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Step * 1 10 of Lemma FOL-proveable-evidence


1. hyps : mFOL() List
2. concl : mFOL()
3. subgoals : mFOL-sequent() List
4. subproofs : ℕ||subgoals|| ⟶ proof-tree(mFOL-sequent();FOLRule();λsr.FOLeffect(sr))
5. ∀b:ℕ||subgoals||. ∀s:mFOL-sequent().
     (correct_proof(mFOL-sequent();λsr.FOLeffect(sr);s;subproofs b) ⇒ FOL-sequent-evidence{i:l}(s))
6. ∀i:ℕ||subgoals||. correct_proof(mFOL-sequent();λsr.FOLeffect(sr);subgoals[i];subproofs i)
7. hypnum : ℕ
8. hypnum < ||hyps||
9. ↑mFOconnect?(hyps[hypnum])
10. mFOconnect-knd(hyps[hypnum]) = "imp" ∈ Atom
11. [<hyps, mFOconnect-left(hyps[hypnum])>; <[mFOconnect-right(hyps[hypnum]) / hyps], concl>] = subgoals ∈ (mFOL-sequent\000C() List)
12. FOL-sequent-evidence{i:l}(<hyps, mFOconnect-left(hyps[hypnum])>)
13. FOL-sequent-evidence{i:l}(<[mFOconnect-right(hyps[hypnum]) / hyps], concl>)
⊢ FOL-sequent-evidence{i:l}(<hyps, concl>)
BY
{ (ThinVar `subproofs' THEN Thin (-3)) }

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

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Latex:

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Latex:

1.  hyps  :  mFOL()  List
2.  concl  :  mFOL()
3.  subgoals  :  mFOL-sequent()  List
4.  subproofs  :  \mBbbN{}||subgoals||  {}\mrightarrow{}  proof-tree(mFOL-sequent();FOLRule();\mlambda{}sr.FOLeffect(sr))
5.  \mforall{}b:\mBbbN{}||subgoals||.  \mforall{}s:mFOL-sequent().
          (correct\_proof(mFOL-sequent();\mlambda{}sr.FOLeffect(sr);s;subproofs  b)  {}\mRightarrow{}  FOL-sequent-evidence\{i:l\}(s))
6.  \mforall{}i:\mBbbN{}||subgoals||.  correct\_proof(mFOL-sequent();\mlambda{}sr.FOLeffect(sr);subgoals[i];subproofs  i)
7.  hypnum  :  \mBbbN{}
8.  hypnum  <  ||hyps||
9.  \muparrow{}mFOconnect?(hyps[hypnum])
10.  mFOconnect-knd(hyps[hypnum])  =  "imp"
11.  [<hyps,  mFOconnect-left(hyps[hypnum])>  <[mFOconnect-right(hyps[hypnum])  /  hyps],  concl>]  =  subg\000Coals
12.  FOL-sequent-evidence\{i:l\}(<hyps,  mFOconnect-left(hyps[hypnum])>)
13.  FOL-sequent-evidence\{i:l\}(<[mFOconnect-right(hyps[hypnum])  /  hyps],  concl>)
\mvdash{}  FOL-sequent-evidence\{i:l\}(<hyps,  concl>)


By

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

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