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Step * 1 3 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. var : ℤ
8. ¬(var ∈ mFOL-sequent-freevars(<hyps, concl>))
9. ↑mFOquant?(concl)
10. mFOquant-isall(concl) = tt
11. [<hyps, mFOquant-body(concl)[var/mFOquant-var(concl)]>] = subgoals ∈ (mFOL-sequent() List)
12. FOL-sequent-evidence{i:l}(<hyps, mFOquant-body(concl)[var/mFOquant-var(concl)]>)
⊢ FOL-sequent-evidence{i:l}(<hyps, concl>)
BY
{ Thin (-2) }

1
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. var : ℤ
8. ¬(var ∈ mFOL-sequent-freevars(<hyps, concl>))
9. ↑mFOquant?(concl)
10. mFOquant-isall(concl) = tt
11. FOL-sequent-evidence{i:l}(<hyps, mFOquant-body(concl)[var/mFOquant-var(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.  var  :  \mBbbZ{}
8.  \mneg{}(var  \mmember{}  mFOL-sequent-freevars(<hyps,  concl>))
9.  \muparrow{}mFOquant?(concl)
10.  mFOquant-isall(concl)  =  tt
11.  [<hyps,  mFOquant-body(concl)[var/mFOquant-var(concl)]>]  =  subgoals
12.  FOL-sequent-evidence\{i:l\}(<hyps,  mFOquant-body(concl)[var/mFOquant-var(concl)]>)
\mvdash{}  FOL-sequent-evidence\{i:l\}(<hyps,  concl>)


By

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Latex:
Thin  (-2)

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