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Step * 2 2 1 1 1 of Lemma satisfies-shadow-inequalities

.....wf..... 
1. n : ℕ
2. ineqs : {L:ℤ List| ||L|| = n ∈ ℤ}  List
3. i : ℕ+n
4. xs : ℤ List
5. (∀as∈ineqs.xs ⋅ as ≥0)
6. ∀L:ℤ List. ((L ∈ ineqs) ⇒ i < ||L||)
7. map(λL.L\i;filter(λL.(L[i] =z 0);ineqs)) ∈ ℤ List List
8. valueall-type(ℤ List List)
9. ∀[P:{L:ℤ List| ||L|| = (n - 1) ∈ ℤ}  ⟶ ℙ]
     ∀L1,L2:{L:ℤ List| ||L|| = (n - 1) ∈ ℤ}  List.  ((∀x∈L1 @ L2.P[x]) ⇐⇒ (∀x∈L1.P[x]) ∧ (∀x∈L2.P[x]))
⊢ eager-product-map(λas,bs. shadow-vec(i;as;bs);filter(λL.0 <z L[i];ineqs);filter(λL.L[i] <z 0;ineqs)) ∈ {L:ℤ List| 
                                                                                                    ||L||
                                                                                                    = (n - 1)
                                                                                                    ∈ ℤ}  List
BY
{ ((InstLemma `eager-product-map_wf` 
      [⌜{L:ℤ List| ||L|| = (n - 1) ∈ ℤ} ⌝;⌜{L:ℤ List| ||L|| = n ∈ ℤ} ⌝;⌜{L:ℤ List| ||L|| = n ∈ ℤ} ⌝]⋅
    THENA Auto
    )
   THEN BHyp -1
   THEN Auto
   THEN BackThruSomeHyp
   THEN ParallelLast
   THEN Auto) }

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Latex:

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Latex:
.....wf..... 
1.  n  :  \mBbbN{}
2.  ineqs  :  \{L:\mBbbZ{}  List|  ||L||  =  n\}    List
3.  i  :  \mBbbN{}\msupplus{}n
4.  xs  :  \mBbbZ{}  List
5.  (\mforall{}as\mmember{}ineqs.xs  \mcdot{}  as  \mgeq{}0)
6.  \mforall{}L:\mBbbZ{}  List.  ((L  \mmember{}  ineqs)  {}\mRightarrow{}  i  <  ||L||)
7.  map(\mlambda{}L.L\mbackslash{}i;filter(\mlambda{}L.(L[i]  =\msubz{}  0);ineqs))  \mmember{}  \mBbbZ{}  List  List
8.  valueall-type(\mBbbZ{}  List  List)
9.  \mforall{}[P:\{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}    {}\mrightarrow{}  \mBbbP{}]
          \mforall{}L1,L2:\{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}    List.    ((\mforall{}x\mmember{}L1  @  L2.P[x])  \mLeftarrow{}{}\mRightarrow{}  (\mforall{}x\mmember{}L1.P[x])  \mwedge{}  (\mforall{}x\mmember{}L2.P[x]))
\mvdash{}  eager-product-map(\mlambda{}as,bs.  shadow-vec(i;as;bs);filter(\mlambda{}L.0  <z  L[i];ineqs);filter(\mlambda{}L.L[i]  <z  0;ineqs\000C))
    \mmember{}  \{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}    List


By

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Latex:
((InstLemma  `eager-product-map\_wf` 
        [\mkleeneopen{}\{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}  \mkleeneclose{};\mkleeneopen{}\{L:\mBbbZ{}  List|  ||L||  =  n\}  \mkleeneclose{};\mkleeneopen{}\{L:\mBbbZ{}  List|  ||L||  =  n\}  \mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  BHyp  -1
  THEN  Auto
  THEN  BackThruSomeHyp
  THEN  ParallelLast
  THEN  Auto)

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