--- a/src/HOL/Matrix_LP/Matrix.thy Wed Aug 21 14:09:44 2024 +0100
+++ b/src/HOL/Matrix_LP/Matrix.thy Thu Aug 22 22:26:28 2024 +0100
@@ -22,6 +22,12 @@
declare Rep_matrix_inverse[simp]
+lemma matrix_eqI:
+ fixes A B :: "'a::zero matrix"
+ assumes "\<And>m n. Rep_matrix A m n = Rep_matrix B m n"
+ shows "A=B"
+ using Rep_matrix_inject assms by blast
+
lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
by (induct A) (simp add: Abs_matrix_inverse matrix_def)
@@ -44,8 +50,8 @@
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)
+ have "m \<in> ?S \<Longrightarrow> m \<le> Max(?S)" by (simp add: Max_ge [OF c])
+ moreover from d have "~(m \<le> 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)
@@ -62,7 +68,7 @@
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)"
+lemma transpose_infmatrix: "transpose_infmatrix (\<lambda>j i. P j i) = (\<lambda>j i. P i j)"
apply (rule ext)+
by simp
@@ -71,7 +77,7 @@
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 ?swap = "\<lambda>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)
@@ -102,43 +108,32 @@
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 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
+ by (metis transpose_infmatrix_twice)
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
+ unfolding transpose_matrix_def o_def
+ by (metis Rep_matrix_inject transpose_infmatrix_closed transpose_infmatrix_inject)
lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
-by (simp add: transpose_matrix_def)
+ by (simp add: transpose_matrix_def)
lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
-by (simp add: transpose_matrix_def)
+ 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)
+ 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)
+ by (metis nrows_transpose transpose_transpose_id)
-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: "ncols A \<le> n \<Longrightarrow> Rep_matrix A m n = 0"
+ by (metis nrows nrows_transpose transpose_matrix)
-lemma ncols_le: "(ncols A <= n) = (\<forall>j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
+lemma ncols_le: "(ncols A \<le> n) \<longleftrightarrow> (\<forall>j i. n \<le> i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
apply (auto)
apply (simp add: ncols)
proof (simp add: ncols_def, auto)
@@ -146,8 +141,8 @@
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)
+ assume "~(Suc (?m) \<le> n)"
+ then have b:"n \<le> ?m" by (simp)
fix a b
assume "(a,b) \<in> ?P"
then have "?p \<noteq> {}" by (auto)
@@ -158,76 +153,60 @@
ultimately show "False" using b by (simp)
qed
-lemma less_ncols: "(n < ncols A) = (\<exists>j i. n <= i & (Rep_matrix A j i) \<noteq> 0)"
+lemma less_ncols: "(n < ncols A) = (\<exists>j i. n \<le> i & (Rep_matrix A j i) \<noteq> 0)"
proof -
- have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+ have a: "!! (a::nat) b. (a < b) = (~(b \<le> 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)"
+lemma le_ncols: "(n \<le> ncols A) = (\<forall> m. (\<forall> j i. m \<le> i \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n \<le> m)"
apply (auto)
-apply (subgoal_tac "ncols A <= m")
+apply (subgoal_tac "ncols A \<le> 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) = (\<forall>j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" (is ?s)
+lemma nrows_le: "(nrows A \<le> n) = (\<forall>j i. n \<le> 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> = (\<forall>j i. n <= i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
- also have "\<dots> = (\<forall>j i. n <= i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
- finally show "(nrows A <= n) = (\<forall>j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
+ have "(nrows A \<le> n) = (ncols (transpose_matrix A) \<le> n)" by (simp)
+ also have "\<dots> = (\<forall>j i. n \<le> i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
+ also have "\<dots> = (\<forall>j i. n \<le> i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
+ finally show "(nrows A \<le> n) = (\<forall>j i. n \<le> j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
qed
-lemma less_nrows: "(m < nrows A) = (\<exists>j i. m <= j & (Rep_matrix A j i) \<noteq> 0)"
+lemma less_nrows: "(m < nrows A) = (\<exists>j i. m \<le> j & (Rep_matrix A j i) \<noteq> 0)"
proof -
