--- a/src/CCL/Type.thy Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/Type.thy Tue Nov 11 15:55:31 2014 +0100
@@ -11,39 +11,39 @@
consts
- Subtype :: "['a set, 'a => o] => 'a set"
+ Subtype :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> 'a set"
Bool :: "i set"
Unit :: "i set"
- Plus :: "[i set, i set] => i set" (infixr "+" 55)
- Pi :: "[i set, i => i set] => i set"
- Sigma :: "[i set, i => i set] => i set"
+ Plus :: "[i set, i set] \<Rightarrow> i set" (infixr "+" 55)
+ Pi :: "[i set, i \<Rightarrow> i set] \<Rightarrow> i set"
+ Sigma :: "[i set, i \<Rightarrow> i set] \<Rightarrow> i set"
Nat :: "i set"
- List :: "i set => i set"
- Lists :: "i set => i set"
- ILists :: "i set => i set"
- TAll :: "(i set => i set) => i set" (binder "TALL " 55)
- TEx :: "(i set => i set) => i set" (binder "TEX " 55)
- Lift :: "i set => i set" ("(3[_])")
+ List :: "i set \<Rightarrow> i set"
+ Lists :: "i set \<Rightarrow> i set"
+ ILists :: "i set \<Rightarrow> i set"
+ TAll :: "(i set \<Rightarrow> i set) \<Rightarrow> i set" (binder "TALL " 55)
+ TEx :: "(i set \<Rightarrow> i set) \<Rightarrow> i set" (binder "TEX " 55)
+ Lift :: "i set \<Rightarrow> i set" ("(3[_])")
- SPLIT :: "[i, [i, i] => i set] => i set"
+ SPLIT :: "[i, [i, i] \<Rightarrow> i set] \<Rightarrow> i set"
syntax
- "_Pi" :: "[idt, i set, i set] => i set" ("(3PROD _:_./ _)"
+ "_Pi" :: "[idt, i set, i set] \<Rightarrow> i set" ("(3PROD _:_./ _)"
[0,0,60] 60)
- "_Sigma" :: "[idt, i set, i set] => i set" ("(3SUM _:_./ _)"
+ "_Sigma" :: "[idt, i set, i set] \<Rightarrow> i set" ("(3SUM _:_./ _)"
[0,0,60] 60)
- "_arrow" :: "[i set, i set] => i set" ("(_ ->/ _)" [54, 53] 53)
- "_star" :: "[i set, i set] => i set" ("(_ */ _)" [56, 55] 55)
- "_Subtype" :: "[idt, 'a set, o] => 'a set" ("(1{_: _ ./ _})")
+ "_arrow" :: "[i set, i set] \<Rightarrow> i set" ("(_ ->/ _)" [54, 53] 53)
+ "_star" :: "[i set, i set] \<Rightarrow> i set" ("(_ */ _)" [56, 55] 55)
+ "_Subtype" :: "[idt, 'a set, o] \<Rightarrow> 'a set" ("(1{_: _ ./ _})")
translations
- "PROD x:A. B" => "CONST Pi(A, %x. B)"
- "A -> B" => "CONST Pi(A, %_. B)"
- "SUM x:A. B" => "CONST Sigma(A, %x. B)"
- "A * B" => "CONST Sigma(A, %_. B)"
- "{x: A. B}" == "CONST Subtype(A, %x. B)"
+ "PROD x:A. B" => "CONST Pi(A, \<lambda>x. B)"
+ "A -> B" => "CONST Pi(A, \<lambda>_. B)"
+ "SUM x:A. B" => "CONST Sigma(A, \<lambda>x. B)"
+ "A * B" => "CONST Sigma(A, \<lambda>_. B)"
+ "{x: A. B}" == "CONST Subtype(A, \<lambda>x. B)"
print_translation {*
[(@{const_syntax Pi},
@@ -53,23 +53,23 @@
*}
defs
