### Nuprl Lemma : sum_functionality_wrt_sqequal

`∀[n:ℕ]. ∀[f,g:Base].  Σ(f[x] | x < n) ~ Σ(g[x] | x < n) supposing ∀i:ℕn. (f[i] ~ g[i])`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` natural_number: `\$n` base: `Base` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` guard: `{T}` so_apply: `x[s]` sum: `Σ(f[x] | x < k)` sum_aux: `sum_aux(k;v;i;x.f[x])` int_seg: `{i..j-}` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` lelt: `i ≤ j < k` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf all_wf int_seg_wf sqequal-wf-base less_than_transitivity1 less_than_irreflexivity base_wf set_subtype_base lelt_wf int_subtype_base decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf decidable__lt eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot sum-unroll
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom baseApply closedConclusion baseClosed applyEquality because_Cache equalityTransitivity equalitySymmetry unionElimination productElimination equalityElimination lessCases imageMemberEquality imageElimination dependent_set_memberEquality promote_hyp instantiate cumulativity

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:Base].    \mSigma{}(f[x]  |  x  <  n)  \msim{}  \mSigma{}(g[x]  |  x  <  n)  supposing  \mforall{}i:\mBbbN{}n.  (f[i]  \msim{}  g[i])

Date html generated: 2017_04_14-AM-09_21_24
Last ObjectModification: 2017_02_27-PM-03_57_26

Theory : int_2

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