- have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+ have a: "!! (a::nat) b. (a < b) = (~(b \<le> 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 le_nrows: "(n \<le> nrows A) = (\<forall> m. (\<forall> j i. m \<le> j \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n \<le> m)"
+ by (meson order.trans nrows nrows_le)
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
+ by (meson leI nrows)
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
+ by (meson leI ncols)
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
+ by (induct n) auto
lemma finite_natarray2: "finite {(x, y). x < (m::nat) & y < (n::nat)}"
-by simp
+ by simp
lemma RepAbs_matrix:
- assumes aem: "\<exists>m. \<forall>j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"\<exists>n. \<forall>j i. (n <= i \<longrightarrow> x j i = 0)" (is ?en)
+ assumes aem: "\<exists>m. \<forall>j i. m \<le> j \<longrightarrow> x j i = 0" (is ?em) and aen:"\<exists>n. \<forall>j i. (n \<le> 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: "\<forall>j i. m <= j \<longrightarrow> x j i = 0" by (blast)
- from aen obtain n where b: "\<forall>j i. n <= i \<longrightarrow> x j i = 0" by (blast)
+ from aem obtain m where a: "\<forall>j i. m \<le> j \<longrightarrow> x j i = 0" by (blast)
+ from aen obtain n where b: "\<forall>j i. n \<le> i \<longrightarrow> x j i = 0" by (blast)
let ?u = "{(i, j). x i j \<noteq> 0}"
let ?v = "{(i, j). i < m & j < n}"
- have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
+ have c: "!! (m::nat) a. ~(m \<le> a) \<Longrightarrow> a < m" by (arith)
from a b have "(?u \<inter> (-?v)) = {}"
apply (simp)
apply (rule set_eqI)
@@ -242,28 +221,28 @@
qed
definition apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix" where
- "apply_infmatrix f == % A. (% j i. f (A j i))"
+ "apply_infmatrix f == \<lambda>A. (\<lambda>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))"
+ "apply_matrix f == \<lambda>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))"
+ "combine_infmatrix f == \<lambda>A B. (\<lambda>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))"
+ "combine_matrix f == \<lambda>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)
+ 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)
+ by (simp add: combine_infmatrix_def)
definition commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool" where
-"commutative f == \<forall>x y. f x y = f y x"
+ "commutative f == \<forall>x y. f x y = f y x"
definition associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
-"associative f == \<forall>x y z. f (f x y) z = f x (f y z)"
+ "associative f == \<forall>x y z. f (f x y) z = f x (f y z)"
text\<open>
To reason about associativity and commutativity of operations on matrices,
@@ -291,10 +270,10 @@
\<close>
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)
+ by (smt (verit) UnCI expand_combine_infmatrix mem_Collect_eq nonzero_positions_def subsetI)
lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
-by (insert Rep_matrix [of A], simp add: matrix_def)
+ by (simp add: finite_nonzero_positions)
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)"
@@ -315,10 +294,10 @@
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)
+ by (simp add: associative_def combine_infmatrix_def)
lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
-by (auto)
+ 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)
@@ -331,16 +310,16 @@
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)"
+lemma combine_nrows_max: "f 0 0 = 0 \<Longrightarrow> nrows (combine_matrix f A B) \<le> 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)"
+lemma combine_ncols_max: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) \<le> 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"
+lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A \<le> q \<Longrightarrow> nrows B \<le> q \<Longrightarrow> nrows(combine_matrix f A B) \<le> 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"
+lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A \<le> q \<Longrightarrow> ncols B \<le> q \<Longrightarrow> ncols(combine_matrix f A B) \<le> q"
by (simp add: ncols_le)
definition zero_r_neutral :: "('a \<Rightarrow> 'b::zero \<Rightarrow> 'a) \<Rightarrow> bool" where
@@ -355,7 +334,7 @@
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)"
