(* Title: ZF/Induct/Comb.thy
Author: Lawrence C Paulson
Copyright 1994 University of Cambridge
*)
header {* Combinatory Logic example: the Church-Rosser Theorem *}
theory Comb imports Main begin
text {*
Curiously, combinators do not include free variables.
Example taken from \cite{camilleri-melham}.
*}
subsection {* Definitions *}
text {* Datatype definition of combinators @{text S} and @{text K}. *}
consts comb :: i
datatype comb =
K
| S
| app ("p \<in> comb", "q \<in> comb") (infixl "@@" 90)
notation (xsymbols)
app (infixl "\<bullet>" 90)
text {*
Inductive definition of contractions, @{text "-1->"} and
(multi-step) reductions, @{text "-\<longrightarrow>"}.
*}
consts
contract :: i
abbreviation
contract_syntax :: "[i,i] => o" (infixl "-1->" 50)
where "p -1-> q == <p,q> \<in> contract"
abbreviation
contract_multi :: "[i,i] => o" (infixl "-\<longrightarrow>" 50)
where "p -\<longrightarrow> q == <p,q> \<in> contract^*"
inductive
domains "contract" \<subseteq> "comb \<times> comb"
intros
K: "[| p \<in> comb; q \<in> comb |] ==> K\<bullet>p\<bullet>q -1-> p"
S: "[| p \<in> comb; q \<in> comb; r \<in> comb |] ==> S\<bullet>p\<bullet>q\<bullet>r -1-> (p\<bullet>r)\<bullet>(q\<bullet>r)"
Ap1: "[| p-1->q; r \<in> comb |] ==> p\<bullet>r -1-> q\<bullet>r"
Ap2: "[| p-1->q; r \<in> comb |] ==> r\<bullet>p -1-> r\<bullet>q"
type_intros comb.intros
text {*
Inductive definition of parallel contractions, @{text "=1=>"} and
(multi-step) parallel reductions, @{text "===>"}.
*}
consts
parcontract :: i
abbreviation
parcontract_syntax :: "[i,i] => o" (infixl "=1=>" 50)
where "p =1=> q == <p,q> \<in> parcontract"
abbreviation
parcontract_multi :: "[i,i] => o" (infixl "===>" 50)
where "p ===> q == <p,q> \<in> parcontract^+"
inductive
domains "parcontract" \<subseteq> "comb \<times> comb"
intros
refl: "[| p \<in> comb |] ==> p =1=> p"
K: "[| p \<in> comb; q \<in> comb |] ==> K\<bullet>p\<bullet>q =1=> p"
S: "[| p \<in> comb; q \<in> comb; r \<in> comb |] ==> S\<bullet>p\<bullet>q\<bullet>r =1=> (p\<bullet>r)\<bullet>(q\<bullet>r)"
Ap: "[| p=1=>q; r=1=>s |] ==> p\<bullet>r =1=> q\<bullet>s"
type_intros comb.intros
text {*
Misc definitions.
*}
definition
I :: i where
"I == S\<bullet>K\<bullet>K"
definition
diamond :: "i => o" where
"diamond(r) ==
\<forall>x y. <x,y>\<in>r \<longrightarrow> (\<forall>y'. <x,y'>\<in>r \<longrightarrow> (\<exists>z. <y,z>\<in>r & <y',z> \<in> r))"
subsection {* Transitive closure preserves the Church-Rosser property *}
lemma diamond_strip_lemmaD [rule_format]:
"[| diamond(r); <x,y>:r^+ |] ==>
\<forall>y'. <x,y'>:r \<longrightarrow> (\<exists>z. <y',z>: r^+ & <y,z>: r)"
apply (unfold diamond_def)
apply (erule trancl_induct)
apply (blast intro: r_into_trancl)
apply clarify
apply (drule spec [THEN mp], assumption)
apply (blast intro: r_into_trancl trans_trancl [THEN transD])
done
lemma diamond_trancl: "diamond(r) ==> diamond(r^+)"
apply (simp (no_asm_simp) add: diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule trancl_induct)
apply auto
apply (best intro: r_into_trancl trans_trancl [THEN transD]
dest: diamond_strip_lemmaD)+
done
inductive_cases Ap_E [elim!]: "p\<bullet>q \<in> comb"
declare comb.intros [intro!]
subsection {* Results about Contraction *}
text {*
For type checking: replaces @{term "a -1-> b"} by @{text "a, b \<in>
comb"}.
