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Step * 1 9 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. hypnum : ℕ@i
8. hypnum < ||hyps||
9. ↑mFOconnect?(hyps[hypnum])
10. mFOconnect-knd(hyps[hypnum]) = "or" ∈ Atom
11. [<[mFOconnect-left(hyps[hypnum]) / hyps], concl>; <[mFOconnect-right(hyps[hypnum]) / hyps], concl>] = subgoals ∈ (mF\000COL-sequent() List)
12. mFOL-sequent-evidence(<[mFOconnect-left(hyps[hypnum]) / hyps], concl>)
13. mFOL-sequent-evidence(<[mFOconnect-right(hyps[hypnum]) / hyps], concl>)
⊢ mFOL-sequent-evidence(<hyps, concl>)
BY
{ (ThinVar `subproofs' THEN Thin (-3)) }

1
1. hyps : mFOL() List@i
2. concl : mFOL()@i
3. subgoals : mFOL-sequent() List@i
4. hypnum : ℕ@i
5. hypnum < ||hyps||
6. ↑mFOconnect?(hyps[hypnum])
7. mFOconnect-knd(hyps[hypnum]) = "or" ∈ Atom
8. mFOL-sequent-evidence(<[mFOconnect-left(hyps[hypnum]) / hyps], concl>)
9. mFOL-sequent-evidence(<[mFOconnect-right(hyps[hypnum]) / hyps], concl>)
⊢ mFOL-sequent-evidence(<hyps, concl>)

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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.  hypnum  :  \mBbbN{}@i
8.  hypnum  <  ||hyps||
9.  \muparrow{}mFOconnect?(hyps[hypnum])
10.  mFOconnect-knd(hyps[hypnum])  =  "or"
11.  [<[mFOconnect-left(hyps[hypnum])  /  hyps],  concl>
          <[mFOconnect-right(hyps[hypnum])  /  hyps],  concl>]
=  subgoals
12.  mFOL-sequent-evidence(<[mFOconnect-left(hyps[hypnum])  /  hyps],  concl>)
13.  mFOL-sequent-evidence(<[mFOconnect-right(hyps[hypnum])  /  hyps],  concl>)
\mvdash{}  mFOL-sequent-evidence(<hyps,  concl>)


By

(ThinVar  `subproofs'  THEN  Thin  (-3))

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