--- a/doc-src/Classes/Thy/document/Classes.tex Fri Aug 26 10:25:13 2011 +0200
+++ b/doc-src/Classes/Thy/document/Classes.tex Fri Aug 26 18:24:22 2011 +0200
@@ -1167,13 +1167,13 @@
mult{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline
mult{\isaliteral{5F}{\isacharunderscore}}int\ i\ j\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ j{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
+neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\isanewline
instance\ Semigroup\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3B}{\isacharsemicolon}}\isanewline
{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Integer{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-\isanewline
instance\ Monoidl\ Integer\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3B}{\isacharsemicolon}}\isanewline
{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
@@ -1231,8 +1231,8 @@
\ \ val\ pow{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
\ \ val\ pow{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ group\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
\ \ val\ mult{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline
+\ \ val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline
\ \ val\ semigroup{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ semigroup\isanewline
-\ \ val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline
\ \ val\ monoidl{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ monoidl\isanewline
\ \ val\ monoid{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ monoid\isanewline
\ \ val\ inverse{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ IntInf{\isaliteral{2E}{\isachardot}}int\isanewline
@@ -1273,9 +1273,9 @@
\isanewline
fun\ mult{\isaliteral{5F}{\isacharunderscore}}int\ i\ j\ {\isaliteral{3D}{\isacharequal}}\ IntInf{\isaliteral{2E}{\isachardot}}{\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-val\ semigroup{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-val\ neutral{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+val\ semigroup{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ IntInf{\isaliteral{2E}{\isachardot}}int\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
val\ monoidl{\isaliteral{5F}{\isacharunderscore}}int\ {\isaliteral{3D}{\isacharequal}}\isanewline
\ \ {\isaliteral{7B}{\isacharbraceleft}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoidl\ {\isaliteral{3D}{\isacharequal}}\ semigroup{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{2C}{\isacharcomma}}\ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\isanewline
@@ -1368,12 +1368,12 @@
\isanewline
def\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ j{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ i\ {\isaliteral{2B}{\isacharplus}}\ j\isanewline
\isanewline
+def\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\isanewline
implicit\ def\ semigroup{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ semigroup{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ semigroup{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline
{\isaliteral{7D}{\isacharbraceright}}\isanewline
\isanewline
-def\ neutral{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ BigInt\ {\isaliteral{3D}{\isacharequal}}\ BigInt{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
implicit\ def\ monoidl{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{3A}{\isacharcolon}}\ monoidl{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ new\ monoidl{\isaliteral{5B}{\isacharbrackleft}}BigInt{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}neutral{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}int\isanewline
\ \ val\ {\isaliteral{60}{\isacharbackquote}}Example{\isaliteral{2E}{\isachardot}}mult{\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{3A}{\isacharcolon}}\ BigInt{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ mult{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\isanewline
--- a/doc-src/Codegen/Thy/document/Introduction.tex Fri Aug 26 10:25:13 2011 +0200
+++ b/doc-src/Codegen/Thy/document/Introduction.tex Fri Aug 26 18:24:22 2011 +0200
@@ -413,13 +413,13 @@
mult{\isaliteral{5F}{\isacharunderscore}}nat\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{3D}{\isacharequal}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
mult{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{28}{\isacharparenleft}}mult{\isaliteral{5F}{\isacharunderscore}}nat\ m\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
+neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3D}{\isacharequal}}\ Suc\ Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\isanewline
instance\ Semigroup\ Nat\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ Nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3D}{\isacharequal}}\ Suc\ Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-\isanewline
instance\ Monoid\ Nat\ where\ {\isaliteral{7B}{\isacharbraceleft}}\isanewline
\ \ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