+| "foldseq f s (Suc n) = f (s 0) (foldseq f (\<lambda>k. s(Suc k)) n)"
primrec foldseq_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
where
@@ -365,28 +344,28 @@
lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
proof -
assume a:"associative f"
- then have sublemma: "\<And>n. \<forall>N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ then have sublemma: "\<And>n. \<forall>N s. N \<le> n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
proof -
fix n
- show "\<forall>N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ show "\<forall>N s. N \<le> n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
proof (induct n)
- show "\<forall>N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
+ show "\<forall>N s. N \<le> 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
next
fix n
- assume b: "\<forall>N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
- have c:"\<And>N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
- show "\<forall>N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
+ assume b: "\<forall>N s. N \<le> n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+ have c:"\<And>N s. N \<le> n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
+ show "\<forall>N t. N \<le> Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
proof (auto)
fix N t
- assume Nsuc: "N <= Suc n"
+ assume Nsuc: "N \<le> Suc n"
show "foldseq f t N = foldseq_transposed f t N"
proof cases
- assume "N <= n"
+ assume "N \<le> n"
then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
next
- assume "~(N <= n)"
+ assume "~(N \<le> n)"
with Nsuc have Nsuceq: "N = Suc n" by simp
- have neqz: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m & Suc m <= n" by arith
+ have neqz: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m & Suc m \<le> 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)
@@ -400,9 +379,9 @@
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")
+ assume "n = Suc m & Suc m \<le> n"
+ then have mless: "Suc m \<le> n" by arith
+ then have step1: "foldseq_transposed f (\<lambda>k. t (Suc k)) m = foldseq f (\<lambda>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
@@ -419,18 +398,18 @@
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)"
+lemma foldseq_distr: "\<lbrakk>associative f; commutative f\<rbrakk> \<Longrightarrow> foldseq f (\<lambda>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 "\<And>n. (\<forall>u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
+ have "\<And>n. (\<forall>u v. foldseq f (\<lambda>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
+ then show "foldseq f (\<lambda>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); \<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 | (\<forall>y. f y x = y) | (\<forall>y. g y x = y)"
@@ -452,10 +431,10 @@
*)
lemma foldseq_zero:
-assumes fz: "f 0 0 = 0" and sz: "\<forall>i. i <= n \<longrightarrow> s i = 0"
+assumes fz: "f 0 0 = 0" and sz: "\<forall>i. i \<le> n \<longrightarrow> s i = 0"
shows "foldseq f s n = 0"
proof -
- have "\<And>n. \<forall>s. (\<forall>i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
+ have "\<And>n. \<forall>s. (\<forall>i. i \<le> n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
apply (induct_tac n)
apply (simp)
by (simp add: fz)
@@ -463,7 +442,7 @@
qed
lemma foldseq_significant_positions:
- assumes p: "\<forall>i. i <= N \<longrightarrow> S i = T i"
+ assumes p: "\<forall>i. i \<le> N \<longrightarrow> S i = T i"
shows "foldseq f S N = foldseq f T N"
proof -
have "\<And>m. \<forall>s t. (\<forall>i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
@@ -485,12 +464,12 @@
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"
+ assumes "M \<le> N"
+ shows "foldseq f S N = foldseq f (\<lambda>k. (if k < M then (S k) else (foldseq f (\<lambda>k. S(k+M)) (N-M)))) M"
proof -
- have suc: "\<And>a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
+ have suc: "\<And>a b. \<lbrakk>a \<le> Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a \<le> b" by arith
have a: "\<And>a b c . a = b \<Longrightarrow> f c a = f c b" by blast
- have "\<And>n. \<forall>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"
+ have "\<And>n. \<forall>m s. m \<le> n \<longrightarrow> foldseq f s n = foldseq f (\<lambda>k. (if k < m then (s k) else (foldseq f (\<lambda>k. s(k+m)) (n-m)))) m"
