Theory Comb

theory Comb
imports Main
(*  Title:      HOL/Induct/Comb.thy
    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 "•" 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) ∈ 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) ∈ 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) ∈ 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) ∈ 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) = (∀x y. (x,y) ∈ r --> 
                  (∀y'. (x,y') ∈ r --> 
                    (∃z. (y,z) ∈ r & (y',z) ∈ 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) ∈ r^* |] ==>   
          ∀y'. (x,y') ∈ r --> (∃z. (y',z) ∈ r^* & (y,z) ∈ 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 ==> (∃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 ==> (∃x'. z = K##x' & x =1=> x')"
by blast

lemma S1_parcontractD [dest!]: "S##x =1=> z ==> (∃x'. z = S##x' & x =1=> x')"
by blast

lemma S2_parcontractD [dest!]:
     "S##x##y =1=> z ==> (∃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