--- a/src/HOL/Induct/Comb.thy Wed Dec 30 20:12:26 2015 +0100
+++ b/src/HOL/Induct/Comb.thy Wed Dec 30 20:24:43 2015 +0100
@@ -18,53 +18,47 @@
subsection \<open>Definitions\<close>
-text \<open>Datatype definition of combinators @{text S} and @{text K}.\<close>
+text \<open>Datatype definition of combinators \<open>S\<close> and \<open>K\<close>.\<close>
datatype comb = K
| S
- | Ap comb comb (infixl "##" 90)
-
-notation (xsymbols)
- Ap (infixl "\<bullet>" 90)
-
+ | Ap comb comb (infixl "\<bullet>" 90)
text \<open>
- Inductive definition of contractions, @{text "-1->"} and
- (multi-step) reductions, @{text "--->"}.
+ Inductive definition of contractions, \<open>\<rightarrow>\<^sup>1\<close> and
+ (multi-step) reductions, \<open>\<rightarrow>\<close>.
\<close>
-inductive_set
- contract :: "(comb*comb) set"
- and contract_rel1 :: "[comb,comb] => bool" (infixl "-1->" 50)
- where
- "x -1-> y == (x,y) \<in> contract"
- | K: "K##x##y -1-> x"
- | S: "S##x##y##z -1-> (x##z)##(y##z)"
- | Ap1: "x-1->y ==> x##z -1-> y##z"
- | Ap2: "x-1->y ==> z##x -1-> z##y"
+inductive_set contract :: "(comb*comb) set"
+ and contract_rel1 :: "[comb,comb] \<Rightarrow> bool" (infixl "\<rightarrow>\<^sup>1" 50)
+where
+ "x \<rightarrow>\<^sup>1 y == (x,y) \<in> contract"
+| K: "K\<bullet>x\<bullet>y \<rightarrow>\<^sup>1 x"
+| S: "S\<bullet>x\<bullet>y\<bullet>z \<rightarrow>\<^sup>1 (x\<bullet>z)\<bullet>(y\<bullet>z)"
+| Ap1: "x\<rightarrow>\<^sup>1y \<Longrightarrow> x\<bullet>z \<rightarrow>\<^sup>1 y\<bullet>z"
+| Ap2: "x\<rightarrow>\<^sup>1y \<Longrightarrow> z\<bullet>x \<rightarrow>\<^sup>1 z\<bullet>y"
abbreviation
- contract_rel :: "[comb,comb] => bool" (infixl "--->" 50) where
- "x ---> y == (x,y) \<in> contract^*"
+ contract_rel :: "[comb,comb] \<Rightarrow> bool" (infixl "\<rightarrow>" 50) where
+ "x \<rightarrow> y == (x,y) \<in> contract\<^sup>*"
text \<open>
- Inductive definition of parallel contractions, @{text "=1=>"} and
- (multi-step) parallel reductions, @{text "===>"}.
+ Inductive definition of parallel contractions, \<open>\<Rrightarrow>\<^sup>1\<close> and
+ (multi-step) parallel reductions, \<open>\<Rrightarrow>\<close>.
\<close>
-inductive_set
- parcontract :: "(comb*comb) set"
- and parcontract_rel1 :: "[comb,comb] => bool" (infixl "=1=>" 50)
- where
- "x =1=> y == (x,y) \<in> parcontract"
- | refl: "x =1=> x"
- | K: "K##x##y =1=> x"
- | S: "S##x##y##z =1=> (x##z)##(y##z)"
- | Ap: "[| x=1=>y; z=1=>w |] ==> x##z =1=> y##w"
+inductive_set parcontract :: "(comb*comb) set"
+ and parcontract_rel1 :: "[comb,comb] \<Rightarrow> bool" (infixl "\<Rrightarrow>\<^sup>1" 50)
+where
+ "x \<Rrightarrow>\<^sup>1 y == (x,y) \<in> parcontract"
+| refl: "x \<Rrightarrow>\<^sup>1 x"
+| K: "K\<bullet>x\<bullet>y \<Rrightarrow>\<^sup>1 x"
+| S: "S\<bullet>x\<bullet>y\<bullet>z \<Rrightarrow>\<^sup>1 (x\<bullet>z)\<bullet>(y\<bullet>z)"
+| Ap: "[| x\<Rrightarrow>\<^sup>1y; z\<Rrightarrow>\<^sup>1w |] ==> x\<bullet>z \<Rrightarrow>\<^sup>1 y\<bullet>w"
abbreviation
- parcontract_rel :: "[comb,comb] => bool" (infixl "===>" 50) where
- "x ===> y == (x,y) \<in> parcontract^*"
+ parcontract_rel :: "[comb,comb] \<Rightarrow> bool" (infixl "\<Rrightarrow>" 50) where
+ "x \<Rrightarrow> y == (x,y) \<in> parcontract\<^sup>*"
text \<open>
Misc definitions.