- Subtype_def: "{x:A. P(x)} == {x. x:A & P(x)}"
+ Subtype_def: "{x:A. P(x)} == {x. x:A \<and> P(x)}"
Unit_def: "Unit == {x. x=one}"
Bool_def: "Bool == {x. x=true | x=false}"
Plus_def: "A+B == {x. (EX a:A. x=inl(a)) | (EX b:B. x=inr(b))}"
- Pi_def: "Pi(A,B) == {x. EX b. x=lam x. b(x) & (ALL x:A. b(x):B(x))}"
+ Pi_def: "Pi(A,B) == {x. EX b. x=lam x. b(x) \<and> (ALL x:A. b(x):B(x))}"
Sigma_def: "Sigma(A,B) == {x. EX a:A. EX b:B(a).x=<a,b>}"
- Nat_def: "Nat == lfp(% X. Unit + X)"
- List_def: "List(A) == lfp(% X. Unit + A*X)"
+ Nat_def: "Nat == lfp(\<lambda>X. Unit + X)"
+ List_def: "List(A) == lfp(\<lambda>X. Unit + A*X)"
- Lists_def: "Lists(A) == gfp(% X. Unit + A*X)"
- ILists_def: "ILists(A) == gfp(% X.{} + A*X)"
+ Lists_def: "Lists(A) == gfp(\<lambda>X. Unit + A*X)"
+ ILists_def: "ILists(A) == gfp(\<lambda>X.{} + A*X)"
Tall_def: "TALL X. B(X) == Inter({X. EX Y. X=B(Y)})"
Tex_def: "TEX X. B(X) == Union({X. EX Y. X=B(Y)})"
Lift_def: "[A] == A Un {bot}"
- SPLIT_def: "SPLIT(p,B) == Union({A. EX x y. p=<x,y> & A=B(x,y)})"
+ SPLIT_def: "SPLIT(p,B) == Union({A. EX x y. p=<x,y> \<and> A=B(x,y)})"
lemmas simp_type_defs =
@@ -78,26 +78,26 @@
and simp_data_defs = one_def inl_def inr_def
and ind_data_defs = zero_def succ_def nil_def cons_def
-lemma subsetXH: "A <= B <-> (ALL x. x:A --> x:B)"
+lemma subsetXH: "A <= B \<longleftrightarrow> (ALL x. x:A \<longrightarrow> x:B)"
by blast
subsection {* Exhaustion Rules *}
-lemma EmptyXH: "!!a. a : {} <-> False"
- and SubtypeXH: "!!a A P. a : {x:A. P(x)} <-> (a:A & P(a))"
- and UnitXH: "!!a. a : Unit <-> a=one"
- and BoolXH: "!!a. a : Bool <-> a=true | a=false"
- and PlusXH: "!!a A B. a : A+B <-> (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))"
- and PiXH: "!!a A B. a : PROD x:A. B(x) <-> (EX b. a=lam x. b(x) & (ALL x:A. b(x):B(x)))"
- and SgXH: "!!a A B. a : SUM x:A. B(x) <-> (EX x:A. EX y:B(x).a=<x,y>)"
+lemma EmptyXH: "\<And>a. a : {} \<longleftrightarrow> False"
+ and SubtypeXH: "\<And>a A P. a : {x:A. P(x)} \<longleftrightarrow> (a:A \<and> P(a))"
+ and UnitXH: "\<And>a. a : Unit \<longleftrightarrow> a=one"
+ and BoolXH: "\<And>a. a : Bool \<longleftrightarrow> a=true | a=false"
+ and PlusXH: "\<And>a A B. a : A+B \<longleftrightarrow> (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))"
+ and PiXH: "\<And>a A B. a : PROD x:A. B(x) \<longleftrightarrow> (EX b. a=lam x. b(x) \<and> (ALL x:A. b(x):B(x)))"
+ and SgXH: "\<And>a A B. a : SUM x:A. B(x) \<longleftrightarrow> (EX x:A. EX y:B(x).a=<x,y>)"
unfolding simp_type_defs by blast+
