Nuprl Lemma : sum-partial-has-value
∀[n:ℕ]. ∀[f:ℕn ⟶ partial(ℕ)]. ∀i:ℕn. (f[i])↓ supposing (Σ(f[x] | x < n))↓
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
partial: partial(T),
int_seg: {i..j-},
nat: ℕ,
has-value: (a)↓,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
has-value: (a)↓,
top: Top,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
uniform-comp-nat-induction,
nat_wf,
int_seg_wf,
partial_wf,
sum-partial-nat,
has-value_wf-partial,
set-value-type,
le_wf,
istype-int,
int-value-type,
sum-unroll,
istype-void,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-top,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
istype-lt,
subtype_partial_sqtype_base,
set_subtype_base,
int_subtype_base,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
less_than_wf,
value-type-has-value,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
int_seg_subtype_nat,
isect_wf,
all_wf,
uall_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
independent_functionElimination,
hypothesis,
lemma_by_obid,
lambdaEquality,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
functionEquality,
applyEquality,
because_Cache,
intEquality,
independent_isectElimination,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :universeIsType,
Error :functionIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomSqleEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
voidElimination,
productElimination,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
addEquality,
minusEquality,
multiplyEquality,
callbyvalueAdd,
instantiate,
cumulativity,
Error :isectIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} partial(\mBbbN{})]. \mforall{}i:\mBbbN{}n. (f[i])\mdownarrow{} supposing (\mSigma{}(f[x] | x < n))\mdownarrow{}
Date html generated:
2019_06_20-PM-01_18_19
Last ObjectModification:
2018_10_05-AM-11_01_46
Theory : int_2
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