---

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

1
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. var : ℤ@i
8. ¬(var ∈ mFOL-freevars(concl))
9. (∀h∈hyps.¬(var ∈ mFOL-freevars(h)))
10. ↑mFOquant?(concl)
11. mFOquant-isall(concl) = tt
12. mFOL-sequent-evidence(<hyps, mFOquant-body(concl)[var/mFOquant-var(concl)]>)
⊢ mFOL-sequent-evidence(<hyps, concl>)

---

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.  var  :  \mBbbZ{}@i
8.  \mneg{}(var  \mmember{}  mFOL-freevars(concl))
9.  (\mforall{}h\mmember{}hyps.\mneg{}(var  \mmember{}  mFOL-freevars(h)))
10.  \muparrow{}mFOquant?(concl)
11.  mFOquant-isall(concl)  =  tt
12.  [<hyps,  mFOquant-body(concl)[var/mFOquant-var(concl)]>]  =  subgoals
13.  mFOL-sequent-evidence(<hyps,  mFOquant-body(concl)[var/mFOquant-var(concl)]>)
\mvdash{}  mFOL-sequent-evidence(<hyps,  concl>)


By

Thin  (-2)

---

Home Index