFile of string;
+exception Load_cplexResult of string;
+exception Save_cplexFile of string;
+
+datatype cplexTerm = cplexVar of string
+ | cplexNum of string
+ | cplexInf
+ | cplexNeg of cplexTerm
+ | cplexProd of cplexTerm * cplexTerm
+ | cplexSum of (cplexTerm list)
+datatype cplexComp = cplexLe | cplexLeq | cplexEq | cplexGe | cplexGeq
+datatype cplexGoal = cplexMinimize of cplexTerm | cplexMaximize of cplexTerm
+datatype cplexConstr = cplexConstr of cplexComp * (cplexTerm * cplexTerm)
+datatype cplexBounds = cplexBounds of cplexTerm * cplexComp * cplexTerm
+ * cplexComp * cplexTerm
+ | cplexBound of cplexTerm * cplexComp * cplexTerm
+datatype cplexProg = cplexProg of string
+ * cplexGoal
+ * ((string option * cplexConstr) list)
+ * cplexBounds list
+
+fun rev_cmp cplexLe = cplexGe
+ | rev_cmp cplexLeq = cplexGeq
+ | rev_cmp cplexGe = cplexLe
+ | rev_cmp cplexGeq = cplexLeq
+ | rev_cmp cplexEq = cplexEq
+
+fun the NONE = raise (Load_cplexFile "SOME expected")
+ | the (SOME x) = x;
+
+fun modulo_signed is_something (cplexNeg u) = is_something u
+ | modulo_signed is_something u = is_something u
+
+fun is_Num (cplexNum _) = true
+ | is_Num _ = false
+
+fun is_Inf cplexInf = true
+ | is_Inf _ = false
+
+fun is_Var (cplexVar _) = true
+ | is_Var _ = false
+
+fun is_Neg (cplexNeg _) = true
+ | is_Neg _ = false
+
+fun is_normed_Prod (cplexProd (t1, t2)) =
+ (is_Num t1) andalso (is_Var t2)
+ | is_normed_Prod x = is_Var x
+
+fun is_normed_Sum (cplexSum ts) =
+ (ts <> []) andalso forall (modulo_signed is_normed_Prod) ts
+ | is_normed_Sum x = modulo_signed is_normed_Prod x
+
+fun is_normed_Constr (cplexConstr (_, (t1, t2))) =
+ (is_normed_Sum t1) andalso (modulo_signed is_Num t2)
+
+fun is_Num_or_Inf x = is_Inf x orelse is_Num x
+
+fun is_normed_Bounds (cplexBounds (t1, c1, t2, c2, t3)) =
+ (c1 = cplexLe orelse c1 = cplexLeq) andalso
+ (c2 = cplexLe orelse c2 = cplexLeq) andalso
+ is_Var t2 andalso
+ modulo_signed is_Num_or_Inf t1 andalso
+ modulo_signed is_Num_or_Inf t3
+ | is_normed_Bounds (cplexBound (t1, c, t2)) =
+ (is_Var t1 andalso (modulo_signed is_Num_or_Inf t2))
+ orelse
+ (c <> cplexEq andalso
+ is_Var t2 andalso (modulo_signed is_Num_or_Inf t1))
+
+fun term_of_goal (cplexMinimize x) = x
+ | term_of_goal (cplexMaximize x) = x
+
+fun is_normed_cplexProg (cplexProg (_, goal, constraints, bounds)) =
+ is_normed_Sum (term_of_goal goal) andalso
+ forall (fn (_,x) => is_normed_Constr x) constraints andalso
+ forall is_normed_Bounds bounds
+
+fun is_NL s = s = "\n"
+
+fun is_blank s = forall (fn c => c <> #"\n" andalso Char.isSpace c) (String.explode s)
+
+fun is_num a =
+ let
+ val b = String.explode a
+ fun num4 cs = forall Char.isDigit cs
+ fun num3 [] = true
+ | num3 (ds as (c::cs)) =
+ if c = #"+" orelse c = #"-" then
+ num4 cs
+ else
+ num4 ds
+ fun num2 [] = true
+ | num2 (c::cs) =
+ if c = #"e" orelse c = #"E" then num3 cs
+ else (Char.isDigit c) andalso num2 cs
+ fun num1 [] = true
+ | num1 (c::cs) =
+ if c = #"." then num2 cs
+ else if c = #"e" orelse c = #"E" then num3 cs
+ else (Char.isDigit c) andalso num1 cs
+ fun num [] = true
+ | num (c::cs) =
+ if c = #"." then num2 cs
+ else (Char.isDigit c) andalso num1 cs
+ in
+ num b
+ end
+
+fun is_delimiter s = s = "+" orelse s = "-" orelse s = ":"
+
+fun is_cmp s = s = "<" orelse s = ">" orelse s = "<="
+ orelse s = ">=" orelse s = "="
+
+fun is_symbol a =
+ let
+ val symbol_char = String.explode "!\"#$%&()/,.;?@_`'{}|~"
+ fun is_symbol_char c = Char.isAlphaNum c orelse
+ exists (fn d => d=c) symbol_char
+ fun is_symbol_start c = is_symbol_char c andalso
+ not (Char.isDigit c) andalso
+ not (c= #".")
+ val b = String.explode a
+ in
+ b <> [] andalso is_symbol_start (hd b) andalso
+ forall is_symbol_char b
+ end
+
+fun to_upper s = String.implode (map Char.toUpper (String.explode s))
+
+fun keyword x =
+ let
+ val a = to_upper x
+ in
+ if a = "BOUNDS" orelse a = "BOUND" then
+ SOME "BOUNDS"
+ else if a = "MINIMIZE" orelse a = "MINIMUM" orelse a = "MIN" then
+ SOME "MINIMIZE"
+ else if a = "MAXIMIZE" orelse a = "MAXIMUM" orelse a = "MAX" then
+ SOME "MAXIMIZE"
+ else if a = "ST" orelse a = "S.T." orelse a = "ST." then
+ SOME "ST"
+ else if a = "FREE" orelse a = "END" then
+ SOME a
+ else if a = "GENERAL" orelse a = "GENERALS" orelse a = "GEN" then
+ SOME "GENERAL"
+ else if a = "INTEGER" orelse a = "INTEGERS" orelse a = "INT" then
+ SOME "INTEGER"
+ else if a = "BINARY" orelse a = "BINARIES" orelse a = "BIN" then
+ SOME "BINARY"
+ else if a = "INF" orelse a = "INFINITY" then
+ SOME "INF"
+ else
+ NONE
+ end
+
+val TOKEN_ERROR = ~1
+val TOKEN_BLANK = 0
+val TOKEN_NUM = 1
+val TOKEN_DELIMITER = 2
+val TOKEN_SYMBOL = 3
+val TOKEN_LABEL = 4
+val TOKEN_CMP = 5
+val TOKEN_KEYWORD = 6
+val TOKEN_NL = 7
+
+(* tokenize takes a list of chars as argument and returns a list of
+ int * string pairs, each string representing a "cplex token",
+ and each int being one of TOKEN_NUM, TOKEN_DELIMITER, TOKEN_CMP
+ or TOKEN_SYMBOL *)
+fun tokenize s =
+ let
+ val flist = [(is_NL, TOKEN_NL),
+ (is_blank, TOKEN_BLANK),
+ (is_num, TOKEN_NUM),
+ (is_delimiter, TOKEN_DELIMITER),
+ (is_cmp, TOKEN_CMP),
+ (is_symbol, TOKEN_SYMBOL)]
+ fun match_helper [] s = (fn _ => false, TOKEN_ERROR)
+ | match_helper (f::fs) s =
+ if ((fst f) s) then f else match_helper fs s
+ fun match s = match_helper flist s
+ fun tok s =
+ if s = "" then [] else
+ let
+ val h = String.substring (s,0,1)
+ val (f, j) = match h
+ fun len i =
+ if size s = i then i
+ else if f (String.substring (s,0,i+1)) then
+ len (i+1)
+ else i
+ in
+ if j < 0 then
+ (if h = "\\" then []
+ else raise (Load_cplexFile ("token expected, found: "
+ ^s)))
+ else
+ let
+ val l = len 1
+ val u = String.substring (s,0,l)
+ val v = String.extract (s,l,NONE)
+ in
+ if j = 0 then tok v else (j, u) :: tok v
+ end
+ end
+ in
+ tok s
+ end
+
+exception Tokenize of string;
+
+fun tokenize_general flist s =
+ let
+ fun match_helper [] s = raise (Tokenize s)
+ | match_helper (f::fs) s =
+ if ((fst f) s) then f else match_helper fs s
+ fun match s = match_helper flist s
+ fun tok s =
+ if s = "" then [] else
+ let
+ val h = String.substring (s,0,1)
+ val (f, j) = match h
+ fun len i =
+ if size s = i then i
+ else if f (String.substring (s,0,i+1)) then
+ len (i+1)
+ else i
+ val l = len 1
+ in
+ (j, String.substring (s,0,l)) :: tok (String.extract (s,l,NONE))
+ end
+ in
+ tok s
+ end
+
+fun load_cplexFile name =
+ let
+ val f = TextIO.openIn name
+ val ignore_NL = Unsynchronized.ref true
+ val rest = Unsynchronized.ref []
+
+ fun is_symbol s c = (fst c) = TOKEN_SYMBOL andalso (to_upper (snd c)) = s
+
+ fun readToken_helper () =
+ if length (!rest) > 0 then
+ let val u = hd (!rest) in
+ (
+ rest := tl (!rest);
+ SOME u
+ )
+ end
+ else
+ (case TextIO.inputLine f of
+ NONE => NONE
+ | SOME s =>
+ let val t = tokenize s in
+ if (length t >= 2 andalso
+ snd(hd (tl t)) = ":")
+ then
+ rest := (TOKEN_LABEL, snd (hd t)) :: (tl (tl t))
+ else if (length t >= 2) andalso is_symbol "SUBJECT" (hd (t))
+ andalso is_symbol "TO" (hd (tl t))
+ then
+ rest := (TOKEN_SYMBOL, "ST") :: (tl (tl t))
+ else
+ rest := t;
+ readToken_helper ()
+ end)
+
+ fun readToken_helper2 () =
+ let val c = readToken_helper () in
+ if c = NONE then NONE
+ else if !ignore_NL andalso fst (the c) = TOKEN_NL then
+ readToken_helper2 ()
+ else if fst (the c) = TOKEN_SYMBOL
+ andalso keyword (snd (the c)) <> NONE
+ then SOME (TOKEN_KEYWORD, the (keyword (snd (the c))))
+ else c
+ end
+
+ fun readToken () = readToken_helper2 ()
+
+ fun pushToken a = rest := (a::(!rest))
+
+ fun is_value token =
+ fst token = TOKEN_NUM orelse (fst token = TOKEN_KEYWORD
+ andalso snd token = "INF")
+
+ fun get_value token =
+ if fst token = TOKEN_NUM then
+ cplexNum (snd token)
+ else if fst token = TOKEN_KEYWORD andalso snd token = "INF"
+ then
+ cplexInf
+ else
+ raise (Load_cplexFile "num expected")
+
+ fun readTerm_Product only_num =
+ let val c = readToken () in
+ if c = NONE then NONE
+ else if fst (the c) = TOKEN_SYMBOL
+ then (
+ if only_num then (pushToken (the c); NONE)
+ else SOME (cplexVar (snd (the c)))
+ )
+ else if only_num andalso is_value (the c) then
+ SOME (get_value (the c))
+ else if is_value (the c) then
+ let val t1 = get_value (the c)
+ val d = readToken ()
+ in
+ if d = NONE then SOME t1
+ else if fst (the d) = TOKEN_SYMBOL then
+ SOME (cplexProd (t1, cplexVar (snd (the d))))
+ else
+ (pushToken (the d); SOME t1)
+ end
+ else (pushToken (the c); NONE)
+ end
+
+ fun readTerm_Signed only_signed only_num =
+ let
+ val c = readToken ()
+ in
+ if c = NONE then NONE
+ else
+ let val d = the c in
+ if d = (TOKEN_DELIMITER, "+") then
+ readTerm_Product only_num
+ else if d = (TOKEN_DELIMITER, "-") then
+ SOME (cplexNeg (the (readTerm_Product
+ only_num)))
+ else (pushToken d;
+ if only_signed then NONE
+ else readTerm_Product only_num)
+ end
+ end
+
+ fun readTerm_Sum first_signed =
+ let val c = readTerm_Signed first_signed false in
+ if c = NONE then [] else (the c)::(readTerm_Sum true)
+ end
+
+ fun readTerm () =
+ let val c = readTerm_Sum false in
+ if c = [] then NONE
+ else if tl c = [] then SOME (hd c)
+ else SOME (cplexSum c)
+ end
+
+ fun readLabeledTerm () =
+ let val c = readToken () in
+ if c = NONE then (NONE, NONE)
+ else if fst (the c) = TOKEN_LABEL then
+ let val t = readTerm () in
+ if t = NONE then
+ raise (Load_cplexFile ("term after label "^
+ (snd (the c))^
+ " expected"))
+ else (SOME (snd (the c)), t)
+ end
+ else (pushToken (the c); (NONE, readTerm ()))
+ end
+
+ fun readGoal () =
+ let
+ val g = readToken ()
+ in
+ if g = SOME (TOKEN_KEYWORD, "MAXIMIZE") then
+ cplexMaximize (the (snd (readLabeledTerm ())))
+ else if g = SOME (TOKEN_KEYWORD, "MINIMIZE") then
+ cplexMinimize (the (snd (readLabeledTerm ())))
+ else raise (Load_cplexFile "MAXIMIZE or MINIMIZE expected")
+ end
+
+ fun str2cmp b =
+ (case b of
+ "<" => cplexLe
+ | "<=" => cplexLeq
+ | ">" => cplexGe
+ | ">=" => cplexGeq
+ | "=" => cplexEq
+ | _ => raise (Load_cplexFile (b^" is no TOKEN_CMP")))
+
+ fun readConstraint () =
+ let
+ val t = readLabeledTerm ()
+ fun make_constraint b t1 t2 =
+ cplexConstr
+ (str2cmp b,
+ (t1, t2))
+ in
+ if snd t = NONE then NONE
+ else
+ let val c = readToken () in
+ if c = NONE orelse fst (the c) <> TOKEN_CMP
+ then raise (Load_cplexFile "TOKEN_CMP expected")
+ else
+ let val n = readTerm_Signed false true in
+ if n = NONE then
+ raise (Load_cplexFile "num expected")
+ else
+ SOME (fst t,
+ make_constraint (snd (the c))
+ (the (snd t))
+ (the n))
+ end
+ end
+ end
+
+ fun readST () =
+ let
+ fun readbody () =
+ let val t = readConstraint () in
+ if t = NONE then []
+ else if (is_normed_Constr (snd (the t))) then
+ (the t)::(readbody ())
+ else if (fst (the t) <> NONE) then
+ raise (Load_cplexFile
+ ("constraint '"^(the (fst (the t)))^
+ "'is not normed"))
+ else
+ raise (Load_cplexFile
+ "constraint is not normed")
+ end
+ in
+ if readToken () = SOME (TOKEN_KEYWORD, "ST")
+ then
+ readbody ()
+ else
+ raise (Load_cplexFile "ST expected")
+ end
+
+ fun readCmp () =
+ let val c = readToken () in
+ if c = NONE then NONE
+ else if fst (the c) = TOKEN_CMP then
+ SOME (str2cmp (snd (the c)))
+ else (pushToken (the c); NONE)
+ end
+
+ fun skip_NL () =
+ let val c = readToken () in
+ if c <> NONE andalso fst (the c) = TOKEN_NL then
+ skip_NL ()
+ else
+ (pushToken (the c); ())
+ end
+
+ fun make_bounds c t1 t2 =
+ cplexBound (t1, c, t2)
+
+ fun readBound () =
+ let
+ val _ = skip_NL ()
+ val t1 = readTerm ()
+ in
+ if t1 = NONE then NONE
+ else
+ let
+ val c1 = readCmp ()
+ in
+ if c1 = NONE then
+ let
+ val c = readToken ()
+ in
+ if c = SOME (TOKEN_KEYWORD, "FREE") then
+ SOME (
+ cplexBounds (cplexNeg cplexInf,
+ cplexLeq,
+ the t1,
+ cplexLeq,
+ cplexInf))
+ else
+ raise (Load_cplexFile "FREE expected")
+ end
+ else
+ let
+ val t2 = readTerm ()
+ in
+ if t2 = NONE then
+ raise (Load_cplexFile "term expected")
+ else
+ let val c2 = readCmp () in
+ if c2 = NONE then
+ SOME (make_bounds (the c1)
+ (the t1)
+ (the t2))
+ else
+ SOME (
+ cplexBounds (the t1,
+ the c1,
+ the t2,
+ the c2,
+ the (readTerm())))
+ end
+ end
+ end
+ end
+
+ fun readBounds () =
+ let
+ fun makestring _ = "?"
