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Copy pathMatrix.thy
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Matrix.thy
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(* Title: HOL/Matrix_LP/Matrix.thy
Author: Steven Obua
*)
theory Matrix
imports Main "~~/src/HOL/Library/Lattice_Algebras" temp Complex
begin
type_synonym infmatrix = "nat \<Rightarrow> nat \<Rightarrow> real"
consts infinite::nat
definition nonzero_positions :: "(nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> (nat \<times> nat) set" where
"nonzero_positions A = {pos. A (fst pos) (snd pos) ~= 0}"
definition "matrix = {(f::(nat \<Rightarrow> nat \<Rightarrow> real)). finite (nonzero_positions f)\<and>
( (nonzero_positions f) ~={}\<longrightarrow> Max (fst`(nonzero_positions f))<infinite\<and>Max (snd`(nonzero_positions f))<infinite ) }"
typedef matrix = "matrix :: (nat \<Rightarrow> nat \<Rightarrow> real) set"
unfolding matrix_def
proof
show "(\<lambda>j i. 0) \<in> {(f::(nat \<Rightarrow> nat \<Rightarrow> real)). finite (nonzero_positions f)\<and>
((nonzero_positions f) ~={}\<longrightarrow>Max (fst`(nonzero_positions f))<infinite\<and>Max (snd`(nonzero_positions f))<infinite ) }"
apply (simp add: nonzero_positions_def)
done
qed
declare Rep_matrix_inverse[simp]
lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A)) \<and>
((nonzero_positions (Rep_matrix A)) ~={}\<longrightarrow>Max (fst`(nonzero_positions (Rep_matrix A)))<infinite\<and>Max (snd`(nonzero_positions (Rep_matrix A)))<infinite )"
apply(induct A)
apply(simp add:Abs_matrix_inverse matrix_def)
using Rep_matrix matrix_def by blast
definition nrows :: " 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 :: " 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_max[simp]:"nrows A\<le>infinite"
apply(simp add:nrows_def)
using finite_nonzero_positions by fastforce
lemma ncols_max[simp]:"ncols A\<le>infinite"
apply(simp add:ncols_def)
using finite_nonzero_positions by fastforce
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 :: " infmatrix \<Rightarrow> infmatrix" where
"transpose_infmatrix A j i == A i j"
definition transpose_matrix :: " matrix \<Rightarrow> matrix" where
"transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"
declare transpose_infmatrix_def[simp]
(*ext (\<forall>x. f x = g x) =\<Rightarrow> f = g*)
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)
have " finite {pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0} "
using `inj_on (\<lambda>pos. (snd pos, fst pos)) {pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}` calculation finite_imageD by blast
have " ((\<exists>a b. Rep_matrix x b a \<noteq> 0) \<longrightarrow> Max {y. \<exists>b. Rep_matrix x b y \<noteq> 0} < infinite
\<and> Max {y. \<exists>a. Rep_matrix x y a \<noteq> 0} < infinite)"
proof -
have " (nonzero_positions (Rep_matrix x)) ~={}\<longrightarrow>Max (fst`(nonzero_positions (Rep_matrix x)))<infinite
\<and>Max (snd`(nonzero_positions (Rep_matrix x)))<infinite"
using finite_nonzero_positions by blast
then have "((nonzero_positions (Rep_matrix x)) ~={}) = (\<exists>a b. Rep_matrix x b a \<noteq> 0)"
by (metis (mono_tags, lifting) Collect_empty_eq le0 nonzero_positions_def nrows nrows_def)
then have " fst ` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = fst ` {(a,b). Rep_matrix x a b \<noteq> 0}"
by (metis (mono_tags, lifting) Collect_cong split_beta')
then have "fst ` {(a,b). Rep_matrix x a b \<noteq> 0} ={y. \<exists>a. Rep_matrix x y a \<noteq> 0}"
apply(auto)
by force
then have "fst ` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}={y. \<exists>a. Rep_matrix x y a \<noteq> 0}"
by (simp add: `fst \` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = fst \` {(a, b). Rep_matrix x a b \<noteq> 0}`)