apply (induct_tac n)
apply (simp)
apply (simp)
@@ -504,12 +483,12 @@
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
+ assume subb:"m \<le> na"
+ from suba have subc:"!! m s. m \<le> 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> \<exists>mm. m = Suc mm & mm <= na" by arith
+ foldseq f (\<lambda>k. s(Suc k)) na"
+ by (rule subc[of m "\<lambda>k. s(Suc k)", THEN sym], simp add: subb)
+ from subb have sube: "m \<noteq> 0 \<Longrightarrow> \<exists>mm. m = Suc mm & mm \<le> 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)
@@ -525,30 +504,20 @@
qed
lemma foldseq_zerotail:
- assumes
- fz: "f 0 0 = 0"
- and sz: "\<forall>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
+ assumes fz: "f 0 0 = 0" and sz: "\<forall>i. n \<le> i \<longrightarrow> s i = 0" and nm: "n \<le> m"
+ shows "foldseq f s n = foldseq f s m"
+ unfolding foldseq_tail[OF nm]
+ by (metis (no_types, lifting) foldseq_zero fz le_add2 linorder_not_le sz)
lemma foldseq_zerotail2:
assumes "\<forall>x. f x 0 = x"
and "\<forall>i. n < i \<longrightarrow> s i = 0"
- and nm: "n <= m"
+ and nm: "n \<le> m"
shows "foldseq f s n = foldseq f s m"
proof -
have "f 0 0 = 0" by (simp add: assms)
- have b: "\<And>m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> \<exists>k. m-n = Suc k" by arith
- have c: "0 <= m" by simp
+ have b: "\<And>m n. n \<le> m \<Longrightarrow> m \<noteq> n \<Longrightarrow> \<exists>k. m-n = Suc k" by arith
+ have c: "0 \<le> m" by simp
have d: "\<And>k. k \<noteq> 0 \<Longrightarrow> \<exists>l. k = Suc l" by arith
show ?thesis
apply (subst foldseq_tail[OF nm])
@@ -567,7 +536,7 @@
qed
lemma foldseq_zerostart:
- "\<forall>x. f 0 (f 0 x) = f 0 x \<Longrightarrow> \<forall>i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
+ "\<forall>x. f 0 (f 0 x) = f 0 x \<Longrightarrow> \<forall>i. i \<le> n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
proof -
assume f00x: "\<forall>x. f 0 (f 0 x) = f 0 x"
have "\<forall>s. (\<forall>i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
@@ -577,16 +546,16 @@
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
+ have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (\<lambda>k. s(Suc k)) (Suc n))" by simp
assume b: "\<forall>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: "\<forall>i. i <= Suc n \<longrightarrow> s i = 0"
+ assume d: "\<forall>i. i \<le> 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 "\<forall>i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
+ then show "\<forall>i. i \<le> n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
qed
lemma foldseq_zerostart2:
@@ -595,7 +564,7 @@
assume a: "\<forall>i. i<n \<longrightarrow> s i = 0"
assume x: "\<forall>x. f 0 x = x"
from x have f00x: "\<forall>x. f 0 (f 0 x) = f 0 x" by blast
- have b: "\<And>i l. i < Suc l = (i <= l)" by arith
+ have b: "\<And>i l. i < Suc l = (i \<le> l)" by arith
have d: "\<And>k. k \<noteq> 0 \<Longrightarrow> \<exists>l. k = Suc l" by arith
show "foldseq f s n = s n"
apply (case_tac "n=0")
@@ -611,7 +580,7 @@
lemma foldseq_almostzero:
assumes f0x: "\<forall>x. f 0 x = x" and fx0: "\<forall>x. f x 0 = x" and s0: "\<forall>i. i \<noteq> j \<longrightarrow> s i = 0"
- shows "foldseq f s n = (if (j <= n) then (s j) else 0)"
+ shows "foldseq f s n = (if (j \<le> n) then (s j) else 0)"
proof -
from s0 have a: "\<forall>i. i < j \<longrightarrow> s i = 0" by simp
from s0 have b: "\<forall>i. j < i \<longrightarrow> s i = 0" by simp
@@ -627,20 +596,20 @@
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"
+ shows "g(foldseq f s n) = foldseq f (\<lambda>x. g(s x)) n"
proof -
- have "\<forall>s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
+ have "\<forall>s. g(foldseq f s n) = foldseq f (\<lambda>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)
+ apply (drule_tac x="\<lambda>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)"
+ "mult_matrix_n n fmul fadd A B == Abs_matrix(\<lambda>j i. foldseq fadd (\<lambda>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"
@@ -657,7 +626,7 @@
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"
+ assumes "ncols A \<le> n" "nrows B \<le> n" "ncols A \<le> m" "nrows B \<le> 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"
@@ -676,8 +645,8 @@