+ fun readbody () =
+ let
+ val b = readBound ()
+ in
+ if b = NONE then []
+ else if (is_normed_Bounds (the b)) then
+ (the b)::(readbody())
+ else (
+ raise (Load_cplexFile
+ ("bounds are not normed in: "^
+ (makestring (the b)))))
+ end
+ in
+ if readToken () = SOME (TOKEN_KEYWORD, "BOUNDS") then
+ readbody ()
+ else raise (Load_cplexFile "BOUNDS expected")
+ end
+
+ fun readEnd () =
+ if readToken () = SOME (TOKEN_KEYWORD, "END") then ()
+ else raise (Load_cplexFile "END expected")
+
+ val result_Goal = readGoal ()
+ val result_ST = readST ()
+ val _ = ignore_NL := false
+ val result_Bounds = readBounds ()
+ val _ = ignore_NL := true
+ val _ = readEnd ()
+ val _ = TextIO.closeIn f
+ in
+ cplexProg (name, result_Goal, result_ST, result_Bounds)
+ end
+
+fun save_cplexFile filename (cplexProg (_, goal, constraints, bounds)) =
+ let
+ val f = TextIO.openOut filename
+
+ fun basic_write s = TextIO.output(f, s)
+
+ val linebuf = Unsynchronized.ref ""
+ fun buf_flushline s =
+ (basic_write (!linebuf);
+ basic_write "\n";
+ linebuf := s)
+ fun buf_add s = linebuf := (!linebuf) ^ s
+
+ fun write s =
+ if (String.size s) + (String.size (!linebuf)) >= 250 then
+ buf_flushline (" "^s)
+ else
+ buf_add s
+
+ fun writeln s = (buf_add s; buf_flushline "")
+
+ fun write_term (cplexVar x) = write x
+ | write_term (cplexNum x) = write x
+ | write_term cplexInf = write "inf"
+ | write_term (cplexProd (cplexNum "1", b)) = write_term b
+ | write_term (cplexProd (a, b)) =
+ (write_term a; write " "; write_term b)
+ | write_term (cplexNeg x) = (write " - "; write_term x)
+ | write_term (cplexSum ts) = write_terms ts
+ and write_terms [] = ()
+ | write_terms (t::ts) =
+ (if (not (is_Neg t)) then write " + " else ();
+ write_term t; write_terms ts)
+
+ fun write_goal (cplexMaximize term) =
+ (writeln "MAXIMIZE"; write_term term; writeln "")
+ | write_goal (cplexMinimize term) =
+ (writeln "MINIMIZE"; write_term term; writeln "")
+
+ fun write_cmp cplexLe = write "<"
+ | write_cmp cplexLeq = write "<="
+ | write_cmp cplexEq = write "="
+ | write_cmp cplexGe = write ">"
+ | write_cmp cplexGeq = write ">="
+
+ fun write_constr (cplexConstr (cmp, (a,b))) =
+ (write_term a;
+ write " ";
+ write_cmp cmp;
+ write " ";
+ write_term b)
+
+ fun write_constraints [] = ()
+ | write_constraints (c::cs) =
+ (if (fst c <> NONE)
+ then
+ (write (the (fst c)); write ": ")
+ else
+ ();
+ write_constr (snd c);
+ writeln "";
+ write_constraints cs)
+
+ fun write_bounds [] = ()
+ | write_bounds ((cplexBounds (t1,c1,t2,c2,t3))::bs) =
+ ((if t1 = cplexNeg cplexInf andalso t3 = cplexInf
+ andalso (c1 = cplexLeq orelse c1 = cplexLe)
+ andalso (c2 = cplexLeq orelse c2 = cplexLe)
+ then
+ (write_term t2; write " free")
+ else
+ (write_term t1; write " "; write_cmp c1; write " ";
+ write_term t2; write " "; write_cmp c2; write " ";
+ write_term t3)
+ ); writeln ""; write_bounds bs)
+ | write_bounds ((cplexBound (t1, c, t2)) :: bs) =
+ (write_term t1; write " ";
+ write_cmp c; write " ";
+ write_term t2; writeln ""; write_bounds bs)
+
+ val _ = write_goal goal
+ val _ = (writeln ""; writeln "ST")
+ val _ = write_constraints constraints
+ val _ = (writeln ""; writeln "BOUNDS")
+ val _ = write_bounds bounds
+ val _ = (writeln ""; writeln "END")
+ val _ = TextIO.closeOut f
+ in
+ ()
+ end
+
+fun norm_Constr (constr as cplexConstr (c, (t1, t2))) =
+ if not (modulo_signed is_Num t2) andalso
+ modulo_signed is_Num t1
+ then
+ [cplexConstr (rev_cmp c, (t2, t1))]
+ else if (c = cplexLe orelse c = cplexLeq) andalso
+ (t1 = (cplexNeg cplexInf) orelse t2 = cplexInf)
+ then
+ []
+ else if (c = cplexGe orelse c = cplexGeq) andalso
+ (t1 = cplexInf orelse t2 = cplexNeg cplexInf)
+ then
+ []
+ else
+ [constr]
+
+fun bound2constr (cplexBounds (t1,c1,t2,c2,t3)) =
+ (norm_Constr(cplexConstr (c1, (t1, t2))))
+ @ (norm_Constr(cplexConstr (c2, (t2, t3))))
+ | bound2constr (cplexBound (t1, cplexEq, t2)) =
+ (norm_Constr(cplexConstr (cplexLeq, (t1, t2))))
+ @ (norm_Constr(cplexConstr (cplexLeq, (t2, t1))))
+ | bound2constr (cplexBound (t1, c1, t2)) =
+ norm_Constr(cplexConstr (c1, (t1,t2)))
+
+val emptyset = Symtab.empty
+
+fun singleton v = Symtab.update (v, ()) emptyset
+
+fun merge a b = Symtab.merge (op =) (a, b)
+
+fun mergemap f ts = fold (fn x => fn table => merge table (f x)) ts Symtab.empty
+
+fun diff a b = Symtab.fold (Symtab.delete_safe o fst) b a
+
+fun collect_vars (cplexVar v) = singleton v
+ | collect_vars (cplexNeg t) = collect_vars t
+ | collect_vars (cplexProd (t1, t2)) =
+ merge (collect_vars t1) (collect_vars t2)
+ | collect_vars (cplexSum ts) = mergemap collect_vars ts
+ | collect_vars _ = emptyset
+
+(* Eliminates all nonfree bounds from the linear program and produces an
+ equivalent program with only free bounds
+ IF for the input program P holds: is_normed_cplexProg P *)
+fun elim_nonfree_bounds (cplexProg (name, goal, constraints, bounds)) =
+ let
+ fun collect_constr_vars (_, cplexConstr (_, (t1,_))) =
+ (collect_vars t1)
+
+ val cvars = merge (collect_vars (term_of_goal goal))
+ (mergemap collect_constr_vars constraints)
+
+ fun collect_lower_bounded_vars
+ (cplexBounds (_, _, cplexVar v, _, _)) =
+ singleton v
+ | collect_lower_bounded_vars
+ (cplexBound (_, cplexLe, cplexVar v)) =
+ singleton v
+ | collect_lower_bounded_vars
+ (cplexBound (_, cplexLeq, cplexVar v)) =
+ singleton v
+ | collect_lower_bounded_vars
+ (cplexBound (cplexVar v, cplexGe,_)) =
+ singleton v
+ | collect_lower_bounded_vars
+ (cplexBound (cplexVar v, cplexGeq, _)) =
+ singleton v
+ | collect_lower_bounded_vars
+ (cplexBound (cplexVar v, cplexEq, _)) =
+ singleton v
+ | collect_lower_bounded_vars _ = emptyset
+
+ val lvars = mergemap collect_lower_bounded_vars bounds
+ val positive_vars = diff cvars lvars
+ val zero = cplexNum "0"
+
+ fun make_pos_constr v =
+ (NONE, cplexConstr (cplexGeq, ((cplexVar v), zero)))
+
+ fun make_free_bound v =
+ cplexBounds (cplexNeg cplexInf, cplexLeq,
+ cplexVar v, cplexLeq,
+ cplexInf)
+
+ val pos_constrs = rev (Symtab.fold
+ (fn (k, _) => cons (make_pos_constr k))
+ positive_vars [])
+ val bound_constrs = map (pair NONE)
+ (maps bound2constr bounds)
+ val constraints' = constraints @ pos_constrs @ bound_constrs
+ val bounds' = rev (Symtab.fold (fn (v, _) => cons (make_free_bound v)) cvars []);
+ in
+ cplexProg (name, goal, constraints', bounds')
+ end
+
+fun relax_strict_ineqs (cplexProg (name, goals, constrs, bounds)) =
+ let
+ fun relax cplexLe = cplexLeq
+ | relax cplexGe = cplexGeq
+ | relax x = x
+
+ fun relax_constr (n, cplexConstr(c, (t1, t2))) =
+ (n, cplexConstr(relax c, (t1, t2)))
+
+ fun relax_bounds (cplexBounds (t1, c1, t2, c2, t3)) =
+ cplexBounds (t1, relax c1, t2, relax c2, t3)
+ | relax_bounds (cplexBound (t1, c, t2)) =
+ cplexBound (t1, relax c, t2)
+ in
+ cplexProg (name,
+ goals,
+ map relax_constr constrs,
+ map relax_bounds bounds)
+ end
+
+datatype cplexResult = Unbounded
+ | Infeasible
+ | Undefined
+ | Optimal of string * ((string * string) list)
+
+fun is_separator x = forall (fn c => c = #"-") (String.explode x)
+
+fun is_sign x = (x = "+" orelse x = "-")
+
+fun is_colon x = (x = ":")
+
+fun is_resultsymbol a =
+ let
+ val symbol_char = String.explode "!\"#$%&()/,.;?@_`'{}|~-"
+ fun is_symbol_char c = Char.isAlphaNum c orelse
+ exists (fn d => d=c) symbol_char
+ fun is_symbol_start c = is_symbol_char c andalso
+ not (Char.isDigit c) andalso
+ not (c= #".") andalso
+ not (c= #"-")
+ val b = String.explode a
+ in
+ b <> [] andalso is_symbol_start (hd b) andalso
+ forall is_symbol_char b
+ end
+
+val TOKEN_SIGN = 100
+val TOKEN_COLON = 101
+val TOKEN_SEPARATOR = 102
+
+fun load_glpkResult name =
+ let
+ val flist = [(is_NL, TOKEN_NL),
+ (is_blank, TOKEN_BLANK),
+ (is_num, TOKEN_NUM),
+ (is_sign, TOKEN_SIGN),
+ (is_colon, TOKEN_COLON),
+ (is_cmp, TOKEN_CMP),
+ (is_resultsymbol, TOKEN_SYMBOL),
+ (is_separator, TOKEN_SEPARATOR)]
+
+ val tokenize = tokenize_general flist
+
+ val f = TextIO.openIn name
+
+ val rest = Unsynchronized.ref []
+
+ fun readToken_helper () =
+ if length (!rest) > 0 then
+ let val u = hd (!rest) in
+ (
+ rest := tl (!rest);
+ SOME u
+ )
+ end
+ else
+ (case TextIO.inputLine f of
+ NONE => NONE
+ | SOME s => (rest := tokenize s; readToken_helper()))
+
+ fun is_tt tok ty = (tok <> NONE andalso (fst (the tok)) = ty)
+
+ fun pushToken a = if a = NONE then () else (rest := ((the a)::(!rest)))
+
+ fun readToken () =
+ let val t = readToken_helper () in
+ if is_tt t TOKEN_BLANK then
+ readToken ()
+ else if is_tt t TOKEN_NL then
+ let val t2 = readToken_helper () in
+ if is_tt t2 TOKEN_SIGN then
+ (pushToken (SOME (TOKEN_SEPARATOR, "-")); t)
+ else
+ (pushToken t2; t)
+ end
+ else if is_tt t TOKEN_SIGN then
+ let val t2 = readToken_helper () in
+ if is_tt t2 TOKEN_NUM then
+ (SOME (TOKEN_NUM, (snd (the t))^(snd (the t2))))
+ else
+ (pushToken t2; t)
+ end
+ else
+ t
+ end
+
+ fun readRestOfLine P =
+ let
+ val t = readToken ()
+ in
+ if is_tt t TOKEN_NL orelse t = NONE
+ then P
+ else readRestOfLine P
+ end
+
+ fun readHeader () =
+ let
+ fun readStatus () = readRestOfLine ("STATUS", snd (the (readToken ())))
+ fun readObjective () = readRestOfLine ("OBJECTIVE", snd (the (readToken (); readToken (); readToken ())))
+ val t1 = readToken ()
+ val t2 = readToken ()
+ in
+ if is_tt t1 TOKEN_SYMBOL andalso is_tt t2 TOKEN_COLON
+ then
+ case to_upper (snd (the t1)) of
+ "STATUS" => (readStatus ())::(readHeader ())
+ | "OBJECTIVE" => (readObjective())::(readHeader ())
+ | _ => (readRestOfLine (); readHeader ())
+ else
+ (pushToken t2; pushToken t1; [])
+ end
+
+ fun skip_until_sep () =
+ let val x = readToken () in
+ if is_tt x TOKEN_SEPARATOR then
+ readRestOfLine ()
+ else
+ skip_until_sep ()
+ end
+
+ fun load_value () =
+ let
+ val t1 = readToken ()
+ val t2 = readToken ()
+ in
+ if is_tt t1 TOKEN_NUM andalso is_tt t2 TOKEN_SYMBOL then
+ let
+ val t = readToken ()
+ val state = if is_tt t TOKEN_NL then readToken () else t
+ val _ = if is_tt state TOKEN_SYMBOL then () else raise (Load_cplexResult "state expected")
+ val k = readToken ()
+ in
+ if is_tt k TOKEN_NUM then
+ readRestOfLine (SOME (snd (the t2), snd (the k)))
+ else
+ raise (Load_cplexResult "number expected")
+ end
+ else
+ (pushToken t2; pushToken t1; NONE)
+ end
+
+ fun load_values () =
+ let val v = load_value () in
+ if v = NONE then [] else (the v)::(load_values ())
+ end
+
+ val header = readHeader ()
+
+ val result =
+ case AList.lookup (op =) header "STATUS" of
+ SOME "INFEASIBLE" => Infeasible
+ | SOME "UNBOUNDED" => Unbounded
+ | SOME "OPTIMAL" => Optimal (the (AList.lookup (op =) header "OBJECTIVE"),
+ (skip_until_sep ();
+ skip_until_sep ();
+ load_values ()))
+ | _ => Undefined
+
+ val _ = TextIO.closeIn f
+ in
+ result
+ end
+ handle (Tokenize s) => raise (Load_cplexResult ("Tokenize: "^s))
+ | Option => raise (Load_cplexResult "Option")
+
+fun load_cplexResult name =
+ let
+ val flist = [(is_NL, TOKEN_NL),
+ (is_blank, TOKEN_BLANK),
+ (is_num, TOKEN_NUM),
+ (is_sign, TOKEN_SIGN),
+ (is_colon, TOKEN_COLON),
+ (is_cmp, TOKEN_CMP),
+ (is_resultsymbol, TOKEN_SYMBOL)]
+
+ val tokenize = tokenize_general flist
+
+ val f = TextIO.openIn name
+
+ val rest = Unsynchronized.ref []
+
+ fun readToken_helper () =
+ if length (!rest) > 0 then
+ let val u = hd (!rest) in
+ (
+ rest := tl (!rest);
+ SOME u
+ )
+ end
+ else
+ (case TextIO.inputLine f of
+ NONE => NONE
+ | SOME s => (rest := tokenize s; readToken_helper()))
+
+ fun is_tt tok ty = (tok <> NONE andalso (fst (the tok)) = ty)
+
+ fun pushToken a = if a = NONE then () else (rest := ((the a)::(!rest)))
+
+ fun readToken () =
+ let val t = readToken_helper () in
+ if is_tt t TOKEN_BLANK then
+ readToken ()
+ else if is_tt t TOKEN_SIGN then
+ let val t2 = readToken_helper () in
+ if is_tt t2 TOKEN_NUM then
+ (SOME (TOKEN_NUM, (snd (the t))^(snd (the t2))))
+ else
+ (pushToken t2; t)
+ end
+ else
+ t
+ end
+
+ fun readRestOfLine P =
+ let
+ val t = readToken ()
+ in
+ if is_tt t TOKEN_NL orelse t = NONE
+ then P
+ else readRestOfLine P
+ end
+
+ fun readHeader () =
+ let
+ fun readStatus () = readRestOfLine ("STATUS", snd (the (readToken ())))
+ fun readObjective () =
+ let
+ val t = readToken ()
+ in
+ if is_tt t TOKEN_SYMBOL andalso to_upper (snd (the t)) = "VALUE" then
+ readRestOfLine ("OBJECTIVE", snd (the (readToken())))
+ else
+ readRestOfLine ("OBJECTIVE_NAME", snd (the t))
+ end
+
+ val t = readToken ()
+ in
+ if is_tt t TOKEN_SYMBOL then
+ case to_upper (snd (the t)) of
+ "STATUS" => (readStatus ())::(readHeader ())
+ | "OBJECTIVE" => (readObjective ())::(readHeader ())
+ | "SECTION" => (pushToken t; [])
+ | _ => (readRestOfLine (); readHeader ())
+ else
+ (readRestOfLine (); readHeader ())
+ end
+
+ fun skip_nls () =
+ let val x = readToken () in
+ if is_tt x TOKEN_NL then
+ skip_nls ()
+ else
+ (pushToken x; ())
+ end
+
+ fun skip_paragraph () =
+ if is_tt (readToken ()) TOKEN_NL then
+ (if is_tt (readToken ()) TOKEN_NL then
+ skip_nls ()
+ else
+ skip_paragraph ())
+ else
+ skip_paragraph ()
+
+ fun load_value () =
+ let
+ val t1 = readToken ()
+ val t1 = if is_tt t1 TOKEN_SYMBOL andalso snd (the t1) = "A" then readToken () else t1
+ in
+ if is_tt t1 TOKEN_NUM then
+ let
+ val name = readToken ()
+ val status = readToken ()
+ val value = readToken ()
+ in
+ if is_tt name TOKEN_SYMBOL andalso
+ is_tt status TOKEN_SYMBOL andalso
+ is_tt value TOKEN_NUM
+ then
+ readRestOfLine (SOME (snd (the name), snd (the value)))
+ else
+ raise (Load_cplexResult "column line expected")
+ end
+ else
+ (pushToken t1; NONE)
+ end
+
+ fun load_values () =
+ let val v = load_value () in
+ if v = NONE then [] else (the v)::(load_values ())
+ end
+
+ val header = readHeader ()
+
+ val result =
+ case AList.lookup (op =) header "STATUS" of
+ SOME "INFEASIBLE" => Infeasible
+ | SOME "NONOPTIMAL" => Unbounded
+ | SOME "OPTIMAL" => Optimal (the (AList.lookup (op =) header "OBJECTIVE"),
+ (skip_paragraph ();
+ skip_paragraph ();
+ skip_paragraph ();
+ skip_paragraph ();
+ skip_paragraph ();
+ load_values ()))
+ | _ => Undefined
+
+ val _ = TextIO.closeIn f
+ in
+ result
+ end
+ handle (Tokenize s) => raise (Load_cplexResult ("Tokenize: "^s))
+ | Option => raise (Load_cplexResult "Option")
+
+exception Execute of string;
+
+fun tmp_file s = Path.implode (Path.expand (File.tmp_path (Path.basic s)));
+fun wrap s = "\""^s^"\"";
+
+fun solve_glpk prog =
+ let
+ val name = string_of_int (Time.toMicroseconds (Time.now ()))
+ val lpname = tmp_file (name^".lp")
+ val resultname = tmp_file (name^".txt")
+ val _ = save_cplexFile lpname prog
+ val cplex_path = getenv "GLPK_PATH"
+ val cplex = if cplex_path = "" then "glpsol" else cplex_path
+ val command = (wrap cplex)^" --lpt "^(wrap lpname)^" --output "^(wrap resultname)
+ val answer = #1 (Isabelle_System.bash_output command)
+ in
+ let
+ val result = load_glpkResult resultname
+ val _ = OS.FileSys.remove lpname
+ val _ = OS.FileSys.remove resultname
+ in
+ result
+ end
+ handle (Load_cplexResult s) => raise (Execute ("Load_cplexResult: "^s^"\nExecute: "^answer))
+ | _ => raise (Execute answer) (* FIXME avoid handle _ *)
+ end
+
+fun solve_cplex prog =
+ let
+ fun write_script s lp r =
+ let
+ val f = TextIO.openOut s
+ val _ = TextIO.output (f, "read\n"^lp^"\noptimize\nwrite\n"^r^"\nquit")
+ val _ = TextIO.closeOut f
+ in
+ ()
+ end
+
+ val name = string_of_int (Time.toMicroseconds (Time.now ()))
+ val lpname = tmp_file (name^".lp")
+ val resultname = tmp_file (name^".txt")
+ val scriptname = tmp_file (name^".script")
+ val _ = save_cplexFile lpname prog
+ val _ = write_script scriptname lpname resultname
+ in
+ let
+ val result = load_cplexResult resultname
+ val _ = OS.FileSys.remove lpname
+ val _ = OS.FileSys.remove resultname
+ val _ = OS.FileSys.remove scriptname
+ in
+ result
+ end
+ end
+
+fun solve prog =
+ case get_solver () of
+ SOLVER_DEFAULT =>
+ (case getenv "LP_SOLVER" of
+ "CPLEX" => solve_cplex prog
+ | "GLPK" => solve_glpk prog
+ | _ => raise (Execute ("LP_SOLVER must be set to CPLEX or to GLPK")))
+ | SOLVER_CPLEX => solve_cplex prog
+ | SOLVER_GLPK => solve_glpk prog
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/FloatSparseMatrixBuilder.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,284 @@
+(* Title: HOL/Matrix/FloatSparseMatrixBuilder.ML
+ Author: Steven Obua
+*)
+
+signature FLOAT_SPARSE_MATRIX_BUILDER =
+sig
+ include MATRIX_BUILDER
+
+ structure cplex : CPLEX
+
+ type float = Float.float
+ val approx_value : int -> (float -> float) -> string -> term * term
+ val approx_vector : int -> (float -> float) -> vector -> term * term
+ val approx_matrix : int -> (float -> float) -> matrix -> term * term
+
+ val mk_spvec_entry : int -> float -> term
+ val mk_spvec_entry' : int -> term -> term
+ val mk_spmat_entry : int -> term -> term
+ val spvecT: typ
+ val spmatT: typ
+
+ val v_elem_at : vector -> int -> string option
+ val m_elem_at : matrix -> int -> vector option
+ val v_only_elem : vector -> int option
+ val v_fold : (int * string -> 'a -> 'a) -> vector -> 'a -> 'a
+ val m_fold : (int * vector -> 'a -> 'a) -> matrix -> 'a -> 'a
+
+ val transpose_matrix : matrix -> matrix
+
+ val cut_vector : int -> vector -> vector
+ val cut_matrix : vector -> int option -> matrix -> matrix
+
+ val delete_matrix : int list -> matrix -> matrix
+ val cut_matrix' : int list -> matrix -> matrix
+ val delete_vector : int list -> vector -> vector
+ val cut_vector' : int list -> vector -> vector
+
+ val indices_of_matrix : matrix -> int list
+ val indices_of_vector : vector -> int list
+
+ (* cplexProg c A b *)
+ val cplexProg : vector -> matrix -> vector -> cplex.cplexProg * (string -> int)
+ (* dual_cplexProg c A b *)
+ val dual_cplexProg : vector -> matrix -> vector -> cplex.cplexProg * (string -> int)
+end;
+
+structure FloatSparseMatrixBuilder : FLOAT_SPARSE_MATRIX_BUILDER =
+struct
+
+type float = Float.float
+structure Inttab = Table(type key = int val ord = rev_order o int_ord);
+
+type vector = string Inttab.table
+type matrix = vector Inttab.table
+
+val spvec_elemT = HOLogic.mk_prodT (HOLogic.natT, HOLogic.realT);
+val spvecT = HOLogic.listT spvec_elemT;
+val spmat_elemT = HOLogic.mk_prodT (HOLogic.natT, spvecT);
+val spmatT = HOLogic.listT spmat_elemT;
+
+fun approx_value prec f =
+ FloatArith.approx_float prec (fn (x, y) => (f x, f y));
+
+fun mk_spvec_entry i f =
+ HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, FloatArith.mk_float f);
+
+fun mk_spvec_entry' i x =
+ HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, x);
+
+fun mk_spmat_entry i e =
+ HOLogic.mk_prod (HOLogic.mk_number HOLogic.natT i, e);
+
+fun approx_vector prec pprt vector =
+ let
+ fun app (index, s) (lower, upper) =
+ let
+ val (flower, fupper) = approx_value prec pprt s
+ val index = HOLogic.mk_number HOLogic.natT index
+ val elower = HOLogic.mk_prod (index, flower)
+ val eupper = HOLogic.mk_prod (index, fupper)
+ in (elower :: lower, eupper :: upper) end;
+ in
+ pairself (HOLogic.mk_list spvec_elemT) (Inttab.fold app vector ([], []))
+ end;
+
+fun approx_matrix prec pprt vector =
+ let
+ fun app (index, v) (lower, upper) =
+ let
+ val (flower, fupper) = approx_vector prec pprt v
+ val index = HOLogic.mk_number HOLogic.natT index
+ val elower = HOLogic.mk_prod (index, flower)
+ val eupper = HOLogic.mk_prod (index, fupper)
+ in (elower :: lower, eupper :: upper) end;
+ in
+ pairself (HOLogic.mk_list spmat_elemT) (Inttab.fold app vector ([], []))
+ end;
+
+exception Nat_expected of int;
+
+val zero_interval = approx_value 1 I "0"
+
+fun set_elem vector index str =
+ if index < 0 then
+ raise (Nat_expected index)
+ else if (approx_value 1 I str) = zero_interval then
+ vector
+ else
+ Inttab.update (index, str) vector
+
+fun set_vector matrix index vector =
+ if index < 0 then
+ raise (Nat_expected index)
+ else if Inttab.is_empty vector then
+ matrix
+ else
+ Inttab.update (index, vector) matrix
+
+val empty_matrix = Inttab.empty
+val empty_vector = Inttab.empty
+
+(* dual stuff *)
+
+structure cplex = Cplex
+
+fun transpose_matrix matrix =
+ let
+ fun upd j (i, s) =
+ Inttab.map_default (i, Inttab.empty) (Inttab.update (j, s));
+ fun updm (j, v) = Inttab.fold (upd j) v;
+ in Inttab.fold updm matrix empty_matrix end;
+
+exception No_name of string;
+
+exception Superfluous_constr_right_hand_sides
+
+fun cplexProg c A b =
+ let
+ val ytable = Unsynchronized.ref Inttab.empty
+ fun indexof s =
+ if String.size s = 0 then raise (No_name s)
+ else case Int.fromString (String.extract(s, 1, NONE)) of
