Nuprl Lemma : countable-nsub-family
∀B:ℕ ⟶ ℕ+. ∃g:ℕ ⟶ (i:ℕ × ℕB[i]). Surj(ℕ;i:ℕ × ℕB[i];g)
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
pi1: fst(t),
so_lambda: λ2x.t[x],
squash: ↓T,
less_than: a < b,
surject: Surj(A;B;f),
less_than': less_than'(a;b),
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
uimplies: b supposing a,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nat: ℕ,
implies: P ⇒ Q,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
biject: Bij(A;B;f),
all: ∀x:A. B[x]
Lemmas referenced :
le_wf,
product_subtype_base,
lelt_wf,
set_subtype_base,
int_subtype_base,
ifthenelse_wf,
int_seg_properties,
int_seg_subtype_nat,
int_formula_prop_less_lemma,
intformless_wf,
nat_plus_properties,
decidable__lt,
istype-false,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
istype-less_than,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
compose_wf,
nat_plus_wf,
int_seg_wf,
compose-surjections,
nat_wf,
surject_wf,
istype-nat,
coded-pair_wf,
code-pair-bijection
Rules used in proof :
baseClosed,
closedConclusion,
baseApply,
independent_pairEquality,
dependent_pairEquality_alt,
spreadEquality,
functionExtensionality,
sqequalBase,
intEquality,
imageElimination,
applyLambdaEquality,
cumulativity,
instantiate,
promote_hyp,
equalityIstype,
equalitySymmetry,
equalityTransitivity,
productIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
independent_isectElimination,
equalityElimination,
unionElimination,
inhabitedIsType,
functionIsType,
because_Cache,
independent_functionElimination,
sqequalRule,
rename,
setElimination,
applyEquality,
natural_numberEquality,
productEquality,
universeIsType,
hypothesis,
hypothesisEquality,
isectElimination,
lambdaEquality_alt,
dependent_pairFormation_alt,
thin,
productElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}. \mexists{}g:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{} \mtimes{} \mBbbN{}B[i]). Surj(\mBbbN{};i:\mBbbN{} \mtimes{} \mBbbN{}B[i];g)
Date html generated:
2019_10_15-AM-10_25_40
Last ObjectModification:
2019_10_08-PM-00_38_37
Theory : num_thy_1
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