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 max1: "!! a x y. (a::nat) \<le> x \<Longrightarrow> a \<le> max x y" by (arith)
+lemma max2: "!! b x y. (b::nat) \<le> y \<Longrightarrow> b \<le> max x y" by (arith)
lemma r_distributive_matrix:
assumes
@@ -775,13 +744,13 @@
lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
proof -
- have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
+ have a:"!! (x::nat). x \<le> 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)
+ have a:"!! (x::nat). x \<le> 0 \<Longrightarrow> x = 0" by (arith)
show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
qed
@@ -821,26 +790,23 @@
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
+ by (simp add: transpose_matrix_def zero_matrix_def RepAbs_matrix transpose_infmatrix)
-lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
+lemma apply_zero_matrix_def[simp]: "apply_matrix (\<lambda>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)"
+ "singleton_matrix j i a == Abs_matrix(\<lambda>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)))"
+ "move_matrix A y x == Abs_matrix(\<lambda>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)"
+ "take_rows A r == Abs_matrix(\<lambda>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)"
+ "take_columns A c == Abs_matrix(\<lambda>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"
@@ -857,17 +823,14 @@
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
+ by (simp add: matrix_eqI)
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+
+ 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)
@@ -909,9 +872,7 @@
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
+ by (simp add: matrix_eqI)
lemma Rep_move_matrix[simp]:
"Rep_matrix (move_matrix A y x) j i =
@@ -926,27 +887,17 @@
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
+ by (simp add: matrix_eqI)
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
+ by (simp add: matrix_eqI)
-lemma move_matrix_singleton[simp]: "move_matrix (singleton_matrix u v x) j i =
+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
+ apply (intro matrix_eqI)
+ apply (split if_split)
+ apply (auto simp: split: if_split_asm)
done
lemma Rep_take_columns[simp]:
@@ -975,15 +926,11 @@
"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 column_of_matrix: "ncols A \<le> n \<Longrightarrow> column_of_matrix A n = 0"
+ by (simp add: matrix_eqI 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 row_of_matrix: "nrows A \<le> n \<Longrightarrow> row_of_matrix A n = 0"
+ by (simp add: matrix_eqI nrows)
lemma mult_matrix_singleton_right[simp]:
assumes
@@ -991,7 +938,7 @@
"\<forall>x. fmul 0 x = 0"
"\<forall>x. fadd 0 x = x"
"\<forall>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))"
+ shows "(mult_matrix fmul fadd A (singleton_matrix j i e)) = apply_matrix (\<lambda>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)
@@ -1012,37 +959,33 @@
"\<forall>a. fmul a 0 = 0"
"\<forall>a. fadd a 0 = a"
"\<forall>a. fadd 0 a = a"
- and contraprems:
- "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
- shows
- "A = B"
+ 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: "\<And>f g. (\<forall>x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
have "\<exists>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)
+ using Rep_matrix_inject a by blast
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:"(\<forall>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)
+ have e: "(\<lambda>x. fmul x e) 0 = 0" by (simp add: assms)
+ have "Rep_matrix (apply_matrix (\<lambda>x. fmul x e) (column_of_matrix A I)) \<noteq>
+ Rep_matrix (apply_matrix (\<lambda>x. fmul x e) (column_of_matrix B I))"
+ using fprems
+ by (metis Rep_apply_matrix Rep_column_of_matrix eprops c)
+ then have "~(?comp A ?S = ?comp B ?S)"
+ by (simp add: fprems eprops Rep_matrix_inject)
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"
+ "foldmatrix f g A m n == foldseq_transposed g (\<lambda>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"
+ "foldmatrix_transposed f g A m n == foldseq g (\<lambda>j. foldseq_transposed f (A j) n) m"
lemma foldmatrix_transpose:
assumes
@@ -1055,13 +998,13 @@
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)
+ by (drule_tac x="(\<lambda>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)
+ apply (drule_tac x="\<lambda>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
