--- a/src/HOL/Library/Permutations.thy Thu Mar 12 15:31:44 2009 +0100
+++ b/src/HOL/Library/Permutations.thy Thu Mar 12 08:57:03 2009 -0700
@@ -18,7 +18,7 @@
(* ------------------------------------------------------------------------- *)
declare swap_self[simp]
-lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id"
+lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id"
by (auto simp add: expand_fun_eq swap_def fun_upd_def)
lemma swap_id_refl: "Fun.swap a a id = id" by simp
lemma swap_id_sym: "Fun.swap a b id = Fun.swap b a id"
@@ -45,18 +45,18 @@
lemma permutes_image: assumes pS: "p permutes S" shows "p ` S = S"
using pS
- unfolding permutes_def
- apply -
- apply (rule set_ext)
+ unfolding permutes_def
+ apply -
+ apply (rule set_ext)
apply (simp add: image_iff)
apply metis
done
-lemma permutes_inj: "p permutes S ==> inj p "
- unfolding permutes_def inj_on_def by blast
+lemma permutes_inj: "p permutes S ==> inj p "
+ unfolding permutes_def inj_on_def by blast
-lemma permutes_surj: "p permutes s ==> surj p"
- unfolding permutes_def surj_def by metis
+lemma permutes_surj: "p permutes s ==> surj p"
+ unfolding permutes_def surj_def by metis
lemma permutes_inv_o: assumes pS: "p permutes S"
shows " p o inv p = id"
@@ -65,7 +65,7 @@
unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
-lemma permutes_inverses:
+lemma permutes_inverses:
fixes p :: "'a \<Rightarrow> 'a"
assumes pS: "p permutes S"
shows "p (inv p x) = x"
@@ -76,11 +76,11 @@
unfolding permutes_def by blast
lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
- unfolding expand_fun_eq permutes_def apply simp by metis
+ unfolding expand_fun_eq permutes_def apply simp by metis
lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
unfolding expand_fun_eq permutes_def apply simp by metis
-
+
lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
unfolding permutes_def by simp
@@ -104,13 +104,13 @@
(* Group properties. *)
(* ------------------------------------------------------------------------- *)
-lemma permutes_id: "id permutes S" unfolding permutes_def by simp
+lemma permutes_id: "id permutes S" unfolding permutes_def by simp
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S ==> q o p permutes S"
unfolding permutes_def o_def by metis
lemma permutes_inv: assumes pS: "p permutes S" shows "inv p permutes S"
- using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis
+ using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis
lemma permutes_inv_inv: assumes pS: "p permutes S" shows "inv (inv p) = p"
unfolding expand_fun_eq permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
@@ -120,7 +120,7 @@
(* The number of permutations on a finite set. *)
(* ------------------------------------------------------------------------- *)
-lemma permutes_insert_lemma:
+lemma permutes_insert_lemma:
assumes pS: "p permutes (insert a S)"
shows "Fun.swap a (p a) id o p permutes S"
apply (rule permutes_superset[where S = "insert a S"])
@@ -134,17 +134,17 @@
(\<lambda>(b,p). Fun.swap a b id o p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
proof-
- {fix p
+ {fix p
{assume pS: "p permutes insert a S"
let ?b = "p a"
let ?q = "Fun.swap a (p a) id o p"
- have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp
- have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp
+ have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp
+ have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp
from permutes_insert_lemma[OF pS] th0 th1
have "\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S" by blast}
moreover
{fix b q assume bq: "p = Fun.swap a b id o q" "b \<in> insert a S" "q permutes S"
- from permutes_subset[OF bq(3), of "insert a S"]
+ from permutes_subset[OF bq(3), of "insert a S"]
have qS: "q permutes insert a S" by auto
have aS: "a \<in> insert a S" by simp
from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]]
@@ -168,8 +168,8 @@
show "\<forall>n. ({} hassize n) \<longrightarrow> ({p. p permutes {}} hassize fact n)"
by (simp add: hassize_def permutes_empty)
next
- fix x F
- assume fF: "finite F" and xF: "x \<notin> F"
+ fix x F
+ assume fF: "finite F" and xF: "x \<notin> F"
and H: "\<forall>n. (F hassize n) \<longrightarrow> ({p. p permutes F} hassize fact n)"
{fix n assume H0: "insert x F hassize n"
let ?xF = "{p. p permutes insert x F}"
@@ -180,7 +180,7 @@
have xfgpF': "?xF = ?g ` ?pF'" .
