Nuprl Lemma : m-TB-product
∀m:ℕ. ∀[X:ℕm ⟶ Type]. ∀[d:i:ℕm ⟶ metric(X[i])]. ((∀i:ℕm. m-TB(X[i];d[i])) ⇒ m-TB(i:ℕm ⟶ X[i];prod-metric(m;d)))
Proof
Definitions occuring in Statement :
m-TB: m-TB(X;d),
prod-metric: prod-metric(k;d),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
subtype_rel: A ⊆r B,
pi1: fst(t),
rleq: x ≤ y,
rnonneg: rnonneg(x),
equipollent: A ~ B,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
true: True,
compose: f o g,
prod-metric: prod-metric(k;d),
mdist: mdist(d;x;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
m-TB-iff,
int_seg_wf,
prod-metric_wf,
m-TB_wf,
metric_wf,
istype-universe,
istype-nat,
mul_bounds_1a,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_wf,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
intformless_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
rless_wf,
le_witness_for_triv,
equipollent-product,
nat_plus_subtype_nat,
int-prod_wf_nat_plus,
equipollent_inversion,
int-prod_wf,
compose_wf,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
set_subtype_base,
le_wf,
int_subtype_base,
subtract_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
rmul_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
rsum_wf,
subtract-add-cancel,
itermSubtract_wf,
int_term_value_subtract_lemma,
uimplies_transitivity,
rleq_functionality,
rsum-constant2,
req_weakening,
rleq_functionality_wrt_implies,
rsum_functionality_wrt_rleq2,
rleq_weakening_equal,
decidable__equal_int,
rinv_wf2,
rleq-int-fractions2,
rleq-int-fractions,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req_transitivity,
rinv-mul-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
productElimination,
independent_functionElimination,
because_Cache,
functionEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
universeIsType,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
imageElimination,
addEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp,
productIsType,
closedConclusion,
inrFormation_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
functionExtensionality,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
baseApply,
intEquality,
sqequalBase,
equalityElimination,
cumulativity
Latex:
\mforall{}m:\mBbbN{}
\mforall{}[X:\mBbbN{}m {}\mrightarrow{} Type]. \mforall{}[d:i:\mBbbN{}m {}\mrightarrow{} metric(X[i])].
((\mforall{}i:\mBbbN{}m. m-TB(X[i];d[i])) {}\mRightarrow{} m-TB(i:\mBbbN{}m {}\mrightarrow{} X[i];prod-metric(m;d)))
Date html generated:
2019_10_30-AM-06_51_38
Last ObjectModification:
2019_10_10-PM-07_00_22
Theory : reals
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