@@ -1071,7 +1014,7 @@
"associative g"
"\<forall>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"
+ "foldseq g (\<lambda>j. foldseq f (A j) n) m = foldseq f (\<lambda>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)
@@ -1123,14 +1066,14 @@
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)
+lemma nrows_move_matrix_le: "nrows (move_matrix A j i) \<le> nat((int (nrows A)) + j)"
+ apply (auto simp: 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)
+lemma ncols_move_matrix_le: "ncols (move_matrix A j i) \<le> nat((int (ncols A)) + i)"
+ apply (auto simp: ncols_le)
apply (rule ncols)
apply (arith)
done
@@ -1152,14 +1095,13 @@
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)
+ have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n) x = foldseq fadd1 (\<lambda>k. fmul2 (s k) x) n"
+ by (rule_tac g1 = "\<lambda>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 (\<lambda>k. fmul1 x (s k)) n"
+ using assms by (rule_tac g1 = "\<lambda>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 (intro matrix_eqI)
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)+
@@ -1236,14 +1178,10 @@
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
+ by (simp add: matrix_eqI)
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
+ by (simp add: matrix_eqI)
lemma Rep_mult_matrix:
assumes
@@ -1252,7 +1190,7 @@
"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))"
+ foldseq fadd (\<lambda>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)
@@ -1268,21 +1206,14 @@
"\<forall>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 ac_simps)
- done
+ by (simp add: matrix_eqI Rep_mult_matrix ac_simps)
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
+ by (simp add: matrix_eqI)
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
+ by (simp add: matrix_eqI)
instantiation matrix :: ("{zero, ord}") ord
begin
@@ -1298,82 +1229,79 @@
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
+proof
+ fix x y z :: "'a matrix"
+ assume "x \<le> y" "y \<le> z"
+ show "x \<le> z"
+ by (meson \<open>x \<le> y\<close> \<open>y \<le> z\<close> le_matrix_def order_trans)
+next
+ fix x y :: "'a matrix"
+ assume "x \<le> y" "y \<le> x"
+ show "x = y"
+ by (meson \<open>x \<le> y\<close> \<open>y \<le> x\<close> le_matrix_def matrix_eqI order_antisym)
+qed (auto simp: less_def le_matrix_def)
lemma le_apply_matrix:
assumes
"f 0 = 0"
- "\<forall>x y. x <= y \<longrightarrow> f x <= f y"
- "(a::('a::{ord, zero}) matrix) <= b"
- shows
- "apply_matrix f a <= apply_matrix f b"
+ "\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y"
+ "(a::('a::{ord, zero}) matrix) \<le> b"
+ shows "apply_matrix f a \<le> apply_matrix f b"
using assms by (simp add: le_matrix_def)
lemma le_combine_matrix:
assumes
"f 0 0 = 0"
- "\<forall>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"
+ "\<forall>a b c d. a \<le> b & c \<le> d \<longrightarrow> f a c \<le> f b d"
+ "A \<le> B"
+ "C \<le> D"
+ shows "combine_matrix f A C \<le> combine_matrix f B D"
using assms by (simp add: le_matrix_def)
lemma le_left_combine_matrix:
assumes
"f 0 0 = 0"
- "\<forall>a b c. a <= b \<longrightarrow> f c a <= f c b"
- "A <= B"
+ "\<forall>a b c. a \<le> b \<longrightarrow> f c a \<le> f c b"
+ "A \<le> B"
shows
- "combine_matrix f C A <= combine_matrix f C B"
+ "combine_matrix f C A \<le> combine_matrix f C B"
using assms by (simp add: le_matrix_def)
lemma le_right_combine_matrix:
assumes
"f 0 0 = 0"
- "\<forall>a b c. a <= b \<longrightarrow> f a c <= f b c"
- "A <= B"
+ "\<forall>a b c. a \<le> b \<longrightarrow> f a c \<le> f b c"
+ "A \<le> B"
shows
- "combine_matrix f A C <= combine_matrix f B C"
+ "combine_matrix f A C \<le> 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)"
+lemma le_transpose_matrix: "(A \<le> B) = (transpose_matrix A \<le> transpose_matrix B)"
by (simp add: le_matrix_def, auto)
lemma le_foldseq:
assumes
- "\<forall>a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
- "\<forall>i. i <= n \<longrightarrow> s i <= t i"
+ "\<forall>a b c d . a \<le> b & c \<le> d \<longrightarrow> f a c \<le> f b d"
+ "\<forall>i. i \<le> n \<longrightarrow> s i \<le> t i"
shows
- "foldseq f s n <= foldseq f t n"
+ "foldseq f s n \<le> foldseq f t n"
proof -
- have "\<forall>s t. (\<forall>i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n"