@@ -72,11 +66,11 @@
definition
I :: comb where
- "I = S##K##K"
+ "I = S\<bullet>K\<bullet>K"
definition
- diamond :: "('a * 'a)set => bool" where
- --\<open>confluence; Lambda/Commutation treats this more abstractly\<close>
+ diamond :: "('a * 'a)set \<Rightarrow> bool" where
+ \<comment>\<open>confluence; Lambda/Commutation treats this more abstractly\<close>
"diamond(r) = (\<forall>x y. (x,y) \<in> r -->
(\<forall>y'. (x,y') \<in> r -->
(\<exists>z. (y,z) \<in> r & (y',z) \<in> r)))"
@@ -89,15 +83,15 @@
text\<open>Strip lemma.
The induction hypothesis covers all but the last diamond of the strip.\<close>
lemma diamond_strip_lemmaE [rule_format]:
- "[| diamond(r); (x,y) \<in> r^* |] ==>
- \<forall>y'. (x,y') \<in> r --> (\<exists>z. (y',z) \<in> r^* & (y,z) \<in> r)"
+ "[| diamond(r); (x,y) \<in> r\<^sup>* |] ==>
+ \<forall>y'. (x,y') \<in> r --> (\<exists>z. (y',z) \<in> r\<^sup>* & (y,z) \<in> r)"
apply (unfold diamond_def)
apply (erule rtrancl_induct)
apply (meson rtrancl_refl)
apply (meson rtrancl_trans r_into_rtrancl)
done
-lemma diamond_rtrancl: "diamond(r) ==> diamond(r^*)"
+lemma diamond_rtrancl: "diamond(r) \<Longrightarrow> diamond(r\<^sup>*)"
apply (simp (no_asm_simp) add: diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule rtrancl_induct, blast)
@@ -110,30 +104,30 @@
text \<open>Derive a case for each combinator constructor.\<close>
inductive_cases
- K_contractE [elim!]: "K -1-> r"
- and S_contractE [elim!]: "S -1-> r"
- and Ap_contractE [elim!]: "p##q -1-> r"
+ K_contractE [elim!]: "K \<rightarrow>\<^sup>1 r"
+ and S_contractE [elim!]: "S \<rightarrow>\<^sup>1 r"
+ and Ap_contractE [elim!]: "p\<bullet>q \<rightarrow>\<^sup>1 r"
declare contract.K [intro!] contract.S [intro!]
declare contract.Ap1 [intro] contract.Ap2 [intro]
-lemma I_contract_E [elim!]: "I -1-> z ==> P"
+lemma I_contract_E [elim!]: "I \<rightarrow>\<^sup>1 z \<Longrightarrow> P"
by (unfold I_def, blast)
-lemma K1_contractD [elim!]: "K##x -1-> z ==> (\<exists>x'. z = K##x' & x -1-> x')"
+lemma K1_contractD [elim!]: "K\<bullet>x \<rightarrow>\<^sup>1 z \<Longrightarrow> (\<exists>x'. z = K\<bullet>x' & x \<rightarrow>\<^sup>1 x')"
by blast
-lemma Ap_reduce1 [intro]: "x ---> y ==> x##z ---> y##z"
+lemma Ap_reduce1 [intro]: "x \<rightarrow> y \<Longrightarrow> x\<bullet>z \<rightarrow> y\<bullet>z"
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_trans)+
done
-lemma Ap_reduce2 [intro]: "x ---> y ==> z##x ---> z##y"
+lemma Ap_reduce2 [intro]: "x \<rightarrow> y \<Longrightarrow> z\<bullet>x \<rightarrow> z\<bullet>y"
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_trans)+
done
-text \<open>Counterexample to the diamond property for @{term "x -1-> y"}\<close>
+text \<open>Counterexample to the diamond property for @{term "x \<rightarrow>\<^sup>1 y"}\<close>
lemma not_diamond_contract: "~ diamond(contract)"
by (unfold diamond_def, metis S_contractE contract.K)
