# Theory Cplex

theory Cplex
imports SparseMatrix LP ComputeNumeral
```(*  Title:      HOL/Matrix_LP/Cplex.thy
Author:     Steven Obua
*)

theory Cplex
imports SparseMatrix LP ComputeFloat ComputeNumeral
begin

ML_file "Cplex_tools.ML"
ML_file "CplexMatrixConverter.ML"
ML_file "FloatSparseMatrixBuilder.ML"
ML_file "fspmlp.ML"

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 ≤ A"
"A ≤ sparse_row_matrix A2"
"sparse_row_matrix c1 ≤ c"
"c ≤ sparse_row_matrix c2"
"sparse_row_matrix r1 ≤ x"
"x ≤ sparse_row_matrix r2"
"A * x ≤ sparse_row_matrix (b::('a::lattice_ring) spmat)"
shows
"c * x ≤ 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 (insert assms)
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 ≤ A"
"A ≤ sparse_row_matrix A2"
"sparse_row_matrix c1 ≤ c"
"c ≤ sparse_row_matrix c2"
"sparse_row_matrix r1 ≤ x"
"x ≤ sparse_row_matrix r2"
"A * x ≤ sparse_row_matrix (b::('a::lattice_ring) spmat)"
shows
"c * x ≤ 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])

ML_file "matrixlp.ML"

end

```