then have " snd ` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = snd ` {(a,b). Rep_matrix x a b \<noteq> 0}"
by (metis (mono_tags, lifting) Collect_cong split_beta')
then have "snd ` {(a,b). Rep_matrix x a b \<noteq> 0} ={y. \<exists>b. Rep_matrix x b y \<noteq> 0}"
apply(auto)
by force
then have "snd ` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}= {y. \<exists>b. Rep_matrix x b y \<noteq> 0}"
by (simp add: `snd \` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = snd \` {(a, b). Rep_matrix x a b \<noteq> 0}`)
show " ((\<exists>a b. Rep_matrix x b a \<noteq> 0) \<longrightarrow> Max {y. \<exists>b. Rep_matrix x b y \<noteq> 0} < infinite
\<and> Max {y. \<exists>a. Rep_matrix x y a \<noteq> 0} < infinite)"
using `fst \` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = {y. \<exists>a. Rep_matrix x y a \<noteq> 0}` `nonzero_positions (Rep_matrix x) \<noteq> {} \<longrightarrow> Max (fst \` nonzero_positions (Rep_matrix x)) < infinite \<and> Max (snd \` nonzero_positions (Rep_matrix x)) < infinite` `snd \` {pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0} = {y. \<exists>b. Rep_matrix x b y \<noteq> 0}` nonzero_positions_def by auto
qed
then show " finite {pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0} \<and>
((\<exists>a b. Rep_matrix x b a \<noteq> 0) \<longrightarrow> Max {y. \<exists>b. Rep_matrix x b y \<noteq> 0} < infinite \<and> Max {y. \<exists>a. Rep_matrix x y a \<noteq> 0} < infinite)"
using `finite {pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}` by blast
qed
lemma infmatrixforward: "(x:: 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",auto)
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 {(x, y). x < (m::nat) & y < (n::nat)}"
by simp
lemma RepAbs_matrix:
assumes aem: "? m<=infinite. ! j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"? n<=infinite. ! 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 = "{(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)
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)
moreover have "{pos. x (fst pos) (snd pos) \<noteq> 0} = ?u" by auto
have "finite {pos. x (fst pos) (snd pos) \<noteq> 0}"
using `{pos. x (fst pos) (snd pos) \<noteq> 0} = {(i, j). x i j \<noteq> 0}` d finite_subset by fastforce
have aa:"! j i. infinite <= j \<longrightarrow> x j i = 0" using aem by auto
have bb:"! j i. infinite <= i \<longrightarrow> x j i = 0" using aen less_le_trans by auto
have "fst ` {pos. x (fst pos) (snd pos) \<noteq> 0}= fst ` {(a,b). x a b \<noteq> 0}"
by (simp add: `{pos. x (fst pos) (snd pos) \<noteq> 0} = {(i, j). x i j \<noteq> 0}`)
have " fst ` {(a,b). x a b \<noteq> 0}={y. \<exists>a. x y a \<noteq> 0}"
apply auto
by force
have "fst ` {pos. x (fst pos) (snd pos) \<noteq> 0}={y. \<exists>a. x y a \<noteq> 0}"
by (simp add: `fst \` {(a, b). x a b \<noteq> 0} = {y. \<exists>a. x y a \<noteq> 0}` `fst \` {pos. x (fst pos) (snd pos) \<noteq> 0} = fst \` {(a, b). x a b \<noteq> 0}`)
have a:"{y. \<exists>a. x y a \<noteq> 0}\<noteq>{}\<and>Max {y. \<exists>a. x y a \<noteq> 0} \<ge> infinite \<Longrightarrow> (\<exists> yy\<ge> infinite. yy\<in>{y. \<exists>a. x y a \<noteq> 0})"
by (metis (no_types, lifting) Max_in `finite {pos. x (fst pos) (snd pos) \<noteq> 0}` `fst \` {(a, b). x a b \<noteq> 0} = {y. \<exists>a. x y a \<noteq> 0}` `{pos. x (fst pos) (snd pos) \<noteq> 0} = {(i, j). x i j \<noteq> 0}` finite_imageI)
have "(\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max ({y. \<exists>a. x y a \<noteq> 0}) < infinite"
apply auto
using a aa c by blast
have "snd ` {pos. x (fst pos) (snd pos) \<noteq> 0}= snd ` {(a,b). x a b \<noteq> 0}"
by (simp add: `{pos. x (fst pos) (snd pos) \<noteq> 0} = {(i, j). x i j \<noteq> 0}`)
have " snd ` {(a,b). x a b \<noteq> 0}={y. \<exists>a. x a y \<noteq> 0}"
apply auto
by force