lemmas XHs = EmptyXH SubtypeXH UnitXH BoolXH PlusXH PiXH SgXH
-lemma LiftXH: "a : [A] <-> (a=bot | a:A)"
- and TallXH: "a : TALL X. B(X) <-> (ALL X. a:B(X))"
- and TexXH: "a : TEX X. B(X) <-> (EX X. a:B(X))"
+lemma LiftXH: "a : [A] \<longleftrightarrow> (a=bot | a:A)"
+ and TallXH: "a : TALL X. B(X) \<longleftrightarrow> (ALL X. a:B(X))"
+ and TexXH: "a : TEX X. B(X) \<longleftrightarrow> (EX X. a:B(X))"
unfolding simp_type_defs by blast+
ML {* ML_Thms.bind_thms ("case_rls", XH_to_Es @{thms XHs}) *}
@@ -108,10 +108,10 @@
lemma oneT: "one : Unit"
and trueT: "true : Bool"
and falseT: "false : Bool"
- and lamT: "!!b B. [| !!x. x:A ==> b(x):B(x) |] ==> lam x. b(x) : Pi(A,B)"
- and pairT: "!!b B. [| a:A; b:B(a) |] ==> <a,b>:Sigma(A,B)"
- and inlT: "a:A ==> inl(a) : A+B"
- and inrT: "b:B ==> inr(b) : A+B"
+ and lamT: "\<And>b B. (\<And>x. x:A \<Longrightarrow> b(x):B(x)) \<Longrightarrow> lam x. b(x) : Pi(A,B)"
+ and pairT: "\<And>b B. \<lbrakk>a:A; b:B(a)\<rbrakk> \<Longrightarrow> <a,b>:Sigma(A,B)"
+ and inlT: "a:A \<Longrightarrow> inl(a) : A+B"
+ and inrT: "b:B \<Longrightarrow> inr(b) : A+B"
by (blast intro: XHs [THEN iffD2])+
lemmas canTs = oneT trueT falseT pairT lamT inlT inrT
@@ -119,7 +119,7 @@
subsection {* Non-Canonical Type Rules *}
-lemma lem: "[| a:B(u); u=v |] ==> a : B(v)"
+lemma lem: "\<lbrakk>a:B(u); u = v\<rbrakk> \<Longrightarrow> a : B(v)"
by blast
@@ -137,22 +137,19 @@
Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms case_rls} @{thms case_rls})
*}
-lemma ifT:
- "[| b:Bool; b=true ==> t:A(true); b=false ==> u:A(false) |] ==>
- if b then t else u : A(b)"
+lemma ifT: "\<lbrakk>b:Bool; b=true \<Longrightarrow> t:A(true); b=false \<Longrightarrow> u:A(false)\<rbrakk> \<Longrightarrow> if b then t else u : A(b)"
by ncanT
-lemma applyT: "[| f : Pi(A,B); a:A |] ==> f ` a : B(a)"
+lemma applyT: "\<lbrakk>f : Pi(A,B); a:A\<rbrakk> \<Longrightarrow> f ` a : B(a)"
by ncanT
-lemma splitT:
- "[| p:Sigma(A,B); !!x y. [| x:A; y:B(x); p=<x,y> |] ==> c(x,y):C(<x,y>) |]
- ==> split(p,c):C(p)"
+lemma splitT: "\<lbrakk>p:Sigma(A,B); \<And>x y. \<lbrakk>x:A; y:B(x); p=<x,y>\<rbrakk> \<Longrightarrow> c(x,y):C(<x,y>)\<rbrakk> \<Longrightarrow> split(p,c):C(p)"
by ncanT
lemma whenT:
- "[| p:A+B; !!x.[| x:A; p=inl(x) |] ==> a(x):C(inl(x)); !!y.[| y:B; p=inr(y) |]
- ==> b(y):C(inr(y)) |] ==> when(p,a,b) : C(p)"
+ "\<lbrakk>p:A+B;
+ \<And>x. \<lbrakk>x:A; p=inl(x)\<rbrakk> \<Longrightarrow> a(x):C(inl(x));
+ \<And>y. \<lbrakk>y:B; p=inr(y)\<rbrakk> \<Longrightarrow> b(y):C(inr(y))\<rbrakk> \<Longrightarrow> when(p,a,b) : C(p)"