from hassize_insert[OF xF H0] have Fs: "F hassize (n - 1)" .
from H Fs have pFs: "?pF hassize fact (n - 1)" by blast
- hence pF'f: "finite ?pF'" using H0 unfolding hassize_def
+ hence pF'f: "finite ?pF'" using H0 unfolding hassize_def
apply (simp only: Collect_split Collect_mem_eq)
apply (rule finite_cartesian_product)
apply simp_all
@@ -189,12 +189,12 @@
have ginj: "inj_on ?g ?pF'"
proof-
{
- fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'"
+ fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'"
and eq: "?g (b,p) = ?g (c,q)"
from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" "p permutes F" "q permutes F" by auto
- from ths(4) xF eq have "b = ?g (b,p) x" unfolding permutes_def
+ from ths(4) xF eq have "b = ?g (b,p) x" unfolding permutes_def
by (auto simp add: swap_def fun_upd_def expand_fun_eq)
- also have "\<dots> = ?g (c,q) x" using ths(5) xF eq
+ also have "\<dots> = ?g (c,q) x" using ths(5) xF eq
by (auto simp add: swap_def fun_upd_def expand_fun_eq)
also have "\<dots> = c"using ths(5) xF unfolding permutes_def
by (auto simp add: swap_def fun_upd_def expand_fun_eq)
@@ -208,16 +208,16 @@
qed
from xF H0 have n0: "n \<noteq> 0 " by (auto simp add: hassize_def)
hence "\<exists>m. n = Suc m" by arith
- then obtain m where n[simp]: "n = Suc m" by blast
- from pFs H0 have xFc: "card ?xF = fact n"
+ then obtain m where n[simp]: "n = Suc m" by blast
+ from pFs H0 have xFc: "card ?xF = fact n"
unfolding xfgpF' card_image[OF ginj] hassize_def
apply (simp only: Collect_split Collect_mem_eq card_cartesian_product)
by simp
- from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp
+ from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp
have "?xF hassize fact n"
- using xFf xFc
+ using xFf xFc
unfolding hassize_def xFf by blast }
- thus "\<forall>n. (insert x F hassize n) \<longrightarrow> ({p. p permutes insert x F} hassize fact n)"
+ thus "\<forall>n. (insert x F hassize n) \<longrightarrow> ({p. p permutes insert x F} hassize fact n)"
by blast
qed
with Sn show ?thesis by blast
@@ -230,11 +230,11 @@
(* Permutations of index set for iterated operations. *)
(* ------------------------------------------------------------------------- *)
-lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
+lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
shows "fold_image times f z S = fold_image times (f o p) z S"
using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
unfolding permutes_image[OF pS] .
-lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
+lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
shows "fold_image plus f z S = fold_image plus (f o p) z S"
proof-
interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc)
@@ -244,22 +244,22 @@
unfolding permutes_image[OF pS] .