+ have "\<forall>s t. (\<forall>i. i<=n \<longrightarrow> s i \<le> t i) \<longrightarrow> foldseq f s n \<le> 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
+ then show "foldseq f s n \<le> foldseq f t n" using assms by simp
qed
lemma le_left_mult:
assumes
- "\<forall>a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
- "\<forall>c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
+ "\<forall>a b c d. a \<le> b & c \<le> d \<longrightarrow> fadd a c \<le> fadd b d"
+ "\<forall>c a b. 0 \<le> c & a \<le> b \<longrightarrow> fmul c a \<le> fmul c b"
"\<forall>a. fmul 0 a = 0"
"\<forall>a. fmul a 0 = 0"
"fadd 0 0 = 0"
- "0 <= C"
- "A <= B"
+ "0 \<le> C"
+ "A \<le> B"
shows
- "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
+ "mult_matrix fmul fadd C A \<le> mult_matrix fmul fadd C B"
using assms
apply (simp add: le_matrix_def Rep_mult_matrix)
apply (auto)
@@ -1384,15 +1312,15 @@
lemma le_right_mult:
assumes
- "\<forall>a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
- "\<forall>c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
+ "\<forall>a b c d. a \<le> b & c \<le> d \<longrightarrow> fadd a c \<le> fadd b d"
+ "\<forall>c a b. 0 \<le> c & a \<le> b \<longrightarrow> fmul a c \<le> fmul b c"
"\<forall>a. fmul 0 a = 0"
"\<forall>a. fmul a 0 = 0"
"fadd 0 0 = 0"
- "0 <= C"
- "A <= B"
+ "0 \<le> C"
+ "A \<le> B"
shows
- "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
+ "mult_matrix fmul fadd A C \<le> mult_matrix fmul fadd B C"
using assms
apply (simp add: le_matrix_def Rep_mult_matrix)
apply (auto)
@@ -1404,10 +1332,10 @@
lemma spec2: "\<forall>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_matrix_le[simp]: "(singleton_matrix j i a \<le> singleton_matrix j i b) = (a \<le> (b::_::order))"
+ by (auto simp: le_matrix_def)
-lemma singleton_le_zero[simp]: "(singleton_matrix j i x <= 0) = (x <= (0::'a::{order,zero}))"
+lemma singleton_le_zero[simp]: "(singleton_matrix j i x \<le> 0) = (x \<le> (0::'a::{order,zero}))"
apply (auto)
apply (simp add: le_matrix_def)
apply (drule_tac j=j and i=i in spec2)
@@ -1415,7 +1343,7 @@
apply (simp add: le_matrix_def)
done
-lemma singleton_ge_zero[simp]: "(0 <= singleton_matrix j i x) = ((0::'a::{order,zero}) <= x)"
+lemma singleton_ge_zero[simp]: "(0 \<le> singleton_matrix j i x) = ((0::'a::{order,zero}) \<le> x)"
apply (auto)
apply (simp add: le_matrix_def)
apply (drule_tac j=j and i=i in spec2)
@@ -1423,20 +1351,20 @@
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)
+lemma move_matrix_le_zero[simp]: "0 \<le> j \<Longrightarrow> 0 \<le> i \<Longrightarrow> (move_matrix A j i \<le> 0) = (A \<le> (0::('a::{order,zero}) matrix))"
+ apply (auto simp: 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)
+lemma move_matrix_zero_le[simp]: "0 \<le> j \<Longrightarrow> 0 \<le> i \<Longrightarrow> (0 \<le> move_matrix A j i) = ((0::('a::{order,zero}) matrix) \<le> A)"
+ apply (auto simp: 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)
+lemma move_matrix_le_move_matrix_iff[simp]: "0 \<le> j \<Longrightarrow> 0 \<le> i \<Longrightarrow> (move_matrix A j i \<le> move_matrix B j i) = (A \<le> (B::('a::{order,zero}) matrix))"
+ apply (auto simp: le_matrix_def)
apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
apply (auto)
done
@@ -1449,7 +1377,7 @@
definition "sup = combine_matrix sup"
instance
- by standard (auto simp add: le_infI le_matrix_def inf_matrix_def sup_matrix_def)
+ by standard (auto simp: le_infI le_matrix_def inf_matrix_def sup_matrix_def)
end
@@ -1542,25 +1470,25 @@
proof
fix A B :: "'a matrix"
show "- A + A = 0"
- by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ by (simp add: plus_matrix_def minus_matrix_def matrix_eqI)
show "A + - B = A - B"
- by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject [symmetric] ext)
+ by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def matrix_eqI)
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)
+ by (simp add: plus_matrix_def minus_matrix_def matrix_eqI)
show "A - B = A + - B"
- by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def matrix_eqI)
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"
+ assume "A \<le> B"
+ then show "C + A \<le> C + B"
apply (simp add: plus_matrix_def)
apply (rule le_left_combine_matrix)
apply (simp_all)
