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Step * 1 of Lemma linearization-value

.....assertion..... 
∀[L:ℤ List List]. ∀[p:iPolynomial()].
  ∀f:ℤ ⟶ ℤ
    (int_term_value(f;ipolynomial-term(filter(λm.snd(m) ∈b L;p)))
    = linearization(p;L) ⋅ map(λvs.accumulate (with value x and list item v):
                                    x * (f v)
                                   over list:
                                     vs
                                   with starting value:
                                    1);L)
    ∈ ℤ) 
  supposing no_repeats(ℤ List;L)
BY
{ TACTIC:(InductionOnList THEN Reduce 0 THEN Auto) }

1
1. p : iPolynomial()
2. no_repeats(ℤ List;[])
3. f : ℤ ⟶ ℤ
⊢ int_term_value(f;ipolynomial-term(filter(λm.ff;p))) = linearization(p;[]) ⋅ [] ∈ ℤ

2
1. u : ℤ List
2. v : ℤ List List
3. ∀[p:iPolynomial()]
     ∀f:ℤ ⟶ ℤ
       (int_term_value(f;ipolynomial-term(filter(λm.snd(m) ∈b v;p)))
       = linearization(p;v) ⋅ map(λvs.accumulate (with value x and list item v):
                                       x * (f v)
                                      over list:
                                        vs
                                      with starting value:
                                       1);v)
       ∈ ℤ) 
     supposing no_repeats(ℤ List;v)
4. p : iPolynomial()
5. no_repeats(ℤ List;[u / v])
6. f : ℤ ⟶ ℤ
⊢ int_term_value(f;ipolynomial-term(filter(λm.((list-deq(IntDeq) u (snd(m))) ∨bsnd(m) ∈b v);p)))
= linearization(p;[u / v]) ⋅ [accumulate (with value x and list item v):
                               x * (f v)
                              over list:
                                u
                              with starting value:
                               1) / 
                              map(λvs.accumulate (with value x and list item v):
                                       x * (f v)
                                      over list:
                                        vs
                                      with starting value:
                                       1);v)]
∈ ℤ

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Latex:

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Latex:
.....assertion..... 
\mforall{}[L:\mBbbZ{}  List  List].  \mforall{}[p:iPolynomial()].
    \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
        (int\_term\_value(f;ipolynomial-term(filter(\mlambda{}m.snd(m)  \mmember{}\msubb{}  L;p)))
        =  linearization(p;L)  \mcdot{}  map(\mlambda{}vs.accumulate  (with  value  x  and  list  item  v):
                                                                        x  *  (f  v)
                                                                      over  list:
                                                                          vs
                                                                      with  starting  value:
                                                                        1);L)) 
    supposing  no\_repeats(\mBbbZ{}  List;L)


By

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Latex:
TACTIC:(InductionOnList  THEN  Reduce  0  THEN  Auto)

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