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Step * 1 7 of Lemma mFOL-proveable-evidence


1. hyps : mFOL() List@i
2. concl : mFOL()@i
3. subgoals : mFOL-sequent() List@i
4. subproofs : ℕ||subgoals|| ─→ proof-tree(mFOL-sequent();mFOLRule();λsr.mFOLeffect(sr))@i'
5. ∀b:ℕ||subgoals||. ∀s:mFOL-sequent().
     (correct_proof(mFOL-sequent();λsr.mFOLeffect(sr);s;subproofs b) ⇒ mFOL-sequent-evidence(s))@i'
6. ∀i:ℕ||subgoals||. correct_proof(mFOL-sequent();λsr.mFOLeffect(sr);subgoals[i];subproofs i)@i'
7. (concl ∈ hyps)
8. [] = subgoals ∈ (mFOL-sequent() List)
⊢ mFOL-sequent-evidence(<hyps, concl>)
BY
{ (ThinVar `subgoals'
   THEN ((RepeatFor 2 (D (-1)) THEN (HypSubst' (-1) 0 THENA Auto))
         THEN BLemma `mFOL-sequent-evidence-trivial`
         THEN Auto)⋅
   ) }

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Latex:



1.  hyps  :  mFOL()  List@i
2.  concl  :  mFOL()@i
3.  subgoals  :  mFOL-sequent()  List@i
4.  subproofs  :  \mBbbN{}||subgoals||  {}\mrightarrow{}  proof-tree(mFOL-sequent();mFOLRule();\mlambda{}sr.mFOLeffect(sr))@i'
5.  \mforall{}b:\mBbbN{}||subgoals||.  \mforall{}s:mFOL-sequent().
          (correct\_proof(mFOL-sequent();\mlambda{}sr.mFOLeffect(sr);s;subproofs  b)  {}\mRightarrow{}  mFOL-sequent-evidence(s))@i'
6.  \mforall{}i:\mBbbN{}||subgoals||.  correct\_proof(mFOL-sequent();\mlambda{}sr.mFOLeffect(sr);subgoals[i];subproofs  i)@i'
7.  (concl  \mmember{}  hyps)
8.  []  =  subgoals
\mvdash{}  mFOL-sequent-evidence(<hyps,  concl>)


By

(ThinVar  `subgoals'
  THEN  ((RepeatFor  2  (D  (-1))  THEN  (HypSubst'  (-1)  0  THENA  Auto))
              THEN  BLemma  `mFOL-sequent-evidence-trivial`
              THEN  Auto)\mcdot{}
  )

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