(* Title: HOL/Induct/Comb.thy
Author: Lawrence C Paulson
Copyright 1996 University of Cambridge
*)
section \<open>Combinatory Logic example: the Church-Rosser Theorem\<close>
theory Comb
imports Main
begin
text \<open>
Curiously, combinators do not include free variables.
Example taken from @{cite camilleri92}.
\<close>
subsection \<open>Definitions\<close>
text \<open>Datatype definition of combinators \<open>S\<close> and \<open>K\<close>.\<close>
datatype comb = K
| S
| Ap comb comb (infixl "\<bullet>" 90)
text \<open>
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] \<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] \<Rightarrow> bool" (infixl "\<rightarrow>" 50) where
"x \<rightarrow> y == (x,y) \<in> contract\<^sup>*"
text \<open>
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] \<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] \<Rightarrow> bool" (infixl "\<Rrightarrow>" 50) where
"x \<Rrightarrow> y == (x,y) \<in> parcontract\<^sup>*"
text \<open>
Misc definitions.
\<close>
definition
I :: comb where
"I = S\<bullet>K\<bullet>K"
definition
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)))"
subsection \<open>Reflexive/Transitive closure preserves Church-Rosser property\<close>
text\<open>So does the Transitive closure, with a similar proof\<close>
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\<^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) \<Longrightarrow> diamond(r\<^sup>*)"
apply (simp (no_asm_simp) add: diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule rtrancl_induct, blast)
apply (meson rtrancl_trans r_into_rtrancl diamond_strip_lemmaE)
done
subsection \<open>Non-contraction results\<close>
text \<open>Derive a case for each combinator constructor.\<close>
inductive_cases
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 \<rightarrow>\<^sup>1 z \<Longrightarrow> P"
by (unfold I_def, blast)
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 \<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 \<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 \<rightarrow>\<^sup>1 y"}\<close>
lemma not_diamond_contract: "~ diamond(contract)"
by (unfold diamond_def, metis S_contractE contract.K)
subsection \<open>Results about Parallel Contraction\<close>
text \<open>Derive a case for each combinator constructor.\<close>
inductive_cases
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]
(*** Basic properties of parallel contraction ***)
subsection \<open>Basic properties of parallel contraction\<close>
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\<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\<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>
text\<open>Church-Rosser property for parallel contraction\<close>
lemma diamond_parcontract: "diamond parcontract"
apply (unfold diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule parcontract.induct, fast+)
done
text \<open>
\<^medskip>
Equivalence of @{prop "p \<rightarrow> q"} and @{prop "p \<Rrightarrow> q"}.
\<close>
lemma contract_subset_parcontract: "contract \<subseteq> parcontract"
by (auto, erule contract.induct, blast+)
text\<open>Reductions: simply throw together reflexivity, transitivity and
the one-step reductions\<close>
declare r_into_rtrancl [intro] rtrancl_trans [intro]
(*Example only: not used*)
lemma reduce_I: "I\<bullet>x \<rightarrow> x"
by (unfold I_def, blast)
lemma parcontract_subset_reduce: "parcontract \<subseteq> contract\<^sup>*"
by (auto, erule parcontract.induct, blast+)
lemma reduce_eq_parreduce: "contract\<^sup>* = parcontract\<^sup>*"
by (metis contract_subset_parcontract parcontract_subset_reduce rtrancl_subset)
theorem diamond_reduce: "diamond(contract\<^sup>*)"
by (simp add: reduce_eq_parreduce diamond_rtrancl diamond_parcontract)
end