*}
lemmas contract_combE2 = contract.dom_subset [THEN subsetD, THEN SigmaE2]
and contract_combD1 = contract.dom_subset [THEN subsetD, THEN SigmaD1]
and contract_combD2 = contract.dom_subset [THEN subsetD, THEN SigmaD2]
lemma field_contract_eq: "field(contract) = comb"
by (blast intro: contract.K elim!: contract_combE2)
lemmas reduction_refl =
field_contract_eq [THEN equalityD2, THEN subsetD, THEN rtrancl_refl]
lemmas rtrancl_into_rtrancl2 =
r_into_rtrancl [THEN trans_rtrancl [THEN transD]]
declare reduction_refl [intro!] contract.K [intro!] contract.S [intro!]
lemmas reduction_rls =
contract.K [THEN rtrancl_into_rtrancl2]
contract.S [THEN rtrancl_into_rtrancl2]
contract.Ap1 [THEN rtrancl_into_rtrancl2]
contract.Ap2 [THEN rtrancl_into_rtrancl2]
lemma "p \<in> comb ==> I\<bullet>p -\<longrightarrow> p"
-- {* Example only: not used *}
by (unfold I_def) (blast intro: reduction_rls)
lemma comb_I: "I \<in> comb"
by (unfold I_def) blast
subsection {* Non-contraction results *}
text {* Derive a case for each combinator constructor. *}
inductive_cases
K_contractE [elim!]: "K -1-> r"
and S_contractE [elim!]: "S -1-> r"
and Ap_contractE [elim!]: "p\<bullet>q -1-> r"
lemma I_contract_E: "I -1-> r ==> P"
by (auto simp add: I_def)
lemma K1_contractD: "K\<bullet>p -1-> r ==> (\<exists>q. r = K\<bullet>q & p -1-> q)"
by auto
lemma Ap_reduce1: "[| p -\<longrightarrow> q; r \<in> comb |] ==> p\<bullet>r -\<longrightarrow> q\<bullet>r"
apply (frule rtrancl_type [THEN subsetD, THEN SigmaD1])
apply (drule field_contract_eq [THEN equalityD1, THEN subsetD])
apply (erule rtrancl_induct)
apply (blast intro: reduction_rls)
apply (erule trans_rtrancl [THEN transD])
apply (blast intro: contract_combD2 reduction_rls)
done
lemma Ap_reduce2: "[| p -\<longrightarrow> q; r \<in> comb |] ==> r\<bullet>p -\<longrightarrow> r\<bullet>q"
apply (frule rtrancl_type [THEN subsetD, THEN SigmaD1])
apply (drule field_contract_eq [THEN equalityD1, THEN subsetD])
apply (erule rtrancl_induct)
apply (blast intro: reduction_rls)
apply (blast intro: trans_rtrancl [THEN transD]
contract_combD2 reduction_rls)
done
text {* Counterexample to the diamond property for @{text "-1->"}. *}
lemma KIII_contract1: "K\<bullet>I\<bullet>(I\<bullet>I) -1-> I"
by (blast intro: comb.intros contract.K comb_I)
lemma KIII_contract2: "K\<bullet>I\<bullet>(I\<bullet>I) -1-> K\<bullet>I\<bullet>((K\<bullet>I)\<bullet>(K\<bullet>I))"
by (unfold I_def) (blast intro: comb.intros contract.intros)
lemma KIII_contract3: "K\<bullet>I\<bullet>((K\<bullet>I)\<bullet>(K\<bullet>I)) -1-> I"
by (blast intro: comb.intros contract.K comb_I)
lemma not_diamond_contract: "\<not> diamond(contract)"
apply (unfold diamond_def)
apply (blast intro: KIII_contract1 KIII_contract2 KIII_contract3
