--- a/src/HOL/Library/Function_Algebras.thy Tue Feb 21 16:42:57 2012 +0100
+++ b/src/HOL/Library/Function_Algebras.thy Tue Feb 21 16:48:10 2012 +0100
@@ -13,9 +13,7 @@
instantiation "fun" :: (type, plus) plus
begin
-definition
- "f + g = (\<lambda>x. f x + g x)"
-
+definition "f + g = (\<lambda>x. f x + g x)"
instance ..
end
@@ -23,9 +21,7 @@
instantiation "fun" :: (type, zero) zero
begin
-definition
- "0 = (\<lambda>x. 0)"
-
+definition "0 = (\<lambda>x. 0)"
instance ..
end
@@ -33,9 +29,7 @@
instantiation "fun" :: (type, times) times
begin
-definition
- "f * g = (\<lambda>x. f x * g x)"
-
+definition "f * g = (\<lambda>x. f x * g x)"
instance ..
end
@@ -43,9 +37,7 @@
instantiation "fun" :: (type, one) one
begin
-definition
- "1 = (\<lambda>x. 1)"
-
+definition "1 = (\<lambda>x. 1)"
instance ..
end
@@ -53,69 +45,70 @@
text {* Additive structures *}
-instance "fun" :: (type, semigroup_add) semigroup_add proof
-qed (simp add: plus_fun_def add.assoc)
+instance "fun" :: (type, semigroup_add) semigroup_add
+ by default (simp add: plus_fun_def add.assoc)
-instance "fun" :: (type, cancel_semigroup_add) cancel_semigroup_add proof
-qed (simp_all add: plus_fun_def fun_eq_iff)
+instance "fun" :: (type, cancel_semigroup_add) cancel_semigroup_add
+ by default (simp_all add: plus_fun_def fun_eq_iff)
-instance "fun" :: (type, ab_semigroup_add) ab_semigroup_add proof
-qed (simp add: plus_fun_def add.commute)
+instance "fun" :: (type, ab_semigroup_add) ab_semigroup_add
+ by default (simp add: plus_fun_def add.commute)
-instance "fun" :: (type, cancel_ab_semigroup_add) cancel_ab_semigroup_add proof
-qed simp
+instance "fun" :: (type, cancel_ab_semigroup_add) cancel_ab_semigroup_add
+ by default simp
-instance "fun" :: (type, monoid_add) monoid_add proof
-qed (simp_all add: plus_fun_def zero_fun_def)
+instance "fun" :: (type, monoid_add) monoid_add
+ by default (simp_all add: plus_fun_def zero_fun_def)
-instance "fun" :: (type, comm_monoid_add) comm_monoid_add proof
-qed simp
+instance "fun" :: (type, comm_monoid_add) comm_monoid_add
+ by default simp
instance "fun" :: (type, cancel_comm_monoid_add) cancel_comm_monoid_add ..
-instance "fun" :: (type, group_add) group_add proof
-qed (simp_all add: plus_fun_def zero_fun_def fun_Compl_def fun_diff_def diff_minus)
+instance "fun" :: (type, group_add) group_add
+ by default
+ (simp_all add: plus_fun_def zero_fun_def fun_Compl_def fun_diff_def diff_minus)
-instance "fun" :: (type, ab_group_add) ab_group_add proof
-qed (simp_all add: diff_minus)
+instance "fun" :: (type, ab_group_add) ab_group_add
+ by default (simp_all add: diff_minus)
text {* Multiplicative structures *}
-instance "fun" :: (type, semigroup_mult) semigroup_mult proof
-qed (simp add: times_fun_def mult.assoc)
+instance "fun" :: (type, semigroup_mult) semigroup_mult
+ by default (simp add: times_fun_def mult.assoc)
-instance "fun" :: (type, ab_semigroup_mult) ab_semigroup_mult proof
-qed (simp add: times_fun_def mult.commute)
+instance "fun" :: (type, ab_semigroup_mult) ab_semigroup_mult
+ by default (simp add: times_fun_def mult.commute)
-instance "fun" :: (type, ab_semigroup_idem_mult) ab_semigroup_idem_mult proof
-qed (simp add: times_fun_def)
+instance "fun" :: (type, ab_semigroup_idem_mult) ab_semigroup_idem_mult
+ by default (simp add: times_fun_def)
-instance "fun" :: (type, monoid_mult) monoid_mult proof
-qed (simp_all add: times_fun_def one_fun_def)
+instance "fun" :: (type, monoid_mult) monoid_mult
+ by default (simp_all add: times_fun_def one_fun_def)
-instance "fun" :: (type, comm_monoid_mult) comm_monoid_mult proof
-qed simp
+instance "fun" :: (type, comm_monoid_mult) comm_monoid_mult
+ by default simp
text {* Misc *}
instance "fun" :: (type, "Rings.dvd") "Rings.dvd" ..
