Nuprl Lemma : tuple-sum-wf-partial
∀[P:Type]. ∀[G:P ⟶ Type]. ∀[f:i:P ⟶ (G i) ⟶ partial(ℕ)]. ∀[as:P List]. ∀[x:tuple-type(map(G;as))].
(tuple-sum(f;as;x) ∈ partial(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
,
partial: partial(T)
,
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
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
cons: [a / b]
,
decidable: Dec(P)
,
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
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
subtype_rel: A ⊆r B
,
bfalse: ff
,
bool: 𝔹
,
unit: Unit
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,
nat-partial-nat,
istype-false,
istype-le,
unit_wf2,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
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,
istype-nat,
list_wf,
partial_wf,
nat_wf,
istype-universe,
add-wf-partial-nat,
tuple-type_wf,
map_wf
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,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIstype,
because_Cache,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
equalityElimination,
Error :functionIsType,
universeEquality,
Error :productIsType,
cumulativity
Latex:
\mforall{}[P:Type]. \mforall{}[G:P {}\mrightarrow{} Type]. \mforall{}[f:i:P {}\mrightarrow{} (G i) {}\mrightarrow{} partial(\mBbbN{})]. \mforall{}[as:P List].
\mforall{}[x:tuple-type(map(G;as))].
(tuple-sum(f;as;x) \mmember{} partial(\mBbbN{}))
Date html generated:
2019_06_20-PM-02_03_24
Last ObjectModification:
2019_02_22-AM-11_13_33
Theory : tuples
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