+ SOME i => i | NONE => raise (No_name s)
+
+ fun nameof i =
+ let
+ val s = "x" ^ string_of_int i
+ val _ = Unsynchronized.change ytable (Inttab.update (i, s))
+ in
+ s
+ end
+
+ fun split_numstr s =
+ if String.isPrefix "-" s then (false,String.extract(s, 1, NONE))
+ else if String.isPrefix "+" s then (true, String.extract(s, 1, NONE))
+ else (true, s)
+
+ fun mk_term index s =
+ let
+ val (p, s) = split_numstr s
+ val prod = cplex.cplexProd (cplex.cplexNum s, cplex.cplexVar (nameof index))
+ in
+ if p then prod else cplex.cplexNeg prod
+ end
+
+ fun vec2sum vector =
+ cplex.cplexSum (Inttab.fold (fn (index, s) => fn list => (mk_term index s) :: list) vector [])
+
+ fun mk_constr index vector c =
+ let
+ val s = case Inttab.lookup c index of SOME s => s | NONE => "0"
+ val (p, s) = split_numstr s
+ val num = if p then cplex.cplexNum s else cplex.cplexNeg (cplex.cplexNum s)
+ in
+ (NONE, cplex.cplexConstr (cplex.cplexLeq, (vec2sum vector, num)))
+ end
+
+ fun delete index c = Inttab.delete index c handle Inttab.UNDEF _ => c
+
+ val (list, b) = Inttab.fold
+ (fn (index, v) => fn (list, c) => ((mk_constr index v c)::list, delete index c))
+ A ([], b)
+ val _ = if Inttab.is_empty b then () else raise Superfluous_constr_right_hand_sides
+
+ fun mk_free y = cplex.cplexBounds (cplex.cplexNeg cplex.cplexInf, cplex.cplexLeq,
+ cplex.cplexVar y, cplex.cplexLeq,
+ cplex.cplexInf)
+
+ val yvars = Inttab.fold (fn (_, y) => fn l => (mk_free y)::l) (!ytable) []
+
+ val prog = cplex.cplexProg ("original", cplex.cplexMaximize (vec2sum c), list, yvars)
+ in
+ (prog, indexof)
+ end
+
+
+fun dual_cplexProg c A b =
+ let
+ fun indexof s =
+ if String.size s = 0 then raise (No_name s)
+ else case Int.fromString (String.extract(s, 1, NONE)) of
+ SOME i => i | NONE => raise (No_name s)
+
+ fun nameof i = "y" ^ string_of_int i
+
+ fun split_numstr s =
+ if String.isPrefix "-" s then (false,String.extract(s, 1, NONE))
+ else if String.isPrefix "+" s then (true, String.extract(s, 1, NONE))
+ else (true, s)
+
+ fun mk_term index s =
+ let
+ val (p, s) = split_numstr s
+ val prod = cplex.cplexProd (cplex.cplexNum s, cplex.cplexVar (nameof index))
+ in
+ if p then prod else cplex.cplexNeg prod
+ end
+
+ fun vec2sum vector =
+ cplex.cplexSum (Inttab.fold (fn (index, s) => fn list => (mk_term index s)::list) vector [])
+
+ fun mk_constr index vector c =
+ let
+ val s = case Inttab.lookup c index of SOME s => s | NONE => "0"
+ val (p, s) = split_numstr s
+ val num = if p then cplex.cplexNum s else cplex.cplexNeg (cplex.cplexNum s)
+ in
+ (NONE, cplex.cplexConstr (cplex.cplexEq, (vec2sum vector, num)))
+ end
+
+ fun delete index c = Inttab.delete index c handle Inttab.UNDEF _ => c
+
+ val (list, c) = Inttab.fold
+ (fn (index, v) => fn (list, c) => ((mk_constr index v c)::list, delete index c))
+ (transpose_matrix A) ([], c)
+ val _ = if Inttab.is_empty c then () else raise Superfluous_constr_right_hand_sides
+
+ val prog = cplex.cplexProg ("dual", cplex.cplexMinimize (vec2sum b), list, [])
+ in
+ (prog, indexof)
+ end
+
+fun cut_vector size v =
+ let
+ val count = Unsynchronized.ref 0;
+ fun app (i, s) = if (!count < size) then
+ (count := !count +1 ; Inttab.update (i, s))
+ else I
+ in
+ Inttab.fold app v empty_vector
+ end
+
+fun cut_matrix vfilter vsize m =
+ let
+ fun app (i, v) =
+ if is_none (Inttab.lookup vfilter i) then I
+ else case vsize
+ of NONE => Inttab.update (i, v)
+ | SOME s => Inttab.update (i, cut_vector s v)
+ in Inttab.fold app m empty_matrix end
+
+fun v_elem_at v i = Inttab.lookup v i
+fun m_elem_at m i = Inttab.lookup m i
+
+fun v_only_elem v =
+ case Inttab.min_key v of
+ NONE => NONE
+ | SOME vmin => (case Inttab.max_key v of
+ NONE => SOME vmin
+ | SOME vmax => if vmin = vmax then SOME vmin else NONE)
+
+fun v_fold f = Inttab.fold f;
+fun m_fold f = Inttab.fold f;
+
+fun indices_of_vector v = Inttab.keys v
+fun indices_of_matrix m = Inttab.keys m
+fun delete_vector indices v = fold Inttab.delete indices v
+fun delete_matrix indices m = fold Inttab.delete indices m
+fun cut_matrix' indices _ = fold (fn i => fn m => (case Inttab.lookup m i of NONE => m | SOME v => Inttab.update (i, v) m)) indices Inttab.empty
+fun cut_vector' indices _ = fold (fn i => fn v => (case Inttab.lookup v i of NONE => v | SOME x => Inttab.update (i, x) v)) indices Inttab.empty
+
+
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/LP.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,164 @@
+(* Title: HOL/Matrix/LP.thy
+ Author: Steven Obua
+*)
+
+theory LP
+imports Main "~~/src/HOL/Library/Lattice_Algebras"
+begin
+
+lemma le_add_right_mono:
+ assumes
+ "a <= b + (c::'a::ordered_ab_group_add)"
+ "c <= d"
+ shows "a <= b + d"
+ apply (rule_tac order_trans[where y = "b+c"])
+ apply (simp_all add: assms)
+ done
+
+lemma linprog_dual_estimate:
+ assumes
+ "A * x \<le> (b::'a::lattice_ring)"
+ "0 \<le> y"
+ "abs (A - A') \<le> \<delta>A"
+ "b \<le> b'"
+ "abs (c - c') \<le> \<delta>c"
+ "abs x \<le> r"
+ shows
+ "c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
+proof -
+ from assms have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
+ from assms have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: algebra_simps)
+ from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
+ have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
+ by (simp only: 4 estimate_by_abs)
+ have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
+ by (simp add: abs_le_mult)
+ have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
+ by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
+ have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <= (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
+ by (simp add: abs_triangle_ineq mult_right_mono)
+ have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
+ by (simp add: abs_le_mult mult_right_mono)
+ have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
+ have 11: "abs (c'-c) = abs (c-c')"
+ by (subst 10, subst abs_minus_cancel, simp)
+ have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
+ by (simp add: 11 assms mult_right_mono)
+ have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
+ by (simp add: assms mult_right_mono mult_left_mono)
+ have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
+ apply (rule mult_left_mono)
+ apply (simp add: assms)
+ apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
+ apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
+ apply (simp_all)
+ apply (rule order_trans[where y="abs (A-A')"], simp_all add: assms)
+ apply (rule order_trans[where y="abs (c-c')"], simp_all add: assms)
+ done
+ from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
+ by (simp)
+ show ?thesis
+ apply (rule le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
+ apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified assms]])
+ done
+qed
+
+lemma le_ge_imp_abs_diff_1:
+ assumes
+ "A1 <= (A::'a::lattice_ring)"
+ "A <= A2"
+ shows "abs (A-A1) <= A2-A1"
+proof -
+ have "0 <= A - A1"
+ proof -
+ have 1: "A - A1 = A + (- A1)" by simp
+ show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified assms])
+ qed
+ then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
+ with assms show "abs (A-A1) <= (A2-A1)" by simp
+qed
+
+lemma mult_le_prts:
+ assumes
+ "a1 <= (a::'a::lattice_ring)"
+ "a <= a2"
+ "b1 <= b"
+ "b <= b2"
+ shows
+ "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
+proof -
+ have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
+ apply (subst prts[symmetric])+
+ apply simp
+ done
+ then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
+ by (simp add: algebra_simps)
+ moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
+ by (simp_all add: assms mult_mono)
+ moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
+ proof -
+ have "pprt a * nprt b <= pprt a * nprt b2"
+ by (simp add: mult_left_mono assms)
+ moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
+ by (simp add: mult_right_mono_neg assms)
+ ultimately show ?thesis
+ by simp
+ qed
+ moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
+ proof -
+ have "nprt a * pprt b <= nprt a2 * pprt b"
+ by (simp add: mult_right_mono assms)
+ moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
+ by (simp add: mult_left_mono_neg assms)
+ ultimately show ?thesis
+ by simp
+ qed
+ moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
+ proof -
+ have "nprt a * nprt b <= nprt a * nprt b1"
+ by (simp add: mult_left_mono_neg assms)
+ moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
+ by (simp add: mult_right_mono_neg assms)
+ ultimately show ?thesis
+ by simp
+ qed
+ ultimately show ?thesis
+ by - (rule add_mono | simp)+
+qed
+
+lemma mult_le_dual_prts:
+ assumes
+ "A * x \<le> (b::'a::lattice_ring)"
+ "0 \<le> y"
+ "A1 \<le> A"
+ "A \<le> A2"
+ "c1 \<le> c"
+ "c \<le> c2"
+ "r1 \<le> x"
+ "x \<le> r2"
+ shows
+ "c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
+ (is "_ <= _ + ?C")
+proof -
+ from assms have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: algebra_simps)
+ ultimately have "c * x + (y * A - c) * x <= y * b" by simp
+ then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
+ then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: algebra_simps)
+ have s2: "c - y * A <= c2 - y * A1"
+ by (simp add: diff_minus assms add_mono mult_left_mono)
+ have s1: "c1 - y * A2 <= c - y * A"
+ by (simp add: diff_minus assms add_mono mult_left_mono)
+ have prts: "(c - y * A) * x <= ?C"
+ apply (simp add: Let_def)
+ apply (rule mult_le_prts)
+ apply (simp_all add: assms s1 s2)
+ done
+ then have "y * b + (c - y * A) * x <= y * b + ?C"
+ by simp
+ with cx show ?thesis
+ by(simp only:)
+qed
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/Matrix.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,1836 @@
+(* Title: HOL/Matrix/Matrix.thy
+ Author: Steven Obua
+*)
+
+theory Matrix
+imports Main "~~/src/HOL/Library/Lattice_Algebras"
+begin
+
+type_synonym 'a infmatrix = "nat \<Rightarrow> nat \<Rightarrow> 'a"
+
+definition nonzero_positions :: "(nat \<Rightarrow> nat \<Rightarrow> 'a::zero) \<Rightarrow> (nat \<times> nat) set" where
+ "nonzero_positions A = {pos. A (fst pos) (snd pos) ~= 0}"
+
+definition "matrix = {(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
+
+typedef (open) 'a matrix = "matrix :: (nat \<Rightarrow> nat \<Rightarrow> 'a::zero) set"
+ unfolding matrix_def
+proof
+ show "(\<lambda>j i. 0) \<in> {(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
+ by (simp add: nonzero_positions_def)
+qed
+
+declare Rep_matrix_inverse[simp]
+
+lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
+ by (induct A) (simp add: Abs_matrix_inverse matrix_def)
+
+definition nrows :: "('a::zero) matrix \<Rightarrow> nat" where
+ "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
+
+definition ncols :: "('a::zero) matrix \<Rightarrow> nat" where
+ "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))"
+
+lemma nrows:
+ assumes hyp: "nrows A \<le> m"
+ shows "(Rep_matrix A m n) = 0"
+proof cases
+ assume "nonzero_positions(Rep_matrix A) = {}"
+ then show "(Rep_matrix A m n) = 0" by (simp add: nonzero_positions_def)
+next
+ assume a: "nonzero_positions(Rep_matrix A) \<noteq> {}"
+ let ?S = "fst`(nonzero_positions(Rep_matrix A))"
+ have c: "finite (?S)" by (simp add: finite_nonzero_positions)
+ from hyp have d: "Max (?S) < m" by (simp add: a nrows_def)
+ have "m \<notin> ?S"
+ proof -
+ have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max_ge [OF c])
+ moreover from d have "~(m <= Max ?S)" by (simp)
+ ultimately show "m \<notin> ?S" by (auto)
+ qed
+ thus "Rep_matrix A m n = 0" by (simp add: nonzero_positions_def image_Collect)
+qed
+
+definition transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix" where
+ "transpose_infmatrix A j i == A i j"
+
+definition transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix" where
+ "transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"
+
+declare transpose_infmatrix_def[simp]
+
+lemma transpose_infmatrix_twice[simp]: "transpose_infmatrix (transpose_infmatrix A) = A"
+by ((rule ext)+, simp)
+
+lemma transpose_infmatrix: "transpose_infmatrix (% j i. P j i) = (% j i. P i j)"
+ apply (rule ext)+
+ by simp
+
+lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
+apply (rule Abs_matrix_inverse)
+apply (simp add: matrix_def nonzero_positions_def image_def)
+proof -
+ let ?A = "{pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}"
+ let ?swap = "% pos. (snd pos, fst pos)"
+ let ?B = "{pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}"
+ have swap_image: "?swap`?A = ?B"
+ apply (simp add: image_def)
+ apply (rule set_eqI)
+ apply (simp)
+ proof
+ fix y
+ assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)"
+ thus "Rep_matrix x (fst y) (snd y) \<noteq> 0"
+ proof -
+ from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
+ then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
+ qed
+ next
+ fix y
+ assume hyp: "Rep_matrix x (fst y) (snd y) \<noteq> 0"
+ show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))"
+ by (rule exI[of _ "snd y"], rule exI[of _ "fst y"]) (simp add: hyp)
+ qed
+ then have "finite (?swap`?A)"
+ proof -
+ have "finite (nonzero_positions (Rep_matrix x))" by (simp add: finite_nonzero_positions)
+ then have "finite ?B" by (simp add: nonzero_positions_def)
+ with swap_image show "finite (?swap`?A)" by (simp)
+ qed
+ moreover
+ have "inj_on ?swap ?A" by (simp add: inj_on_def)
+ ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A])
+qed
+
+lemma infmatrixforward: "(x::'a infmatrix) = y \<Longrightarrow> \<forall> a b. x a b = y a b" by auto
+
+lemma transpose_infmatrix_inject: "(transpose_infmatrix A = transpose_infmatrix B) = (A = B)"
+apply (auto)
+apply (rule ext)+
+apply (simp add: transpose_infmatrix)
+apply (drule infmatrixforward)
+apply (simp)
+done
+
+lemma transpose_matrix_inject: "(transpose_matrix A = transpose_matrix B) = (A = B)"
+apply (simp add: transpose_matrix_def)
+apply (subst Rep_matrix_inject[THEN sym])+
+apply (simp only: transpose_infmatrix_closed transpose_infmatrix_inject)
+done
+
+lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
+by (simp add: transpose_matrix_def)
+
+lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
+by (simp add: transpose_matrix_def)
+
+lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A"
+by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
+
+lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A"
+by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
+
+lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0"
+proof -
+ assume "ncols A <= n"
+ then have "nrows (transpose_matrix A) <= n" by (simp)
+ then have "Rep_matrix (transpose_matrix A) n m = 0" by (rule nrows)
+ thus "Rep_matrix A m n = 0" by (simp add: transpose_matrix_def)
+qed
+
+lemma ncols_le: "(ncols A <= n) = (! j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
+apply (auto)
+apply (simp add: ncols)
+proof (simp add: ncols_def, auto)
+ let ?P = "nonzero_positions (Rep_matrix A)"
+ let ?p = "snd`?P"
+ have a:"finite ?p" by (simp add: finite_nonzero_positions)
+ let ?m = "Max ?p"
+ assume "~(Suc (?m) <= n)"
+ then have b:"n <= ?m" by (simp)
+ fix a b
+ assume "(a,b) \<in> ?P"
+ then have "?p \<noteq> {}" by (auto)
+ with a have "?m \<in> ?p" by (simp)
+ moreover have "!x. (x \<in> ?p \<longrightarrow> (? y. (Rep_matrix A y x) \<noteq> 0))" by (simp add: nonzero_positions_def image_def)
+ ultimately have "? y. (Rep_matrix A y ?m) \<noteq> 0" by (simp)
+ moreover assume ?st
+ ultimately show "False" using b by (simp)
+qed
+
+lemma less_ncols: "(n < ncols A) = (? j i. n <= i & (Rep_matrix A j i) \<noteq> 0)"
+proof -
+ have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+ show ?thesis by (simp add: a ncols_le)
+qed
+
+lemma le_ncols: "(n <= ncols A) = (\<forall> m. (\<forall> j i. m <= i \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
+apply (auto)
+apply (subgoal_tac "ncols A <= m")
+apply (simp)
+apply (simp add: ncols_le)
+apply (drule_tac x="ncols A" in spec)
+by (simp add: ncols)
+
+lemma nrows_le: "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" (is ?s)
+proof -
+ have "(nrows A <= n) = (ncols (transpose_matrix A) <= n)" by (simp)
+ also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
+ also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
+ finally show "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
+qed
+
+lemma less_nrows: "(m < nrows A) = (? j i. m <= j & (Rep_matrix A j i) \<noteq> 0)"
+proof -
+ have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+ show ?thesis by (simp add: a nrows_le)
+qed
+
+lemma le_nrows: "(n <= nrows A) = (\<forall> m. (\<forall> j i. m <= j \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
+apply (auto)
+apply (subgoal_tac "nrows A <= m")
+apply (simp)
+apply (simp add: nrows_le)
+apply (drule_tac x="nrows A" in spec)
+by (simp add: nrows)
+
+lemma nrows_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> m < nrows A"
+apply (case_tac "nrows A <= m")
+apply (simp_all add: nrows)
+done
+
+lemma ncols_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> n < ncols A"
+apply (case_tac "ncols A <= n")
+apply (simp_all add: ncols)
+done
+
+lemma finite_natarray1: "finite {x. x < (n::nat)}"
+apply (induct n)
+apply (simp)
+proof -
+ fix n
+ have "{x. x < Suc n} = insert n {x. x < n}" by (rule set_eqI, simp, arith)
+ moreover assume "finite {x. x < n}"
+ ultimately show "finite {x. x < Suc n}" by (simp)
+qed
+
+lemma finite_natarray2: "finite {pos. (fst pos) < (m::nat) & (snd pos) < (n::nat)}"
+ apply (induct m)
+ apply (simp+)
+ proof -
+ fix m::nat
+ let ?s0 = "{pos. fst pos < m & snd pos < n}"
+ let ?s1 = "{pos. fst pos < (Suc m) & snd pos < n}"
+ let ?sd = "{pos. fst pos = m & snd pos < n}"
+ assume f0: "finite ?s0"
+ have f1: "finite ?sd"
+ proof -
+ let ?f = "% x. (m, x)"
+ have "{pos. fst pos = m & snd pos < n} = ?f ` {x. x < n}" by (rule set_eqI, simp add: image_def, auto)
+ moreover have "finite {x. x < n}" by (simp add: finite_natarray1)
+ ultimately show "finite {pos. fst pos = m & snd pos < n}" by (simp)
+ qed
+ have su: "?s0 \<union> ?sd = ?s1" by (rule set_eqI, simp, arith)
+ from f0 f1 have "finite (?s0 \<union> ?sd)" by (rule finite_UnI)
+ with su show "finite ?s1" by (simp)
+qed
+
+lemma RepAbs_matrix:
+ assumes aem: "? m. ! j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"? n. ! j i. (n <= i \<longrightarrow> x j i = 0)" (is ?en)
+ shows "(Rep_matrix (Abs_matrix x)) = x"
+apply (rule Abs_matrix_inverse)
+apply (simp add: matrix_def nonzero_positions_def)
+proof -
+ from aem obtain m where a: "! j i. m <= j \<longrightarrow> x j i = 0" by (blast)
+ from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)
+ let ?u = "{pos. x (fst pos) (snd pos) \<noteq> 0}"
+ let ?v = "{pos. fst pos < m & snd pos < n}"
+ have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
+ from a b have "(?u \<inter> (-?v)) = {}"
+ apply (simp)
+ apply (rule set_eqI)
+ apply (simp)
+ apply auto
+ by (rule c, auto)+
+ then have d: "?u \<subseteq> ?v" by blast
+ moreover have "finite ?v" by (simp add: finite_natarray2)
+ ultimately show "finite ?u" by (rule finite_subset)
+qed
+
+definition apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix" where
+ "apply_infmatrix f == % A. (% j i. f (A j i))"
+
+definition apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix" where
+ "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
+
+definition combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix" where
+ "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
+
+definition combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix" where
+ "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
+
+lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
+by (simp add: apply_infmatrix_def)
+
+lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
+by (simp add: combine_infmatrix_def)
+
+definition commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool" where
+"commutative f == ! x y. f x y = f y x"
+
+definition associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
+"associative f == ! x y z. f (f x y) z = f x (f y z)"
+
+text{*
+To reason about associativity and commutativity of operations on matrices,
+let's take a step back and look at the general situtation: Assume that we have
+sets $A$ and $B$ with $B \subset A$ and an abstraction $u: A \rightarrow B$. This abstraction has to fulfill $u(b) = b$ for all $b \in B$, but is arbitrary otherwise.
+Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
+It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
+*}
+
+lemma combine_infmatrix_commute:
+ "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
+by (simp add: commutative_def combine_infmatrix_def)
+
+lemma combine_matrix_commute:
+"commutative f \<Longrightarrow> commutative (combine_matrix f)"
+by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
+
+text{*
+On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
+as $f$ we take addition on $\ganz$, which is clearly associative. The abstraction is given by $u(a) = 0$ for $a \notin B$. Then we have
+\[ f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1, \]
+but on the other hand we have
+\[ f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.\]
+A way out of this problem is to assume that $f(A\times A)\subset A$ holds, and this is what we are going to do:
+*}
+
+lemma nonzero_positions_combine_infmatrix[simp]: "f 0 0 = 0 \<Longrightarrow> nonzero_positions (combine_infmatrix f A B) \<subseteq> (nonzero_positions A) \<union> (nonzero_positions B)"
+by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
+
+lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
+by (insert Rep_matrix [of A], simp add: matrix_def)
+
+lemma combine_infmatrix_closed [simp]:
+ "f 0 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) = combine_infmatrix f (Rep_matrix A) (Rep_matrix B)"
+apply (rule Abs_matrix_inverse)
+apply (simp add: matrix_def)
+apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
+by (simp_all)
+
+text {* We need the next two lemmas only later, but it is analog to the above one, so we prove them now: *}
+lemma nonzero_positions_apply_infmatrix[simp]: "f 0 = 0 \<Longrightarrow> nonzero_positions (apply_infmatrix f A) \<subseteq> nonzero_positions A"
+by (rule subsetI, simp add: nonzero_positions_def apply_infmatrix_def, auto)
+
+lemma apply_infmatrix_closed [simp]:
+ "f 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
+apply (rule Abs_matrix_inverse)
+apply (simp add: matrix_def)
+apply (rule finite_subset[of _ "nonzero_positions (Rep_matrix A)"])
+by (simp_all)
+
+lemma combine_infmatrix_assoc[simp]: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_infmatrix f)"
+by (simp add: associative_def combine_infmatrix_def)
+
+lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
+by (auto)
+
+lemma combine_matrix_assoc: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_matrix f)"
+apply (simp(no_asm) add: associative_def combine_matrix_def, auto)
+apply (rule comb [of Abs_matrix Abs_matrix])
+by (auto, insert combine_infmatrix_assoc[of f], simp add: associative_def)
+
+lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
+by (simp add: apply_matrix_def)
+
+lemma Rep_combine_matrix[simp]: "f 0 0 = 0 \<Longrightarrow> Rep_matrix (combine_matrix f A B) j i = f (Rep_matrix A j i) (Rep_matrix B j i)"
+ by(simp add: combine_matrix_def)
+
+lemma combine_nrows_max: "f 0 0 = 0 \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
+by (simp add: nrows_le)
+
+lemma combine_ncols_max: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"
+by (simp add: ncols_le)
+
+lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A <= q \<Longrightarrow> nrows B <= q \<Longrightarrow> nrows(combine_matrix f A B) <= q"
+ by (simp add: nrows_le)
+
+lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A <= q \<Longrightarrow> ncols B <= q \<Longrightarrow> ncols(combine_matrix f A B) <= q"
+ by (simp add: ncols_le)
+
+definition zero_r_neutral :: "('a \<Rightarrow> 'b::zero \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "zero_r_neutral f == ! a. f a 0 = a"
+
+definition zero_l_neutral :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
+ "zero_l_neutral f == ! a. f 0 a = a"
+
+definition zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool" where
+ "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
+
+primrec foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
+where
+ "foldseq f s 0 = s 0"
+| "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
+
+primrec foldseq_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
+where
+ "foldseq_transposed f s 0 = s 0"
+| "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
+
+lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
+proof -
+ assume a:"associative f"
+ then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ proof -
+ fix n
+ show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ proof (induct n)
+ show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
+ next
+ fix n
+ assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
+ show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
+ proof (auto)
+ fix N t
+ assume Nsuc: "N <= Suc n"
+ show "foldseq f t N = foldseq_transposed f t N"
+ proof cases
+ assume "N <= n"
+ then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
+ next
+ assume "~(N <= n)"
+ with Nsuc have Nsuceq: "N = Suc n" by simp
+ have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
+ have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
+ show "foldseq f t N = foldseq_transposed f t N"
+ apply (simp add: Nsuceq)
+ apply (subst c)
+ apply (simp)
+ apply (case_tac "n = 0")
+ apply (simp)
+ apply (drule neqz)
+ apply (erule exE)
+ apply (simp)
+ apply (subst assocf)
+ proof -
+ fix m
+ assume "n = Suc m & Suc m <= n"
+ then have mless: "Suc m <= n" by arith
+ then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
+ apply (subst c)
+ by simp+
+ have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
+ have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
+ apply (subst c)
+ by (simp add: mless)+
+ have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
+ from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
+ qed
+ qed
+ qed
+ qed
+ qed
+ show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
+ qed
+
+lemma foldseq_distr: "\<lbrakk>associative f; commutative f\<rbrakk> \<Longrightarrow> foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)"
+proof -
+ assume assoc: "associative f"
+ assume comm: "commutative f"
+ from assoc have a:"!! x y z. f (f x y) z = f x (f y z)" by (simp add: associative_def)
+ from comm have b: "!! x y. f x y = f y x" by (simp add: commutative_def)
+ from assoc comm have c: "!! x y z. f x (f y z) = f y (f x z)" by (simp add: commutative_def associative_def)
+ have "!! n. (! u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
+ apply (induct_tac n)
+ apply (simp+, auto)
+ by (simp add: a b c)
+ then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
+qed
+
+theorem "\<lbrakk>associative f; associative g; \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); ? x y. (f x) \<noteq> (f y); ? x y. (g x) \<noteq> (g y); f x x = x; g x x = x\<rbrakk> \<Longrightarrow> f=g | (! y. f y x = y) | (! y. g y x = y)"
+oops
+(* Model found
+
+Trying to find a model that refutes: \<lbrakk>associative f; associative g;
+ \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); \<exists>x y. f x \<noteq> f y;
+ \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
+\<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)
+Searching for a model of size 1, translating term... invoking SAT solver... no model found.
+Searching for a model of size 2, translating term... invoking SAT solver... no model found.
+Searching for a model of size 3, translating term... invoking SAT solver...
+Model found:
+Size of types: 'a: 3
+x: a1
+g: (a0\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1), a1\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a1, a2\<mapsto>a0), a2\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1))
+f: (a0\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0), a1\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a1, a2\<mapsto>a1), a2\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0))
+*)
+
+lemma foldseq_zero:
+assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
+shows "foldseq f s n = 0"
+proof -
+ have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
+ apply (induct_tac n)
+ apply (simp)
+ by (simp add: fz)
+ then show "foldseq f s n = 0" by (simp add: sz)
+qed
+
+lemma foldseq_significant_positions:
+ assumes p: "! i. i <= N \<longrightarrow> S i = T i"
+ shows "foldseq f S N = foldseq f T N"
+proof -
+ have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
+ apply (induct_tac m)
+ apply (simp)
+ apply (simp)
+ apply (auto)
+ proof -
+ fix n
+ fix s::"nat\<Rightarrow>'a"
+ fix t::"nat\<Rightarrow>'a"
+ assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
+ assume b: "\<forall>i\<le>Suc n. s i = t i"
+ have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
+ have d:"!! s t. (\<forall>i\<le>n. s i = t i) \<Longrightarrow> foldseq f s n = foldseq f t n" by (simp add: a)
+ show "f (t 0) (foldseq f (\<lambda>k. s (Suc k)) n) = f (t 0) (foldseq f (\<lambda>k. t (Suc k)) n)" by (rule c, simp add: d b)
+ qed
+ with p show ?thesis by simp
+qed
+
+lemma foldseq_tail:
+ assumes "M <= N"
+ shows "foldseq f S N = foldseq f (% k. (if k < M then (S k) else (foldseq f (% k. S(k+M)) (N-M)))) M"
+proof -
+ have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
+ have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
+ have "!! n. ! m s. m <= n \<longrightarrow> foldseq f s n = foldseq f (% k. (if k < m then (s k) else (foldseq f (% k. s(k+m)) (n-m)))) m"
+ apply (induct_tac n)
+ apply (simp)
+ apply (simp)
+ apply (auto)
+ apply (case_tac "m = Suc na")
+ apply (simp)
+ apply (rule a)
+ apply (rule foldseq_significant_positions)
+ apply (auto)
+ apply (drule suc, simp+)
+ proof -
+ fix na m s
+ assume suba:"\<forall>m\<le>na. \<forall>s. foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m"
+ assume subb:"m <= na"
+ from suba have subc:"!! m s. m <= na \<Longrightarrow>foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m" by simp
+ have subd: "foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m =
+ foldseq f (% k. s(Suc k)) na"
+ by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
+ from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
+ show "f (s 0) (foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m) =
+ foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
+ apply (simp add: subd)
+ apply (cases "m = 0")
+ apply (simp)
+ apply (drule sube)
+ apply (auto)
+ apply (rule a)
+ by (simp add: subc cong del: if_cong)
+ qed
+ then show ?thesis using assms by simp
+qed
+
+lemma foldseq_zerotail:
+ assumes
+ fz: "f 0 0 = 0"
+ and sz: "! i. n <= i \<longrightarrow> s i = 0"
+ and nm: "n <= m"
+ shows
+ "foldseq f s n = foldseq f s m"
+proof -
+ show "foldseq f s n = foldseq f s m"
+ apply (simp add: foldseq_tail[OF nm, of f s])
+ apply (rule foldseq_significant_positions)
+ apply (auto)
+ apply (subst foldseq_zero)
+ by (simp add: fz sz)+
+qed
+
+lemma foldseq_zerotail2:
+ assumes "! x. f x 0 = x"
+ and "! i. n < i \<longrightarrow> s i = 0"
+ and nm: "n <= m"
+ shows "foldseq f s n = foldseq f s m"
+proof -
+ have "f 0 0 = 0" by (simp add: assms)
+ have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
+ have c: "0 <= m" by simp
+ have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
+ show ?thesis
+ apply (subst foldseq_tail[OF nm])
+ apply (rule foldseq_significant_positions)
+ apply (auto)
+ apply (case_tac "m=n")
+ apply (simp+)
+ apply (drule b[OF nm])
+ apply (auto)
+ apply (case_tac "k=0")
+ apply (simp add: assms)
+ apply (drule d)
+ apply (auto)
+ apply (simp add: assms foldseq_zero)
+ done
+qed
+
+lemma foldseq_zerostart:
+ "! x. f 0 (f 0 x) = f 0 x \<Longrightarrow> ! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
+proof -
+ assume f00x: "! x. f 0 (f 0 x) = f 0 x"
+ have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
+ apply (induct n)
+ apply (simp)
+ apply (rule allI, rule impI)
+ proof -
+ fix n
+ fix s
+ have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
+ assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
+ from b have c:"!! s. (\<forall>i\<le>n. s i = 0) \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
+ assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
+ show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
+ apply (subst a)
+ apply (subst c)
+ by (simp add: d f00x)+
+ qed
+ then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
+qed
+
+lemma foldseq_zerostart2:
+ "! x. f 0 x = x \<Longrightarrow> ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
+proof -
+ assume a:"! i. i<n \<longrightarrow> s i = 0"
+ assume x:"! x. f 0 x = x"
+ from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
+ have b: "!! i l. i < Suc l = (i <= l)" by arith
+ have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
+ show "foldseq f s n = s n"
+ apply (case_tac "n=0")
+ apply (simp)
+ apply (insert a)
+ apply (drule d)
+ apply (auto)
+ apply (simp add: b)
+ apply (insert f00x)
+ apply (drule foldseq_zerostart)
+ by (simp add: x)+
+qed
+
+lemma foldseq_almostzero:
+ assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
+ shows "foldseq f s n = (if (j <= n) then (s j) else 0)"
+proof -
+ from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
+ from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
+ show ?thesis
+ apply auto
+ apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
+ apply simp
+ apply (subst foldseq_zerostart2)
+ apply (simp add: f0x a)+
+ apply (subst foldseq_zero)
+ by (simp add: s0 f0x)+
+qed
+
+lemma foldseq_distr_unary:
+ assumes "!! a b. g (f a b) = f (g a) (g b)"
+ shows "g(foldseq f s n) = foldseq f (% x. g(s x)) n"
+proof -
+ have "! s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
+ apply (induct_tac n)
+ apply (simp)
+ apply (simp)
+ apply (auto)
+ apply (drule_tac x="% k. s (Suc k)" in spec)
+ by (simp add: assms)
+ then show ?thesis by simp
+qed
+
+definition mult_matrix_n :: "nat \<Rightarrow> (('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix" where
+ "mult_matrix_n n fmul fadd A B == Abs_matrix(% j i. foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
+
+definition mult_matrix :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix" where
+ "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
+
+lemma mult_matrix_n:
+ assumes "ncols A \<le> n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
+ shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B"
+proof -
+ show ?thesis using assms
+ apply (simp add: mult_matrix_def mult_matrix_n_def)
+ apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+ apply (rule foldseq_zerotail, simp_all add: nrows_le ncols_le assms)
+ done
+qed
+
+lemma mult_matrix_nm:
+ assumes "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
+ shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
+proof -
+ from assms have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B"
+ by (simp add: mult_matrix_n)
+ also from assms have "\<dots> = mult_matrix_n m fmul fadd A B"
+ by (simp add: mult_matrix_n[THEN sym])
+ finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
+qed
+
+definition r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
+ "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
+
+definition l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
+
+definition distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd"
+
+lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
+lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
+
+lemma r_distributive_matrix:
+ assumes
+ "r_distributive fmul fadd"
+ "associative fadd"
+ "commutative fadd"
+ "fadd 0 0 = 0"
+ "! a. fmul a 0 = 0"
+ "! a. fmul 0 a = 0"
+ shows "r_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
+proof -
+ from assms show ?thesis
+ apply (simp add: r_distributive_def mult_matrix_def, auto)
+ proof -
+ fix a::"'a matrix"
+ fix u::"'b matrix"
+ fix v::"'b matrix"
+ let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
+ from assms show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
+ combine_matrix fadd (mult_matrix_n (max (ncols a) (nrows u)) fmul fadd a u) (mult_matrix_n (max (ncols a) (nrows v)) fmul fadd a v)"
+ apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (subst mult_matrix_nm[of _ _ v ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (subst mult_matrix_nm[of _ _ u ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
+ apply (simp add: combine_matrix_def combine_infmatrix_def)
+ apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+ apply (simplesubst RepAbs_matrix)
+ apply (simp, auto)
+ apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+ apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
+ apply (subst RepAbs_matrix)
+ apply (simp, auto)
+ apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+ apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
+ done
+ qed
+qed
+
+lemma l_distributive_matrix:
+ assumes
+ "l_distributive fmul fadd"
+ "associative fadd"
+ "commutative fadd"
+ "fadd 0 0 = 0"
+ "! a. fmul a 0 = 0"
+ "! a. fmul 0 a = 0"
+ shows "l_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
+proof -
+ from assms show ?thesis
+ apply (simp add: l_distributive_def mult_matrix_def, auto)
+ proof -
+ fix a::"'b matrix"
+ fix u::"'a matrix"
+ fix v::"'a matrix"
+ let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
+ from assms show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
+ combine_matrix fadd (mult_matrix_n (max (ncols u) (nrows a)) fmul fadd u a) (mult_matrix_n (max (ncols v) (nrows a)) fmul fadd v a)"
+ apply (subst mult_matrix_nm[of v _ _ ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (subst mult_matrix_nm[of u _ _ ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+ apply (simp add: max1 max2 combine_nrows combine_ncols)+
+ apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
+ apply (simp add: combine_matrix_def combine_infmatrix_def)
+ apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+ apply (simplesubst RepAbs_matrix)
+ apply (simp, auto)
+ apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
+ apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+ apply (subst RepAbs_matrix)
+ apply (simp, auto)
+ apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
+ apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+ done
+ qed
+qed
+
+instantiation matrix :: (zero) zero
+begin
+
+definition zero_matrix_def: "0 = Abs_matrix (\<lambda>j i. 0)"
+
+instance ..
