---

Step * 1 2 2 2 1 2 1 of Lemma shadow-inequalities_wf


1. n : ℕ
2. ineqs : {L:ℤ List| ||L|| = n ∈ ℤ}  List
3. i : ℕn
4. ∀L:ℤ List. ((L ∈ ineqs) ⇒ i < ||L||)
5. map(λL.L\i;filter(λL.(L[i] =z 0);ineqs)) ∈ {L:ℤ List| ||L|| = (n - 1) ∈ ℤ}  List
6. ∀[f:{L:ℤ List| ||L|| = n ∈ ℤ}  ⟶ {L:ℤ List| ||L|| = n ∈ ℤ}  ⟶ {L:ℤ List| ||L|| = (n - 1) ∈ ℤ} ]. ∀[as,bs:{L:ℤ List|\000C 
                                                                                                            ||L||
                                                                                                            = n
                                                                                                            ∈ ℤ}  List].
     (eager-product-map(f;as;bs) ∈ {L:ℤ List| ||L|| = (n - 1) ∈ ℤ}  List)
⊢ 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
{ TACTIC:((BHyp -1 THEN Auto) THEN BackThruSomeHyp THEN Auto THEN ParallelLast THEN Auto) }

---

Latex:

---

Latex:

1.  n  :  \mBbbN{}
2.  ineqs  :  \{L:\mBbbZ{}  List|  ||L||  =  n\}    List
3.  i  :  \mBbbN{}n
4.  \mforall{}L:\mBbbZ{}  List.  ((L  \mmember{}  ineqs)  {}\mRightarrow{}  i  <  ||L||)
5.  map(\mlambda{}L.L\mbackslash{}i;filter(\mlambda{}L.(L[i]  =\msubz{}  0);ineqs))  \mmember{}  \{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}    List
6.  \mforall{}[f:\{L:\mBbbZ{}  List|  ||L||  =  n\}    {}\mrightarrow{}  \{L:\mBbbZ{}  List|  ||L||  =  n\}    {}\mrightarrow{}  \{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}  ].
      \mforall{}[as,bs:\{L:\mBbbZ{}  List|  ||L||  =  n\}    List].
          (eager-product-map(f;as;bs)  \mmember{}  \{L:\mBbbZ{}  List|  ||L||  =  (n  -  1)\}    List)
\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

---

Latex:
TACTIC:((BHyp  -1  THEN  Auto)  THEN  BackThruSomeHyp  THEN  Auto  THEN  ParallelLast  THEN  Auto)

---

Home Index