qed
-lemma setsum_permute: assumes pS: "p permutes S"
+lemma setsum_permute: assumes pS: "p permutes S"
shows "setsum f S = setsum (f o p) S"
unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp
-lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}"
+lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}"
shows "setsum f {m .. n} = setsum (f o p) {m .. n}"
- using setsum_permute[OF pS, of f ] pS by blast
+ using setsum_permute[OF pS, of f ] pS by blast
-lemma setprod_permute: assumes pS: "p permutes S"
+lemma setprod_permute: assumes pS: "p permutes S"
shows "setprod f S = setprod (f o p) S"
- unfolding setprod_def
+ unfolding setprod_def
using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp
-lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}"
+lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}"
shows "setprod f {m .. n} = setprod (f o p) {m .. n}"
- using setprod_permute[OF pS, of f ] pS by blast
+ using setprod_permute[OF pS, of f ] pS by blast
(* ------------------------------------------------------------------------- *)
(* Various combinations of transpositions with 2, 1 and 0 common elements. *)
@@ -298,16 +298,16 @@
lemma permutation_swap_id: "permutation (Fun.swap a b id)"
apply (cases "a=b", simp_all)
- unfolding permutation_def using swapidseq_swap[of a b] by blast
+ unfolding permutation_def using swapidseq_swap[of a b] by blast
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q ==> swapidseq (n + m) (p o q)"
proof (induct n p arbitrary: m q rule: swapidseq.induct)
case (id m q) thus ?case by simp
next
- case (comp_Suc n p a b m q)
+ case (comp_Suc n p a b m q)
have th: "Suc n + m = Suc (n + m)" by arith
- show ?case unfolding th o_assoc[symmetric]
- apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) by blast+
+ show ?case unfolding th o_assoc[symmetric]
+ apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) by blast+
qed
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q ==> permutation(p o q)"
@@ -321,17 +321,17 @@
lemma swapidseq_inverse_exists: "swapidseq n p ==> \<exists>q. swapidseq n q \<and> p o q = id \<and> q o p = id"
proof(induct n p rule: swapidseq.induct)
case id thus ?case by (rule exI[where x=id], simp)
-next
+next
case (comp_Suc n p a b)
from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
let ?q = "q o Fun.swap a b id"
note H = comp_Suc.hyps
from swapidseq_swap[of a b] H(3) have th0: "swapidseq 1 (Fun.swap a b id)" by simp
- from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp
+ from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp
have "Fun.swap a b id o p o ?q = Fun.swap a b id o (p o q) o Fun.swap a b id" by (simp add: o_assoc)
also have "\<dots> = id" by (simp add: q(2))
finally have th2: "Fun.swap a b id o p o ?q = id" .
- have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc)
+ have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc)
hence "?q \<circ> (Fun.swap a b id \<circ> p) = id" by (simp add: q(3))
with th1 th2 show ?case by blast
qed
@@ -351,15 +351,15 @@
(\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> (a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d) ==> P a b c d)
==> (\<And>a b c d. a \<noteq> b --> c \<noteq> d \<longrightarrow> P a b c d)" by metis
-lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or>
- (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)"
+lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or>
+ (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)"
proof-
assume H: "a\<noteq>b" "c\<noteq>d"
-have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
-( Fun.swap a b id o Fun.swap c d id = id \<or>
- (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))"
+have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
+( Fun.swap a b id o Fun.swap c d id = id \<or>
+ (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))"
apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
- apply (simp_all only: swapid_sym)
+ apply (simp_all only: swapid_sym)
apply (case_tac "a = c \<and> b = d", clarsimp simp only: swapid_sym swap_id_idempotent)
apply (case_tac "a = c \<and> b \<noteq> d")
apply (rule disjI2)
@@ -379,7 +379,7 @@
apply (rule_tac x="b" in exI)
apply (clarsimp simp add: expand_fun_eq swap_def)
done
-with H show ?thesis by metis
+with H show ?thesis by metis
qed
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
@@ -411,12 +411,12 @@
c d q m where cdqm: "Suc n = Suc m" "p = Fun.swap c d id o q" "swapidseq m q" "c \<noteq> d" "n = m"
by auto
{assume H: "Fun.swap a b id o Fun.swap c d id = id"
-
- have ?case apply (simp only: cdqm o_assoc H)
+
+ have ?case apply (simp only: cdqm o_assoc H)
by (simp add: cdqm)}
moreover
{ fix x y z
- assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y"
+ assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y"
"Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id"
from H have az: "a \<noteq> z" by simp
@@ -430,23 +430,23 @@
hence th1: "(Fun.swap a z id o q) a = a" unfolding th3 .