+
+end
+
+lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
+ apply (simp add: zero_matrix_def)
+ apply (subst RepAbs_matrix)
+ by (auto)
+
+lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
+proof -
+ have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
+ show "nrows 0 = 0" by (rule a, subst nrows_le, simp)
+qed
+
+lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
+proof -
+ have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
+ show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
+qed
+
+lemma combine_matrix_zero_l_neutral: "zero_l_neutral f \<Longrightarrow> zero_l_neutral (combine_matrix f)"
+ by (simp add: zero_l_neutral_def combine_matrix_def combine_infmatrix_def)
+
+lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
+ by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
+
+lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
+ apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)
+ apply (auto)
+ by (subst foldseq_zero, (simp add: zero_matrix_def)+)+
+
+lemma mult_matrix_n_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd A 0 = 0"
+ apply (simp add: mult_matrix_n_def)
+ apply (subst foldseq_zero)
+ by (simp_all add: zero_matrix_def)
+
+lemma mult_matrix_n_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd 0 A = 0"
+ apply (simp add: mult_matrix_n_def)
+ apply (subst foldseq_zero)
+ by (simp_all add: zero_matrix_def)
+
+lemma mult_matrix_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd 0 A = 0"
+by (simp add: mult_matrix_def)
+
+lemma mult_matrix_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd A 0 = 0"
+by (simp add: mult_matrix_def)
+
+lemma apply_matrix_zero[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f 0 = 0"
+ apply (simp add: apply_matrix_def apply_infmatrix_def)
+ by (simp add: zero_matrix_def)
+
+lemma combine_matrix_zero: "f 0 0 = 0 \<Longrightarrow> combine_matrix f 0 0 = 0"
+ apply (simp add: combine_matrix_def combine_infmatrix_def)
+ by (simp add: zero_matrix_def)
+
+lemma transpose_matrix_zero[simp]: "transpose_matrix 0 = 0"
+apply (simp add: transpose_matrix_def zero_matrix_def RepAbs_matrix)
+apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
+apply (simp add: RepAbs_matrix)
+done
+
+lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
+ apply (simp add: apply_matrix_def apply_infmatrix_def)
+ by (simp add: zero_matrix_def)
+
+definition singleton_matrix :: "nat \<Rightarrow> nat \<Rightarrow> ('a::zero) \<Rightarrow> 'a matrix" where
+ "singleton_matrix j i a == Abs_matrix(% m n. if j = m & i = n then a else 0)"
+
+definition move_matrix :: "('a::zero) matrix \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a matrix" where
+ "move_matrix A y x == Abs_matrix(% j i. if (((int j)-y) < 0) | (((int i)-x) < 0) then 0 else Rep_matrix A (nat ((int j)-y)) (nat ((int i)-x)))"
+
+definition take_rows :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
+ "take_rows A r == Abs_matrix(% j i. if (j < r) then (Rep_matrix A j i) else 0)"
+
+definition take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
+ "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
+
+definition column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
+ "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
+
+definition row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
+ "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
+
+lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
+apply (simp add: singleton_matrix_def)
+apply (auto)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc m"], simp)
+apply (rule exI[of _ "Suc n"], simp+)
+by (subst RepAbs_matrix, rule exI[of _ "Suc j"], simp, rule exI[of _ "Suc i"], simp+)+
+
+lemma apply_singleton_matrix[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f (singleton_matrix j i x) = (singleton_matrix j i (f x))"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma singleton_matrix_zero[simp]: "singleton_matrix j i 0 = 0"
+ by (simp add: singleton_matrix_def zero_matrix_def)
+
+lemma nrows_singleton[simp]: "nrows(singleton_matrix j i e) = (if e = 0 then 0 else Suc j)"
+proof-
+have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
+from th show ?thesis
+apply (auto)
+apply (rule le_antisym)
+apply (subst nrows_le)
+apply (simp add: singleton_matrix_def, auto)
+apply (subst RepAbs_matrix)
+apply auto
+apply (simp add: Suc_le_eq)
+apply (rule not_leE)
+apply (subst nrows_le)
+by simp
+qed
+
+lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
+proof-
+have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
+from th show ?thesis
+apply (auto)
+apply (rule le_antisym)
+apply (subst ncols_le)
+apply (simp add: singleton_matrix_def, auto)
+apply (subst RepAbs_matrix)
+apply auto
+apply (simp add: Suc_le_eq)
+apply (rule not_leE)
+apply (subst ncols_le)
+by simp
+qed
+
+lemma combine_singleton: "f 0 0 = 0 \<Longrightarrow> combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
+apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc j"], simp)
+apply (rule exI[of _ "Suc i"], simp)
+apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc j"], simp)
+apply (rule exI[of _ "Suc i"], simp)
+by simp
+
+lemma transpose_singleton[simp]: "transpose_matrix (singleton_matrix j i a) = singleton_matrix i j a"
+apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
+apply (simp)
+done
+
+lemma Rep_move_matrix[simp]:
+ "Rep_matrix (move_matrix A y x) j i =
+ (if (((int j)-y) < 0) | (((int i)-x) < 0) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
+apply (simp add: move_matrix_def)
+apply (auto)
+by (subst RepAbs_matrix,
+ rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
+ rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
+
+lemma move_matrix_0_0[simp]: "move_matrix A 0 0 = A"
+by (simp add: move_matrix_def)
+
+lemma move_matrix_ortho: "move_matrix A j i = move_matrix (move_matrix A j 0) 0 i"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma transpose_move_matrix[simp]:
+ "transpose_matrix (move_matrix A x y) = move_matrix (transpose_matrix A) y x"
+apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
+apply (simp)
+done
+
+lemma move_matrix_singleton[simp]: "move_matrix (singleton_matrix u v x) j i =
+ (if (j + int u < 0) | (i + int v < 0) then 0 else (singleton_matrix (nat (j + int u)) (nat (i + int v)) x))"
+ apply (subst Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply (case_tac "j + int u < 0")
+ apply (simp, arith)
+ apply (case_tac "i + int v < 0")
+ apply (simp, arith)
+ apply simp
+ apply arith
+ done
+
+lemma Rep_take_columns[simp]:
+ "Rep_matrix (take_columns A c) j i =
+ (if i < c then (Rep_matrix A j i) else 0)"
+apply (simp add: take_columns_def)
+apply (simplesubst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
+apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
+done
+
+lemma Rep_take_rows[simp]:
+ "Rep_matrix (take_rows A r) j i =
+ (if j < r then (Rep_matrix A j i) else 0)"
+apply (simp add: take_rows_def)
+apply (simplesubst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
+apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
+done
+
+lemma Rep_column_of_matrix[simp]:
+ "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then (Rep_matrix A j c) else 0)"
+ by (simp add: column_of_matrix_def)
+
+lemma Rep_row_of_matrix[simp]:
+ "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then (Rep_matrix A r i) else 0)"
+ by (simp add: row_of_matrix_def)
+
+lemma column_of_matrix: "ncols A <= n \<Longrightarrow> column_of_matrix A n = 0"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by (simp add: ncols)
+
+lemma row_of_matrix: "nrows A <= n \<Longrightarrow> row_of_matrix A n = 0"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by (simp add: nrows)
+
+lemma mult_matrix_singleton_right[simp]:
+ assumes
+ "! x. fmul x 0 = 0"
+ "! x. fmul 0 x = 0"
+ "! x. fadd 0 x = x"
+ "! x. fadd x 0 = x"
+ shows "(mult_matrix fmul fadd A (singleton_matrix j i e)) = apply_matrix (% x. fmul x e) (move_matrix (column_of_matrix A j) 0 (int i))"
+ apply (simp add: mult_matrix_def)
+ apply (subst mult_matrix_nm[of _ _ _ "max (ncols A) (Suc j)"])
+ apply (auto)
+ apply (simp add: assms)+
+ apply (simp add: mult_matrix_n_def apply_matrix_def apply_infmatrix_def)
+ apply (rule comb[of "Abs_matrix" "Abs_matrix"], auto, (rule ext)+)
+ apply (subst foldseq_almostzero[of _ j])
+ apply (simp add: assms)+
+ apply (auto)
+ done
+
+lemma mult_matrix_ext:
+ assumes
+ eprem:
+ "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
+ and fprems:
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "! a. fadd a 0 = a"
+ "! a. fadd 0 a = a"
+ and contraprems:
+ "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
+ shows
+ "A = B"
+proof(rule contrapos_np[of "False"], simp)
+ assume a: "A \<noteq> B"
+ have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
+ have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
+ apply (rule contrapos_np[of "False"], simp+)
+ apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
+ by (simp add: Rep_matrix_inject a)
+ then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
+ from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
+ let ?S = "singleton_matrix I 0 e"
+ let ?comp = "mult_matrix fmul fadd"
+ have d: "!!x f g. f = g \<Longrightarrow> f x = g x" by blast
+ have e: "(% x. fmul x e) 0 = 0" by (simp add: assms)
+ have "~(?comp A ?S = ?comp B ?S)"
+ apply (rule notI)
+ apply (simp add: fprems eprops)
+ apply (simp add: Rep_matrix_inject[THEN sym])
+ apply (drule d[of _ _ "J"], drule d[of _ _ "0"])
+ by (simp add: e c eprops)
+ with contraprems show "False" by simp
+qed
+
+definition foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
+ "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
+
+definition foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
+ "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
+
+lemma foldmatrix_transpose:
+ assumes
+ "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
+ shows
+ "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
+proof -
+ have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
+ have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
+ apply (induct n)
+ apply (simp add: foldmatrix_def foldmatrix_transposed_def assms)+
+ apply (auto)
+ by (drule_tac x="(% j i. A j (Suc i))" in forall, simp)
+ show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
+ apply (simp add: foldmatrix_def foldmatrix_transposed_def)
+ apply (induct m, simp)
+ apply (simp)
+ apply (insert tworows)
+ apply (drule_tac x="% j i. (if j = 0 then (foldseq_transposed g (\<lambda>u. A u i) m) else (A (Suc m) i))" in spec)
+ by (simp add: foldmatrix_def foldmatrix_transposed_def)
+qed
+
+lemma foldseq_foldseq:
+assumes
+ "associative f"
+ "associative g"
+ "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
+shows
+ "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
+ apply (insert foldmatrix_transpose[of g f A m n])
+ by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] assms)
+
+lemma mult_n_nrows:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+"fadd 0 0 = 0"
+shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
+apply (subst nrows_le)
+apply (simp add: mult_matrix_n_def)
+apply (subst RepAbs_matrix)
+apply (rule_tac x="nrows A" in exI)
+apply (simp add: nrows assms foldseq_zero)
+apply (rule_tac x="ncols B" in exI)
+apply (simp add: ncols assms foldseq_zero)
+apply (simp add: nrows assms foldseq_zero)
+done
+
+lemma mult_n_ncols:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+"fadd 0 0 = 0"
+shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
+apply (subst ncols_le)
+apply (simp add: mult_matrix_n_def)
+apply (subst RepAbs_matrix)
+apply (rule_tac x="nrows A" in exI)
+apply (simp add: nrows assms foldseq_zero)
+apply (rule_tac x="ncols B" in exI)
+apply (simp add: ncols assms foldseq_zero)
+apply (simp add: ncols assms foldseq_zero)
+done
+
+lemma mult_nrows:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+"fadd 0 0 = 0"
+shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
+by (simp add: mult_matrix_def mult_n_nrows assms)
+
+lemma mult_ncols:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+"fadd 0 0 = 0"
+shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
+by (simp add: mult_matrix_def mult_n_ncols assms)
+
+lemma nrows_move_matrix_le: "nrows (move_matrix A j i) <= nat((int (nrows A)) + j)"
+ apply (auto simp add: nrows_le)
+ apply (rule nrows)
+ apply (arith)
+ done
+
+lemma ncols_move_matrix_le: "ncols (move_matrix A j i) <= nat((int (ncols A)) + i)"
+ apply (auto simp add: ncols_le)
+ apply (rule ncols)
+ apply (arith)
+ done
+
+lemma mult_matrix_assoc:
+ assumes
+ "! a. fmul1 0 a = 0"
+ "! a. fmul1 a 0 = 0"
+ "! a. fmul2 0 a = 0"
+ "! a. fmul2 a 0 = 0"
+ "fadd1 0 0 = 0"
+ "fadd2 0 0 = 0"
+ "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
+ "associative fadd1"
+ "associative fadd2"
+ "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
+ "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
+ "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
+ shows "mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B) C = mult_matrix fmul1 fadd1 A (mult_matrix fmul2 fadd2 B C)"
+proof -
+ have comb_left: "!! A B x y. A = B \<Longrightarrow> (Rep_matrix (Abs_matrix A)) x y = (Rep_matrix(Abs_matrix B)) x y" by blast
+ have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n) x = foldseq fadd1 (% k. fmul2 (s k) x) n"
+ by (rule_tac g1 = "% y. fmul2 y x" in ssubst [OF foldseq_distr_unary], insert assms, simp_all)
+ have fmul1fadd2fold: "!! x s n. fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (% k. fmul1 x (s k)) n"
+ using assms by (rule_tac g1 = "% y. fmul1 x y" in ssubst [OF foldseq_distr_unary], simp_all)
+ let ?N = "max (ncols A) (max (ncols B) (max (nrows B) (nrows C)))"
+ show ?thesis
+ apply (simp add: Rep_matrix_inject[THEN sym])
+ apply (rule ext)+
+ apply (simp add: mult_matrix_def)
+ apply (simplesubst mult_matrix_nm[of _ "max (ncols (mult_matrix_n (max (ncols A) (nrows B)) fmul1 fadd1 A B)) (nrows C)" _ "max (ncols B) (nrows C)"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simplesubst mult_matrix_nm[of _ "max (ncols A) (nrows (mult_matrix_n (max (ncols B) (nrows C)) fmul2 fadd2 B C))" _ "max (ncols A) (nrows B)"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
+ apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
+ apply (simp add: mult_matrix_n_def)
+ apply (rule comb_left)
+ apply ((rule ext)+, simp)
+ apply (simplesubst RepAbs_matrix)
+ apply (rule exI[of _ "nrows B"])
+ apply (simp add: nrows assms foldseq_zero)
+ apply (rule exI[of _ "ncols C"])
+ apply (simp add: assms ncols foldseq_zero)
+ apply (subst RepAbs_matrix)
+ apply (rule exI[of _ "nrows A"])
+ apply (simp add: nrows assms foldseq_zero)
+ apply (rule exI[of _ "ncols B"])
+ apply (simp add: assms ncols foldseq_zero)
+ apply (simp add: fmul2fadd1fold fmul1fadd2fold assms)
+ apply (subst foldseq_foldseq)
+ apply (simp add: assms)+
+ apply (simp add: transpose_infmatrix)
+ done
+qed
+
+lemma
+ assumes
+ "! a. fmul1 0 a = 0"
+ "! a. fmul1 a 0 = 0"
+ "! a. fmul2 0 a = 0"
+ "! a. fmul2 a 0 = 0"
+ "fadd1 0 0 = 0"
+ "fadd2 0 0 = 0"
+ "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
+ "associative fadd1"
+ "associative fadd2"
+ "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
+ "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
+ "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
+ shows
+ "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)"
+apply (rule ext)+
+apply (simp add: comp_def )
+apply (simp add: mult_matrix_assoc assms)
+done
+
+lemma mult_matrix_assoc_simple:
+ assumes
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "fadd 0 0 = 0"
+ "associative fadd"
+ "commutative fadd"
+ "associative fmul"
+ "distributive fmul fadd"
+ shows "mult_matrix fmul fadd (mult_matrix fmul fadd A B) C = mult_matrix fmul fadd A (mult_matrix fmul fadd B C)"
+proof -
+ have "!! a b c d. fadd (fadd a b) (fadd c d) = fadd (fadd a c) (fadd b d)"
+ using assms by (simp add: associative_def commutative_def)
+ then show ?thesis
+ apply (subst mult_matrix_assoc)
+ using assms
+ apply simp_all
+ apply (simp_all add: associative_def distributive_def l_distributive_def r_distributive_def)
+ done
+qed
+
+lemma transpose_apply_matrix: "f 0 = 0 \<Longrightarrow> transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
+apply (simp add: Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by simp
+
+lemma transpose_combine_matrix: "f 0 0 = 0 \<Longrightarrow> transpose_matrix (combine_matrix f A B) = combine_matrix f (transpose_matrix A) (transpose_matrix B)"
+apply (simp add: Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by simp
+
+lemma Rep_mult_matrix:
+ assumes
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "fadd 0 0 = 0"
+ shows
+ "Rep_matrix(mult_matrix fmul fadd A B) j i =
+ foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
+apply (simp add: mult_matrix_def mult_matrix_n_def)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], insert assms, simp add: nrows foldseq_zero)
+apply (rule exI[of _ "ncols B"], insert assms, simp add: ncols foldseq_zero)
+apply simp
+done
+
+lemma transpose_mult_matrix:
+ assumes
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "fadd 0 0 = 0"
+ "! x y. fmul y x = fmul x y"
+ shows
+ "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
+ apply (simp add: Rep_matrix_inject[THEN sym])
+ apply (rule ext)+
+ using assms
+ apply (simp add: Rep_mult_matrix max_ac)
+ done
+
+lemma column_transpose_matrix: "column_of_matrix (transpose_matrix A) n = transpose_matrix (row_of_matrix A n)"
+apply (simp add: Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by simp
+
+lemma take_columns_transpose_matrix: "take_columns (transpose_matrix A) n = transpose_matrix (take_rows A n)"
+apply (simp add: Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+by simp
+
+instantiation matrix :: ("{zero, ord}") ord
+begin
+
+definition
+ le_matrix_def: "A \<le> B \<longleftrightarrow> (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
+
+definition
+ less_def: "A < (B\<Colon>'a matrix) \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
+
+instance ..
+
+end
+
+instance matrix :: ("{zero, order}") order
+apply intro_classes
+apply (simp_all add: le_matrix_def less_def)
+apply (auto)
+apply (drule_tac x=j in spec, drule_tac x=j in spec)
+apply (drule_tac x=i in spec, drule_tac x=i in spec)
+apply (simp)
+apply (simp add: Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
+apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
+apply simp
+done
+
+lemma le_apply_matrix:
+ assumes
+ "f 0 = 0"
+ "! x y. x <= y \<longrightarrow> f x <= f y"
+ "(a::('a::{ord, zero}) matrix) <= b"
+ shows
+ "apply_matrix f a <= apply_matrix f b"
+ using assms by (simp add: le_matrix_def)
+
+lemma le_combine_matrix:
+ assumes
+ "f 0 0 = 0"
+ "! a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
+ "A <= B"
+ "C <= D"
+ shows
+ "combine_matrix f A C <= combine_matrix f B D"
+ using assms by (simp add: le_matrix_def)
+
+lemma le_left_combine_matrix:
+ assumes
+ "f 0 0 = 0"
+ "! a b c. a <= b \<longrightarrow> f c a <= f c b"
+ "A <= B"
+ shows
+ "combine_matrix f C A <= combine_matrix f C B"
+ using assms by (simp add: le_matrix_def)
+
+lemma le_right_combine_matrix:
+ assumes
+ "f 0 0 = 0"
+ "! a b c. a <= b \<longrightarrow> f a c <= f b c"
+ "A <= B"
+ shows
+ "combine_matrix f A C <= combine_matrix f B C"
+ using assms by (simp add: le_matrix_def)
+
+lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
+ by (simp add: le_matrix_def, auto)
+
+lemma le_foldseq:
+ assumes
+ "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
+ "! i. i <= n \<longrightarrow> s i <= t i"
+ shows
+ "foldseq f s n <= foldseq f t n"
+proof -
+ have "! s t. (! i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n"
+ by (induct n) (simp_all add: assms)
+ then show "foldseq f s n <= foldseq f t n" using assms by simp
+qed
+
+lemma le_left_mult:
+ assumes
+ "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
+ "! c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "fadd 0 0 = 0"
+ "0 <= C"
+ "A <= B"
+ shows
+ "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
+ using assms
+ apply (simp add: le_matrix_def Rep_mult_matrix)
+ apply (auto)
+ apply (simplesubst foldseq_zerotail[of _ _ _ "max (ncols C) (max (nrows A) (nrows B))"], simp_all add: nrows ncols max1 max2)+
+ apply (rule le_foldseq)
+ apply (auto)
+ done
+
+lemma le_right_mult:
+ assumes
+ "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
+ "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
+ "! a. fmul 0 a = 0"
+ "! a. fmul a 0 = 0"
+ "fadd 0 0 = 0"
+ "0 <= C"
+ "A <= B"
+ shows
+ "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
+ using assms
+ apply (simp add: le_matrix_def Rep_mult_matrix)
+ apply (auto)
+ apply (simplesubst foldseq_zerotail[of _ _ _ "max (nrows C) (max (ncols A) (ncols B))"], simp_all add: nrows ncols max1 max2)+
+ apply (rule le_foldseq)
+ apply (auto)
+ done
+
+lemma spec2: "! j i. P j i \<Longrightarrow> P j i" by blast
+lemma neg_imp: "(\<not> Q \<longrightarrow> \<not> P) \<Longrightarrow> P \<longrightarrow> Q" by blast
+
+lemma singleton_matrix_le[simp]: "(singleton_matrix j i a <= singleton_matrix j i b) = (a <= (b::_::order))"
+ by (auto simp add: le_matrix_def)
+
+lemma singleton_le_zero[simp]: "(singleton_matrix j i x <= 0) = (x <= (0::'a::{order,zero}))"
+ apply (auto)
+ apply (simp add: le_matrix_def)
+ apply (drule_tac j=j and i=i in spec2)
+ apply (simp)
+ apply (simp add: le_matrix_def)
+ done
+
+lemma singleton_ge_zero[simp]: "(0 <= singleton_matrix j i x) = ((0::'a::{order,zero}) <= x)"
+ apply (auto)
+ apply (simp add: le_matrix_def)
+ apply (drule_tac j=j and i=i in spec2)
+ apply (simp)
+ apply (simp add: le_matrix_def)
+ done
+
+lemma move_matrix_le_zero[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= 0) = (A <= (0::('a::{order,zero}) matrix))"
+ apply (auto simp add: le_matrix_def)
+ apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
+ apply (auto)
+ done
+
+lemma move_matrix_zero_le[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (0 <= move_matrix A j i) = ((0::('a::{order,zero}) matrix) <= A)"
+ apply (auto simp add: le_matrix_def)
+ apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
+ apply (auto)
+ done
+
+lemma move_matrix_le_move_matrix_iff[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= move_matrix B j i) = (A <= (B::('a::{order,zero}) matrix))"
+ apply (auto simp add: le_matrix_def)
+ apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
+ apply (auto)
+ done
+
+instantiation matrix :: ("{lattice, zero}") lattice
+begin
+
+definition "inf = combine_matrix inf"
+
+definition "sup = combine_matrix sup"
+
+instance
+ by default (auto simp add: le_infI le_matrix_def inf_matrix_def sup_matrix_def)
+
+end
+
+instantiation matrix :: ("{plus, zero}") plus
+begin
+
+definition
+ plus_matrix_def: "A + B = combine_matrix (op +) A B"
+
+instance ..
+
+end
+
+instantiation matrix :: ("{uminus, zero}") uminus
+begin
+
+definition
+ minus_matrix_def: "- A = apply_matrix uminus A"
+
+instance ..
+
+end
+
+instantiation matrix :: ("{minus, zero}") minus
+begin
+
+definition
+ diff_matrix_def: "A - B = combine_matrix (op -) A B"
+
+instance ..
+
+end
+
+instantiation matrix :: ("{plus, times, zero}") times
+begin
+
+definition
+ times_matrix_def: "A * B = mult_matrix (op *) (op +) A B"
+
+instance ..
+
+end
+
+instantiation matrix :: ("{lattice, uminus, zero}") abs
+begin
+
+definition
+ abs_matrix_def: "abs (A \<Colon> 'a matrix) = sup A (- A)"
+
+instance ..
+
+end
+
+instance matrix :: (monoid_add) monoid_add
+proof
+ fix A B C :: "'a matrix"
+ show "A + B + C = A + (B + C)"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: add_assoc)
+ done
+ show "0 + A = A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
+ apply (simp)
+ done
+ show "A + 0 = A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_zero_r_neutral [simplified zero_r_neutral_def, THEN spec])
+ apply (simp)
+ done
+qed
+
+instance matrix :: (comm_monoid_add) comm_monoid_add
+proof
+ fix A B :: "'a matrix"
+ show "A + B = B + A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
+ apply (simp_all add: add_commute)
+ done
+ show "0 + A = A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
+ apply (simp)
+ done
+qed
+
+instance matrix :: (group_add) group_add
+proof
+ fix A B :: "'a matrix"
+ show "- A + A = 0"
+ by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ show "A - B = A + - B"
+ by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject [symmetric] ext diff_minus)
+qed
+
+instance matrix :: (ab_group_add) ab_group_add
+proof
+ fix A B :: "'a matrix"
+ show "- A + A = 0"
+ by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ show "A - B = A + - B"
+ by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+qed
+
+instance matrix :: (ordered_ab_group_add) ordered_ab_group_add
+proof
+ fix A B C :: "'a matrix"
+ assume "A <= B"
+ then show "C + A <= C + B"
+ apply (simp add: plus_matrix_def)
+ apply (rule le_left_combine_matrix)
+ apply (simp_all)
+ done
+qed
+
+instance matrix :: (lattice_ab_group_add) semilattice_inf_ab_group_add ..
+instance matrix :: (lattice_ab_group_add) semilattice_sup_ab_group_add ..
+
+instance matrix :: (semiring_0) semiring_0
+proof
+ fix A B C :: "'a matrix"
+ show "A * B * C = A * (B * C)"
+ apply (simp add: times_matrix_def)
+ apply (rule mult_matrix_assoc)
+ apply (simp_all add: associative_def algebra_simps)
+ done
+ show "(A + B) * C = A * C + B * C"
+ apply (simp add: times_matrix_def plus_matrix_def)
+ apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: associative_def commutative_def algebra_simps)
+ done
+ show "A * (B + C) = A * B + A * C"
+ apply (simp add: times_matrix_def plus_matrix_def)
+ apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: associative_def commutative_def algebra_simps)
+ done
+ show "0 * A = 0" by (simp add: times_matrix_def)
+ show "A * 0 = 0" by (simp add: times_matrix_def)
+qed
+
+instance matrix :: (ring) ring ..