-instance "fun" :: (type, mult_zero) mult_zero proof
-qed (simp_all add: zero_fun_def times_fun_def)
+instance "fun" :: (type, mult_zero) mult_zero
+ by default (simp_all add: zero_fun_def times_fun_def)
-instance "fun" :: (type, zero_neq_one) zero_neq_one proof
-qed (simp add: zero_fun_def one_fun_def fun_eq_iff)
+instance "fun" :: (type, zero_neq_one) zero_neq_one
+ by default (simp add: zero_fun_def one_fun_def fun_eq_iff)
text {* Ring structures *}
-instance "fun" :: (type, semiring) semiring proof
-qed (simp_all add: plus_fun_def times_fun_def algebra_simps)
+instance "fun" :: (type, semiring) semiring
+ by default (simp_all add: plus_fun_def times_fun_def algebra_simps)
-instance "fun" :: (type, comm_semiring) comm_semiring proof
-qed (simp add: plus_fun_def times_fun_def algebra_simps)
+instance "fun" :: (type, comm_semiring) comm_semiring
+ by default (simp add: plus_fun_def times_fun_def algebra_simps)
instance "fun" :: (type, semiring_0) semiring_0 ..
@@ -127,8 +120,7 @@
instance "fun" :: (type, semiring_1) semiring_1 ..
-lemma of_nat_fun:
- shows "of_nat n = (\<lambda>x::'a. of_nat n)"
+lemma of_nat_fun: "of_nat n = (\<lambda>x::'a. of_nat n)"
proof -
have comp: "comp = (\<lambda>f g x. f (g x))"
by (rule ext)+ simp
@@ -147,7 +139,8 @@
instance "fun" :: (type, comm_semiring_1_cancel) comm_semiring_1_cancel ..
-instance "fun" :: (type, semiring_char_0) semiring_char_0 proof
+instance "fun" :: (type, semiring_char_0) semiring_char_0
+proof
from inj_of_nat have "inj (\<lambda>n (x::'a). of_nat n :: 'b)"
by (rule inj_fun)
then have "inj (\<lambda>n. of_nat n :: 'a \<Rightarrow> 'b)"
@@ -168,23 +161,24 @@
text {* Ordereded structures *}
-instance "fun" :: (type, ordered_ab_semigroup_add) ordered_ab_semigroup_add proof
-qed (auto simp add: plus_fun_def le_fun_def intro: add_left_mono)
+instance "fun" :: (type, ordered_ab_semigroup_add) ordered_ab_semigroup_add
+ by default (auto simp add: plus_fun_def le_fun_def intro: add_left_mono)
instance "fun" :: (type, ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
-instance "fun" :: (type, ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le proof
-qed (simp add: plus_fun_def le_fun_def)
+instance "fun" :: (type, ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le
+ by default (simp add: plus_fun_def le_fun_def)
instance "fun" :: (type, ordered_comm_monoid_add) ordered_comm_monoid_add ..
instance "fun" :: (type, ordered_ab_group_add) ordered_ab_group_add ..
-instance "fun" :: (type, ordered_semiring) ordered_semiring proof
-qed (auto simp add: zero_fun_def times_fun_def le_fun_def intro: mult_left_mono mult_right_mono)
+instance "fun" :: (type, ordered_semiring) ordered_semiring
+ by default
+ (auto simp add: zero_fun_def times_fun_def le_fun_def intro: mult_left_mono mult_right_mono)
-instance "fun" :: (type, ordered_comm_semiring) ordered_comm_semiring proof
-qed (fact mult_left_mono)
+instance "fun" :: (type, ordered_comm_semiring) ordered_comm_semiring
+ by default (fact mult_left_mono)
instance "fun" :: (type, ordered_cancel_semiring) ordered_cancel_semiring ..
