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
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