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Step * 1 1 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. ↑mFOconnect?(concl)
8. mFOconnect-knd(concl) = "and" ∈ Atom
9. [<hyps, mFOconnect-left(concl)>; <hyps, mFOconnect-right(concl)>] = subgoals ∈ (mFOL-sequent() List)
10. mFOL-sequent-evidence(<hyps, mFOconnect-left(concl)>)
11. mFOL-sequent-evidence(<hyps, mFOconnect-right(concl)>)
⊢ mFOL-sequent-evidence(<hyps, concl>)
BY
{ ((Subst ⌈concl = mFOconnect-left(concl) ∧ mFOconnect-right(concl) ∈ mFOL()⌉ 0⋅
    THENA (Auto THEN MoveToConcl 2 THEN BLemmaUp `mFOL-induction` ⋅ THEN Reduce 0 THEN Auto)
    )
   THEN BLemma `mFOL-sequent-evidence_and`
   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.  \muparrow{}mFOconnect?(concl)
8.  mFOconnect-knd(concl)  =  "and"
9.  [<hyps,  mFOconnect-left(concl)>  <hyps,  mFOconnect-right(concl)>]  =  subgoals
10.  mFOL-sequent-evidence(<hyps,  mFOconnect-left(concl)>)
11.  mFOL-sequent-evidence(<hyps,  mFOconnect-right(concl)>)
\mvdash{}  mFOL-sequent-evidence(<hyps,  concl>)


By

((Subst  \mkleeneopen{}concl  =  mFOconnect-left(concl)  \mwedge{}  mFOconnect-right(concl)\mkleeneclose{}  0\mcdot{}
    THENA  (Auto  THEN  MoveToConcl  2  THEN  BLemmaUp  `mFOL-induction`  \mcdot{}  THEN  Reduce  0  THEN  Auto)
    )
  THEN  BLemma  `mFOL-sequent-evidence\_and`
  THEN  Auto)

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