Nuprl Lemma : surjection-cantor-finite-branching
∀b:ℕ ⟶ ℕ+. ∃F:(ℕ ⟶ 𝔹) ⟶ n:ℕ ⟶ ℕb n. Surj(ℕ ⟶ 𝔹;n:ℕ ⟶ ℕb n;F)
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
surject: Surj(A;B;f),
prop: ℙ,
subtype_rel: A ⊆r B,
cantor-to-fb: cantor-to-fb(b;g;n),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
not: ¬A,
implies: P ⇒ Q,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
has-value: (a)↓,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
fb-to-cantor: fb-to-cantor(b;f;n),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
true: True,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
cantor-to-fb_wf,
nat_wf,
bool_wf,
fb-to-cantor_wf,
equal_wf,
int_seg_wf,
surject_wf,
nat_plus_wf,
sum_wf,
non_neg_sum,
le_weakening2,
int_seg_properties,
nat_properties,
decidable__lt,
le_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
value-type-has-value,
set-value-type,
int-value-type,
subtract_wf,
mu-bound-unique,
add_nat_wf,
int_seg_subtype_nat,
false_wf,
decidable__le,
add-is-int-iff,
intformle_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
bor_wf,
lt_int_wf,
lelt_wf,
iff_transitivity,
assert_wf,
or_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_lt_int,
subtype_base_sq,
int_subtype_base,
mu-unique,
itermSubtract_wf,
int_term_value_subtract_lemma,
sum-unroll,
eqtt_to_assert,
top_wf,
add-subtract-cancel,
decidable__equal_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
not_wf,
sum_split,
set_subtype_base,
assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
functionEquality,
rename,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
setElimination,
independent_isectElimination,
dependent_functionElimination,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
voidElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
callbyvalueReduce,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
orFunctionality,
instantiate,
cumulativity,
inlFormation,
imageElimination,
inrFormation,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
imageMemberEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}. \mexists{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} n:\mBbbN{} {}\mrightarrow{} \mBbbN{}b n. Surj(\mBbbN{} {}\mrightarrow{} \mBbbB{};n:\mBbbN{} {}\mrightarrow{} \mBbbN{}b n;F)
Date html generated:
2018_05_21-PM-07_58_21
Last ObjectModification:
2017_07_26-PM-05_35_40
Theory : general
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