+
+instance matrix :: (ordered_ring) ordered_ring
+proof
+ fix A B C :: "'a matrix"
+ assume a: "A \<le> B"
+ assume b: "0 \<le> C"
+ from a b show "C * A \<le> C * B"
+ apply (simp add: times_matrix_def)
+ apply (rule le_left_mult)
+ apply (simp_all add: add_mono mult_left_mono)
+ done
+ from a b show "A * C \<le> B * C"
+ apply (simp add: times_matrix_def)
+ apply (rule le_right_mult)
+ apply (simp_all add: add_mono mult_right_mono)
+ done
+qed
+
+instance matrix :: (lattice_ring) lattice_ring
+proof
+ fix A B C :: "('a :: lattice_ring) matrix"
+ show "abs A = sup A (-A)"
+ by (simp add: abs_matrix_def)
+qed
+
+lemma Rep_matrix_add[simp]:
+ "Rep_matrix ((a::('a::monoid_add)matrix)+b) j i = (Rep_matrix a j i) + (Rep_matrix b j i)"
+ by (simp add: plus_matrix_def)
+
+lemma Rep_matrix_mult: "Rep_matrix ((a::('a::semiring_0) matrix) * b) j i =
+ foldseq (op +) (% k. (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
+apply (simp add: times_matrix_def)
+apply (simp add: Rep_mult_matrix)
+done
+
+lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a)
+ \<Longrightarrow> apply_matrix f ((a::('a::monoid_add) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma singleton_matrix_add: "singleton_matrix j i ((a::_::monoid_add)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma nrows_mult: "nrows ((A::('a::semiring_0) matrix) * B) <= nrows A"
+by (simp add: times_matrix_def mult_nrows)
+
+lemma ncols_mult: "ncols ((A::('a::semiring_0) matrix) * B) <= ncols B"
+by (simp add: times_matrix_def mult_ncols)
+
+definition
+ one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix" where
+ "one_matrix n = Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
+
+lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
+apply (simp add: one_matrix_def)
+apply (simplesubst RepAbs_matrix)
+apply (rule exI[of _ n], simp add: split_if)+
+by (simp add: split_if)
+
+lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
+proof -
+ have "?r <= n" by (simp add: nrows_le)
+ moreover have "n <= ?r" by (simp add:le_nrows, arith)
+ ultimately show "?r = n" by simp
+qed
+
+lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
+proof -
+ have "?r <= n" by (simp add: ncols_le)
+ moreover have "n <= ?r" by (simp add: le_ncols, arith)
+ ultimately show "?r = n" by simp
+qed
+
+lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{semiring_1}) matrix) * (one_matrix n) = A"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+apply (simp add: times_matrix_def Rep_mult_matrix)
+apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
+apply (simp_all)
+by (simp add: ncols)
+
+lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::semiring_1) matrix)"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+apply (simp add: times_matrix_def Rep_mult_matrix)
+apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
+apply (simp_all)
+by (simp add: nrows)
+
+lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
+apply (simp add: times_matrix_def)
+apply (subst transpose_mult_matrix)
+apply (simp_all add: mult_commute)
+done
+
+lemma transpose_matrix_add: "transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B"
+by (simp add: plus_matrix_def transpose_combine_matrix)
+
+lemma transpose_matrix_diff: "transpose_matrix ((A::('a::group_add) matrix)-B) = transpose_matrix A - transpose_matrix B"
+by (simp add: diff_matrix_def transpose_combine_matrix)
+
+lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::group_add) matrix)) = - transpose_matrix (A::'a matrix)"
+by (simp add: minus_matrix_def transpose_apply_matrix)
+
+definition right_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
+ "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A"
+
+definition left_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
+ "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A"
+
+definition inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
+ "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
+
+lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
+apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
+by (simp add: right_inverse_matrix_def)
+
+lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
+apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A])
+by (simp add: left_inverse_matrix_def)
+
+lemma left_right_inverse_matrix_unique:
+ assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
+ shows "X = Y"
+proof -
+ have "Y = Y * one_matrix (nrows A)"
+ apply (subst one_matrix_mult_right)
+ using assms
+ apply (simp_all add: left_inverse_matrix_def)
+ done
+ also have "\<dots> = Y * (A * X)"
+ apply (insert assms)
+ apply (frule right_inverse_matrix_dim)
+ by (simp add: right_inverse_matrix_def)
+ also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
+ also have "\<dots> = X"
+ apply (insert assms)
+ apply (frule left_inverse_matrix_dim)
+ apply (simp_all add: left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
+ done
+ ultimately show "X = Y" by (simp)
+qed
+
+lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
+ by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
+
+lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
+ by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
+
+lemma zero_imp_mult_zero: "(a::'a::semiring_0) = 0 | b = 0 \<Longrightarrow> a * b = 0"
+by auto
+
+lemma Rep_matrix_zero_imp_mult_zero:
+ "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0 \<Longrightarrow> A * B = (0::('a::lattice_ring) matrix)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
+done
+
+lemma add_nrows: "nrows (A::('a::monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
+apply (simp add: plus_matrix_def)
+apply (rule combine_nrows)
+apply (simp_all)
+done
+
+lemma move_matrix_row_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) j 0 = (move_matrix A j 0) * B"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero)
+apply (rule_tac foldseq_zerotail[symmetric])
+apply (auto simp add: nrows zero_imp_mult_zero max2)
+apply (rule order_trans)
+apply (rule ncols_move_matrix_le)
+apply (simp add: max1)
+done
+
+lemma move_matrix_col_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) 0 i = A * (move_matrix B 0 i)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero)
+apply (rule_tac foldseq_zerotail[symmetric])
+apply (auto simp add: ncols zero_imp_mult_zero max1)
+apply (rule order_trans)
+apply (rule nrows_move_matrix_le)
+apply (simp add: max2)
+done
+
+lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::monoid_add) matrix)) = (move_matrix A j i) + (move_matrix B j i)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma move_matrix_mult: "move_matrix ((A::('a::semiring_0) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
+by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
+
+definition scalar_mult :: "('a::ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix" where
+ "scalar_mult a m == apply_matrix (op * a) m"
+
+lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0"
+by (simp add: scalar_mult_def)
+
+lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
+by (simp add: scalar_mult_def apply_matrix_add algebra_simps)
+
+lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)"
+by (simp add: scalar_mult_def)
+
+lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto)
+done
+
+lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
+by (simp add: minus_matrix_def)
+
+lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lattice_ab_group_add)) x y = abs (Rep_matrix A x y)"
+by (simp add: abs_lattice sup_matrix_def)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/ROOT.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,2 @@
+
+use_thy "Cplex";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/SparseMatrix.thy Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,1070 @@
+(* Title: HOL/Matrix/SparseMatrix.thy
+ Author: Steven Obua
+*)
+
+theory SparseMatrix
+imports Matrix
+begin
+
+type_synonym 'a spvec = "(nat * 'a) list"
+type_synonym 'a spmat = "'a spvec spvec"
+
+definition sparse_row_vector :: "('a::ab_group_add) spvec \<Rightarrow> 'a matrix"
+ where "sparse_row_vector arr = foldl (% m x. m + (singleton_matrix 0 (fst x) (snd x))) 0 arr"
+
+definition sparse_row_matrix :: "('a::ab_group_add) spmat \<Rightarrow> 'a matrix"
+ where "sparse_row_matrix arr = foldl (% m r. m + (move_matrix (sparse_row_vector (snd r)) (int (fst r)) 0)) 0 arr"
+
+code_datatype sparse_row_vector sparse_row_matrix
+
+lemma sparse_row_vector_empty [simp]: "sparse_row_vector [] = 0"
+ by (simp add: sparse_row_vector_def)
+
+lemma sparse_row_matrix_empty [simp]: "sparse_row_matrix [] = 0"
+ by (simp add: sparse_row_matrix_def)
+
+lemmas [code] = sparse_row_vector_empty [symmetric]
+
+lemma foldl_distrstart: "! a x y. (f (g x y) a = g x (f y a)) \<Longrightarrow> (foldl f (g x y) l = g x (foldl f y l))"
+ by (induct l arbitrary: x y, auto)
+
+lemma sparse_row_vector_cons[simp]:
+ "sparse_row_vector (a # arr) = (singleton_matrix 0 (fst a) (snd a)) + (sparse_row_vector arr)"
+ apply (induct arr)
+ apply (auto simp add: sparse_row_vector_def)
+ apply (simp add: foldl_distrstart [of "\<lambda>m x. m + singleton_matrix 0 (fst x) (snd x)" "\<lambda>x m. singleton_matrix 0 (fst x) (snd x) + m"])
+ done
+
+lemma sparse_row_vector_append[simp]:
+ "sparse_row_vector (a @ b) = (sparse_row_vector a) + (sparse_row_vector b)"
+ by (induct a) auto
+
+lemma nrows_spvec[simp]: "nrows (sparse_row_vector x) <= (Suc 0)"
+ apply (induct x)
+ apply (simp_all add: add_nrows)
+ done
+
+lemma sparse_row_matrix_cons: "sparse_row_matrix (a#arr) = ((move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0)) + sparse_row_matrix arr"
+ apply (induct arr)
+ apply (auto simp add: sparse_row_matrix_def)
+ apply (simp add: foldl_distrstart[of "\<lambda>m x. m + (move_matrix (sparse_row_vector (snd x)) (int (fst x)) 0)"
+ "% a m. (move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0) + m"])
+ done
+
+lemma sparse_row_matrix_append: "sparse_row_matrix (arr@brr) = (sparse_row_matrix arr) + (sparse_row_matrix brr)"
+ apply (induct arr)
+ apply (auto simp add: sparse_row_matrix_cons)
+ done
+
+primrec sorted_spvec :: "'a spvec \<Rightarrow> bool"
+where
+ "sorted_spvec [] = True"
+| sorted_spvec_step: "sorted_spvec (a#as) = (case as of [] \<Rightarrow> True | b#bs \<Rightarrow> ((fst a < fst b) & (sorted_spvec as)))"
+
+primrec sorted_spmat :: "'a spmat \<Rightarrow> bool"
+where
+ "sorted_spmat [] = True"
+| "sorted_spmat (a#as) = ((sorted_spvec (snd a)) & (sorted_spmat as))"
+
+declare sorted_spvec.simps [simp del]
+
+lemma sorted_spvec_empty[simp]: "sorted_spvec [] = True"
+by (simp add: sorted_spvec.simps)
+
+lemma sorted_spvec_cons1: "sorted_spvec (a#as) \<Longrightarrow> sorted_spvec as"
+apply (induct as)
+apply (auto simp add: sorted_spvec.simps)
+done
+
+lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) \<Longrightarrow> sorted_spvec (a#t)"
+apply (induct t)
+apply (auto simp add: sorted_spvec.simps)
+done
+
+lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) \<Longrightarrow> fst a < fst b"
+apply (auto simp add: sorted_spvec.simps)
+done
+
+lemma sorted_sparse_row_vector_zero[rule_format]: "m <= n \<Longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_vector arr) j m = 0"
+apply (induct arr)
+apply (auto)
+apply (frule sorted_spvec_cons2,simp)+
+apply (frule sorted_spvec_cons3, simp)
+done
+
+lemma sorted_sparse_row_matrix_zero[rule_format]: "m <= n \<Longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_matrix arr) m j = 0"
+ apply (induct arr)
+ apply (auto)
+ apply (frule sorted_spvec_cons2, simp)
+ apply (frule sorted_spvec_cons3, simp)
+ apply (simp add: sparse_row_matrix_cons)
+ done
+
+primrec minus_spvec :: "('a::ab_group_add) spvec \<Rightarrow> 'a spvec"
+where
+ "minus_spvec [] = []"
+| "minus_spvec (a#as) = (fst a, -(snd a))#(minus_spvec as)"
+
+primrec abs_spvec :: "('a::lattice_ab_group_add_abs) spvec \<Rightarrow> 'a spvec"
+where
+ "abs_spvec [] = []"
+| "abs_spvec (a#as) = (fst a, abs (snd a))#(abs_spvec as)"
+
+lemma sparse_row_vector_minus:
+ "sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"
+ apply (induct v)
+ apply (simp_all add: sparse_row_vector_cons)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ done
+
+instance matrix :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
+apply default
+unfolding abs_matrix_def .. (*FIXME move*)
+
+lemma sparse_row_vector_abs:
+ "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"
+ apply (induct v)
+ apply simp_all
+ apply (frule_tac sorted_spvec_cons1, simp)
+ apply (simp only: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply auto
+ apply (subgoal_tac "Rep_matrix (sparse_row_vector v) 0 a = 0")
+ apply (simp)
+ apply (rule sorted_sparse_row_vector_zero)
+ apply auto
+ done
+
+lemma sorted_spvec_minus_spvec:
+ "sorted_spvec v \<Longrightarrow> sorted_spvec (minus_spvec v)"
+ apply (induct v)
+ apply (simp)
+ apply (frule sorted_spvec_cons1, simp)
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spvec_abs_spvec:
+ "sorted_spvec v \<Longrightarrow> sorted_spvec (abs_spvec v)"
+ apply (induct v)
+ apply (simp)
+ apply (frule sorted_spvec_cons1, simp)
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+definition "smult_spvec y = map (% a. (fst a, y * snd a))"
+
+lemma smult_spvec_empty[simp]: "smult_spvec y [] = []"
+ by (simp add: smult_spvec_def)
+
+lemma smult_spvec_cons: "smult_spvec y (a#arr) = (fst a, y * (snd a)) # (smult_spvec y arr)"
+ by (simp add: smult_spvec_def)
+
+fun addmult_spvec :: "('a::ring) \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
+where
+ "addmult_spvec y arr [] = arr"
+| "addmult_spvec y [] brr = smult_spvec y brr"
+| "addmult_spvec y ((i,a)#arr) ((j,b)#brr) = (
+ if i < j then ((i,a)#(addmult_spvec y arr ((j,b)#brr)))
+ else (if (j < i) then ((j, y * b)#(addmult_spvec y ((i,a)#arr) brr))
+ else ((i, a + y*b)#(addmult_spvec y arr brr))))"
+(* Steven used termination "measure (% (y, a, b). length a + (length b))" *)
+
+lemma addmult_spvec_empty1[simp]: "addmult_spvec y [] a = smult_spvec y a"
+ by (induct a) auto
+
+lemma addmult_spvec_empty2[simp]: "addmult_spvec y a [] = a"
+ by (induct a) auto
+
+lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lattice_ring)) 0 = 0 \<Longrightarrow>
+ sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
+ apply (induct a)
+ apply (simp_all add: apply_matrix_add)
+ done
+
+lemma sparse_row_vector_smult: "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"
+ apply (induct a)
+ apply (simp_all add: smult_spvec_cons scalar_mult_add)
+ done
+
+lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lattice_ring) a b) =
+ (sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
+ apply (induct y a b rule: addmult_spvec.induct)
+ apply (simp add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)+
+ done
+
+lemma sorted_smult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec (smult_spvec y a)"
+ apply (auto simp add: smult_spvec_def)
+ apply (induct a)
+ apply (auto simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spvec_addmult_spvec_helper: "\<lbrakk>sorted_spvec (addmult_spvec y ((a, b) # arr) brr); aa < a; sorted_spvec ((a, b) # arr);
+ sorted_spvec ((aa, ba) # brr)\<rbrakk> \<Longrightarrow> sorted_spvec ((aa, y * ba) # addmult_spvec y ((a, b) # arr) brr)"
+ apply (induct brr)
+ apply (auto simp add: sorted_spvec.simps)
+ done
+
+lemma sorted_spvec_addmult_spvec_helper2:
+ "\<lbrakk>sorted_spvec (addmult_spvec y arr ((aa, ba) # brr)); a < aa; sorted_spvec ((a, b) # arr); sorted_spvec ((aa, ba) # brr)\<rbrakk>
+ \<Longrightarrow> sorted_spvec ((a, b) # addmult_spvec y arr ((aa, ba) # brr))"
+ apply (induct arr)
+ apply (auto simp add: smult_spvec_def sorted_spvec.simps)
+ done
+
+lemma sorted_spvec_addmult_spvec_helper3[rule_format]:
+ "sorted_spvec (addmult_spvec y arr brr) \<longrightarrow> sorted_spvec ((aa, b) # arr) \<longrightarrow> sorted_spvec ((aa, ba) # brr)
+ \<longrightarrow> sorted_spvec ((aa, b + y * ba) # (addmult_spvec y arr brr))"
+ apply (induct y arr brr rule: addmult_spvec.induct)
+ apply (simp_all add: sorted_spvec.simps smult_spvec_def split:list.split)
+ done
+
+lemma sorted_addmult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> sorted_spvec (addmult_spvec y a b)"
+ apply (induct y a b rule: addmult_spvec.induct)
+ apply (simp_all add: sorted_smult_spvec)
+ apply (rule conjI, intro strip)
+ apply (case_tac "~(i < j)")
+ apply (simp_all)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp add: sorted_spvec_addmult_spvec_helper)
+ apply (intro strip | rule conjI)+
+ apply (frule_tac as=arr in sorted_spvec_cons1)
+ apply (simp add: sorted_spvec_addmult_spvec_helper2)
+ apply (intro strip)
+ apply (frule_tac as=arr in sorted_spvec_cons1)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp)
+ apply (simp_all add: sorted_spvec_addmult_spvec_helper3)
+ done
+
+fun mult_spvec_spmat :: "('a::lattice_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec"
+where
+(* recdef mult_spvec_spmat "measure (% (c, arr, brr). (length arr) + (length brr))" *)
+ "mult_spvec_spmat c [] brr = c"
+| "mult_spvec_spmat c arr [] = c"
+| "mult_spvec_spmat c ((i,a)#arr) ((j,b)#brr) = (
+ if (i < j) then mult_spvec_spmat c arr ((j,b)#brr)
+ else if (j < i) then mult_spvec_spmat c ((i,a)#arr) brr
+ else mult_spvec_spmat (addmult_spvec a c b) arr brr)"
+
+lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lattice_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow>
+ sparse_row_vector (mult_spvec_spmat c a B) = (sparse_row_vector c) + (sparse_row_vector a) * (sparse_row_matrix B)"
+proof -
+ have comp_1: "!! a b. a < b \<Longrightarrow> Suc 0 <= nat ((int b)-(int a))" by arith
+ have not_iff: "!! a b. a = b \<Longrightarrow> (~ a) = (~ b)" by simp
+ have max_helper: "!! a b. ~ (a <= max (Suc a) b) \<Longrightarrow> False"
+ by arith
+ {
+ fix a
+ fix v
+ assume a:"a < nrows(sparse_row_vector v)"
+ have b:"nrows(sparse_row_vector v) <= 1" by simp
+ note dummy = less_le_trans[of a "nrows (sparse_row_vector v)" 1, OF a b]
+ then have "a = 0" by simp
+ }
+ note nrows_helper = this
+ show ?thesis
+ apply (induct c a B rule: mult_spvec_spmat.induct)
+ apply simp+
+ apply (rule conjI)
+ apply (intro strip)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp add: algebra_simps sparse_row_matrix_cons)
+ apply (simplesubst Rep_matrix_zero_imp_mult_zero)
+ apply (simp)
+ apply (rule disjI2)
+ apply (intro strip)
+ apply (subst nrows)
+ apply (rule order_trans[of _ 1])
+ apply (simp add: comp_1)+
+ apply (subst Rep_matrix_zero_imp_mult_zero)
+ apply (intro strip)
+ apply (case_tac "k <= j")
+ apply (rule_tac m1 = k and n1 = i and a1 = a in ssubst[OF sorted_sparse_row_vector_zero])
+ apply (simp_all)
+ apply (rule disjI2)
+ apply (rule nrows)
+ apply (rule order_trans[of _ 1])
+ apply (simp_all add: comp_1)
+
+ apply (intro strip | rule conjI)+
+ apply (frule_tac as=arr in sorted_spvec_cons1)
+ apply (simp add: algebra_simps)
+ apply (subst Rep_matrix_zero_imp_mult_zero)
+ apply (simp)
+ apply (rule disjI2)
+ apply (intro strip)
+ apply (simp add: sparse_row_matrix_cons)
+ apply (case_tac "i <= j")
+ apply (erule sorted_sparse_row_matrix_zero)
+ apply (simp_all)
+ apply (intro strip)
+ apply (case_tac "i=j")
+ apply (simp_all)
+ apply (frule_tac as=arr in sorted_spvec_cons1)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp add: sparse_row_matrix_cons algebra_simps sparse_row_vector_addmult_spvec)
+ apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
+ apply (auto)
+ apply (rule sorted_sparse_row_matrix_zero)
+ apply (simp_all)
+ apply (rule_tac A1 = "sparse_row_vector arr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
+ apply (auto)
+ apply (rule_tac m=k and n = j and a = a and arr=arr in sorted_sparse_row_vector_zero)
