src/HOL/Matrix/Matrix.thy
author nipkow
Sat Jun 23 19:33:22 2007 +0200 (2007-06-23 ago)
changeset 23477 f4b83f03cac9
parent 22452 8a86fd2a1bf0
child 23879 4776af8be741
permissions -rw-r--r--
tuned and renamed group_eq_simps and ring_eq_simps
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(*  Title:      HOL/Matrix/Matrix.thy
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    ID:         $Id$
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    Author:     Steven Obua
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*)
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theory Matrix
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imports MatrixGeneral
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begin
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instance matrix :: ("{zero, lattice}") lattice
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  "inf \<equiv> combine_matrix inf"
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  "sup \<equiv> combine_matrix sup"
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  by default (auto simp add: inf_le1 inf_le2 le_infI le_matrix_def inf_matrix_def sup_matrix_def)
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instance matrix :: ("{plus, zero}") plus
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  plus_matrix_def: "A + B \<equiv> combine_matrix (op +) A B" ..
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instance matrix :: ("{minus, zero}") minus
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  minus_matrix_def: "- A \<equiv> apply_matrix uminus A"
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  diff_matrix_def: "A - B \<equiv> combine_matrix (op -) A B" ..
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instance matrix :: ("{plus, times, zero}") times
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  times_matrix_def: "A * B \<equiv> mult_matrix (op *) (op +) A B" ..
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instance matrix :: (lordered_ab_group) lordered_ab_group_meet
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  abs_matrix_def: "abs (A::('a::lordered_ab_group) matrix) == sup A (- A)"
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proof 
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  fix A B C :: "('a::lordered_ab_group) matrix"
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  show "A + B + C = A + (B + C)"    
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    apply (simp add: plus_matrix_def)
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    apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
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    apply (simp_all add: add_assoc)
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    done
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  show "A + B = B + A"
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    apply (simp add: plus_matrix_def)
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    apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
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    apply (simp_all add: add_commute)
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    done
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  show "0 + A = A"
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    apply (simp add: plus_matrix_def)
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    apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
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    apply (simp)
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    done
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  show "- A + A = 0" 
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    by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
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  show "A - B = A + - B" 
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    by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
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  assume "A <= B"
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  then show "C + A <= C + B"
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    apply (simp add: plus_matrix_def)
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    apply (rule le_left_combine_matrix)
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    apply (simp_all)
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    done
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qed
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instance matrix :: (lordered_ring) lordered_ring
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proof
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  fix A B C :: "('a :: lordered_ring) matrix"
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  show "A * B * C = A * (B * C)"
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    apply (simp add: times_matrix_def)
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    apply (rule mult_matrix_assoc)
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    apply (simp_all add: associative_def ring_simps)
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    done
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  show "(A + B) * C = A * C + B * C"
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    apply (simp add: times_matrix_def plus_matrix_def)
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    apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
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    apply (simp_all add: associative_def commutative_def ring_simps)
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    done
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  show "A * (B + C) = A * B + A * C"
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    apply (simp add: times_matrix_def plus_matrix_def)
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    apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
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    apply (simp_all add: associative_def commutative_def ring_simps)
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    done  
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  show "abs A = sup A (-A)" 
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    by (simp add: abs_matrix_def)
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  assume a: "A \<le> B"
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  assume b: "0 \<le> C"
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  from a b show "C * A \<le> C * B"
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    apply (simp add: times_matrix_def)
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    apply (rule le_left_mult)
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    apply (simp_all add: add_mono mult_left_mono)
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    done
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  from a b show "A * C \<le> B * C"
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    apply (simp add: times_matrix_def)
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    apply (rule le_right_mult)
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    apply (simp_all add: add_mono mult_right_mono)
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    done
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qed 
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lemma Rep_matrix_add[simp]: "Rep_matrix ((a::('a::lordered_ab_group)matrix)+b) j i  = (Rep_matrix a j i) + (Rep_matrix b j i)"
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by (simp add: plus_matrix_def)
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lemma Rep_matrix_mult: "Rep_matrix ((a::('a::lordered_ring) matrix) * b) j i = 
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  foldseq (op +) (% k.  (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
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apply (simp add: times_matrix_def)
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apply (simp add: Rep_mult_matrix)
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done
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lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a) \<Longrightarrow> apply_matrix f ((a::('a::lordered_ab_group) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (simp)
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done
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lemma singleton_matrix_add: "singleton_matrix j i ((a::_::lordered_ab_group)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (simp)
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done
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lemma nrows_mult: "nrows ((A::('a::lordered_ring) matrix) * B) <= nrows A"
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by (simp add: times_matrix_def mult_nrows)
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lemma ncols_mult: "ncols ((A::('a::lordered_ring) matrix) * B) <= ncols B"
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by (simp add: times_matrix_def mult_ncols)
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definition
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  one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix" where
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  "one_matrix n = Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
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lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
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apply (simp add: one_matrix_def)
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apply (simplesubst RepAbs_matrix)
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apply (rule exI[of _ n], simp add: split_if)+
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by (simp add: split_if)
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lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
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proof -
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  have "?r <= n" by (simp add: nrows_le)
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  moreover have "n <= ?r" by (simp add:le_nrows, arith)
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  ultimately show "?r = n" by simp
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qed
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lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
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proof -
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  have "?r <= n" by (simp add: ncols_le)
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  moreover have "n <= ?r" by (simp add: le_ncols, arith)
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  ultimately show "?r = n" by simp
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qed
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lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{lordered_ring,ring_1}) matrix) * (one_matrix n) = A"