@@ -462,8 +462,8 @@
\ \ val\ neutral\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
\ \ val\ pow\ {\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ monoid\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
\ \ val\ mult{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\isanewline
+\ \ val\ neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
\ \ val\ semigroup{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ semigroup\isanewline
-\ \ val\ neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\isanewline
\ \ val\ monoid{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ monoid\isanewline
\ \ val\ bexp\ {\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}\ nat\isanewline
end\ {\isaliteral{3D}{\isacharequal}}\ struct\isanewline
@@ -486,9 +486,9 @@
fun\ mult{\isaliteral{5F}{\isacharunderscore}}nat\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{3D}{\isacharequal}}\ Zero{\isaliteral{5F}{\isacharunderscore}}nat\isanewline
\ \ {\isaliteral{7C}{\isacharbar}}\ mult{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ plus{\isaliteral{5F}{\isacharunderscore}}nat\ n\ {\isaliteral{28}{\isacharparenleft}}mult{\isaliteral{5F}{\isacharunderscore}}nat\ m\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-val\ semigroup{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ nat\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+val\ neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\ Suc\ Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
-val\ neutral{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{3D}{\isacharequal}}\ Suc\ Zero{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+val\ semigroup{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}mult\ {\isaliteral{3D}{\isacharequal}}\ mult{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3A}{\isacharcolon}}\ nat\ semigroup{\isaliteral{3B}{\isacharsemicolon}}\isanewline
\isanewline
val\ monoid{\isaliteral{5F}{\isacharunderscore}}nat\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}semigroup{\isaliteral{5F}{\isacharunderscore}}monoid\ {\isaliteral{3D}{\isacharequal}}\ semigroup{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{2C}{\isacharcomma}}\ neutral\ {\isaliteral{3D}{\isacharequal}}\ neutral{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{7D}{\isacharbraceright}}\isanewline
\ \ {\isaliteral{3A}{\isacharcolon}}\ nat\ monoid{\isaliteral{3B}{\isacharsemicolon}}\isanewline
--- a/src/HOL/Quotient.thy Fri Aug 26 10:25:13 2011 +0200
+++ b/src/HOL/Quotient.thy Fri Aug 26 18:24:22 2011 +0200
@@ -35,12 +35,11 @@
definition
Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
where
- "Respects R x = R x x"
+ "Respects R = {x. R x x}"
lemma in_respects:
shows "x \<in> Respects R \<longleftrightarrow> R x x"
- unfolding mem_def Respects_def
- by simp
+ unfolding Respects_def by simp
subsection {* Function map and function relation *}
@@ -268,14 +267,14 @@
by (auto simp add: in_respects)
lemma ball_reg_right:
- assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
+ assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
shows "All P \<longrightarrow> Ball R Q"
- using a by (metis Collect_def Collect_mem_eq)
+ using a by (metis Collect_mem_eq)
lemma bex_reg_left:
- assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
+ assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
shows "Bex R Q \<longrightarrow> Ex P"
- using a by (metis Collect_def Collect_mem_eq)
+ using a by (metis Collect_mem_eq)
lemma ball_reg_left:
assumes a: "equivp R"
@@ -327,16 +326,16 @@
using a b by metis
lemma ball_reg:
- assumes a: "!x :: 'a. (R x --> P x --> Q x)"
+ assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
and b: "Ball R P"
shows "Ball R Q"
- using a b by (metis Collect_def Collect_mem_eq)
+ using a b by (metis Collect_mem_eq)
lemma bex_reg:
- assumes a: "!x :: 'a. (R x --> P x --> Q x)"
+ assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
and b: "Bex R P"
shows "Bex R Q"
- using a b by (metis Collect_def Collect_mem_eq)
+ using a b by (metis Collect_mem_eq)
lemma ball_all_comm:
@@ -599,16 +598,6 @@
shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
by (auto intro!: fun_relI elim: fun_relE)
-lemma mem_rsp:
- shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
- by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)
-
-lemma mem_prs:
- assumes a1: "Quotient R1 Abs1 Rep1"
- and a2: "Quotient R2 Abs2 Rep2"
- shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
- by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
-
lemma id_rsp:
shows "(R ===> R) id id"
by (auto intro: fun_relI)
@@ -686,8 +675,8 @@
declare [[map set = (vimage, set_rel)]]
lemmas [quot_thm] = fun_quotient
-lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
-lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
+lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
+lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
lemmas [quot_equiv] = identity_equivp