(* Title: HOL/Induct/Comb.thy
ID: $Id$
Author: Lawrence C Paulson
Copyright 1996 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}.
HOL system proofs may be found in the HOL distribution at
.../contrib/rule-induction/cl.ml
*}
subsection {* Definitions *}
text {* Datatype definition of combinators @{text S} and @{text K}. *}
datatype comb = K
| S
| Ap comb comb (infixl "##" 90)
notation (xsymbols)
Ap (infixl "\<bullet>" 90)
text {*
Inductive definition of contractions, @{text "-1->"} and
(multi-step) reductions, @{text "--->"}.
*}
consts
contract :: "(comb*comb) set"
abbreviation
contract_rel1 :: "[comb,comb] => bool" (infixl "-1->" 50) where
"x -1-> y == (x,y) \<in> contract"
abbreviation
contract_rel :: "[comb,comb] => bool" (infixl "--->" 50) where
"x ---> y == (x,y) \<in> contract^*"
inductive contract
intros
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"
text {*
Inductive definition of parallel contractions, @{text "=1=>"} and
(multi-step) parallel reductions, @{text "===>"}.
*}
consts
parcontract :: "(comb*comb) set"
abbreviation
parcontract_rel1 :: "[comb,comb] => bool" (infixl "=1=>" 50) where
"x =1=> y == (x,y) \<in> parcontract"
abbreviation
parcontract_rel :: "[comb,comb] => bool" (infixl "===>" 50) where
"x ===> y == (x,y) \<in> parcontract^*"
inductive parcontract
intros
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"
text {*
Misc definitions.
*}
definition
I :: comb where
"I = S##K##K"
definition
diamond :: "('a * 'a)set => bool" where
--{*confluence; Lambda/Commutation treats this more abstractly*}
"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 {*Reflexive/Transitive closure preserves Church-Rosser property*}
text{*So does the Transitive closure, with a similar proof*}
text{*Strip lemma.
The induction hypothesis covers all but the last diamond of the strip.*}
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)"
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^*)"
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 {* 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##q -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"
by (unfold I_def, blast)
lemma K1_contractD [elim!]: "K##x -1-> z ==> (\<exists>x'. z = K##x' & x -1-> x')"
by blast
lemma Ap_reduce1 [intro]: "x ---> y ==> x##z ---> y##z"
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_trans)+
done
lemma Ap_reduce2 [intro]: "x ---> y ==> z##x ---> z##y"
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_trans)+
done
(** Counterexample to the diamond property for -1-> **)
lemma KIII_contract1: "K##I##(I##I) -1-> I"
by (rule contract.K)
lemma KIII_contract2: "K##I##(I##I) -1-> K##I##((K##I)##(K##I))"
by (unfold I_def, blast)
lemma KIII_contract3: "K##I##((K##I)##(K##I)) -1-> I"
by blast
lemma not_diamond_contract: "~ diamond(contract)"
apply (unfold diamond_def)
apply (best intro: KIII_contract1 KIII_contract2 KIII_contract3)
done
subsection {* Results about Parallel Contraction *}
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##q =1=> r"
declare parcontract.intros [intro]
(*** Basic properties of parallel contraction ***)
subsection {* Basic properties of parallel contraction *}
lemma K1_parcontractD [dest!]: "K##x =1=> z ==> (\<exists>x'. z = K##x' & x =1=> x')"
by blast
lemma S1_parcontractD [dest!]: "S##x =1=> z ==> (\<exists>x'. z = S##x' & x =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')"
by blast
text{*The rules above are not essential but make proofs much faster*}
text{*Church-Rosser property for parallel contraction*}
lemma diamond_parcontract: "diamond parcontract"
apply (unfold diamond_def)
apply (rule impI [THEN allI, THEN allI])
apply (erule parcontract.induct, fast+)
done
text {*
\medskip Equivalence of @{prop "p ---> q"} and @{prop "p ===> q"}.
*}
lemma contract_subset_parcontract: "contract <= parcontract"
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule contract.induct, blast+)
done
text{*Reductions: simply throw together reflexivity, transitivity and
the one-step reductions*}
declare r_into_rtrancl [intro] rtrancl_trans [intro]
(*Example only: not used*)
lemma reduce_I: "I##x ---> x"
by (unfold I_def, blast)
lemma parcontract_subset_reduce: "parcontract <= contract^*"
apply (rule subsetI)
apply (simp only: split_tupled_all)
apply (erule parcontract.induct, blast+)
done
lemma reduce_eq_parreduce: "contract^* = parcontract^*"
by (rule equalityI contract_subset_parcontract [THEN rtrancl_mono]
parcontract_subset_reduce [THEN rtrancl_subset_rtrancl])+
lemma diamond_reduce: "diamond(contract^*)"
by (simp add: reduce_eq_parreduce diamond_rtrancl diamond_parcontract)
end