@@ -144,9 +138,9 @@
text \<open>Derive a case for each combinator constructor.\<close>
inductive_cases
- K_parcontractE [elim!]: "K =1=> r"
- and S_parcontractE [elim!]: "S =1=> r"
- and Ap_parcontractE [elim!]: "p##q =1=> r"
+ K_parcontractE [elim!]: "K \<Rrightarrow>\<^sup>1 r"
+ and S_parcontractE [elim!]: "S \<Rrightarrow>\<^sup>1 r"
+ and Ap_parcontractE [elim!]: "p\<bullet>q \<Rrightarrow>\<^sup>1 r"
declare parcontract.intros [intro]
@@ -154,14 +148,14 @@
subsection \<open>Basic properties of parallel contraction\<close>
-lemma K1_parcontractD [dest!]: "K##x =1=> z ==> (\<exists>x'. z = K##x' & x =1=> x')"
+lemma K1_parcontractD [dest!]: "K\<bullet>x \<Rrightarrow>\<^sup>1 z \<Longrightarrow> (\<exists>x'. z = K\<bullet>x' & x \<Rrightarrow>\<^sup>1 x')"
by blast
-lemma S1_parcontractD [dest!]: "S##x =1=> z ==> (\<exists>x'. z = S##x' & x =1=> x')"
+lemma S1_parcontractD [dest!]: "S\<bullet>x \<Rrightarrow>\<^sup>1 z \<Longrightarrow> (\<exists>x'. z = S\<bullet>x' & x \<Rrightarrow>\<^sup>1 x')"
by blast
lemma S2_parcontractD [dest!]:
- "S##x##y =1=> z ==> (\<exists>x' y'. z = S##x'##y' & x =1=> x' & y =1=> y')"
+ "S\<bullet>x\<bullet>y \<Rrightarrow>\<^sup>1 z \<Longrightarrow> (\<exists>x' y'. z = S\<bullet>x'\<bullet>y' & x \<Rrightarrow>\<^sup>1 x' & y \<Rrightarrow>\<^sup>1 y')"
by blast
text\<open>The rules above are not essential but make proofs much faster\<close>
@@ -174,10 +168,11 @@
done
text \<open>
- \medskip Equivalence of @{prop "p ---> q"} and @{prop "p ===> q"}.
+ \<^medskip>
+ Equivalence of @{prop "p \<rightarrow> q"} and @{prop "p \<Rrightarrow> q"}.
\<close>
-lemma contract_subset_parcontract: "contract <= parcontract"
+lemma contract_subset_parcontract: "contract \<subseteq> parcontract"
by (auto, erule contract.induct, blast+)
text\<open>Reductions: simply throw together reflexivity, transitivity and
@@ -186,16 +181,16 @@
declare r_into_rtrancl [intro] rtrancl_trans [intro]
(*Example only: not used*)
-lemma reduce_I: "I##x ---> x"
+lemma reduce_I: "I\<bullet>x \<rightarrow> x"
by (unfold I_def, blast)
-lemma parcontract_subset_reduce: "parcontract <= contract^*"
+lemma parcontract_subset_reduce: "parcontract \<subseteq> contract\<^sup>*"
by (auto, erule parcontract.induct, blast+)
-lemma reduce_eq_parreduce: "contract^* = parcontract^*"
+lemma reduce_eq_parreduce: "contract\<^sup>* = parcontract\<^sup>*"
by (metis contract_subset_parcontract parcontract_subset_reduce rtrancl_subset)
-theorem diamond_reduce: "diamond(contract^*)"
+theorem diamond_reduce: "diamond(contract\<^sup>*)"
by (simp add: reduce_eq_parreduce diamond_rtrancl diamond_parcontract)
end
--- a/src/HOL/Induct/document/root.tex Wed Dec 30 20:12:26 2015 +0100
+++ b/src/HOL/Induct/document/root.tex Wed Dec 30 20:24:43 2015 +0100
@@ -1,5 +1,5 @@
-
\documentclass[11pt,a4paper]{article}
+\usepackage{amssymb}
\usepackage{isabelle,isabellesym,pdfsetup}
%for best-style documents ...