Nuprl Lemma : sum-partial-list-has-value
∀[T:Type]. ∀[L:T List]. ∀[f:T ⟶ partial(ℕ)]. ∀x:T. (f[x])↓ supposing (x ∈ L) supposing (Σ(f[L[i]] | i < ||L||))↓
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
partial: partial(T),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
has-value: (a)↓,
prop: ℙ,
squash: ↓T,
less_than: a < b,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
le: A ≤ B,
nat: ℕ,
cand: A c∧ B,
l_member: (x ∈ l)
Lemmas referenced :
partial_wf,
nat_wf,
list_wf,
l_member_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
sum-partial-has-value,
lelt_wf,
int-value-type,
le_wf,
set-value-type,
has-value_wf-partial,
sum-partial-nat,
full-omega-unsat,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
Error :isect_memberEquality_alt,
isectElimination,
axiomSqleEquality,
hypothesis,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :functionIsType,
Error :universeIsType,
extract_by_obid,
universeEquality,
cumulativity,
isect_memberFormation,
lambdaFormation,
imageElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
productElimination,
natural_numberEquality,
independent_isectElimination,
rename,
setElimination,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_set_memberEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
approximateComputation,
independent_functionElimination,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f:T {}\mrightarrow{} partial(\mBbbN{})].
\mforall{}x:T. (f[x])\mdownarrow{} supposing (x \mmember{} L) supposing (\mSigma{}(f[L[i]] | i < ||L||))\mdownarrow{}
Date html generated:
2019_06_20-PM-01_48_49
Last ObjectModification:
2018_10_15-PM-01_44_59
Theory : list_1
Home
Index