src/HOL/Induct/Comb.thy
 author paulson Sat, 06 Mar 2010 17:19:29 +0000 changeset 35621 1c084dda4c3c parent 23746 a455e69c31cc child 58249 180f1b3508ed permissions -rw-r--r--
simplified
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(*  Title:      HOL/Induct/Comb.thy
Author:     Lawrence C Paulson
*)

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 "--->"}.
*}

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"

abbreviation
contract_rel :: "[comb,comb] => bool"   (infixl "--->" 50) where
"x ---> y == (x,y) \<in> contract^*"

text {*
Inductive definition of parallel contractions, @{text "=1=>"} and
(multi-step) parallel reductions, @{text "===>"}.
*}

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"

abbreviation
parcontract_rel :: "[comb,comb] => bool"   (infixl "===>" 50) where
"x ===> y == (x,y) \<in> parcontract^*"

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 (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

text {*Counterexample to the diamond property for @{term "x -1-> y"}*}

lemma not_diamond_contract: "~ diamond(contract)"
by (unfold diamond_def, metis S_contractE contract.K)

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"
by (auto, erule contract.induct, blast+)

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^*"
by (auto, erule parcontract.induct, blast+)

lemma reduce_eq_parreduce: "contract^* = parcontract^*"
by (metis contract_subset_parcontract parcontract_subset_reduce rtrancl_subset)

theorem diamond_reduce: "diamond(contract^*)"
by (simp add: reduce_eq_parreduce diamond_rtrancl diamond_parcontract)

end
```