Theory Comb

theory Comb
imports Main
(*  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 ∈ comb", "q ∈ comb") (infixl "@@" 90)

notation (xsymbols)
app (infixl "•" 90)

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


consts
contract :: i

abbreviation
contract_syntax :: "[i,i] => o" (infixl "-1->" 50)
where "p -1-> q == <p,q> ∈ contract"

abbreviation
contract_multi :: "[i,i] => o" (infixl "--->" 50)
where "p ---> q == <p,q> ∈ contract^*"

inductive
domains "contract" "comb × comb"
intros
K: "[| p ∈ comb; q ∈ comb |] ==> K•p•q -1-> p"
S: "[| p ∈ comb; q ∈ comb; r ∈ comb |] ==> S•p•q•r -1-> (p•r)•(q•r)"
Ap1: "[| p-1->q; r ∈ comb |] ==> p•r -1-> q•r"
Ap2: "[| p-1->q; r ∈ comb |] ==> r•p -1-> r•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> ∈ parcontract"

abbreviation
parcontract_multi :: "[i,i] => o" (infixl "===>" 50)
where "p ===> q == <p,q> ∈ parcontract^+"

inductive
domains "parcontract" "comb × comb"
intros
refl: "[| p ∈ comb |] ==> p =1=> p"
K: "[| p ∈ comb; q ∈ comb |] ==> K•p•q =1=> p"
S: "[| p ∈ comb; q ∈ comb; r ∈ comb |] ==> S•p•q•r =1=> (p•r)•(q•r)"
Ap: "[| p=1=>q; r=1=>s |] ==> p•r =1=> q•s"
type_intros comb.intros

text {*
Misc definitions.
*}


definition
I :: i where
"I == S•K•K"

definition
diamond :: "i => o" where
"diamond(r) ==
∀x y. <x,y>∈r --> (∀y'. <x,y'>∈r --> (∃z. <y,z>∈r & <y',z> ∈ r))"



subsection {* Transitive closure preserves the Church-Rosser property *}

lemma diamond_strip_lemmaD [rule_format]:
"[| diamond(r); <x,y>:r^+ |] ==>
∀y'. <x,y'>:r --> (∃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•q ∈ comb"


subsection {* Results about Contraction *}

text {*
For type checking: replaces @{term "a -1-> b"} by @{text "a, b ∈
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 ∈ comb ==> I•p ---> p"
-- {* Example only: not used *}
by (unfold I_def) (blast intro: reduction_rls)

lemma comb_I: "I ∈ 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•q -1-> r"

lemma I_contract_E: "I -1-> r ==> P"
by (auto simp add: I_def)

lemma K1_contractD: "K•p -1-> r ==> (∃q. r = K•q & p -1-> q)"
by auto

lemma Ap_reduce1: "[| p ---> q; r ∈ comb |] ==> p•r ---> q•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 ---> q; r ∈ comb |] ==> r•p ---> r•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•I•(I•I) -1-> I"
by (blast intro: comb_I)

lemma KIII_contract2: "K•I•(I•I) -1-> K•I•((K•I)•(K•I))"
by (unfold I_def) (blast intro: contract.intros)

lemma KIII_contract3: "K•I•((K•I)•(K•I)) -1-> I"
by (blast intro: comb_I)

lemma not_diamond_contract: "¬ 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
∈ 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•q =1=> r"

declare parcontract.intros [intro]


subsection {* Basic properties of parallel contraction *}

lemma K1_parcontractD [dest!]:
"K•p =1=> r ==> (∃p'. r = K•p' & p =1=> p')"
by auto

lemma S1_parcontractD [dest!]:
"S•p =1=> r ==> (∃p'. r = S•p' & p =1=> p')"
by auto

lemma S2_parcontractD [dest!]:
"S•p•q =1=> r ==> (∃p' q'. r = S•p'•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 ---> 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--->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--->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--->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 <-> p--->q"
by (blast intro: parreduce_imp_reduce reduce_imp_parreduce)

end