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Step * 1 11 1 2 2 2 of Lemma FOL-proveable-evidence


1. hyps : mFOL() List
2. concl : mFOL()
3. subgoals : mFOL-sequent() List
4. hypnum : ℕ
5. var : ℤ
6. hypnum < ||hyps||
7. (var ∈ mFOL-sequent-freevars(<hyps, concl>))
8. ↑mFOquant?(hyps[hypnum])
9. mFOquant-isall(hyps[hypnum]) = tt
10. FOL-sequent-evidence{i:l}(<[mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])] / hyps], concl>)
11. hyps[hypnum] = ∀mFOquant-var(hyps[hypnum]);mFOquant-body(hyps[hypnum])) ∈ mFOL()
12. mFOL-freevars(mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]) ⊆ mFOL-sequent-freevars(<hyps, concl>)
13. mFOL-freevars(hyps[hypnum]) ⊆ mFOL-sequent-freevars(<hyps
                                                        , mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]
                                                        >)
14. ¬(mFOquant-var(hyps[hypnum]) ∈ mFOL-freevars(mFOquant-body(hyps[hypnum])))
⊢ FOL-sequent-evidence{i:l}(<hyps, concl>)
BY
{ ((InstLemma `FOL-subst-trivial` [⌜var⌝;⌜mFOquant-var(hyps[hypnum])⌝;⌜mFOquant-body(hyps[hypnum])⌝]⋅ THENA Auto)
   THEN RepeatFor 2 (ParallelOp -6)
   THEN All (Fold `uall`)
   THEN ParallelOp -6
   THEN RepeatFor 2 (ParallelLast')) }

1
1. hyps : mFOL() List
2. concl : mFOL()
3. subgoals : mFOL-sequent() List
4. hypnum : ℕ
5. var : ℤ
6. hypnum < ||hyps||
7. (var ∈ mFOL-sequent-freevars(<hyps, concl>))
8. ↑mFOquant?(hyps[hypnum])
9. mFOquant-isall(hyps[hypnum]) = tt
10. ∀[Dom:Type]. ∀[S:FOStruct+{i:l}(Dom)].
      ∀a:FOAssignment(mFOL-sequent-freevars(<[mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])] / hyps]
                                            , concl
                                            >),Dom)
        Dom,S,a +|= FOL-sequent-abstract(<[mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])] / hyps], concl>)
11. hyps[hypnum] = ∀mFOquant-var(hyps[hypnum]);mFOquant-body(hyps[hypnum])) ∈ mFOL()
12. mFOL-freevars(mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]) ⊆ mFOL-sequent-freevars(<hyps, concl>)
13. mFOL-freevars(hyps[hypnum]) ⊆ mFOL-sequent-freevars(<hyps
                                                        , mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]
                                                        >)
14. ¬(mFOquant-var(hyps[hypnum]) ∈ mFOL-freevars(mFOquant-body(hyps[hypnum])))
15. FOL-abstract(mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])])
= FOL-abstract(mFOquant-body(hyps[hypnum]))
∈ AbstractFOFormula+(mFOL-freevars(mFOquant-body(hyps[hypnum])))
16. [Dom] : Type
17. [S] : FOStruct+{i:l}(Dom)
18. a : FOAssignment(mFOL-sequent-freevars(<hyps, concl>),Dom)
19. Dom,S,a +|= FOL-sequent-abstract(<[mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])] / hyps], concl>)
⊢ Dom,S,a +|= FOL-sequent-abstract(<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.  hypnum  :  \mBbbN{}
5.  var  :  \mBbbZ{}
6.  hypnum  <  ||hyps||
7.  (var  \mmember{}  mFOL-sequent-freevars(<hyps,  concl>))
8.  \muparrow{}mFOquant?(hyps[hypnum])
9.  mFOquant-isall(hyps[hypnum])  =  tt
10.  FOL-sequent-evidence\{i:l\}(<[mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]  /  hyps]
                                                            ,  concl
                                                            >)
11.  hyps[hypnum]  =  \mforall{}mFOquant-var(hyps[hypnum]);mFOquant-body(hyps[hypnum]))
12.  mFOL-freevars(mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])])
        \msubseteq{}  mFOL-sequent-freevars(<hyps,  concl>)
13.  mFOL-freevars(hyps[hypnum])
        \msubseteq{}  mFOL-sequent-freevars(<hyps,  mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]>)
14.  \mneg{}(mFOquant-var(hyps[hypnum])  \mmember{}  mFOL-freevars(mFOquant-body(hyps[hypnum])))
\mvdash{}  FOL-sequent-evidence\{i:l\}(<hyps,  concl>)


By

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Latex:
((InstLemma  `FOL-subst-trivial`  [\mkleeneopen{}var\mkleeneclose{};\mkleeneopen{}mFOquant-var(hyps[hypnum])\mkleeneclose{};\mkleeneopen{}mFOquant-body(hyps[hypnum])\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  RepeatFor  2  (ParallelOp  -6)
  THEN  All  (Fold  `uall`)
  THEN  ParallelOp  -6
  THEN  RepeatFor  2  (ParallelLast'))

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