elim!: I_contract_E)
done
subsection {* Results about Parallel Contraction *}
text {* For type checking: replaces @{text "a =1=> b"} by @{text "a, b
\<in> comb"} *}
lemmas parcontract_combE2 = parcontract.dom_subset [THEN subsetD, THEN SigmaE2]
and parcontract_combD1 = parcontract.dom_subset [THEN subsetD, THEN SigmaD1]
and parcontract_combD2 = parcontract.dom_subset [THEN subsetD, THEN SigmaD2]
lemma field_parcontract_eq: "field(parcontract) = comb"
by (blast intro: parcontract.K elim!: parcontract_combE2)
text {* Derive a case for each combinator constructor. *}
inductive_cases
K_parcontractE [elim!]: "K =1=> r"
and S_parcontractE [elim!]: "S =1=> r"
and Ap_parcontractE [elim!]: "p\<bullet>q =1=> r"
declare parcontract.intros [intro]
subsection {* Basic properties of parallel contraction *}
lemma K1_parcontractD [dest!]:
"K\<bullet>p =1=> r ==> (\<exists>p'. r = K\<bullet>p' & p =1=> p')"
by auto
lemma S1_parcontractD [dest!]:
"S\<bullet>p =1=> r ==> (\<exists>p'. r = S\<bullet>p' & p =1=> p')"
by auto
lemma S2_parcontractD [dest!]:
"S\<bullet>p\<bullet>q =1=> r ==> (\<exists>p' q'. r = S\<bullet>p'\<bullet>q' & p =1=> p' & q =1=> q')"
by auto
lemma diamond_parcontract: "diamond(parcontract)"
-- {* Church-Rosser property for parallel contraction *}
apply (unfold diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule parcontract.induct)
apply (blast elim!: comb.free_elims intro: parcontract_combD2)+
done
text {*
\medskip Equivalence of @{prop "p -\<longrightarrow> q"} and @{prop "p ===> q"}.
*}
lemma contract_imp_parcontract: "p-1->q ==> p=1=>q"
by (induct set: contract) auto
lemma reduce_imp_parreduce: "p-\<longrightarrow>q ==> p===>q"
apply (frule rtrancl_type [THEN subsetD, THEN SigmaD1])
apply (drule field_contract_eq [THEN equalityD1, THEN subsetD])
apply (erule rtrancl_induct)
apply (blast intro: r_into_trancl)
apply (blast intro: contract_imp_parcontract r_into_trancl
trans_trancl [THEN transD])
done
lemma parcontract_imp_reduce: "p=1=>q ==> p-\<longrightarrow>q"
apply (induct set: parcontract)
apply (blast intro: reduction_rls)
apply (blast intro: reduction_rls)
apply (blast intro: reduction_rls)
apply (blast intro: trans_rtrancl [THEN transD]
Ap_reduce1 Ap_reduce2 parcontract_combD1 parcontract_combD2)
done
lemma parreduce_imp_reduce: "p===>q ==> p-\<longrightarrow>q"
apply (frule trancl_type [THEN subsetD, THEN SigmaD1])
apply (drule field_parcontract_eq [THEN equalityD1, THEN subsetD])
apply (erule trancl_induct, erule parcontract_imp_reduce)
apply (erule trans_rtrancl [THEN transD])
apply (erule parcontract_imp_reduce)
done
lemma parreduce_iff_reduce: "p===>q \<longleftrightarrow> p-\<longrightarrow>q"
by (blast intro: parreduce_imp_reduce reduce_imp_parreduce)
end