by ncanT
lemmas ncanTs = ifT applyT splitT whenT
@@ -160,30 +157,30 @@
subsection {* Subtypes *}
-lemma SubtypeD1: "a : Subtype(A, P) ==> a : A"
- and SubtypeD2: "a : Subtype(A, P) ==> P(a)"
+lemma SubtypeD1: "a : Subtype(A, P) \<Longrightarrow> a : A"
+ and SubtypeD2: "a : Subtype(A, P) \<Longrightarrow> P(a)"
by (simp_all add: SubtypeXH)
-lemma SubtypeI: "[| a:A; P(a) |] ==> a : {x:A. P(x)}"
+lemma SubtypeI: "\<lbrakk>a:A; P(a)\<rbrakk> \<Longrightarrow> a : {x:A. P(x)}"
by (simp add: SubtypeXH)
-lemma SubtypeE: "[| a : {x:A. P(x)}; [| a:A; P(a) |] ==> Q |] ==> Q"
+lemma SubtypeE: "\<lbrakk>a : {x:A. P(x)}; \<lbrakk>a:A; P(a)\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
by (simp add: SubtypeXH)
subsection {* Monotonicity *}
-lemma idM: "mono (%X. X)"
+lemma idM: "mono (\<lambda>X. X)"
apply (rule monoI)
apply assumption
done
-lemma constM: "mono(%X. A)"
+lemma constM: "mono(\<lambda>X. A)"
apply (rule monoI)
apply (rule subset_refl)
done
-lemma "mono(%X. A(X)) ==> mono(%X.[A(X)])"
+lemma "mono(\<lambda>X. A(X)) \<Longrightarrow> mono(\<lambda>X.[A(X)])"
apply (rule subsetI [THEN monoI])
apply (drule LiftXH [THEN iffD1])
apply (erule disjE)
@@ -194,18 +191,16 @@
done
lemma SgM:
- "[| mono(%X. A(X)); !!x X. x:A(X) ==> mono(%X. B(X,x)) |] ==>
- mono(%X. Sigma(A(X),B(X)))"
+ "\<lbrakk>mono(\<lambda>X. A(X)); \<And>x X. x:A(X) \<Longrightarrow> mono(\<lambda>X. B(X,x))\<rbrakk> \<Longrightarrow>
+ mono(\<lambda>X. Sigma(A(X),B(X)))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])
-lemma PiM:
- "[| !!x. x:A ==> mono(%X. B(X,x)) |] ==> mono(%X. Pi(A,B(X)))"
+lemma PiM: "(\<And>x. x:A \<Longrightarrow> mono(\<lambda>X. B(X,x))) \<Longrightarrow> mono(\<lambda>X. Pi(A,B(X)))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])
-lemma PlusM:
- "[| mono(%X. A(X)); mono(%X. B(X)) |] ==> mono(%X. A(X)+B(X))"
+lemma PlusM: "\<lbrakk>mono(\<lambda>X. A(X)); mono(\<lambda>X. B(X))\<rbrakk> \<Longrightarrow> mono(\<lambda>X. A(X)+B(X))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])
@@ -214,7 +209,7 @@
subsubsection {* Conversion Rules for Fixed Points via monotonicity and Tarski *}
-lemma NatM: "mono(%X. Unit+X)"
+lemma NatM: "mono(\<lambda>X. Unit+X)"
apply (rule PlusM constM idM)+
done
@@ -223,7 +218,7 @@
apply (rule NatM)
done
-lemma ListM: "mono(%X.(Unit+Sigma(A,%y. X)))"
+lemma ListM: "mono(\<lambda>X.(Unit+Sigma(A,\<lambda>y. X)))"
apply (rule PlusM SgM constM idM)+
done
@@ -237,7 +232,7 @@
apply (rule ListM)
done
-lemma IListsM: "mono(%X.({} + Sigma(A,%y. X)))"