from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1]
have th2: "swapidseq (n - 1) (Fun.swap a z id o q)" "n \<noteq> 0" by blast+
- have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto
+ have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto
have ?case unfolding cdqm(2) H o_assoc th
apply (simp only: Suc_not_Zero simp_thms o_assoc[symmetric])
apply (rule comp_Suc)
using th2 H apply blast+
done}
- ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis
+ ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis
qed
-lemma swapidseq_identity_even:
+lemma swapidseq_identity_even:
assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" shows "even n"
using `swapidseq n id`
proof(induct n rule: nat_less_induct)
fix n
assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)"
- {assume "n = 0" hence "even n" by arith}
- moreover
+ {assume "n = 0" hence "even n" by arith}
+ moreover
{fix a b :: 'a and q m
assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
@@ -462,13 +462,13 @@
definition "evenperm p = even (SOME n. swapidseq n p)"
-lemma swapidseq_even_even: assumes
+lemma swapidseq_even_even: assumes
m: "swapidseq m p" and n: "swapidseq n p"
shows "even m \<longleftrightarrow> even n"
proof-
from swapidseq_inverse_exists[OF n]
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
-
+
from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]]
show ?thesis by arith
qed
@@ -491,12 +491,12 @@
apply (rule evenperm_unique[where n="if a = b then 0 else 1"])
by (simp_all add: swapidseq_swap)
-lemma evenperm_comp:
+lemma evenperm_comp:
assumes p: "permutation p" and q:"permutation q"
shows "evenperm (p o q) = (evenperm p = evenperm q)"
proof-
- from p q obtain
- n m where n: "swapidseq n p" and m: "swapidseq m q"
+ from p q obtain
+ n m where n: "swapidseq n p" and m: "swapidseq m q"
unfolding permutation_def by blast
note nm = swapidseq_comp_add[OF n m]
have th: "even (n + m) = (even n \<longleftrightarrow> even m)" by arith
@@ -525,15 +525,15 @@
apply metis
done
-lemma permutation_bijective:
- assumes p: "permutation p"
+lemma permutation_bijective:
+ assumes p: "permutation p"
shows "bij p"
proof-
from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
- from swapidseq_inverse_exists[OF n] obtain q where
+ from swapidseq_inverse_exists[OF n] obtain q where
q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
thus ?thesis unfolding bij_iff apply (auto simp add: expand_fun_eq) apply metis done
-qed
+qed
lemma permutation_finite_support: assumes p: "permutation p"
shows "finite {x. p x \<noteq> x}"
@@ -555,7 +555,7 @@
lemma bij_inv_eq_iff: "bij p ==> x = inv p y \<longleftrightarrow> p x = y"
using surj_f_inv_f[of p] inv_f_f[of f] by (auto simp add: bij_def)
-lemma bij_swap_comp:
+lemma bij_swap_comp:
assumes bp: "bij p" shows "Fun.swap a b id o p = Fun.swap (inv p a) (inv p b) p"
using surj_f_inv_f[OF bij_is_surj[OF bp]]
by (simp add: expand_fun_eq swap_def bij_inv_eq_iff[OF bp])
@@ -563,12 +563,12 @@
lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id o p)"
proof-
assume H: "bij p"
- show ?thesis
+ show ?thesis
unfolding bij_swap_comp[OF H] bij_swap_iff
using H .
qed
-lemma permutation_lemma:
+lemma permutation_lemma:
assumes fS: "finite S" and p: "bij p" and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
shows "permutation p"
using fS p pS
@@ -580,9 +580,9 @@
let ?q = "Fun.swap a (p a) id o ?r "
have raa: "?r a = a" by (simp add: swap_def)
from bij_swap_ompose_bij[OF insert(4)]
- have br: "bij ?r" .
-
- from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
+ have br: "bij ?r" .
+
+ from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
apply (clarsimp simp add: swap_def)
apply (erule_tac x="x" in allE)
apply auto
@@ -594,7 +594,7 @@
thus ?case by (simp add: o_assoc)
qed
-lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
+lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
(is "?lhs \<longleftrightarrow> ?b \<and> ?f")
proof
assume p: ?lhs
@@ -620,7 +620,7 @@
finally have th0: "p o q o (inv q o inv p) = id" .
have "inv q o inv p o (p o q) = inv q o (inv p o p) o q" by (simp add: o_assoc)
also have "\<dots> = id" by (simp add: ps qs)
- finally have th1: "inv q o inv p o (p o q) = id" .