--- a/src/HOL/Library/Ramsey.thy Tue Feb 21 16:42:57 2012 +0100
+++ b/src/HOL/Library/Ramsey.thy Tue Feb 21 16:48:10 2012 +0100
@@ -47,11 +47,11 @@
qed
} moreover
{ assume "m\<noteq>0" "n\<noteq>0"
- hence "k = (m - 1) + n" "k = m + (n - 1)" using `Suc k = m+n` by auto
- from Suc(1)[OF this(1)] Suc(1)[OF this(2)]
+ then have "k = (m - 1) + n" "k = m + (n - 1)" using `Suc k = m+n` by auto
+ from Suc(1)[OF this(1)] Suc(1)[OF this(2)]
obtain r1 r2 where "r1\<ge>1" "r2\<ge>1" "?R (m - 1) n r1" "?R m (n - 1) r2"
by auto
- hence "r1+r2 \<ge> 1" by arith
+ then have "r1+r2 \<ge> 1" by arith
moreover
have "?R m n (r1+r2)" (is "ALL V E. _ \<longrightarrow> ?EX V E m n")
proof clarify
@@ -62,12 +62,12 @@
let ?M = "{w : V. w\<noteq>v & {v,w} : E}"
let ?N = "{w : V. w\<noteq>v & {v,w} ~: E}"
have "V = insert v (?M \<union> ?N)" using `v : V` by auto
- hence "card V = card(insert v (?M \<union> ?N))" by metis
+ then have "card V = card(insert v (?M \<union> ?N))" by metis
also have "\<dots> = card ?M + card ?N + 1" using `finite V`
by(fastforce intro: card_Un_disjoint)
finally have "card V = card ?M + card ?N + 1" .
- hence "r1+r2 \<le> card ?M + card ?N + 1" using `r1+r2 \<le> card V` by simp
- hence "r1 \<le> card ?M \<or> r2 \<le> card ?N" by arith
+ then have "r1+r2 \<le> card ?M + card ?N + 1" using `r1+r2 \<le> card V` by simp
+ then have "r1 \<le> card ?M \<or> r2 \<le> card ?N" by arith
moreover
{ assume "r1 \<le> card ?M"
moreover have "finite ?M" using `finite V` by auto
@@ -82,7 +82,7 @@
with `R <= V` have "?EX V E m n" by blast
} moreover
{ assume "?C"
- hence "clique (insert v R) E" using `R <= ?M`
+ then have "clique (insert v R) E" using `R <= ?M`
by(auto simp:clique_def insert_commute)
moreover have "card(insert v R) = m"
using `?C` `finite R` `v ~: R` `m\<noteq>0` by simp
@@ -102,7 +102,7 @@
with `R <= V` have "?EX V E m n" by blast
} moreover
{ assume "?I"
- hence "indep (insert v R) E" using `R <= ?N`
+ then have "indep (insert v R) E" using `R <= ?N`
by(auto simp:indep_def insert_commute)
moreover have "card(insert v R) = n"
using `?I` `finite R` `v ~: R` `n\<noteq>0` by simp
@@ -124,17 +124,17 @@
choice_0: "choice P r 0 = (SOME x. P x)"
| choice_Suc: "choice P r (Suc n) = (SOME y. P y & (choice P r n, y) \<in> r)"
-lemma choice_n:
+lemma choice_n:
assumes P0: "P x0"
and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
shows "P (choice P r n)"
proof (induct n)
- case 0 show ?case by (force intro: someI P0)
+ case 0 show ?case by (force intro: someI P0)
next
- case Suc thus ?case by (auto intro: someI2_ex [OF Pstep])
+ case Suc then show ?case by (auto intro: someI2_ex [OF Pstep])
qed
-lemma dependent_choice:
+lemma dependent_choice:
assumes trans: "trans r"
and P0: "P x0"
and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
@@ -144,7 +144,7 @@
fix n
show "P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
next
- have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
+ have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
using Pstep [OF choice_n [OF P0 Pstep]]
by (auto intro: someI2_ex)
fix n m :: nat
@@ -156,8 +156,7 @@
subsubsection {* Partitions of a Set *}
-definition
- part :: "nat => nat => 'a set => ('a set => nat) => bool"
+definition part :: "nat => nat => 'a set => ('a set => nat) => bool"
--{*the function @{term f} partitions the @{term r}-subsets of the typically
infinite set @{term Y} into @{term s} distinct categories.*}
where
@@ -165,52 +164,52 @@
text{*For induction, we decrease the value of @{term r} in partitions.*}
lemma part_Suc_imp_part:
- "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
+ "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
==> part r s (Y - {y}) (%u. f (insert y u))"
apply(simp add: part_def, clarify)
apply(drule_tac x="insert y X" in spec)
apply(force)
done
-lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
+lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
unfolding part_def by blast
-
+
subsection {* Ramsey's Theorem: Infinitary Version *}
-lemma Ramsey_induction:
+lemma Ramsey_induction:
fixes s and r::nat
shows
- "!!(YY::'a set) (f::'a set => nat).
