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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