Nuprl Lemma : general-fan-theorem-troelstra
∀X:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ. ((∀f:ℕ ⟶ 𝔹. ∃n:ℕ. X[n;f]) ⇒ ⇃(∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. X[n;f]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
function: x:A ⟶ B[x],
natural_number: $n
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[s1;s2],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
exists: ∃x:A. B[x],
true: True,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
lelt: i ≤ j < k,
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
outl: outl(x),
isl: isl(x),
less_than: a < b,
bfalse: ff,
pi1: fst(t),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
assert: ↑b,
bnot: ¬bb
Lemmas referenced :
all_wf,
nat_wf,
bool_wf,
exists_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
strong-continuity2-no-inner-squash-unique-bool,
pi1_wf,
equal_wf,
unit_wf2,
assert_wf,
isl_wf,
true_wf,
quotient_wf,
equiv_rel_true,
le_wf,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf,
fan_theorem,
b-exists_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
sq_stable__le,
less-iff-le,
add-associates,
zero-add,
add_functionality_wrt_le,
le-add-cancel2,
eqtt_to_assert,
assert_elim,
bfalse_wf,
and_wf,
btrue_neq_bfalse,
top_wf,
less_than_wf,
btrue_wf,
decidable__assert,
imax_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert-b-exists,
add_nat_wf,
add-is-int-iff,
lelt_wf,
assert_of_band,
subtype_base_sq,
bool_subtype_base,
imax_ub,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
union_subtype_base,
set_subtype_base,
int_subtype_base,
unit_subtype_base,
lt_int_wf,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
int_seg_properties,
outl_wf,
subtype_rel_union,
assert_functionality_wrt_uiff,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
independent_pairFormation,
cumulativity,
universeEquality,
dependent_functionElimination,
functionExtensionality,
instantiate,
productElimination,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionEquality,
productEquality,
inlEquality,
promote_hyp,
dependent_set_memberEquality,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
equalityElimination,
applyLambdaEquality,
lessCases,
isect_memberFormation,
axiomSqEquality,
baseApply,
closedConclusion,
inrFormation,
inlFormation
Latex:
\mforall{}X:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbP{}. ((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}. X[n;f]) {}\mRightarrow{} \00D9(\mexists{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}k. X[n;f]))
Date html generated:
2019_06_20-PM-02_59_58
Last ObjectModification:
2018_08_20-PM-09_40_43
Theory : continuity
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