@@ -1618,38 +1546,38 @@
by (simp add: abs_matrix_def)
qed
+instance matrix :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
+proof
+ show "\<And>a:: 'a matrix. \<bar>a\<bar> = 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 (+) (% k. (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
+ foldseq (+) (\<lambda>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: "\<forall>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
+ by (simp add: matrix_eqI)
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
+ by (simp add: matrix_eqI)
-lemma nrows_mult: "nrows ((A::('a::semiring_0) matrix) * B) <= nrows A"
+lemma nrows_mult: "nrows ((A::('a::semiring_0) matrix) * B) \<le> nrows A"
by (simp add: times_matrix_def mult_nrows)
-lemma ncols_mult: "ncols ((A::('a::semiring_0) matrix) * B) <= ncols B"
+lemma ncols_mult: "ncols ((A::('a::semiring_0) matrix) * B) \<le> 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)"
+ "one_matrix n = Abs_matrix (\<lambda>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)
@@ -1659,33 +1587,29 @@
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)
+ have "?r \<le> n" by (simp add: nrows_le)
+ moreover have "n \<le> ?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)
+ have "?r \<le> n" by (simp add: ncols_le)
+ moreover have "n \<le> ?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_right[simp]: "ncols A \<le> n \<Longrightarrow> (A::('a::{semiring_1}) matrix) * (one_matrix n) = A"
+ apply (intro matrix_eqI)
+ apply (simp add: times_matrix_def Rep_mult_matrix)
+ apply (subst foldseq_almostzero, auto simp: ncols)
+ done
-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 one_matrix_mult_left[simp]: "nrows A \<le> n \<Longrightarrow> (one_matrix n) * A = (A::('a::semiring_1) matrix)"
+ apply (intro matrix_eqI)
+ apply (simp add: times_matrix_def Rep_mult_matrix)
+ apply (subst foldseq_almostzero, auto simp: nrows)
+ done
lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
apply (simp add: times_matrix_def)
@@ -1742,7 +1666,7 @@
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)
+ by (auto simp: 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)
@@ -1752,44 +1676,40 @@
lemma Rep_matrix_zero_imp_mult_zero:
"\<forall>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
+ by (simp add: matrix_eqI Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
-lemma add_nrows: "nrows (A::('a::monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
+lemma add_nrows: "nrows (A::('a::monoid_add) matrix) \<le> u \<Longrightarrow> nrows B \<le> u \<Longrightarrow> nrows (A + B) \<le> 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
+proof -
+ have "\<And>m. \<not> int m < j \<Longrightarrow> ncols (move_matrix A j 0) \<le> max (ncols A) (nrows B)"
+ by (smt (verit, best) max1 nat_int ncols_move_matrix_le)
+ then show ?thesis
+ apply (intro matrix_eqI)
+ apply (auto simp: Rep_matrix_mult foldseq_zero)
+ apply (rule_tac foldseq_zerotail[symmetric])
+ apply (auto simp: nrows zero_imp_mult_zero max2)
+ done
+qed
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
+proof -
+ have "\<And>n. \<not> int n < i \<Longrightarrow> nrows (move_matrix B 0 i) \<le> max (ncols A) (nrows B)"
+ by (smt (verit, del_insts) max2 nat_int nrows_move_matrix_le)
+ then show ?thesis
+ apply (intro matrix_eqI)
+ apply (auto simp: Rep_matrix_mult foldseq_zero)
+ apply (rule_tac foldseq_zerotail[symmetric])
+ apply (auto simp: ncols zero_imp_mult_zero max1)
+ done
+ qed
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
+ by (simp add: matrix_eqI)
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)
@@ -1798,24 +1718,21 @@
"scalar_mult a m == apply_matrix ((*) a) m"
lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0"
-by (simp add: scalar_mult_def)
+ 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)
+ 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)
+ 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
+ by (simp add: scalar_mult_def)
lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
-by (simp add: minus_matrix_def)
+ by (simp add: minus_matrix_def)
lemma Rep_abs[simp]: "Rep_matrix \<bar>A::_::lattice_ab_group_add\<bar> x y = \<bar>Rep_matrix A x y\<bar>"
-by (simp add: abs_lattice sup_matrix_def)
+ by (simp add: abs_lattice sup_matrix_def)
end