have "(snd ` {pos. x (fst pos) (snd pos) \<noteq> 0}) = {y. \<exists>a. x a y \<noteq> 0}"
by (simp add: `snd \` {(a, b). x a b \<noteq> 0} = {y. \<exists>a. x a y \<noteq> 0}` `snd \` {pos. x (fst pos) (snd pos) \<noteq> 0} = snd \` {(a, b). x a b \<noteq> 0}`)
have cc:" {y. \<exists>a. x a y \<noteq> 0}\<noteq>{}\<and>Max {y. \<exists>a. x a y \<noteq> 0} \<ge> infinite \<Longrightarrow> ? yy. yy\<ge>infinite\<and>yy\<in>{y. \<exists>a. x a y \<noteq> 0}" sledgehammer
by (metis (no_types, lifting) Max_in `finite {pos. x (fst pos) (snd pos) \<noteq> 0}` `snd \` {(a, b). x a b \<noteq> 0} = {y. \<exists>a. x a y \<noteq> 0}` `{pos. x (fst pos) (snd pos) \<noteq> 0} = {(i, j). x i j \<noteq> 0}` finite_imageI)
have "(\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max ({y. \<exists>a. x a y \<noteq> 0}) < infinite"
using bb c cc by blast
have " ((\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max (fst ` {pos. x (fst pos) (snd pos) \<noteq> 0}) < infinite
\<and> Max (snd ` {pos. x (fst pos) (snd pos) \<noteq> 0}) < infinite)"
by (simp add: `(\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max {y. \<exists>a. x a y \<noteq> 0} < infinite` `(\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max {y. \<exists>a. x y a \<noteq> 0} < infinite` `fst \` {pos. x (fst pos) (snd pos) \<noteq> 0} = {y. \<exists>a. x y a \<noteq> 0}` `snd \` {(a, b). x a b \<noteq> 0} = {y. \<exists>a. x a y \<noteq> 0}` `snd \` {pos. x (fst pos) (snd pos) \<noteq> 0} = snd \` {(a, b). x a b \<noteq> 0}`)
ultimately show " finite {pos. x (fst pos) (snd pos) \<noteq> 0} \<and>
((\<exists>a b. x a b \<noteq> 0) \<longrightarrow> Max (fst ` {pos. x (fst pos) (snd pos) \<noteq> 0}) < infinite \<and> Max (snd ` {pos. x (fst pos) (snd pos) \<noteq> 0}) < infinite)"
using `finite {pos. x (fst pos) (snd pos) \<noteq> 0}` by blast
qed
definition apply_infmatrix :: "(real \<Rightarrow> real) \<Rightarrow> infmatrix \<Rightarrow> infmatrix" where
"apply_infmatrix f == % A. (% j i. f (A j i))"
definition apply_matrix :: "(real \<Rightarrow> real) \<Rightarrow> matrix \<Rightarrow> matrix" where
"apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
definition combine_infmatrix :: "(real \<Rightarrow> real \<Rightarrow> real) \<Rightarrow> infmatrix \<Rightarrow>infmatrix \<Rightarrow>infmatrix" where
"combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
definition combine_matrix :: "(real \<Rightarrow> real \<Rightarrow> real) \<Rightarrow>matrix \<Rightarrow>matrix \<Rightarrow> 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> 'a) \<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)"
apply(rule subsetI)
apply(simp add:nonzero_positions_def)
apply auto
done
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)
proof -
have g:" f 0 0 = 0 \<Longrightarrow> finite (nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))"
apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
by (simp_all)
have d:" (fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) =
fst ` {(a,b). (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) a b \<noteq> 0}"
by (metis (no_types, lifting) Collect_cong nonzero_positions_def split_beta')
have e:" fst ` {(a,b). (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) a b \<noteq> 0}=
{y. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) y a\<noteq>0 }"
apply auto
apply force
done
have c:"{y. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) y a\<noteq>0 }=
(fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))"
using d e by auto
have a: "f 0 0=0\<Longrightarrow>{y. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) y a\<noteq>0 }\<noteq>{}
\<and> Max {y. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) y a\<noteq>0 } \<ge> infinite\<Longrightarrow>
\<exists>yy\<ge>infinite. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) yy a\<noteq>0"
