Nuprl Lemma : prod-metric_wf
∀[k:ℕ]. ∀[X:ℕk ⟶ Type]. ∀[d:i:ℕk ⟶ metric(X[i])]. (prod-metric(k;d) ∈ metric(i:ℕk ⟶ X[i]))
Proof
Definitions occuring in Statement :
prod-metric: prod-metric(k;d),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
prod-metric: prod-metric(k;d),
uall: ∀[x:A]. B[x],
member: t ∈ T,
metric: metric(X),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
cand: A c∧ B,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rsum_wf,
subtract_wf,
mdist_wf,
subtract-add-cancel,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
rsum-of-nonneg-zero-iff,
mdist-nonneg,
int_seg_properties,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
mdist-same,
rleq_wf,
radd_wf,
req_wf,
int-to-real_wf,
metric_wf,
istype-universe,
istype-nat,
rsum_functionality_wrt_rleq,
mdist-triangle-inequality1,
rleq_functionality,
req_weakening,
req_inversion,
rsum_linearity1
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
hypothesisEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
addEquality,
inhabitedIsType,
functionIsType,
lambdaFormation_alt,
imageElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} Type]. \mforall{}[d:i:\mBbbN{}k {}\mrightarrow{} metric(X[i])]. (prod-metric(k;d) \mmember{} metric(i:\mBbbN{}k {}\mrightarrow{} X[i]))
Date html generated:
2019_10_29-AM-11_09_34
Last ObjectModification:
2019_10_02-AM-09_50_39
Theory : reals
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