Theory Comb

(*  Title:      HOL/Induct/Comb.thy
    Author:     Lawrence C Paulson
    Copyright   1996  University of Cambridge

section ‹Combinatory Logic example: the Church-Rosser Theorem›

theory Comb
imports Main

text ‹
  Combinator terms do not have free variables.
  Example taken from citecamilleri92.

subsection ‹Definitions›

text ‹Datatype definition of combinators S› and K›.›

datatype comb = K
              | S
              | Ap comb comb (infixl "" 90)

text ‹
  Inductive definition of contractions, 1 and
  (multi-step) reductions, →›.

inductive contract1 :: "[comb,comb]  bool"  (infixl "1" 50)
    K:     "Kxy 1 x"
  | S:     "Sxyz 1 (xz)(yz)"
  | Ap1:   "x 1 y  xz 1 yz"
  | Ap2:   "x 1 y  zx 1 zy"

  contract :: "[comb,comb]  bool"   (infixl "" 50) where
  "contract  contract1**"

text ‹
  Inductive definition of parallel contractions, 1 and
  (multi-step) parallel reductions, ⇛›.

inductive parcontract1 :: "[comb,comb]  bool"  (infixl "1" 50)
    refl:  "x 1 x"
  | K:     "Kxy 1 x"
  | S:     "Sxyz 1 (xz)(yz)"
  | Ap:    "x 1 y; z 1 w  xz 1 yw"

  parcontract :: "[comb,comb]  bool"   (infixl "" 50) where
  "parcontract  parcontract1**"

text ‹
  Misc definitions.

  I :: comb where
  "I  SKK"

  diamond   :: "([comb,comb]  bool)  bool" where
    ― ‹confluence; Lambda/Commutation treats this more abstractly›
  "diamond r  x y. r x y 
                  (y'. r x y'  
                    (z. r y z  r y' z))"

subsection ‹Reflexive/Transitive closure preserves Church-Rosser property›

text‹Remark: 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 strip_lemma [rule_format]: 
  assumes "diamond r" and r: "r** x y" "r x y'"
  shows "z. r** y' z  r y z"
  using r
proof (induction rule: rtranclp_induct)
  case base
  then show ?case
    by blast
  case (step y z)
  then show ?case
    using diamond r unfolding diamond_def
    by (metis rtranclp.rtrancl_into_rtrancl)

proposition diamond_rtrancl:
  assumes "diamond r" 
  shows "diamond(r**)"
  unfolding diamond_def
proof (intro strip)
  fix x y y'
  assume "r** x y" "r** x y'"
  then show "z. r** y z  r** y' z"
  proof (induction rule: rtranclp_induct)
    case base
    then show ?case
      by blast
    case (step y z)
    then show ?case
      by (meson assms strip_lemma rtranclp.rtrancl_into_rtrancl)

subsection ‹Non-contraction results›

text ‹Derive a case for each combinator constructor.›

  K_contractE [elim!]: "K 1 r"
  and S_contractE [elim!]: "S 1 r"
  and Ap_contractE [elim!]: "pq 1 r"

declare contract1.K [intro!] contract1.S [intro!]
declare contract1.Ap1 [intro] contract1.Ap2 [intro]

lemma I_contract_E [iff]: "¬ I 1 z"
  unfolding I_def by blast

lemma K1_contractD [elim!]: "Kx 1 z  (x'. z = Kx'  x 1 x')"
  by blast

lemma Ap_reduce1 [intro]: "x  y  xz  yz"
  by (induction rule: rtranclp_induct; blast intro: rtranclp_trans)

lemma Ap_reduce2 [intro]: "x  y  zx  zy"
  by (induction rule: rtranclp_induct; blast intro: rtranclp_trans)

text ‹Counterexample to the diamond property for termx 1 y

lemma not_diamond_contract: "¬ diamond(contract1)"
  unfolding diamond_def by (metis S_contractE contract1.K) 

subsection ‹Results about Parallel Contraction›

text ‹Derive a case for each combinator constructor.›

      K_parcontractE [elim!]: "K 1 r"
  and S_parcontractE [elim!]: "S 1 r"
  and Ap_parcontractE [elim!]: "pq 1 r"

declare parcontract1.intros [intro]

subsection ‹Basic properties of parallel contraction›
text‹The rules below are not essential but make proofs much faster›

lemma K1_parcontractD [dest!]: "Kx 1 z  (x'. z = Kx'  x 1 x')"
  by blast

lemma S1_parcontractD [dest!]: "Sx 1 z  (x'. z = Sx'  x 1 x')"
  by blast

lemma S2_parcontractD [dest!]: "Sxy 1 z  (x' y'. z = Sx'y'  x 1 x'  y 1 y')"
  by blast

text‹Church-Rosser property for parallel contraction›
proposition diamond_parcontract: "diamond parcontract1"
proof -
  have "(z. w 1 z  y' 1 z)" if "y 1 w" "y 1 y'" for w y y'
    using that by (induction arbitrary: y' rule: parcontract1.induct) fast+
  then show ?thesis
    by (auto simp: diamond_def)

subsection ‹Equivalence of propp  q and propp  q.›

lemma contract_imp_parcontract: "x 1 y  x 1 y"
  by (induction rule: contract1.induct; blast)

text‹Reductions: simply throw together reflexivity, transitivity and
  the one-step reductions›

proposition reduce_I: "Ix  x"
  unfolding I_def
  by (meson contract1.K contract1.S r_into_rtranclp rtranclp.rtrancl_into_rtrancl)

lemma parcontract_imp_reduce: "x 1 y  x  y"
proof (induction rule: parcontract1.induct)
  case (Ap x y z w)
  then show ?case
    by (meson Ap_reduce1 Ap_reduce2 rtranclp_trans)
qed auto

lemma reduce_eq_parreduce: "x  y    x  y"
  by (metis contract_imp_parcontract parcontract_imp_reduce predicate2I rtranclp_subset)

theorem diamond_reduce: "diamond(contract)"
  using diamond_parcontract diamond_rtrancl reduce_eq_parreduce by presburger