+ "!!(YY::'a set) (f::'a set => nat).
[|infinite YY; part r s YY f|]
- ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
+ ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
(\<forall>X. X \<subseteq> Y' & finite X & card X = r --> f X = t')"
proof (induct r)
case 0
- thus ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
+ then show ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
next
- case (Suc r)
+ case (Suc r)
show ?case
proof -
from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY" by blast
let ?ramr = "{((y,Y,t),(y',Y',t')). y' \<in> Y & Y' \<subseteq> Y}"
- let ?propr = "%(y,Y,t).
+ let ?propr = "%(y,Y,t).
y \<in> YY & y \<notin> Y & Y \<subseteq> YY & infinite Y & t < s
& (\<forall>X. X\<subseteq>Y & finite X & card X = r --> (f o insert y) X = t)"
have infYY': "infinite (YY-{yy})" using Suc.prems by auto
have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
by (simp add: o_def part_Suc_imp_part yy Suc.prems)
- have transr: "trans ?ramr" by (force simp add: trans_def)
+ have transr: "trans ?ramr" by (force simp add: trans_def)
from Suc.hyps [OF infYY' partf']
obtain Y0 and t0
where "Y0 \<subseteq> YY - {yy}" "infinite Y0" "t0 < s"
"\<forall>X. X\<subseteq>Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0"
- by blast
+ by blast
with yy have propr0: "?propr(yy,Y0,t0)" by blast
- have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
+ have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
proof -
fix x
- assume px: "?propr x" thus "?thesis x"
+ assume px: "?propr x" then show "?thesis x"
proof (cases x)
case (fields yx Yx tx)
then obtain yx' where yx': "yx' \<in> Yx" using px
@@ -223,7 +222,7 @@
obtain Y' and t'
where Y': "Y' \<subseteq> Yx - {yx'}" "infinite Y'" "t' < s"
"\<forall>X. X\<subseteq>Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'"
- by blast
+ by blast
show ?thesis
proof
show "?propr (yx',Y',t') & (x, (yx',Y',t')) \<in> ?ramr"
@@ -258,51 +257,51 @@
show ?thesis
proof (intro exI conjI)
show "?gy ` {n. ?gt n = s'} \<subseteq> YY" using pg
- by (auto simp add: Let_def split_beta)
+ by (auto simp add: Let_def split_beta)
show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
- by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
+ by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
show "s' < s" by (rule less')
- show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
+ show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
--> f X = s'"
proof -
- {fix X
+ {fix X
assume "X \<subseteq> ?gy ` {n. ?gt n = s'}"
and cardX: "finite X" "card X = Suc r"
- then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
- by (auto simp add: subset_image_iff)
+ then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
+ by (auto simp add: subset_image_iff)
with cardX have "AA\<noteq>{}" by auto
- hence AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
+ then have AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
have "f X = s'"
- proof (cases "g (LEAST x. x \<in> AA)")
+ proof (cases "g (LEAST x. x \<in> AA)")
case (fields ya Ya ta)
- with AAleast Xeq
- have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
- hence "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
- also have "... = ta"
+ with AAleast Xeq
+ have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
+ then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
+ also have "... = ta"
proof -
have "X - {ya} \<subseteq> Ya"
- proof
+ proof
fix x assume x: "x \<in> X - {ya}"
- then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
- by (auto simp add: Xeq)
- hence "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
- hence lessa': "(LEAST x. x \<in> AA) < a'"
+ then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
+ by (auto simp add: Xeq)
+ then have "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
+ then have lessa': "(LEAST x. x \<in> AA) < a'"
using Least_le [of "%x. x \<in> AA", OF a'] by arith
show "x \<in> Ya" using xeq fields rg [OF lessa'] by auto
qed
moreover
have "card (X - {ya}) = r"
by (simp add: cardX ya)
- ultimately show ?thesis
+ ultimately show ?thesis
using pg [of "LEAST x. x \<in> AA"] fields cardX
by (clarsimp simp del:insert_Diff_single)
qed
also have "... = s'" using AA AAleast fields by auto
finally show ?thesis .
qed}
- thus ?thesis by blast
- qed
- qed
+ then show ?thesis by blast
+ qed
+ qed
qed
qed
@@ -312,7 +311,7 @@
shows
"[|infinite Z;
\<forall>X. X \<subseteq> Z & finite X & card X = r --> f X < s|]
- ==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
+ ==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
& (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X = t)"
by (blast intro: Ramsey_induction [unfolded part_def])
@@ -326,7 +325,7 @@
proof -
have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
using part by (fastforce simp add: eval_nat_numeral card_Suc_eq)
- obtain Y t
+ obtain Y t
where "Y \<subseteq> Z" "infinite Y" "t < s"
"(\<forall>X. X \<subseteq> Y & finite X & card X = 2 --> f X = t)"
by (insert Ramsey [OF infZ part2]) auto
@@ -342,39 +341,36 @@
\cite{Podelski-Rybalchenko}.