+ apply (simp_all)
+ apply (drule nrows_notzero)
+ apply (drule nrows_helper)
+ apply (arith)
+
+ apply (subst Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply (simp)
+ apply (subst Rep_matrix_mult)
+ apply (rule_tac j1=j in ssubst[OF foldseq_almostzero])
+ apply (simp_all)
+ apply (intro strip, rule conjI)
+ apply (intro strip)
+ apply (drule_tac max_helper)
+ apply (simp)
+ apply (auto)
+ apply (rule zero_imp_mult_zero)
+ apply (rule disjI2)
+ apply (rule nrows)
+ apply (rule order_trans[of _ 1])
+ apply (simp)
+ apply (simp)
+ done
+qed
+
+lemma sorted_mult_spvec_spmat[rule_format]:
+ "sorted_spvec (c::('a::lattice_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
+ apply (induct c a B rule: mult_spvec_spmat.induct)
+ apply (simp_all add: sorted_addmult_spvec)
+ done
+
+primrec mult_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+where
+ "mult_spmat [] A = []"
+| "mult_spmat (a#as) A = (fst a, mult_spvec_spmat [] (snd a) A)#(mult_spmat as A)"
+
+lemma sparse_row_mult_spmat:
+ "sorted_spmat A \<Longrightarrow> sorted_spvec B \<Longrightarrow>
+ sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
+ apply (induct A)
+ apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
+ done
+
+lemma sorted_spvec_mult_spmat[rule_format]:
+ "sorted_spvec (A::('a::lattice_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"
+ apply (induct A)
+ apply (auto)
+ apply (drule sorted_spvec_cons1, simp)
+ apply (case_tac A)
+ apply (auto simp add: sorted_spvec.simps)
+ done
+
+lemma sorted_spmat_mult_spmat:
+ "sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
+ apply (induct A)
+ apply (auto simp add: sorted_mult_spvec_spmat)
+ done
+
+
+fun add_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
+where
+(* "measure (% (a, b). length a + (length b))" *)
+ "add_spvec arr [] = arr"
+| "add_spvec [] brr = brr"
+| "add_spvec ((i,a)#arr) ((j,b)#brr) = (
+ if i < j then (i,a)#(add_spvec arr ((j,b)#brr))
+ else if (j < i) then (j,b) # add_spvec ((i,a)#arr) brr
+ else (i, a+b) # add_spvec arr brr)"
+
+lemma add_spvec_empty1[simp]: "add_spvec [] a = a"
+by (cases a, auto)
+
+lemma sparse_row_vector_add: "sparse_row_vector (add_spvec a b) = (sparse_row_vector a) + (sparse_row_vector b)"
+ apply (induct a b rule: add_spvec.induct)
+ apply (simp_all add: singleton_matrix_add)
+ done
+
+fun add_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+where
+(* "measure (% (A,B). (length A)+(length B))" *)
+ "add_spmat [] bs = bs"
+| "add_spmat as [] = as"
+| "add_spmat ((i,a)#as) ((j,b)#bs) = (
+ if i < j then
+ (i,a) # add_spmat as ((j,b)#bs)
+ else if j < i then
+ (j,b) # add_spmat ((i,a)#as) bs
+ else
+ (i, add_spvec a b) # add_spmat as bs)"
+
+lemma add_spmat_Nil2[simp]: "add_spmat as [] = as"
+by(cases as) auto
+
+lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat A B) = (sparse_row_matrix A) + (sparse_row_matrix B)"
+ apply (induct A B rule: add_spmat.induct)
+ apply (auto simp add: sparse_row_matrix_cons sparse_row_vector_add move_matrix_add)
+ done
+
+lemmas [code] = sparse_row_add_spmat [symmetric]
+lemmas [code] = sparse_row_vector_add [symmetric]
+
+lemma sorted_add_spvec_helper1[rule_format]: "add_spvec ((a,b)#arr) brr = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"
+ proof -
+ have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spvec x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
+ by (induct brr rule: add_spvec.induct) (auto split:if_splits)
+ then show ?thesis
+ by (case_tac brr, auto)
+ qed
+
+lemma sorted_add_spmat_helper1[rule_format]: "add_spmat ((a,b)#arr) brr = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"
+ proof -
+ have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spmat x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
+ by (rule add_spmat.induct) (auto split:if_splits)
+ then show ?thesis
+ by (case_tac brr, auto)
+ qed
+
+lemma sorted_add_spvec_helper: "add_spvec arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
+ apply (induct arr brr rule: add_spvec.induct)
+ apply (auto split:if_splits)
+ done
+
+lemma sorted_add_spmat_helper: "add_spmat arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
+ apply (induct arr brr rule: add_spmat.induct)
+ apply (auto split:if_splits)
+ done
+
+lemma add_spvec_commute: "add_spvec a b = add_spvec b a"
+by (induct a b rule: add_spvec.induct) auto
+
+lemma add_spmat_commute: "add_spmat a b = add_spmat b a"
+ apply (induct a b rule: add_spmat.induct)
+ apply (simp_all add: add_spvec_commute)
+ done
+
+lemma sorted_add_spvec_helper2: "add_spvec ((a,b)#arr) brr = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"
+ apply (drule sorted_add_spvec_helper1)
+ apply (auto)
+ apply (case_tac brr)
+ apply (simp_all)
+ apply (drule_tac sorted_spvec_cons3)
+ apply (simp)
+ done
+
+lemma sorted_add_spmat_helper2: "add_spmat ((a,b)#arr) brr = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"
+ apply (drule sorted_add_spmat_helper1)
+ apply (auto)
+ apply (case_tac brr)
+ apply (simp_all)
+ apply (drule_tac sorted_spvec_cons3)
+ apply (simp)
+ done
+
+lemma sorted_spvec_add_spvec[rule_format]: "sorted_spvec a \<longrightarrow> sorted_spvec b \<longrightarrow> sorted_spvec (add_spvec a b)"
+ apply (induct a b rule: add_spvec.induct)
+ apply (simp_all)
+ apply (rule conjI)
+ apply (clarsimp)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp)
+ apply (subst sorted_spvec_step)
+ apply (clarsimp simp: sorted_add_spvec_helper2 split: list.split)
+ apply (clarify)
+ apply (rule conjI)
+ apply (clarify)
+ apply (frule_tac as=arr in sorted_spvec_cons1, simp)
+ apply (subst sorted_spvec_step)
+ apply (clarsimp simp: sorted_add_spvec_helper2 add_spvec_commute split: list.split)
+ apply (clarify)
+ apply (frule_tac as=arr in sorted_spvec_cons1)
+ apply (frule_tac as=brr in sorted_spvec_cons1)
+ apply (simp)
+ apply (subst sorted_spvec_step)
+ apply (simp split: list.split)
+ apply (clarsimp)
+ apply (drule_tac sorted_add_spvec_helper)
+ apply (auto simp: neq_Nil_conv)
+ apply (drule sorted_spvec_cons3)
+ apply (simp)
+ apply (drule sorted_spvec_cons3)
+ apply (simp)
+ done
+
+lemma sorted_spvec_add_spmat[rule_format]: "sorted_spvec A \<longrightarrow> sorted_spvec B \<longrightarrow> sorted_spvec (add_spmat A B)"
+ apply (induct A B rule: add_spmat.induct)
+ apply (simp_all)
+ apply (rule conjI)
+ apply (intro strip)
+ apply (simp)
+ apply (frule_tac as=bs in sorted_spvec_cons1)
+ apply (simp)
+ apply (subst sorted_spvec_step)
+ apply (simp split: list.split)
+ apply (clarify, simp)
+ apply (simp add: sorted_add_spmat_helper2)
+ apply (clarify)
+ apply (rule conjI)
+ apply (clarify)
+ apply (frule_tac as=as in sorted_spvec_cons1, simp)
+ apply (subst sorted_spvec_step)
+ apply (clarsimp simp: sorted_add_spmat_helper2 add_spmat_commute split: list.split)
+ apply (clarsimp)
+ apply (frule_tac as=as in sorted_spvec_cons1)
+ apply (frule_tac as=bs in sorted_spvec_cons1)
+ apply (simp)
+ apply (subst sorted_spvec_step)
+ apply (simp split: list.split)
+ apply (clarify, simp)
+ apply (drule_tac sorted_add_spmat_helper)
+ apply (auto simp:neq_Nil_conv)
+ apply (drule sorted_spvec_cons3)
+ apply (simp)
+ apply (drule sorted_spvec_cons3)
+ apply (simp)
+ done
+
+lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (add_spmat A B)"
+ apply (induct A B rule: add_spmat.induct)
+ apply (simp_all add: sorted_spvec_add_spvec)
+ done
+
+fun le_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool"
+where
+(* "measure (% (a,b). (length a) + (length b))" *)
+ "le_spvec [] [] = True"
+| "le_spvec ((_,a)#as) [] = (a <= 0 & le_spvec as [])"
+| "le_spvec [] ((_,b)#bs) = (0 <= b & le_spvec [] bs)"
+| "le_spvec ((i,a)#as) ((j,b)#bs) = (
+ if (i < j) then a <= 0 & le_spvec as ((j,b)#bs)
+ else if (j < i) then 0 <= b & le_spvec ((i,a)#as) bs
+ else a <= b & le_spvec as bs)"
+
+fun le_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool"
+where
+(* "measure (% (a,b). (length a) + (length b))" *)
+ "le_spmat [] [] = True"
+| "le_spmat ((i,a)#as) [] = (le_spvec a [] & le_spmat as [])"
+| "le_spmat [] ((j,b)#bs) = (le_spvec [] b & le_spmat [] bs)"
+| "le_spmat ((i,a)#as) ((j,b)#bs) = (
+ if i < j then (le_spvec a [] & le_spmat as ((j,b)#bs))
+ else if j < i then (le_spvec [] b & le_spmat ((i,a)#as) bs)
+ else (le_spvec a b & le_spmat as bs))"
+
+definition disj_matrices :: "('a::zero) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
+ "disj_matrices A B \<longleftrightarrow>
+ (! j i. (Rep_matrix A j i \<noteq> 0) \<longrightarrow> (Rep_matrix B j i = 0)) & (! j i. (Rep_matrix B j i \<noteq> 0) \<longrightarrow> (Rep_matrix A j i = 0))"
+
+declare [[simp_depth_limit = 6]]
+
+lemma disj_matrices_contr1: "disj_matrices A B \<Longrightarrow> Rep_matrix A j i \<noteq> 0 \<Longrightarrow> Rep_matrix B j i = 0"
+ by (simp add: disj_matrices_def)
+
+lemma disj_matrices_contr2: "disj_matrices A B \<Longrightarrow> Rep_matrix B j i \<noteq> 0 \<Longrightarrow> Rep_matrix A j i = 0"
+ by (simp add: disj_matrices_def)
+
+
+lemma disj_matrices_add: "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D \<Longrightarrow> disj_matrices B C \<Longrightarrow>
+ (A + B <= C + D) = (A <= C & B <= (D::('a::lattice_ab_group_add) matrix))"
+ apply (auto)
+ apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
+ apply (intro strip)
+ apply (erule conjE)+
+ apply (drule_tac j=j and i=i in spec2)+
+ apply (case_tac "Rep_matrix B j i = 0")
+ apply (case_tac "Rep_matrix D j i = 0")
+ apply (simp_all)
+ apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
+ apply (intro strip)
+ apply (erule conjE)+
+ apply (drule_tac j=j and i=i in spec2)+
+ apply (case_tac "Rep_matrix A j i = 0")
+ apply (case_tac "Rep_matrix C j i = 0")
+ apply (simp_all)
+ apply (erule add_mono)
+ apply (assumption)
+ done
+
+lemma disj_matrices_zero1[simp]: "disj_matrices 0 B"
+by (simp add: disj_matrices_def)
+
+lemma disj_matrices_zero2[simp]: "disj_matrices A 0"
+by (simp add: disj_matrices_def)
+
+lemma disj_matrices_commute: "disj_matrices A B = disj_matrices B A"
+by (auto simp add: disj_matrices_def)
+
+lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>
+ (A + B <= 0) = (A <= 0 & (B::('a::lattice_ab_group_add) matrix) <= 0)"
+by (rule disj_matrices_add[of A B 0 0, simplified])
+
+lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>
+ (0 <= A + B) = (0 <= A & 0 <= (B::('a::lattice_ab_group_add) matrix))"
+by (rule disj_matrices_add[of 0 0 A B, simplified])
+
+lemma disj_matrices_add_x_le: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
+ (A <= B + C) = (A <= C & 0 <= (B::('a::lattice_ab_group_add) matrix))"
+by (auto simp add: disj_matrices_add[of 0 A B C, simplified])
+
+lemma disj_matrices_add_le_x: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
+ (B + A <= C) = (A <= C & (B::('a::lattice_ab_group_add) matrix) <= 0)"
+by (auto simp add: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)
+
+lemma disj_sparse_row_singleton: "i <= j \<Longrightarrow> sorted_spvec((j,y)#v) \<Longrightarrow> disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"
+ apply (simp add: disj_matrices_def)
+ apply (rule conjI)
+ apply (rule neg_imp)
+ apply (simp)
+ apply (intro strip)
+ apply (rule sorted_sparse_row_vector_zero)
+ apply (simp_all)
+ apply (intro strip)
+ apply (rule sorted_sparse_row_vector_zero)
+ apply (simp_all)
+ done
+
+lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lattice_ab_group_add) matrix) (B+C)"
+ apply (simp add: disj_matrices_def)
+ apply (auto)
+ apply (drule_tac j=j and i=i in spec2)+
+ apply (case_tac "Rep_matrix B j i = 0")
+ apply (case_tac "Rep_matrix C j i = 0")
+ apply (simp_all)
+ done
+
+lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lattice_ab_group_add) matrix)"
+ by (simp add: disj_matrices_x_add disj_matrices_commute)
+
+lemma disj_singleton_matrices[simp]: "disj_matrices (singleton_matrix j i x) (singleton_matrix u v y) = (j \<noteq> u | i \<noteq> v | x = 0 | y = 0)"
+ by (auto simp add: disj_matrices_def)
+
+lemma disj_move_sparse_vec_mat[simplified disj_matrices_commute]:
+ "j <= a \<Longrightarrow> sorted_spvec((a,c)#as) \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector b) (int j) i) (sparse_row_matrix as)"
+ apply (auto simp add: disj_matrices_def)
+ apply (drule nrows_notzero)
+ apply (drule less_le_trans[OF _ nrows_spvec])
+ apply (subgoal_tac "ja = j")
+ apply (simp add: sorted_sparse_row_matrix_zero)
+ apply (arith)
+ apply (rule nrows)
+ apply (rule order_trans[of _ 1 _])
+ apply (simp)
+ apply (case_tac "nat (int ja - int j) = 0")
+ apply (case_tac "ja = j")
+ apply (simp add: sorted_sparse_row_matrix_zero)
+ apply arith+
+ done
+
+lemma disj_move_sparse_row_vector_twice:
+ "j \<noteq> u \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"
+ apply (auto simp add: disj_matrices_def)
+ apply (rule nrows, rule order_trans[of _ 1], simp, drule nrows_notzero, drule less_le_trans[OF _ nrows_spvec], arith)+
+ done
+
+lemma le_spvec_iff_sparse_row_le[rule_format]: "(sorted_spvec a) \<longrightarrow> (sorted_spvec b) \<longrightarrow> (le_spvec a b) = (sparse_row_vector a <= sparse_row_vector b)"
+ apply (induct a b rule: le_spvec.induct)
+ apply (simp_all add: sorted_spvec_cons1 disj_matrices_add_le_zero disj_matrices_add_zero_le
+ disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
+ apply (rule conjI, intro strip)
+ apply (simp add: sorted_spvec_cons1)
+ apply (subst disj_matrices_add_x_le)
+ apply (simp add: disj_sparse_row_singleton[OF less_imp_le] disj_matrices_x_add disj_matrices_commute)
+ apply (simp add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
+ apply (simp, blast)
+ apply (intro strip, rule conjI, intro strip)
+ apply (simp add: sorted_spvec_cons1)
+ apply (subst disj_matrices_add_le_x)
+ apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_sparse_row_singleton[OF less_imp_le] disj_matrices_commute disj_matrices_x_add)
+ apply (blast)
+ apply (intro strip)
+ apply (simp add: sorted_spvec_cons1)
+ apply (case_tac "a=b", simp_all)
+ apply (subst disj_matrices_add)
+ apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
+ done
+
+lemma le_spvec_empty2_sparse_row[rule_format]: "sorted_spvec b \<longrightarrow> le_spvec b [] = (sparse_row_vector b <= 0)"
+ apply (induct b)
+ apply (simp_all add: sorted_spvec_cons1)
+ apply (intro strip)
+ apply (subst disj_matrices_add_le_zero)
+ apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
+ done
+
+lemma le_spvec_empty1_sparse_row[rule_format]: "(sorted_spvec b) \<longrightarrow> (le_spvec [] b = (0 <= sparse_row_vector b))"
+ apply (induct b)
+ apply (simp_all add: sorted_spvec_cons1)
+ apply (intro strip)
+ apply (subst disj_matrices_add_zero_le)
+ apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
+ done
+
+lemma le_spmat_iff_sparse_row_le[rule_format]: "(sorted_spvec A) \<longrightarrow> (sorted_spmat A) \<longrightarrow> (sorted_spvec B) \<longrightarrow> (sorted_spmat B) \<longrightarrow>
+ le_spmat A B = (sparse_row_matrix A <= sparse_row_matrix B)"
+ apply (induct A B rule: le_spmat.induct)
+ apply (simp add: sparse_row_matrix_cons disj_matrices_add_le_zero disj_matrices_add_zero_le disj_move_sparse_vec_mat[OF order_refl]
+ disj_matrices_commute sorted_spvec_cons1 le_spvec_empty2_sparse_row le_spvec_empty1_sparse_row)+
+ apply (rule conjI, intro strip)
+ apply (simp add: sorted_spvec_cons1)
+ apply (subst disj_matrices_add_x_le)
+ apply (rule disj_matrices_add_x)
+ apply (simp add: disj_move_sparse_row_vector_twice)
+ apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
+ apply (simp add: disj_move_sparse_vec_mat[OF order_refl] disj_matrices_commute)
+ apply (simp, blast)
+ apply (intro strip, rule conjI, intro strip)
+ apply (simp add: sorted_spvec_cons1)
+ apply (subst disj_matrices_add_le_x)
+ apply (simp add: disj_move_sparse_vec_mat[OF order_refl])
+ apply (rule disj_matrices_x_add)
+ apply (simp add: disj_move_sparse_row_vector_twice)
+ apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
+ apply (simp, blast)
+ apply (intro strip)
+ apply (case_tac "i=j")
+ apply (simp_all)
+ apply (subst disj_matrices_add)
+ apply (simp_all add: disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl])
+ apply (simp add: sorted_spvec_cons1 le_spvec_iff_sparse_row_le)
+ done
+
+declare [[simp_depth_limit = 999]]
+
+primrec abs_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
+where
+ "abs_spmat [] = []"
+| "abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"
+
+primrec minus_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
+where
+ "minus_spmat [] = []"
+| "minus_spmat (a#as) = (fst a, minus_spvec (snd a))#(minus_spmat as)"
+
+lemma sparse_row_matrix_minus:
+ "sparse_row_matrix (minus_spmat A) = - (sparse_row_matrix A)"
+ apply (induct A)
+ apply (simp_all add: sparse_row_vector_minus sparse_row_matrix_cons)
+ apply (subst Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ done
+
+lemma Rep_sparse_row_vector_zero: "x \<noteq> 0 \<Longrightarrow> Rep_matrix (sparse_row_vector v) x y = 0"
+proof -
+ assume x:"x \<noteq> 0"
+ have r:"nrows (sparse_row_vector v) <= Suc 0" by (rule nrows_spvec)
+ show ?thesis
+ apply (rule nrows)
+ apply (subgoal_tac "Suc 0 <= x")
+ apply (insert r)
+ apply (simp only:)
+ apply (insert x)
+ apply arith
+ done
+qed
+
+lemma sparse_row_matrix_abs:
+ "sorted_spvec A \<Longrightarrow> sorted_spmat A \<Longrightarrow> sparse_row_matrix (abs_spmat A) = abs (sparse_row_matrix A)"
+ apply (induct A)
+ apply (simp_all add: sparse_row_vector_abs sparse_row_matrix_cons)
+ apply (frule_tac sorted_spvec_cons1, simp)
+ apply (simplesubst Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply auto
+ apply (case_tac "x=a")
+ apply (simp)
+ apply (simplesubst sorted_sparse_row_matrix_zero)
+ apply auto
+ apply (simplesubst Rep_sparse_row_vector_zero)
+ apply simp_all
+ done
+
+lemma sorted_spvec_minus_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (minus_spmat A)"
+ apply (induct A)
+ apply (simp)
+ apply (frule sorted_spvec_cons1, simp)
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spvec_abs_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (abs_spmat A)"
+ apply (induct A)
+ apply (simp)
+ apply (frule sorted_spvec_cons1, simp)
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"
+ apply (induct A)
+ apply (simp_all add: sorted_spvec_minus_spvec)
+ done
+
+lemma sorted_spmat_abs_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (abs_spmat A)"
+ apply (induct A)
+ apply (simp_all add: sorted_spvec_abs_spvec)
+ done
+
+definition diff_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
+ where "diff_spmat A B = add_spmat A (minus_spmat B)"
+
+lemma sorted_spmat_diff_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (diff_spmat A B)"
+ by (simp add: diff_spmat_def sorted_spmat_minus_spmat sorted_spmat_add_spmat)
+
+lemma sorted_spvec_diff_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec B \<Longrightarrow> sorted_spvec (diff_spmat A B)"
+ by (simp add: diff_spmat_def sorted_spvec_minus_spmat sorted_spvec_add_spmat)
+
+lemma sparse_row_diff_spmat: "sparse_row_matrix (diff_spmat A B ) = (sparse_row_matrix A) - (sparse_row_matrix B)"
+ by (simp add: diff_spmat_def sparse_row_add_spmat sparse_row_matrix_minus)
+
+definition sorted_sparse_matrix :: "'a spmat \<Rightarrow> bool"
+ where "sorted_sparse_matrix A \<longleftrightarrow> sorted_spvec A & sorted_spmat A"
+
+lemma sorted_sparse_matrix_imp_spvec: "sorted_sparse_matrix A \<Longrightarrow> sorted_spvec A"
+ by (simp add: sorted_sparse_matrix_def)
+
+lemma sorted_sparse_matrix_imp_spmat: "sorted_sparse_matrix A \<Longrightarrow> sorted_spmat A"
+ by (simp add: sorted_sparse_matrix_def)
+
+lemmas sorted_sp_simps =
+ sorted_spvec.simps
+ sorted_spmat.simps