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apply (subst Rep_matrix_inject[THEN sym])
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apply (rule ext)+
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apply (simp add: times_matrix_def Rep_mult_matrix)
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apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
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apply (simp_all)
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by (simp add: max_def ncols)
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lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::{lordered_ring, ring_1}) matrix)"
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apply (subst Rep_matrix_inject[THEN sym])
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apply (rule ext)+
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apply (simp add: times_matrix_def Rep_mult_matrix)
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apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
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apply (simp_all)
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by (simp add: max_def nrows)
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lemma transpose_matrix_mult: "transpose_matrix ((A::('a::{lordered_ring,comm_ring}) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
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apply (simp add: times_matrix_def)
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apply (subst transpose_mult_matrix)
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apply (simp_all add: mult_commute)
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done
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lemma transpose_matrix_add: "transpose_matrix ((A::('a::lordered_ab_group) matrix)+B) = transpose_matrix A + transpose_matrix B"
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by (simp add: plus_matrix_def transpose_combine_matrix)
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lemma transpose_matrix_diff: "transpose_matrix ((A::('a::lordered_ab_group) matrix)-B) = transpose_matrix A - transpose_matrix B"
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by (simp add: diff_matrix_def transpose_combine_matrix)
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lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::lordered_ring) matrix)) = - transpose_matrix (A::('a::lordered_ring) matrix)"
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by (simp add: minus_matrix_def transpose_apply_matrix)
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constdefs 
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  right_inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
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  "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A" 
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  left_inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
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  "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A" 
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  inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
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  "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
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lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
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apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
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by (simp add: right_inverse_matrix_def)
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lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
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apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A]) 
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by (simp add: left_inverse_matrix_def)
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lemma left_right_inverse_matrix_unique: 
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  assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
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  shows "X = Y"
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proof -
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  have "Y = Y * one_matrix (nrows A)" 
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    apply (subst one_matrix_mult_right)
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    apply (insert prems)
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    by (simp_all add: left_inverse_matrix_def)
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  also have "\<dots> = Y * (A * X)" 
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    apply (insert prems)
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    apply (frule right_inverse_matrix_dim)
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    by (simp add: right_inverse_matrix_def)
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  also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
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  also have "\<dots> = X" 
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    apply (insert prems)
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    apply (frule left_inverse_matrix_dim)
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    apply (simp_all add:  left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
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    done
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  ultimately show "X = Y" by (simp)
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qed
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lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
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  by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
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lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
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  by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
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lemma zero_imp_mult_zero: "(a::'a::ring) = 0 | b = 0 \<Longrightarrow> a * b = 0"
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by auto
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lemma Rep_matrix_zero_imp_mult_zero:
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  "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0  \<Longrightarrow> A * B = (0::('a::lordered_ring) matrix)"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
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done
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lemma add_nrows: "nrows (A::('a::comm_monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
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apply (simp add: plus_matrix_def)
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apply (rule combine_nrows)
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apply (simp_all)
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done
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lemma move_matrix_row_mult: "move_matrix ((A::('a::lordered_ring) matrix) * B) j 0 = (move_matrix A j 0) * B"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (auto simp add: Rep_matrix_mult foldseq_zero)
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apply (rule_tac foldseq_zerotail[symmetric])
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apply (auto simp add: nrows zero_imp_mult_zero max2)
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apply (rule order_trans)
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apply (rule ncols_move_matrix_le)
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apply (simp add: max1)
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done
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lemma move_matrix_col_mult: "move_matrix ((A::('a::lordered_ring) matrix) * B) 0 i = A * (move_matrix B 0 i)"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (auto simp add: Rep_matrix_mult foldseq_zero)
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apply (rule_tac foldseq_zerotail[symmetric])
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apply (auto simp add: ncols zero_imp_mult_zero max1)
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apply (rule order_trans)
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apply (rule nrows_move_matrix_le)
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apply (simp add: max2)
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done
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lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::lordered_ab_group) matrix)) = (move_matrix A j i) + (move_matrix B j i)" 
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (simp)
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done
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lemma move_matrix_mult: "move_matrix ((A::('a::lordered_ring) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
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by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
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constdefs
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  scalar_mult :: "('a::lordered_ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix"
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  "scalar_mult a m == apply_matrix (op * a) m"
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lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0" 
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by (simp add: scalar_mult_def)
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lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
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by (simp add: scalar_mult_def apply_matrix_add ring_simps)
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lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)" 
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by (simp add: scalar_mult_def)
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lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
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apply (subst Rep_matrix_inject[symmetric])
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apply (rule ext)+
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apply (auto)
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done
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lemma Rep_minus[simp]: "Rep_matrix (-(A::_::lordered_ab_group)) x y = - (Rep_matrix A x y)"
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by (simp add: minus_matrix_def)
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lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lordered_ring)) x y = abs (Rep_matrix A x y)"
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by (simp add: abs_lattice sup_matrix_def)
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end