+ finally have th1: "inv q o inv p o (p o q) = id" .
from inv_unique_comp[OF th0 th1] show ?thesis .
qed
@@ -646,20 +646,20 @@
==> (\<And>p. p permutes S ==> P p)"
proof(induct S rule: finite_induct)
case empty thus ?case by auto
-next
+next
case (insert x F p)
let ?r = "Fun.swap x (p x) id o p"
let ?q = "Fun.swap x (p x) id o ?r"
have qp: "?q = p" by (simp add: o_assoc)
from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast
- from permutes_in_image[OF insert.prems(3), of x]
+ from permutes_in_image[OF insert.prems(3), of x]
have pxF: "p x \<in> insert x F" by simp
have xF: "x \<in> insert x F" by simp
have rp: "permutation ?r"
- unfolding permutation_permutes using insert.hyps(1)
+ unfolding permutation_permutes using insert.hyps(1)
permutes_insert_lemma[OF insert.prems(3)] by blast
- from insert.prems(2)[OF xF pxF Pr Pr rp]
- show ?case unfolding qp .
+ from insert.prems(2)[OF xF pxF Pr Pr rp]
+ show ?case unfolding qp .
qed
(* ------------------------------------------------------------------------- *)
@@ -668,7 +668,7 @@
definition "sign p = (if evenperm p then (1::int) else -1)"
-lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def)
+lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def)
lemma sign_id: "sign id = 1" by (simp add: sign_def)
lemma sign_inverse: "permutation p ==> sign (inv p) = sign p"
by (simp add: sign_def evenperm_inv)
@@ -685,16 +685,16 @@
assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S. p i <= i" shows "p = id"
proof-
{fix n
- have "p n = n"
+ have "p n = n"
using p le
proof(induct n arbitrary: S rule: less_induct)
- fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m"
+ fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m"
"p permutes S" "\<forall>i \<in>S. p i \<le> i"
{assume "n \<notin> S"
with H(2) have "p n = n" unfolding permutes_def by metis}
moreover
{assume ns: "n \<in> S"
- from H(3) ns have "p n < n \<or> p n = n" by auto
+ from H(3) ns have "p n < n \<or> p n = n" by auto
moreover{assume h: "p n < n"
from H h have "p (p n) = p n" by metis
with permutes_inj[OF H(2)] have "p n = n" unfolding inj_on_def by blast
@@ -713,9 +713,9 @@
with le have "p (inv p i) \<ge> inv p i" by blast
with permutes_inverses[OF p] have "i \<ge> inv p i" by simp}
then have th: "\<forall>i\<in>S. inv p i \<le> i" by blast
- from permutes_natset_le[OF permutes_inv[OF p] th]
+ from permutes_natset_le[OF permutes_inv[OF p] th]
have "inv p = inv id" by simp
- then show ?thesis
+ then show ?thesis
apply (subst permutes_inv_inv[OF p, symmetric])
apply (rule inv_unique_comp)
apply simp_all
@@ -730,7 +730,7 @@
apply auto
done
-lemma image_compose_permutations_left:
+lemma image_compose_permutations_left:
assumes q: "q permutes S" shows "{q o p | p. p permutes S} = {p . p permutes S}"
apply (rule set_ext)
apply auto
@@ -759,7 +759,7 @@
lemma setsum_permutations_inverse: "setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs")
proof-
let ?S = "{p . p permutes S}"
-have th0: "inj_on inv ?S"
+have th0: "inj_on inv ?S"
proof(auto simp add: inj_on_def)
fix q r
assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r"
@@ -828,16 +828,16 @@
by (simp add: expand_fun_eq)
have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" by blast
have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" by blast
- show ?thesis
- unfolding permutes_insert
+ show ?thesis
+ unfolding permutes_insert
unfolding setsum_cartesian_product
unfolding th1[symmetric]
unfolding th0
proof(rule setsum_reindex)
let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
let ?P = "{p. p permutes S}"
- {fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S"
- and p: "p permutes S" and q: "q permutes S"
+ {fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S"
+ and p: "p permutes S" and q: "q permutes S"
and eq: "Fun.swap a b id o p = Fun.swap a c id o q"
from p q aS have pa: "p a = a" and qa: "q a = a"
unfolding permutes_def by metis+
@@ -851,8 +851,8 @@
by (simp add: o_def)
with bc have "b = c \<and> p = q" by blast
}
-
- then show "inj_on ?f (insert a S \<times> ?P)"
+
+ then show "inj_on ?f (insert a S \<times> ?P)"
unfolding inj_on_def
apply clarify by metis
qed