Nuprl Lemma : le-tuple-sum
∀[P:Type]. ∀[as:P List]. ∀[G:P ⟶ Type]. ∀[x:tuple-type(map(G;as))]. ∀[f:i:P ⟶ (G i) ⟶ ℕ].
∀k:ℕ||as||. ((f as[k] x.k) ≤ tuple-sum(f;as;x))
Proof
Definitions occuring in Statement :
select-tuple: x.n,
tuple-sum: tuple-sum(f;L;x),
tuple-type: tuple-type(L),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
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],
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
or: P ∨ Q,
tuple-sum: tuple-sum(f;L;x),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
bfalse: ff,
bool: 𝔹,
unit: Unit,
subtract: n - m,
eq_int: (i =z j),
select-tuple: x.n,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
true: True,
pi1: fst(t),
pi2: snd(t),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
cand: A c∧ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
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,
le_witness_for_triv,
list-cases,
map_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
null_nil_lemma,
tupletype_nil_lemma,
int_seg_properties,
int_seg_wf,
unit_wf2,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
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,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
map_cons_lemma,
length_of_cons_lemma,
null_cons_lemma,
tupletype_cons_lemma,
null-map,
null_wf,
length_wf,
tuple-type_wf,
map_wf,
istype-nat,
list_wf,
istype-universe,
btrue_neq_bfalse,
int_seg_cases,
int_seg_subtype_special,
bool_wf,
bool_subtype_base,
bfalse_wf,
length_wf_nat,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
non_neg_length,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
tuple-sum_wf,
select-cons,
le_int_wf,
assert_of_le_int,
iff_weakening_uiff,
assert_wf,
decidable__lt,
add-subtract-cancel,
select_wf,
select-tuple_wf,
map-length,
map_length,
subtype_rel-equal,
map_select
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
unionElimination,
baseClosed,
functionIsType,
applyEquality,
because_Cache,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
equalityElimination,
addEquality,
productIsType,
cumulativity,
universeEquality,
imageMemberEquality,
minusEquality
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{}].
\mforall{}k:\mBbbN{}||as||. ((f as[k] x.k) \mleq{} tuple-sum(f;as;x))
Date html generated:
2020_05_19-PM-10_00_19
Last ObjectModification:
2020_01_01-AM-10_58_21
Theory : tuples
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