### Nuprl Lemma : sum1

`∀[f:ℕ1 ⟶ ℤ]. (Σ(f[x] | x < 1) ~ f[0])`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` sum: `Σ(f[x] | x < k)` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` so_apply: `x[s]` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_lambda: `λ2x.t[x]` subtract: `n - m` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` true: `True` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]`
Lemmas referenced :  int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma itermVar_wf itermConstant_wf itermAdd_wf intformeq_wf intformnot_wf satisfiable-full-omega-tt lelt_wf decidable__equal_int primrec1_lemma sum_aux-as-primrec false_wf int_seg_wf int_subtype_base subtype_base_sq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate lemma_by_obid sqequalHypSubstitution isectElimination because_Cache independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination sqequalAxiom functionEquality natural_numberEquality intEquality lambdaEquality applyEquality hypothesisEquality sqequalRule independent_pairFormation lambdaFormation isect_memberEquality voidElimination voidEquality unionElimination dependent_set_memberEquality imageMemberEquality baseClosed dependent_pairFormation int_eqEquality computeAll

Latex:
\mforall{}[f:\mBbbN{}1  {}\mrightarrow{}  \mBbbZ{}].  (\mSigma{}(f[x]  |  x  <  1)  \msim{}  f[0])

Date html generated: 2016_05_14-AM-07_31_24
Last ObjectModification: 2016_01_14-PM-09_56_23

Theory : int_2

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