Nuprl Lemma : extended-fan-theorem
∀C:ℕ ⟶ (ℕ ⟶ 𝔹) ⟶ ℙ
((∀a:ℕ ⟶ 𝔹. ∃n:ℕ. (C n a)) ⇒ ⇃(∃m:ℕ. ∀a:ℕ ⟶ 𝔹. ∃n:ℕ. ∀b:ℕ ⟶ 𝔹. ((a = b ∈ (ℕm ⟶ 𝔹)) ⇒ (C n b))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
nat: ℕ,
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T,
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi1: fst(t),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
all_wf,
nat_wf,
bool_wf,
exists_wf,
strong-continuity2-no-inner-squash-unique-bool,
pi1_wf,
equal_wf,
int_seg_wf,
unit_wf2,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
assert_wf,
isl_wf,
true_wf,
quotient_wf,
equiv_rel_true,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf,
fan_theorem,
decidable__assert,
and_wf,
btrue_wf,
subtype_base_sq,
bool_subtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_subtype,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
assert_functionality_wrt_uiff,
itermConstant_wf,
int_term_value_constant_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
because_Cache,
productElimination,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
setElimination,
unionEquality,
productEquality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
intEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}C:\mBbbN{} {}\mrightarrow{} (\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbP{}
((\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}. (C n a)) {}\mRightarrow{} \00D9(\mexists{}m:\mBbbN{}. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((a = b) {}\mRightarrow{} (C n b))))
Date html generated:
2017_04_20-AM-07_22_19
Last ObjectModification:
2017_02_27-PM-05_57_48
Theory : continuity
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