### Nuprl Lemma : int-prod-isolate

`∀[n:ℕ]. ∀[m:ℕn]. ∀[f:ℕn ⟶ ℤ].  (Π(f[x] | x < n) = (Π(if (x =z m) then 1 else f[x] fi  | x < n) * f[m]) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int-prod: `Π(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` multiply: `n * m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  assert: `↑b` bnot: `¬bb` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` less_than': `less_than'(a;b)` le: `A ≤ B` uiff: `uiff(P;Q)` so_apply: `x[s]` so_lambda: `λ2x.t[x]` guard: `{T}` sq_type: `SQType(T)` prop: `ℙ` top: `Top` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` implies: `P `` Q` not: `¬A` uimplies: `b supposing a` or: `P ∨ Q` decidable: `Dec(P)` all: `∀x:A. B[x]` ge: `i ≥ j ` nat: `ℕ` and: `P ∧ Q` lelt: `i ≤ j < k` int_seg: `{i..j-}` member: `t ∈ T` uall: `∀[x:A]. B[x]` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` subtype_rel: `A ⊆r B` true: `True` squash: `↓T` nequal: `a ≠ b ∈ T `
Lemmas referenced :  nat_wf neg_assert_of_eq_int assert-bnot bool_subtype_base bool_cases_sqequal equal_wf eqff_to_assert assert_of_eq_int eqtt_to_assert bool_wf false_wf add-member-int_seg1 le_wf int_term_value_subtract_lemma int_formula_prop_le_lemma itermSubtract_wf intformle_wf decidable__le int_seg_properties subtract_wf int_seg_wf eq_int_wf ifthenelse_wf int_subtype_base subtype_base_sq lelt_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermAdd_wf itermVar_wf intformless_wf intformnot_wf intformand_wf full-omega-unsat decidable__lt nat_properties int-prod-split iff_weakening_equal btrue_wf eq_int_eq_true true_wf squash_wf zero-add int-prod-single int_formula_prop_eq_lemma intformeq_wf int_seg_subtype_nat int-prod_wf int_term_value_mul_lemma itermMultiply_wf decidable__equal_int
Rules used in proof :  axiomEquality functionEquality promote_hyp equalityElimination lambdaFormation functionExtensionality applyEquality equalitySymmetry equalityTransitivity cumulativity instantiate because_Cache sqequalRule voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation independent_functionElimination approximateComputation independent_isectElimination unionElimination natural_numberEquality addEquality dependent_functionElimination hypothesis independent_pairFormation productElimination dependent_set_memberEquality rename setElimination hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution baseClosed imageMemberEquality universeEquality imageElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[m:\mBbbN{}n].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].
(\mPi{}(f[x]  |  x  <  n)  =  (\mPi{}(if  (x  =\msubz{}  m)  then  1  else  f[x]  fi    |  x  <  n)  *  f[m]))

Date html generated: 2018_05_21-PM-00_29_36
Last ObjectModification: 2017_12_10-PM-11_43_03

Theory : int_2

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