*}
-definition
- disj_wf :: "('a * 'a)set => bool"
-where
- "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
+definition disj_wf :: "('a * 'a)set => bool"
+ where "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
-definition
- transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
-where
- "transition_idx s T A =
- (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
+definition transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
+ where
+ "transition_idx s T A =
+ (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
lemma transition_idx_less:
"[|i<j; (s j, s i) \<in> T k; k<n|] ==> transition_idx s T {i,j} < n"
-apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
-apply (simp add: transition_idx_def, blast intro: Least_le)
+apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
+apply (simp add: transition_idx_def, blast intro: Least_le)
done
lemma transition_idx_in:
"[|i<j; (s j, s i) \<in> T k|] ==> (s j, s i) \<in> T (transition_idx s T {i,j})"
-apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
- cong: conj_cong)
-apply (erule LeastI)
+apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
+ cong: conj_cong)
+apply (erule LeastI)
done
text{*To be equal to the union of some well-founded relations is equivalent
to being the subset of such a union.*}
lemma disj_wf:
"disj_wf(r) = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r \<subseteq> (\<Union>i<n. T i))"
-apply (auto simp add: disj_wf_def)
-apply (rule_tac x="%i. T i Int r" in exI)
-apply (rule_tac x=n in exI)
-apply (force simp add: wf_Int1)
+apply (auto simp add: disj_wf_def)
+apply (rule_tac x="%i. T i Int r" in exI)
+apply (rule_tac x=n in exI)
+apply (force simp add: wf_Int1)
done
theorem trans_disj_wf_implies_wf:
@@ -388,13 +384,13 @@
proof -
fix i and j::nat
assume less: "i<j"
- thus "(s j, s i) \<in> r"
+ then show "(s j, s i) \<in> r"
proof (rule less_Suc_induct)
- show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
+ show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
show "\<And>i j k. \<lbrakk>(s j, s i) \<in> r; (s k, s j) \<in> r\<rbrakk> \<Longrightarrow> (s k, s i) \<in> r"
- using transr by (unfold trans_def, blast)
+ using transr by (unfold trans_def, blast)
qed
- qed
+ qed
from dwf
obtain T and n::nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
by (auto simp add: disj_wf_def)
@@ -402,20 +398,20 @@
proof -
fix i and j::nat
assume less: "i<j"
- hence "(s j, s i) \<in> r" by (rule s [of i j])
- thus "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
- qed
+ then have "(s j, s i) \<in> r" by (rule s [of i j])
+ then show "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
+ qed
have trless: "!!i j. i\<noteq>j ==> transition_idx s T {i,j} < n"
apply (auto simp add: linorder_neq_iff)
- apply (blast dest: s_in_T transition_idx_less)
- apply (subst insert_commute)
- apply (blast dest: s_in_T transition_idx_less)
+ apply (blast dest: s_in_T transition_idx_less)
+ apply (subst insert_commute)
+ apply (blast dest: s_in_T transition_idx_less)
done
have
- "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
+ "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
(\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k)"
- by (rule Ramsey2) (auto intro: trless nat_infinite)
- then obtain K and k
+ by (rule Ramsey2) (auto intro: trless nat_infinite)
+ then obtain K and k
where infK: "infinite K" and less: "k < n" and
allk: "\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k"
by auto
@@ -424,18 +420,18 @@
fix m::nat
let ?j = "enumerate K (Suc m)"
let ?i = "enumerate K m"
- have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
- have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
- have ij: "?i < ?j" by (simp add: enumerate_step infK)
- have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
+ have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
+ have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
+ have ij: "?i < ?j" by (simp add: enumerate_step infK)
+ have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
by (simp add: allk)
- obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
+ obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
using s_in_T [OF ij] by blast
- thus "(s ?j, s ?i) \<in> T k"
- by (simp add: ijk [symmetric] transition_idx_in ij)
+ then show "(s ?j, s ?i) \<in> T k"
+ by (simp add: ijk [symmetric] transition_idx_in ij)
qed
- hence "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
- thus False using wfT less by blast
+ then have "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
+ then show False using wfT less by blast
qed
end