proof -
assume a1: "{y. \<exists>a. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) y a \<noteq> 0} \<noteq> {} \<and> infinite \<le> Max {y. \<exists>a. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) y a \<noteq> 0}"
assume a2: "f 0 0 = 0"
have f3: "infinite \<le> Max {n. \<exists>na. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) n na \<noteq> 0}"
using a1 by linarith
have "finite (fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))"
using a2 by (simp add: g)
hence "Max {n. \<exists>na. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) n na \<noteq> 0} \<in> {n. \<exists>na. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) n na \<noteq> 0}"
by (metis Max_in a1 c)
thus ?thesis
using f3 by blast
qed
have b: " f 0 0=0\<Longrightarrow>(snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))\<noteq>{}
\<and>Max (snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) \<ge> infinite\<Longrightarrow>
\<exists>yy. yy\<ge>infinite\<and> yy\<in>{y. \<exists>a. (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) a y \<noteq>0}"
proof -
assume a1: "f 0 0 = 0"
assume a2: "snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) \<noteq> {} \<and> infinite \<le> Max (snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))"
have f3: "\<forall>f r. f ` r = {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
by blast
hence "(snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) \<noteq> {}) = ({n. \<exists>p. p \<in> {p. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) (fst p) (snd p) \<noteq> 0} \<and> n = snd p} \<noteq> {})"
using nonzero_positions_def by presburger
hence f4: "{n. \<exists>p. p \<in> {p. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) (fst p) (snd p) \<noteq> 0} \<and> n = snd p} \<noteq> {}"
using a2 by blast
have f5: "\<forall>r f. \<not> finite r \<or> finite {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
using f3 by (metis (no_types) finite_imageI)
have "finite {p. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) (fst p) (snd p) \<noteq> 0}"
using a1 g nonzero_positions_def by fastforce
hence f6: "finite {n. \<exists>p. p \<in> {p. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) (fst p) (snd p) \<noteq> 0} \<and> n = snd p}"
using f5 by blast
have "Max {n. \<exists>p. p \<in> {p. combine_infmatrix f (Rep_matrix A) (Rep_matrix B) (fst p) (snd p) \<noteq> 0} \<and> n = snd p} \<in> Collect (op \<le> infinite)"
using f3 a2 by (simp add: nonzero_positions_def)
thus ?thesis
using f6 f4 Max_in by fastforce
qed
have c :"f 0 0=0\<Longrightarrow>\<forall>a\<ge>infinite.\<forall>b.(combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) a b=0"
using dual_order.trans nrows by force
have d:"f 0 0=0\<Longrightarrow>\<forall>a\<ge>infinite.\<forall>b.(combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) b a=0"
using dual_order.trans ncols by force
have e:" f 0 0 = 0 \<Longrightarrow>(fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))\<noteq>{}\<Longrightarrow> Max (fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) < infinite"
by (smt Collect_cong a c e nonzero_positions_def not_less split_beta')
have f:" f 0 0 = 0 \<Longrightarrow>
(nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) \<noteq> {} \<longrightarrow>
Max (fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) < infinite \<and>
Max (snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) < infinite)"
using b d e not_less by auto
show " f 0 0 = 0 \<Longrightarrow>
finite (nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) \<and>
(nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)) \<noteq> {} \<longrightarrow>
Max (fst ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) < infinite \<and>
