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Step * of Lemma correct_proof_wf

∀[Sequent,Rule:Type]. ∀[effect:(Sequent × Rule) ⟶ (Sequent List?)]. ∀[s:Sequent].
∀[pf:proof-tree(Sequent;Rule;effect)].
  (correct_proof(Sequent;effect;s;pf) ∈ ℙ)
BY
{ (Auto
   THEN MoveToConcl (-2)
   THEN Unfold `proof-tree` -1
   THEN WElim (-1)
   THEN (D 0 THENA Auto)
   THEN RecUnfold `correct_proof` 0
   THEN RepUR  ``Wsup`` 0
   THEN MemCD
   THEN ((GenConclAtAddr [2;1] THEN D -2 THEN Reduce 0 THEN MemCD THEN Try (InstHyp [⌜i⌝;⌜x[i]⌝] (-5)⋅) THEN Auto)
   ORELSE Auto
   )) }

1
.....wf..... 
1. Sequent : Type
2. Rule : Type
3. effect : (Sequent × Rule) ⟶ (Sequent List?)
4. a : Sequent × Rule
5. f : case effect a of inl(subgoals) => ℕ||subgoals|| | inr(x) => Void ⟶ W(Sequent × Rule;a.case effect a
                                                                            of inl(subgoals) =>
                                                                            ℕ||subgoals||
                                                                            | inr(x) =>
                                                                            Void)
6. ∀b:case effect a of inl(subgoals) => ℕ||subgoals|| | inr(x) => Void. ∀s:Sequent.
     (correct_proof(Sequent;effect;s;f b) ∈ ℙ)
7. s : Sequent@i
8. x : Sequent List@i
9. (effect a) = (inl x) ∈ (Sequent List?)
10. i : ℕ||x||@i
⊢ i ∈ case effect a of inl(subgoals) => ℕ||subgoals|| | inr(x) => Void

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Latex:

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Latex:
\mforall{}[Sequent,Rule:Type].  \mforall{}[effect:(Sequent  \mtimes{}  Rule)  {}\mrightarrow{}  (Sequent  List?)].  \mforall{}[s:Sequent].
\mforall{}[pf:proof-tree(Sequent;Rule;effect)].
    (correct\_proof(Sequent;effect;s;pf)  \mmember{}  \mBbbP{})


By

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Latex:
(Auto
  THEN  MoveToConcl  (-2)
  THEN  Unfold  `proof-tree`  -1
  THEN  WElim  (-1)
  THEN  (D  0  THENA  Auto)
  THEN  RecUnfold  `correct\_proof`  0
  THEN  RepUR    ``Wsup``  0
  THEN  MemCD
  THEN  ((GenConclAtAddr  [2;1]
                THEN  D  -2
                THEN  Reduce  0
                THEN  MemCD
                THEN  Try  (InstHyp  [\mkleeneopen{}i\mkleeneclose{};\mkleeneopen{}x[i]\mkleeneclose{}]  (-5)\mcdot{})
                THEN  Auto)
  ORELSE  Auto
  ))

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