+lemma IListsM: "mono(\<lambda>X.({} + Sigma(A,\<lambda>y. X)))"
apply (rule PlusM SgM constM idM)+
done
@@ -251,10 +246,10 @@
subsection {* Exhaustion Rules *}
-lemma NatXH: "a : Nat <-> (a=zero | (EX x:Nat. a=succ(x)))"
- and ListXH: "a : List(A) <-> (a=[] | (EX x:A. EX xs:List(A).a=x$xs))"
- and ListsXH: "a : Lists(A) <-> (a=[] | (EX x:A. EX xs:Lists(A).a=x$xs))"
- and IListsXH: "a : ILists(A) <-> (EX x:A. EX xs:ILists(A).a=x$xs)"
+lemma NatXH: "a : Nat \<longleftrightarrow> (a=zero | (EX x:Nat. a=succ(x)))"
+ and ListXH: "a : List(A) \<longleftrightarrow> (a=[] | (EX x:A. EX xs:List(A).a=x$xs))"
+ and ListsXH: "a : Lists(A) \<longleftrightarrow> (a=[] | (EX x:A. EX xs:Lists(A).a=x$xs))"
+ and IListsXH: "a : ILists(A) \<longleftrightarrow> (EX x:A. EX xs:ILists(A).a=x$xs)"
unfolding ind_data_defs
by (rule ind_type_eqs [THEN XHlemma1], blast intro!: canTs elim!: case_rls)+
@@ -266,9 +261,9 @@
subsection {* Type Rules *}
lemma zeroT: "zero : Nat"
- and succT: "n:Nat ==> succ(n) : Nat"
+ and succT: "n:Nat \<Longrightarrow> succ(n) : Nat"
and nilT: "[] : List(A)"
- and consT: "[| h:A; t:List(A) |] ==> h$t : List(A)"
+ and consT: "\<lbrakk>h:A; t:List(A)\<rbrakk> \<Longrightarrow> h$t : List(A)"
by (blast intro: iXHs [THEN iffD2])+
lemmas icanTs = zeroT succT nilT consT
@@ -278,14 +273,12 @@
Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms icase_rls} @{thms case_rls})
*}
-lemma ncaseT:
- "[| n:Nat; n=zero ==> b:C(zero); !!x.[| x:Nat; n=succ(x) |] ==> c(x):C(succ(x)) |]
- ==> ncase(n,b,c) : C(n)"
+lemma ncaseT: "\<lbrakk>n:Nat; n=zero \<Longrightarrow> b:C(zero); \<And>x. \<lbrakk>x:Nat; n=succ(x)\<rbrakk> \<Longrightarrow> c(x):C(succ(x))\<rbrakk>
+ \<Longrightarrow> ncase(n,b,c) : C(n)"
by incanT
-lemma lcaseT:
- "[| l:List(A); l=[] ==> b:C([]); !!h t.[| h:A; t:List(A); l=h$t |] ==>
- c(h,t):C(h$t) |] ==> lcase(l,b,c) : C(l)"
+lemma lcaseT: "\<lbrakk>l:List(A); l = [] \<Longrightarrow> b:C([]); \<And>h t. \<lbrakk>h:A; t:List(A); l=h$t\<rbrakk> \<Longrightarrow> c(h,t):C(h$t)\<rbrakk>
+ \<Longrightarrow> lcase(l,b,c) : C(l)"
by incanT
lemmas incanTs = ncaseT lcaseT
@@ -295,14 +288,13 @@
lemmas ind_Ms = NatM ListM
-lemma Nat_ind: "[| n:Nat; P(zero); !!x.[| x:Nat; P(x) |] ==> P(succ(x)) |] ==> P(n)"
+lemma Nat_ind: "\<lbrakk>n:Nat; P(zero); \<And>x. \<lbrakk>x:Nat; P(x)\<rbrakk> \<Longrightarrow> P(succ(x))\<rbrakk> \<Longrightarrow> P(n)"
apply (unfold ind_data_defs)
apply (erule def_induct [OF Nat_def _ NatM])
apply (blast intro: canTs elim!: case_rls)
done
-lemma List_ind:
- "[| l:List(A); P([]); !!x xs.[| x:A; xs:List(A); P(xs) |] ==> P(x$xs) |] ==> P(l)"
+lemma List_ind: "\<lbrakk>l:List(A); P([]); \<And>x xs. \<lbrakk>x:A; xs:List(A); P(xs)\<rbrakk> \<Longrightarrow> P(x$xs)\<rbrakk> \<Longrightarrow> P(l)"
apply (unfold ind_data_defs)
apply (erule def_induct [OF List_def _ ListM])
apply (blast intro: canTs elim!: case_rls)
@@ -313,16 +305,12 @@
subsection {* Primitive Recursive Rules *}
-lemma nrecT:
- "[| n:Nat; b:C(zero);
- !!x g.[| x:Nat; g:C(x) |] ==> c(x,g):C(succ(x)) |] ==>
- nrec(n,b,c) : C(n)"
+lemma nrecT: "\<lbrakk>n:Nat; b:C(zero); \<And>x g. \<lbrakk>x:Nat; g:C(x)\<rbrakk> \<Longrightarrow> c(x,g):C(succ(x))\<rbrakk>
+ \<Longrightarrow> nrec(n,b,c) : C(n)"
by (erule Nat_ind) auto
-lemma lrecT:
- "[| l:List(A); b:C([]);
- !!x xs g.[| x:A; xs:List(A); g:C(xs) |] ==> c(x,xs,g):C(x$xs) |] ==>
- lrec(l,b,c) : C(l)"
+lemma lrecT: "\<lbrakk>l:List(A); b:C([]); \<And>x xs g. \<lbrakk>x:A; xs:List(A); g:C(xs)\<rbrakk> \<Longrightarrow> c(x,xs,g):C(x$xs) \<rbrakk>
+ \<Longrightarrow> lrec(l,b,c) : C(l)"
by (erule List_ind) auto
lemmas precTs = nrecT lrecT
@@ -330,8 +318,7 @@
subsection {* Theorem proving *}
-lemma SgE2:
- "[| <a,b> : Sigma(A,B); [| a:A; b:B(a) |] ==> P |] ==> P"
+lemma SgE2: "\<lbrakk><a,b> : Sigma(A,B); \<lbrakk>a:A; b:B(a)\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
unfolding SgXH by blast
(* General theorem proving ignores non-canonical term-formers, *)
@@ -346,7 +333,7 @@
subsection {* Infinite Data Types *}
-lemma lfp_subset_gfp: "mono(f) ==> lfp(f) <= gfp(f)"
+lemma lfp_subset_gfp: "mono(f) \<Longrightarrow> lfp(f) <= gfp(f)"
apply (rule lfp_lowerbound [THEN subset_trans])
apply (erule gfp_lemma3)
apply (rule subset_refl)
@@ -354,16 +341,14 @@
lemma gfpI:
assumes "a:A"
- and "!!x X.[| x:A; ALL y:A. t(y):X |] ==> t(x) : B(X)"
+ and "\<And>x X. \<lbrakk>x:A; ALL y:A. t(y):X\<rbrakk> \<Longrightarrow> t(x) : B(X)"
shows "t(a) : gfp(B)"
apply (rule coinduct)
- apply (rule_tac P = "%x. EX y:A. x=t (y)" in CollectI)
+ apply (rule_tac P = "\<lambda>x. EX y:A. x=t (y)" in CollectI)
apply (blast intro!: assms)+
done
-lemma def_gfpI:
- "[| C==gfp(B); a:A; !!x X.[| x:A; ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==>
- t(a) : C"
+lemma def_gfpI: "\<lbrakk>C == gfp(B); a:A; \<And>x X. \<lbrakk>x:A; ALL y:A. t(y):X\<rbrakk> \<Longrightarrow> t(x) : B(X)\<rbrakk> \<Longrightarrow> t(a) : C"
apply unfold
apply (erule gfpI)
apply blast
@@ -381,15 +366,15 @@
subsection {* Lemmas and tactics for using the rule @{text
"coinduct3"} on @{text "[="} and @{text "="} *}
