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