Max (snd ` nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) < infinite)"
using `f 0 0 = 0 \<Longrightarrow> finite (nonzero_positions (combine_infmatrix f (Rep_matrix A) (Rep_matrix B)))` f by blast
qed
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)
proof -
have a:"f 0 = 0 \<Longrightarrow> finite (nonzero_positions (apply_infmatrix f (Rep_matrix A)))"
by (meson finite_nonzero_positions_Rep nonzero_positions_apply_infmatrix rev_finite_subset)
have bb: " f 0=0\<Longrightarrow>(snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A) ))\<noteq>{}
\<and>Max (snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A) )) \<ge> infinite\<Longrightarrow>
\<exists>yy. yy\<ge>infinite\<and> yy\<in>{y. \<exists>a. (apply_infmatrix f (Rep_matrix A)) a y \<noteq>0}"
proof -
assume a1: "f 0 = 0"
assume a2: "snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {} \<and> infinite \<le> Max (snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A)))"
have f3: "\<forall>f r. f ` r = {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
by blast
hence "(snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {}) = ({n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = snd p} \<noteq> {})"
using nonzero_positions_def by presburger
hence f4: "{n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = snd p} \<noteq> {}"
using a2 by blast
have f5: "\<forall>r f. \<not> finite r \<or> finite {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
using f3 by (metis (no_types) finite_imageI)
have "finite {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0}"
using a1 a nonzero_positions_def by fastforce
hence "finite {n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = snd p}"
using f5 by blast
hence "\<exists>n. apply_infmatrix f (Rep_matrix A) n (Max {n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = snd p}) \<noteq> 0"
using f4 Max_in by fastforce
thus ?thesis
using f3 a2 nonzero_positions_def by force
qed
have b: " f 0=0\<Longrightarrow>(fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A) ))\<noteq>{}
\<and>Max (fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A) )) \<ge> infinite\<Longrightarrow>
\<exists>yy. yy\<ge>infinite\<and> yy\<in>{y. \<exists>a. (apply_infmatrix f (Rep_matrix A)) y a \<noteq>0}"
proof -
assume a1: "f 0 = 0"
assume a2: "fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {} \<and> infinite \<le> Max (fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A)))"
have f3: "\<forall>f r. f ` r = {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
by blast
hence "(fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {}) = ({n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = fst p} \<noteq> {})"
using nonzero_positions_def by presburger
hence f4: "{n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = fst p} \<noteq> {}"
using a2 by blast
have f5: "\<forall>r f. \<not> finite r \<or> finite {n. \<exists>p. (p\<Colon>nat \<times> nat) \<in> r \<and> (n\<Colon>nat) = f p}"
using f3 by (metis (no_types) finite_imageI)
have "finite {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0}"
using a1 a nonzero_positions_def by fastforce
hence "finite {n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = fst p}"
using f5 by blast
hence "\<exists>n. apply_infmatrix f (Rep_matrix A) (Max {n. \<exists>p. p \<in> {p. apply_infmatrix f (Rep_matrix A) (fst p) (snd p) \<noteq> 0} \<and> n = fst p}) n \<noteq> 0"
using f4 Max_in by fastforce
thus ?thesis
using f3 a2 nonzero_positions_def by auto
qed
have cc:"f 0=0\<Longrightarrow>\<forall>a.\<forall>b\<ge>infinite. (apply_infmatrix f (Rep_matrix A)) b a=0"
by (metis expand_apply_infmatrix ncols_max ncols_transpose nrows_le)
have c:"f 0 = 0 \<Longrightarrow>nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {} \<Longrightarrow> Max (fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A))) < infinite"
