(* Title: HOL/Matrix/cplex/Cplex.thy
ID: $Id$
Author: Steven Obua
*)
theory Cplex
imports SparseMatrix LP "~~/src/HOL/Tools/ComputeFloat" "~~/src/HOL/Tools/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::lordered_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::lordered_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