+ sorted_sparse_matrix_def
+
+lemma bool1: "(\<not> True) = False" by blast
+lemma bool2: "(\<not> False) = True" by blast
+lemma bool3: "((P\<Colon>bool) \<and> True) = P" by blast
+lemma bool4: "(True \<and> (P\<Colon>bool)) = P" by blast
+lemma bool5: "((P\<Colon>bool) \<and> False) = False" by blast
+lemma bool6: "(False \<and> (P\<Colon>bool)) = False" by blast
+lemma bool7: "((P\<Colon>bool) \<or> True) = True" by blast
+lemma bool8: "(True \<or> (P\<Colon>bool)) = True" by blast
+lemma bool9: "((P\<Colon>bool) \<or> False) = P" by blast
+lemma bool10: "(False \<or> (P\<Colon>bool)) = P" by blast
+lemmas boolarith = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10
+
+lemma if_case_eq: "(if b then x else y) = (case b of True => x | False => y)" by simp
+
+primrec pprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
+where
+ "pprt_spvec [] = []"
+| "pprt_spvec (a#as) = (fst a, pprt (snd a)) # (pprt_spvec as)"
+
+primrec nprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
+where
+ "nprt_spvec [] = []"
+| "nprt_spvec (a#as) = (fst a, nprt (snd a)) # (nprt_spvec as)"
+
+primrec pprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
+where
+ "pprt_spmat [] = []"
+| "pprt_spmat (a#as) = (fst a, pprt_spvec (snd a))#(pprt_spmat as)"
+
+primrec nprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
+where
+ "nprt_spmat [] = []"
+| "nprt_spmat (a#as) = (fst a, nprt_spvec (snd a))#(nprt_spmat as)"
+
+
+lemma pprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
+ apply (simp add: pprt_def sup_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ apply (case_tac "Rep_matrix A x xa \<noteq> 0")
+ apply (simp_all add: disj_matrices_contr1)
+ done
+
+lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
+ apply (simp add: nprt_def inf_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ apply (case_tac "Rep_matrix A x xa \<noteq> 0")
+ apply (simp_all add: disj_matrices_contr1)
+ done
+
+lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (pprt x)"
+ apply (simp add: pprt_def sup_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ done
+
+lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (nprt x)"
+ apply (simp add: nprt_def inf_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply simp
+ done
+
+lemma less_imp_le: "a < b \<Longrightarrow> a <= (b::_::order)" by (simp add: less_def)
+
+lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
+ apply (induct v)
+ apply (simp_all)
+ apply (frule sorted_spvec_cons1, auto)
+ apply (subst pprt_add)
+ apply (subst disj_matrices_commute)
+ apply (rule disj_sparse_row_singleton)
+ apply auto
+ done
+
+lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
+ apply (induct v)
+ apply (simp_all)
+ apply (frule sorted_spvec_cons1, auto)
+ apply (subst nprt_add)
+ apply (subst disj_matrices_commute)
+ apply (rule disj_sparse_row_singleton)
+ apply auto
+ done
+
+
+lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
+ apply (simp add: pprt_def)
+ apply (simp add: sup_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply (simp)
+ done
+
+lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
+ apply (simp add: nprt_def)
+ apply (simp add: inf_matrix_def)
+ apply (simp add: Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply (simp)
+ done
+
+lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
+ apply (induct m)
+ apply simp
+ apply simp
+ apply (frule sorted_spvec_cons1)
+ apply (simp add: sparse_row_matrix_cons sparse_row_vector_pprt)
+ apply (subst pprt_add)
+ apply (subst disj_matrices_commute)
+ apply (rule disj_move_sparse_vec_mat)
+ apply auto
+ apply (simp add: sorted_spvec.simps)
+ apply (simp split: list.split)
+ apply auto
+ apply (simp add: pprt_move_matrix)
+ done
+
+lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
+ apply (induct m)
+ apply simp
+ apply simp
+ apply (frule sorted_spvec_cons1)
+ apply (simp add: sparse_row_matrix_cons sparse_row_vector_nprt)
+ apply (subst nprt_add)
+ apply (subst disj_matrices_commute)
+ apply (rule disj_move_sparse_vec_mat)
+ apply auto
+ apply (simp add: sorted_spvec.simps)
+ apply (simp split: list.split)
+ apply auto
+ apply (simp add: nprt_move_matrix)
+ done
+
+lemma sorted_pprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (pprt_spvec v)"
+ apply (induct v)
+ apply (simp)
+ apply (frule sorted_spvec_cons1)
+ apply simp
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_nprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (nprt_spvec v)"
+ apply (induct v)
+ apply (simp)
+ apply (frule sorted_spvec_cons1)
+ apply simp
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spvec_pprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (pprt_spmat m)"
+ apply (induct m)
+ apply (simp)
+ apply (frule sorted_spvec_cons1)
+ apply simp
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spvec_nprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (nprt_spmat m)"
+ apply (induct m)
+ apply (simp)
+ apply (frule sorted_spvec_cons1)
+ apply simp
+ apply (simp add: sorted_spvec.simps split:list.split_asm)
+ done
+
+lemma sorted_spmat_pprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (pprt_spmat m)"
+ apply (induct m)
+ apply (simp_all add: sorted_pprt_spvec)
+ done
+
+lemma sorted_spmat_nprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (nprt_spmat m)"
+ apply (induct m)
+ apply (simp_all add: sorted_nprt_spvec)
+ done
+
+definition mult_est_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
+ "mult_est_spmat r1 r2 s1 s2 =
+ add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
+ (add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"
+
+lemmas sparse_row_matrix_op_simps =
+ sorted_sparse_matrix_imp_spmat sorted_sparse_matrix_imp_spvec
+ sparse_row_add_spmat sorted_spvec_add_spmat sorted_spmat_add_spmat
+ sparse_row_diff_spmat sorted_spvec_diff_spmat sorted_spmat_diff_spmat
+ sparse_row_matrix_minus sorted_spvec_minus_spmat sorted_spmat_minus_spmat
+ sparse_row_mult_spmat sorted_spvec_mult_spmat sorted_spmat_mult_spmat
+ sparse_row_matrix_abs sorted_spvec_abs_spmat sorted_spmat_abs_spmat
+ le_spmat_iff_sparse_row_le
+ sparse_row_matrix_pprt sorted_spvec_pprt_spmat sorted_spmat_pprt_spmat
+ sparse_row_matrix_nprt sorted_spvec_nprt_spmat sorted_spmat_nprt_spmat
+
+lemma zero_eq_Numeral0: "(0::_::number_ring) = Numeral0" by simp
+
+lemmas sparse_row_matrix_arith_simps[simplified zero_eq_Numeral0] =
+ mult_spmat.simps mult_spvec_spmat.simps
+ addmult_spvec.simps
+ smult_spvec_empty smult_spvec_cons
+ add_spmat.simps add_spvec.simps
+ minus_spmat.simps minus_spvec.simps
+ abs_spmat.simps abs_spvec.simps
+ diff_spmat_def
+ le_spmat.simps le_spvec.simps
+ pprt_spmat.simps pprt_spvec.simps
+ nprt_spmat.simps nprt_spvec.simps
+ mult_est_spmat_def
+
+
+(*lemma spm_linprog_dual_estimate_1:
+ assumes
+ "sorted_sparse_matrix A1"
+ "sorted_sparse_matrix A2"
+ "sorted_sparse_matrix c1"
+ "sorted_sparse_matrix c2"
+ "sorted_sparse_matrix y"
+ "sorted_spvec b"
+ "sorted_spvec r"
+ "le_spmat ([], y)"
+ "A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
+ "sparse_row_matrix A1 <= A"
+ "A <= sparse_row_matrix A2"
+ "sparse_row_matrix c1 <= c"
+ "c <= sparse_row_matrix c2"
+ "abs x \<le> sparse_row_matrix r"
+ shows
+ "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b, mult_spmat (add_spmat (add_spmat (mult_spmat y (diff_spmat A2 A1),
+ abs_spmat (diff_spmat (mult_spmat y A1) c1)), diff_spmat c2 c1)) r))"
+ by (insert prems, simp add: sparse_row_matrix_op_simps linprog_dual_estimate_1[where A=A])
+*)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/document/root.tex Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,25 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage{isabelle,isabellesym}
+
+% this should be the last package used
+\usepackage{pdfsetup}
+
+% urls in roman style, theory text in math-similar italics
+\urlstyle{rm}
+\isabellestyle{it}
+
+\newcommand{\ganz}{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}
+
+\begin{document}
+
+\title{Matrix}
+\author{Steven Obua}
+\maketitle
+
+%\tableofcontents
+
+\parindent 0pt\parskip 0.5ex
+
+\input{session}
+
+\end{document}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/fspmlp.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,313 @@
+(* Title: HOL/Matrix/fspmlp.ML
+ Author: Steven Obua
+*)
+
+signature FSPMLP =
+sig
+ type linprog
+ type vector = FloatSparseMatrixBuilder.vector
+ type matrix = FloatSparseMatrixBuilder.matrix
+
+ val y : linprog -> term
+ val A : linprog -> term * term
+ val b : linprog -> term
+ val c : linprog -> term * term
+ val r12 : linprog -> term * term
+
+ exception Load of string
+
+ val load : string -> int -> bool -> linprog
+end
+
+structure Fspmlp : FSPMLP =
+struct
+
+type vector = FloatSparseMatrixBuilder.vector
+type matrix = FloatSparseMatrixBuilder.matrix
+
+type linprog = term * (term * term) * term * (term * term) * (term * term)
+
+fun y (c1, _, _, _, _) = c1
+fun A (_, c2, _, _, _) = c2
+fun b (_, _, c3, _, _) = c3
+fun c (_, _, _, c4, _) = c4
+fun r12 (_, _, _, _, c6) = c6
+
+structure CplexFloatSparseMatrixConverter =
+MAKE_CPLEX_MATRIX_CONVERTER(structure cplex = Cplex and matrix_builder = FloatSparseMatrixBuilder);
+
+datatype bound_type = LOWER | UPPER
+
+fun intbound_ord ((i1: int, b1),(i2,b2)) =
+ if i1 < i2 then LESS
+ else if i1 = i2 then
+ (if b1 = b2 then EQUAL else if b1=LOWER then LESS else GREATER)
+ else GREATER
+
+structure Inttab = Table(type key = int val ord = (rev_order o int_ord));
+
+structure VarGraph = Table(type key = int*bound_type val ord = intbound_ord);
+(* key -> (float option) * (int -> (float * (((float * float) * key) list)))) *)
+(* dest_key -> (sure_bound * (row_index -> (row_bound * (((coeff_lower * coeff_upper) * src_key) list)))) *)
+
+exception Internal of string;
+
+fun add_row_bound g dest_key row_index row_bound =
+ let
+ val x =
+ case VarGraph.lookup g dest_key of
+ NONE => (NONE, Inttab.update (row_index, (row_bound, [])) Inttab.empty)
+ | SOME (sure_bound, f) =>
+ (sure_bound,
+ case Inttab.lookup f row_index of
+ NONE => Inttab.update (row_index, (row_bound, [])) f
+ | SOME _ => raise (Internal "add_row_bound"))
+ in
+ VarGraph.update (dest_key, x) g
+ end
+
+fun update_sure_bound g (key as (_, btype)) bound =
+ let
+ val x =
+ case VarGraph.lookup g key of
+ NONE => (SOME bound, Inttab.empty)
+ | SOME (NONE, f) => (SOME bound, f)
+ | SOME (SOME old_bound, f) =>
+ (SOME ((case btype of
+ UPPER => Float.min
+ | LOWER => Float.max)
+ old_bound bound), f)
+ in
+ VarGraph.update (key, x) g
+ end
+
+fun get_sure_bound g key =
+ case VarGraph.lookup g key of
+ NONE => NONE
+ | SOME (sure_bound, _) => sure_bound
+
+(*fun get_row_bound g key row_index =
+ case VarGraph.lookup g key of
+ NONE => NONE
+ | SOME (sure_bound, f) =>
+ (case Inttab.lookup f row_index of
+ NONE => NONE
+ | SOME (row_bound, _) => (sure_bound, row_bound))*)
+
+fun add_edge g src_key dest_key row_index coeff =
+ case VarGraph.lookup g dest_key of
+ NONE => raise (Internal "add_edge: dest_key not found")
+ | SOME (sure_bound, f) =>
+ (case Inttab.lookup f row_index of
+ NONE => raise (Internal "add_edge: row_index not found")
+ | SOME (row_bound, sources) =>
+ VarGraph.update (dest_key, (sure_bound, Inttab.update (row_index, (row_bound, (coeff, src_key) :: sources)) f)) g)
+
+fun split_graph g =
+ let
+ fun split (key, (sure_bound, _)) (r1, r2) = case sure_bound
+ of NONE => (r1, r2)
+ | SOME bound => (case key
+ of (u, UPPER) => (r1, Inttab.update (u, bound) r2)
+ | (u, LOWER) => (Inttab.update (u, bound) r1, r2))
+ in VarGraph.fold split g (Inttab.empty, Inttab.empty) end
+
+(* If safe is true, termination is guaranteed, but the sure bounds may be not optimal (relative to the algorithm).
+ If safe is false, termination is not guaranteed, but on termination the sure bounds are optimal (relative to the algorithm) *)
+fun propagate_sure_bounds safe names g =
+ let
+ (* returns NONE if no new sure bound could be calculated, otherwise the new sure bound is returned *)
+ fun calc_sure_bound_from_sources g (key as (_, btype)) =
+ let
+ fun mult_upper x (lower, upper) =
+ if Float.sign x = LESS then
+ Float.mult x lower
+ else
+ Float.mult x upper
+
+ fun mult_lower x (lower, upper) =
+ if Float.sign x = LESS then
+ Float.mult x upper
+ else
+ Float.mult x lower
+
+ val mult_btype = case btype of UPPER => mult_upper | LOWER => mult_lower
+
+ fun calc_sure_bound (_, (row_bound, sources)) sure_bound =
+ let
+ fun add_src_bound (coeff, src_key) sum =
+ case sum of
+ NONE => NONE
+ | SOME x =>
+ (case get_sure_bound g src_key of
+ NONE => NONE
+ | SOME src_sure_bound => SOME (Float.add x (mult_btype src_sure_bound coeff)))
+ in
+ case fold add_src_bound sources (SOME row_bound) of
+ NONE => sure_bound
+ | new_sure_bound as (SOME new_bound) =>
+ (case sure_bound of
+ NONE => new_sure_bound
+ | SOME old_bound =>
+ SOME (case btype of
+ UPPER => Float.min old_bound new_bound
+ | LOWER => Float.max old_bound new_bound))
+ end
+ in
+ case VarGraph.lookup g key of
+ NONE => NONE
+ | SOME (sure_bound, f) =>
+ let
+ val x = Inttab.fold calc_sure_bound f sure_bound
+ in
+ if x = sure_bound then NONE else x
+ end
+ end
+
+ fun propagate (key, _) (g, b) =
+ case calc_sure_bound_from_sources g key of
+ NONE => (g,b)
+ | SOME bound => (update_sure_bound g key bound,
+ if safe then
+ case get_sure_bound g key of
+ NONE => true
+ | _ => b
+ else
+ true)
+
+ val (g, b) = VarGraph.fold propagate g (g, false)
+ in
+ if b then propagate_sure_bounds safe names g else g
+ end
+
+exception Load of string;
+
+val empty_spvec = @{term "Nil :: real spvec"};
+fun cons_spvec x xs = @{term "Cons :: nat * real => real spvec => real spvec"} $ x $ xs;
+val empty_spmat = @{term "Nil :: real spmat"};
+fun cons_spmat x xs = @{term "Cons :: nat * real spvec => real spmat => real spmat"} $ x $ xs;
+
+fun calcr safe_propagation xlen names prec A b =
+ let
+ fun test_1 (lower, upper) =
+ if lower = upper then
+ (if Float.eq (lower, (~1, 0)) then ~1
+ else if Float.eq (lower, (1, 0)) then 1
+ else 0)
+ else 0
+
+ fun calcr (row_index, a) g =
+ let
+ val b = FloatSparseMatrixBuilder.v_elem_at b row_index
+ val (_, b2) = FloatArith.approx_decstr_by_bin prec (case b of NONE => "0" | SOME b => b)
+ val approx_a = FloatSparseMatrixBuilder.v_fold (fn (i, s) => fn l =>
+ (i, FloatArith.approx_decstr_by_bin prec s)::l) a []
+
+ fun fold_dest_nodes (dest_index, dest_value) g =
+ let
+ val dest_test = test_1 dest_value
+ in
+ if dest_test = 0 then
+ g
+ else let
+ val (dest_key as (_, dest_btype), row_bound) =
+ if dest_test = ~1 then
+ ((dest_index, LOWER), Float.neg b2)
+ else
+ ((dest_index, UPPER), b2)
+
+ fun fold_src_nodes (src_index, src_value as (src_lower, src_upper)) g =
+ if src_index = dest_index then g
+ else
+ let
+ val coeff = case dest_btype of
+ UPPER => (Float.neg src_upper, Float.neg src_lower)
+ | LOWER => src_value
+ in
+ if Float.sign src_lower = LESS then
+ add_edge g (src_index, UPPER) dest_key row_index coeff
+ else
+ add_edge g (src_index, LOWER) dest_key row_index coeff
+ end
+ in
+ fold fold_src_nodes approx_a (add_row_bound g dest_key row_index row_bound)
+ end
+ end
+ in
+ case approx_a of
+ [] => g
+ | [(u, a)] =>
+ let
+ val atest = test_1 a
+ in
+ if atest = ~1 then
+ update_sure_bound g (u, LOWER) (Float.neg b2)
+ else if atest = 1 then
+ update_sure_bound g (u, UPPER) b2
+ else
+ g
+ end
+ | _ => fold fold_dest_nodes approx_a g
+ end
+
+ val g = FloatSparseMatrixBuilder.m_fold calcr A VarGraph.empty
+
+ val g = propagate_sure_bounds safe_propagation names g
+
+ val (r1, r2) = split_graph g
+
+ fun add_row_entry m index f vname value =
+ let
+ val v = (case value of
+ SOME value => FloatSparseMatrixBuilder.mk_spvec_entry 0 value
+ | NONE => FloatSparseMatrixBuilder.mk_spvec_entry' 0 (f $ (Var ((vname,0), HOLogic.realT))))
+ val vec = cons_spvec v empty_spvec
+ in
+ cons_spmat (FloatSparseMatrixBuilder.mk_spmat_entry index vec) m
+ end
+
+ fun abs_estimate i r1 r2 =
+ if i = 0 then
+ let val e = empty_spmat in (e, e) end
+ else
+ let
+ val index = xlen-i
+ val (r12_1, r12_2) = abs_estimate (i-1) r1 r2
+ val b1 = Inttab.lookup r1 index
+ val b2 = Inttab.lookup r2 index
+ in
+ (add_row_entry r12_1 index @{term "lbound :: real => real"} ((names index)^"l") b1,
+ add_row_entry r12_2 index @{term "ubound :: real => real"} ((names index)^"u") b2)
+ end
+
+ val (r1, r2) = abs_estimate xlen r1 r2
+
+ in
+ (r1, r2)
+ end
+
+fun load filename prec safe_propagation =
+ let
+ val prog = Cplex.load_cplexFile filename
+ val prog = Cplex.elim_nonfree_bounds prog
+ val prog = Cplex.relax_strict_ineqs prog
+ val (maximize, c, A, b, (xlen, names, _)) = CplexFloatSparseMatrixConverter.convert_prog prog
+ val (r1, r2) = calcr safe_propagation xlen names prec A b
+ val _ = if maximize then () else raise Load "sorry, cannot handle minimization problems"
+ val (dualprog, indexof) = FloatSparseMatrixBuilder.dual_cplexProg c A b
+ val results = Cplex.solve dualprog
+ val (_, v) = CplexFloatSparseMatrixConverter.convert_results results indexof
+ (*val A = FloatSparseMatrixBuilder.cut_matrix v NONE A*)
+ fun id x = x
+ val v = FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 v
+ val b = FloatSparseMatrixBuilder.transpose_matrix (FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 b)
+ val c = FloatSparseMatrixBuilder.set_vector FloatSparseMatrixBuilder.empty_matrix 0 c
+ val (y1, _) = FloatSparseMatrixBuilder.approx_matrix prec Float.positive_part v
+ val A = FloatSparseMatrixBuilder.approx_matrix prec id A
+ val (_,b2) = FloatSparseMatrixBuilder.approx_matrix prec id b
+ val c = FloatSparseMatrixBuilder.approx_matrix prec id c
+ in
+ (y1, A, b2, c, (r1, r2))
+ end handle CplexFloatSparseMatrixConverter.Converter s => (raise (Load ("Converter: "^s)))
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix_LP/matrixlp.ML Sat Mar 17 12:52:40 2012 +0100
@@ -0,0 +1,59 @@
+(* Title: HOL/Matrix/matrixlp.ML
+ Author: Steven Obua
+*)
+
+signature MATRIX_LP =
+sig
+ val matrix_compute : cterm -> thm
+ val matrix_simplify : thm -> thm
+ val prove_bound : string -> int -> thm
+ val float2real : string * string -> Real.real
+end
+
+structure MatrixLP : MATRIX_LP =
+struct
+
+val compute_thms = ComputeHOL.prep_thms @{thms "ComputeHOL.compute_list_case" "ComputeHOL.compute_let"
+ "ComputeHOL.compute_if" "ComputeFloat.arith" "SparseMatrix.sparse_row_matrix_arith_simps"
+ "ComputeHOL.compute_bool" "ComputeHOL.compute_pair"
+ "SparseMatrix.sorted_sp_simps"
+ "ComputeNumeral.natnorm"}; (*"ComputeNumeral.number_norm"*)
+
+val spm_mult_le_dual_prts_no_let_real = @{thm "spm_mult_le_dual_prts_no_let" [where ?'a = real]}
+
+fun lp_dual_estimate_prt lptfile prec =
+ let
+ val cert = cterm_of @{theory}
+ fun var s x = (cert (Var ((s, 0), FloatSparseMatrixBuilder.spmatT)), x)
+ val l = Fspmlp.load lptfile prec false
+ val (y, (A1, A2), (c1, c2), b, (r1, r2)) =
+ let
+ open Fspmlp
+ in
+ (y l |> cert, A l |> pairself cert, c l |> pairself cert, b l |> cert, r12 l |> pairself cert)
+ end
+ in
+ Thm.instantiate ([],
+ [var "A1" A1, var "A2" A2, var "y" y, var "c1" c1, var "c2" c2, var "r1" r1, var "r2" r2, var "b" b])
+ spm_mult_le_dual_prts_no_let_real
+ end
+
+val computer = PCompute.make Compute.SML @{theory} compute_thms []
+
+fun matrix_compute c = hd (PCompute.rewrite computer [c])
+
+fun matrix_simplify th =
+ let
+ val simp_th = matrix_compute (cprop_of th)
+ val th = Thm.strip_shyps (Thm.equal_elim simp_th th)
+ fun removeTrue th = removeTrue (Thm.implies_elim th TrueI) handle THM _ => th
+ in
+ removeTrue th
+ end
+
+val prove_bound = matrix_simplify oo lp_dual_estimate_prt;
+
+val realFromStr = the o Real.fromString;
+fun float2real (x, y) = realFromStr x * Math.pow (2.0, realFromStr y);
+
+end