Theory InductiveInvariant

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theory InductiveInvariant
imports Main
begin

(*  ID:         $Id: InductiveInvariant.thy,v 1.6 2006/11/17 01:20:34 wenzelm Exp $
    Author:     Sava Krsti\'{c} and John Matthews
*)

header {* Some of the results in Inductive Invariants for Nested Recursion *}

theory InductiveInvariant imports Main begin

text {* A formalization of some of the results in \emph{Inductive
  Invariants for Nested Recursion}, by Sava Krsti\'{c} and John
  Matthews.  Appears in the proceedings of TPHOLs 2003, LNCS
  vol. 2758, pp. 253-269. *}


text "S is an inductive invariant of the functional F with respect to the wellfounded relation r."

definition
  indinv :: "('a * 'a) set => ('a => 'b => bool) => (('a => 'b) => ('a => 'b)) => bool" where
  "indinv r S F = (∀f x. (∀y. (y,x) : r --> S y (f y)) --> S x (F f x))"


text "S is an inductive invariant of the functional F on set D with respect to the wellfounded relation r."

definition
  indinv_on :: "('a * 'a) set => 'a set => ('a => 'b => bool) => (('a => 'b) => ('a => 'b)) => bool" where
  "indinv_on r D S F = (∀f. ∀x∈D. (∀y∈D. (y,x) ∈ r --> S y (f y)) --> S x (F f x))"


text "The key theorem, corresponding to theorem 1 of the paper. All other results
      in this theory are proved using instances of this theorem, and theorems
      derived from this theorem."

theorem indinv_wfrec:
  assumes wf:  "wf r" and
          inv: "indinv r S F"
  shows        "S x (wfrec r F x)"
  using wf
proof (induct x)
  fix x
  assume  IHYP: "!!y. (y,x) ∈ r ==> S y (wfrec r F y)"
  then have     "!!y. (y,x) ∈ r ==> S y (cut (wfrec r F) r x y)" by (simp add: tfl_cut_apply)
  with inv have "S x (F (cut (wfrec r F) r x) x)" by (unfold indinv_def, blast)
  thus "S x (wfrec r F x)" using wf by (simp add: wfrec)
qed

theorem indinv_on_wfrec:
  assumes WF:  "wf r" and
          INV: "indinv_on r D S F" and
          D:   "x∈D"
  shows        "S x (wfrec r F x)"
apply (insert INV D indinv_wfrec [OF WF, of "% x y. x∈D --> S x y"])
by (simp add: indinv_on_def indinv_def)

theorem ind_fixpoint_on_lemma:
  assumes WF:  "wf r" and
         INV: "∀f. ∀x∈D. (∀y∈D. (y,x) ∈ r --> S y (wfrec r F y) & f y = wfrec r F y)
                               --> S x (wfrec r F x) & F f x = wfrec r F x" and
           D: "x∈D"
  shows "F (wfrec r F) x = wfrec r F x & S x (wfrec r F x)"
proof (rule indinv_on_wfrec [OF WF _ D, of "% a b. F (wfrec r F) a = b & wfrec r F a = b & S a b" F, simplified])
  show "indinv_on r D (%a b. F (wfrec r F) a = b & wfrec r F a = b & S a b) F"
  proof (unfold indinv_on_def, clarify)
    fix f x
    assume A1: "∀y∈D. (y, x) ∈ r --> F (wfrec r F) y = f y & wfrec r F y = f y & S y (f y)"
    assume D': "x∈D"
    from A1 INV [THEN spec, of f, THEN bspec, OF D']
      have "S x (wfrec r F x)" and
           "F f x = wfrec r F x" by auto
    moreover
    from A1 have "∀y∈D. (y, x) ∈ r --> S y (wfrec r F y)" by auto
    with D' INV [THEN spec, of "wfrec r F", simplified]
      have "F (wfrec r F) x = wfrec r F x" by blast
    ultimately show "F (wfrec r F) x = F f x & wfrec r F x = F f x & S x (F f x)" by auto
  qed
qed

theorem ind_fixpoint_lemma:
  assumes WF:  "wf r" and
         INV: "∀f x. (∀y. (y,x) ∈ r --> S y (wfrec r F y) & f y = wfrec r F y)
                         --> S x (wfrec r F x) & F f x = wfrec r F x"
  shows "F (wfrec r F) x = wfrec r F x & S x (wfrec r F x)"
apply (rule ind_fixpoint_on_lemma [OF WF _ UNIV_I, simplified])
by (rule INV)

theorem tfl_indinv_wfrec:
"[| f == wfrec r F; wf r; indinv r S F |]
 ==> S x (f x)"
by (simp add: indinv_wfrec)

theorem tfl_indinv_on_wfrec:
"[| f == wfrec r F; wf r; indinv_on r D S F; x∈D |]
 ==> S x (f x)"
by (simp add: indinv_on_wfrec)

end

theorem indinv_wfrec:

  [| wf r; indinv r S F |] ==> S x (wfrec r F x)

theorem indinv_on_wfrec:

  [| wf r; indinv_on r D S F; xD |] ==> S x (wfrec r F x)

theorem ind_fixpoint_on_lemma:

  [| wf r;
     ∀f. ∀xD. (∀yD. (y, x) ∈ r --> S y (wfrec r F y) ∧ f y = wfrec r F y) -->
               S x (wfrec r F x) ∧ F f x = wfrec r F x;
     xD |]
  ==> F (wfrec r F) x = wfrec r F xS x (wfrec r F x)

theorem ind_fixpoint_lemma:

  [| wf r;
     ∀f x. (∀y. (y, x) ∈ r --> S y (wfrec r F y) ∧ f y = wfrec r F y) -->
           S x (wfrec r F x) ∧ F f x = wfrec r F x |]
  ==> F (wfrec r F) x = wfrec r F xS x (wfrec r F x)

theorem tfl_indinv_wfrec:

  [| f == wfrec r F; wf r; indinv r S F |] ==> S x (f x)

theorem tfl_indinv_on_wfrec:

  [| f == wfrec r F; wf r; indinv_on r D S F; xD |] ==> S x (f x)