-lemma lfpI: "[| mono(f); a : f(lfp(f)) |] ==> a : lfp(f)"
+lemma lfpI: "\<lbrakk>mono(f); a : f(lfp(f))\<rbrakk> \<Longrightarrow> a : lfp(f)"
apply (erule lfp_Tarski [THEN ssubst])
apply assumption
done
-lemma ssubst_single: "[| a=a'; a' : A |] ==> a : A"
+lemma ssubst_single: "\<lbrakk>a = a'; a' : A\<rbrakk> \<Longrightarrow> a : A"
by simp
-lemma ssubst_pair: "[| a=a'; b=b'; <a',b'> : A |] ==> <a,b> : A"
+lemma ssubst_pair: "\<lbrakk>a = a'; b = b'; <a',b'> : A\<rbrakk> \<Longrightarrow> <a,b> : A"
by simp
@@ -400,14 +385,14 @@
method_setup coinduct3 = {* Scan.succeed (SIMPLE_METHOD' o coinduct3_tac) *}
-lemma ci3_RI: "[| mono(Agen); a : R |] ==> a : lfp(%x. Agen(x) Un R Un A)"
+lemma ci3_RI: "\<lbrakk>mono(Agen); a : R\<rbrakk> \<Longrightarrow> a : lfp(\<lambda>x. Agen(x) Un R Un A)"
by coinduct3
-lemma ci3_AgenI: "[| mono(Agen); a : Agen(lfp(%x. Agen(x) Un R Un A)) |] ==>
- a : lfp(%x. Agen(x) Un R Un A)"
+lemma ci3_AgenI: "\<lbrakk>mono(Agen); a : Agen(lfp(\<lambda>x. Agen(x) Un R Un A))\<rbrakk> \<Longrightarrow>
+ a : lfp(\<lambda>x. Agen(x) Un R Un A)"
by coinduct3
-lemma ci3_AI: "[| mono(Agen); a : A |] ==> a : lfp(%x. Agen(x) Un R Un A)"
+lemma ci3_AI: "\<lbrakk>mono(Agen); a : A\<rbrakk> \<Longrightarrow> a : lfp(\<lambda>x. Agen(x) Un R Un A)"
by coinduct3
ML {*
@@ -432,19 +417,19 @@
lemma POgenIs:
"<true,true> : POgen(R)"
"<false,false> : POgen(R)"
- "[| <a,a'> : R; <b,b'> : R |] ==> <<a,b>,<a',b'>> : POgen(R)"
- "!!b b'. [|!!x. <b(x),b'(x)> : R |] ==><lam x. b(x),lam x. b'(x)> : POgen(R)"
+ "\<lbrakk><a,a'> : R; <b,b'> : R\<rbrakk> \<Longrightarrow> <<a,b>,<a',b'>> : POgen(R)"
+ "\<And>b b'. (\<And>x. <b(x),b'(x)> : R) \<Longrightarrow> <lam x. b(x),lam x. b'(x)> : POgen(R)"
"<one,one> : POgen(R)"
- "<a,a'> : lfp(%x. POgen(x) Un R Un PO) ==>
- <inl(a),inl(a')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
- "<b,b'> : lfp(%x. POgen(x) Un R Un PO) ==>
- <inr(b),inr(b')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
- "<zero,zero> : POgen(lfp(%x. POgen(x) Un R Un PO))"
- "<n,n'> : lfp(%x. POgen(x) Un R Un PO) ==>
- <succ(n),succ(n')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
- "<[],[]> : POgen(lfp(%x. POgen(x) Un R Un PO))"
- "[| <h,h'> : lfp(%x. POgen(x) Un R Un PO); <t,t'> : lfp(%x. POgen(x) Un R Un PO) |]
- ==> <h$t,h'$t'> : POgen(lfp(%x. POgen(x) Un R Un PO))"
+ "<a,a'> : lfp(\<lambda>x. POgen(x) Un R Un PO) \<Longrightarrow>
+ <inl(a),inl(a')> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
+ "<b,b'> : lfp(\<lambda>x. POgen(x) Un R Un PO) \<Longrightarrow>
