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Step * 1 11 1 2 2 1 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
{ (FLemma `FOL-sequent-evidence_transitivity2` [-5]
   THEN Auto
   THEN Using [`x', ⌜hyps[hypnum]⌝ ] (BLemma `better-FOL-sequent-evidence_transitivity1`)⋅
   THEN Auto
   THEN StrongHypSubst (-8) (-1)
   THEN Auto) }

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. 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])))
15. [Dom] : Type
16. [S] : FOStruct+{i:l}(Dom)
17. a : FOAssignment(mFOL-sequent-freevars(<hyps, mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])]>),Dom)
18. Dom,S,a +|= FOL-abstract(∀mFOquant-var(hyps[hypnum]);mFOquant-body(hyps[hypnum])))
⊢ Dom,S,a +|= FOL-abstract(mFOquant-body(hyps[hypnum])[var/mFOquant-var(hyps[hypnum])])

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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.  (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:
(FLemma  `FOL-sequent-evidence\_transitivity2`  [-5]
  THEN  Auto
  THEN  Using  [`x',  \mkleeneopen{}hyps[hypnum]\mkleeneclose{}  ]  (BLemma  `better-FOL-sequent-evidence\_transitivity1`)\mcdot{}
  THEN  Auto
  THEN  StrongHypSubst  (-8)  (-1)
  THEN  Auto)

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