Nuprl Lemma : tuple-sum_wf
∀[P:Type]. ∀[as:P List]. ∀[G:P ⟶ Type]. ∀[x:tuple-type(map(G;as))]. ∀[f:i:P ⟶ (G i) ⟶ ℕ].  (tuple-sum(f;as;x) ∈ ℕ)
Proof
Definitions occuring in Statement : 
tuple-sum: tuple-sum(f;L;x)
, 
tuple-type: tuple-type(L)
, 
map: map(f;as)
, 
list: T List
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
or: P ∨ Q
, 
tuple-sum: tuple-sum(f;L;x)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
decidable: Dec(P)
, 
cons: [a / b]
, 
colength: colength(L)
, 
nil: []
, 
it: ⋅
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
subtype_rel: A ⊆r B
, 
bfalse: ff
, 
bool: 𝔹
, 
unit: Unit
, 
uiff: uiff(P;Q)
Lemmas referenced : 
nat_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
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, 
istype-less_than, 
list-cases, 
map_nil_lemma, 
null_nil_lemma, 
tupletype_nil_lemma, 
decidable__le, 
intformnot_wf, 
int_formula_prop_not_lemma, 
istype-le, 
unit_wf2, 
product_subtype_list, 
colength-cons-not-zero, 
colength_wf_list, 
subtract-1-ge-0, 
subtype_base_sq, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
set_subtype_base, 
int_subtype_base, 
spread_cons_lemma, 
decidable__equal_int, 
subtract_wf, 
itermSubtract_wf, 
itermAdd_wf, 
int_term_value_subtract_lemma, 
int_term_value_add_lemma, 
le_wf, 
map_cons_lemma, 
null_cons_lemma, 
tupletype_cons_lemma, 
null-map, 
null_wf, 
add_nat_wf, 
add-is-int-iff, 
false_wf, 
tuple-type_wf, 
map_wf, 
istype-nat, 
list_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
thin, 
Error :lambdaFormation_alt, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
sqequalRule, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
Error :dependent_pairFormation_alt, 
Error :lambdaEquality_alt, 
int_eqEquality, 
dependent_functionElimination, 
Error :isect_memberEquality_alt, 
voidElimination, 
independent_pairFormation, 
Error :universeIsType, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
Error :isectIsTypeImplies, 
Error :inhabitedIsType, 
Error :functionIsTypeImplies, 
unionElimination, 
Error :dependent_set_memberEquality_alt, 
Error :functionIsType, 
applyEquality, 
because_Cache, 
promote_hyp, 
hypothesis_subsumption, 
productElimination, 
Error :equalityIstype, 
instantiate, 
applyLambdaEquality, 
imageElimination, 
baseApply, 
closedConclusion, 
baseClosed, 
intEquality, 
sqequalBase, 
equalityElimination, 
addEquality, 
pointwiseFunctionality, 
Error :productIsType, 
cumulativity, 
universeEquality
Latex:
\mforall{}[P:Type].  \mforall{}[as:P  List].  \mforall{}[G:P  {}\mrightarrow{}  Type].  \mforall{}[x:tuple-type(map(G;as))].  \mforall{}[f:i:P  {}\mrightarrow{}  (G  i)  {}\mrightarrow{}  \mBbbN{}].
    (tuple-sum(f;as;x)  \mmember{}  \mBbbN{})
Date html generated:
2019_06_20-PM-02_03_21
Last ObjectModification:
2019_02_22-AM-11_03_09
Theory : tuples
Home
Index