+ <inr(b),inr(b')> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
+ "<zero,zero> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
+ "<n,n'> : lfp(\<lambda>x. POgen(x) Un R Un PO) \<Longrightarrow>
+ <succ(n),succ(n')> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
+ "<[],[]> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
+ "\<lbrakk><h,h'> : lfp(\<lambda>x. POgen(x) Un R Un PO); <t,t'> : lfp(\<lambda>x. POgen(x) Un R Un PO)\<rbrakk>
+ \<Longrightarrow> <h$t,h'$t'> : POgen(lfp(\<lambda>x. POgen(x) Un R Un PO))"
unfolding data_defs by (genIs POgenXH POgen_mono)+
ML {*
@@ -466,19 +451,19 @@
lemma EQgenIs:
"<true,true> : EQgen(R)"
"<false,false> : EQgen(R)"
- "[| <a,a'> : R; <b,b'> : R |] ==> <<a,b>,<a',b'>> : EQgen(R)"
- "!!b b'. [|!!x. <b(x),b'(x)> : R |] ==> <lam x. b(x),lam x. b'(x)> : EQgen(R)"
+ "\<lbrakk><a,a'> : R; <b,b'> : R\<rbrakk> \<Longrightarrow> <<a,b>,<a',b'>> : EQgen(R)"
+ "\<And>b b'. (\<And>x. <b(x),b'(x)> : R) \<Longrightarrow> <lam x. b(x),lam x. b'(x)> : EQgen(R)"
"<one,one> : EQgen(R)"
- "<a,a'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
- <inl(a),inl(a')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
- "<b,b'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
- <inr(b),inr(b')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
- "<zero,zero> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
- "<n,n'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
- <succ(n),succ(n')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
- "<[],[]> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
- "[| <h,h'> : lfp(%x. EQgen(x) Un R Un EQ); <t,t'> : lfp(%x. EQgen(x) Un R Un EQ) |]
- ==> <h$t,h'$t'> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
+ "<a,a'> : lfp(\<lambda>x. EQgen(x) Un R Un EQ) \<Longrightarrow>
+ <inl(a),inl(a')> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
+ "<b,b'> : lfp(\<lambda>x. EQgen(x) Un R Un EQ) \<Longrightarrow>
+ <inr(b),inr(b')> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
+ "<zero,zero> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
+ "<n,n'> : lfp(\<lambda>x. EQgen(x) Un R Un EQ) \<Longrightarrow>
+ <succ(n),succ(n')> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
+ "<[],[]> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
+ "\<lbrakk><h,h'> : lfp(\<lambda>x. EQgen(x) Un R Un EQ); <t,t'> : lfp(\<lambda>x. EQgen(x) Un R Un EQ)\<rbrakk>
+ \<Longrightarrow> <h$t,h'$t'> : EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))"
unfolding data_defs by (genIs EQgenXH EQgen_mono)+
ML {*