using b cc by auto
have c:"f 0=0\<Longrightarrow>\<forall>a.\<forall>b\<ge>infinite. (apply_infmatrix f (Rep_matrix A)) a b=0"
by (metis expand_apply_infmatrix le_trans ncols ncols_max)
have d:"f 0 = 0 \<Longrightarrow>nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {} \<Longrightarrow> Max (snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A))) < infinite"
using bb c by auto
show " f 0 = 0 \<Longrightarrow>
finite (nonzero_positions (apply_infmatrix f (Rep_matrix A))) \<and>
(nonzero_positions (apply_infmatrix f (Rep_matrix A)) \<noteq> {} \<longrightarrow>
Max (fst ` nonzero_positions (apply_infmatrix f (Rep_matrix A))) < infinite \<and>
Max (snd ` nonzero_positions (apply_infmatrix f (Rep_matrix A))) < infinite) "
using a b cc d by fastforce
qed
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)"
(* calculate A*B *)
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 foldseq1_significant_positions:
assumes p: "! i. i <= N \<longrightarrow> S i = T i"
shows "foldseq_transposed f S N = foldseq_transposed f T N"
proof -
have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq_transposed f s m = foldseq_transposed 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_transposed f s n = foldseq_transposed 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_transposed f s n = foldseq_transposed f t n" by (simp add: a)
show " f (foldseq_transposed f s n) (t (Suc n)) = f (foldseq_transposed f t n) (t (Suc n))" by (metis a b le_SucI)
qed
with p show ?thesis by simp
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> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow>matrix \<Rightarrow>matrix \<Rightarrow>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_n1 :: "nat \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow>matrix \<Rightarrow>matrix \<Rightarrow>matrix" where
"mult_matrix_n1 n fmul fadd A B == Abs_matrix(% j i. foldseq_transposed fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
definition mult_matrix1 :: "(real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> matrix \<Rightarrow>matrix \<Rightarrow>matrix" where
"mult_matrix1 fmul fadd A B == mult_matrix_n1 (max (ncols A) (nrows B)) fmul fadd A B"
definition mult_matrix :: "(real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> matrix \<Rightarrow>matrix \<Rightarrow>matrix" where
"mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
consts infinity::"nat"
definition tr_n:: "nat \<Rightarrow> (real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> matrix\<Rightarrow>real" where
"tr_n n fadd A==foldseq fadd (%k. Rep_matrix A k k) n"
definition tr::"(real\<Rightarrow>real\<Rightarrow>real) \<Rightarrow> matrix\<Rightarrow>real"where
"tr fadd A==foldseq fadd (%k. Rep_matrix A k k) (max (nrows A) (ncols A)) "
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::"matrix"
fix u::" matrix"
fix v::" 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 a _ _ ?mx fadd fmul])
apply (simp add: max1 max2 combine_nrows combine_ncols)+
apply (subst mult_matrix_nm[of a _ v ?mx fadd fmul])
apply (simp add: max1 max2 combine_nrows combine_ncols)+
apply (subst mult_matrix_nm[of a _ 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 (meson dual_order.trans nrows nrows_max)
apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
apply (meson dual_order.trans ncols ncols_max)
apply (subst RepAbs_matrix)
apply (simp, auto)
apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
apply (meson dual_order.trans nrows nrows_max)
apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
apply (meson dual_order.trans ncols ncols_max)
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::"matrix"
fix u::" matrix"
fix v::" 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)