Nuprl Lemma : simple_more_general_fan_theorem
∀[T:ℕ ⟶ Type]
(∀i:ℕ. Bounded(T[i]))
⇒ (∀[X:n:ℕ ⟶ (i:ℕn ⟶ T[i]) ⟶ ℙ]
(∀n:ℕ. ∀s:i:ℕn ⟶ T[i]. Dec(X[n;s])) ⇒ (∃k:ℕ [(∀f:i:ℕ ⟶ T[i]. ∃n:ℕk. X[n;f])])
supposing ∀f:i:ℕ ⟶ T[i]. (↓∃n:ℕ. X[n;f]))
supposing ∀i:ℕ. T[i]
Proof
Definitions occuring in Statement :
bounded-type: Bounded(T),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
ge: i ≥ j ,
seq-adjoin: s++t,
pi2: snd(t),
project-seq: project-seq(s),
pi1: fst(t),
istype: istype(T),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
subtract: n - m,
gt: i > j,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
true: True,
top: Top,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
seq-append: seq-append(n;m;s1;s2),
not: ¬A,
false: False,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
guard: {T},
so_apply: x[s1;s2],
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
squash: ↓T,
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
bounded-type: Bounded(T)
Lemmas referenced :
less_than_irreflexivity,
less_than_transitivity1,
Error :neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
nat_properties,
le-add-cancel-alt,
mul-commutes,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
minus-zero,
le_reflexive,
le_weakening2,
subtype_rel_sets_simple,
minus-minus,
add-member-int_seg2,
decidable__exists_int_seg,
sq_stable_from_decidable,
le_weakening,
le_transitivity,
pi2_wf,
less_than_anti-reflexive,
le-add-cancel2,
not-equal-2,
less-iff-le,
not-lt-2,
decidable__lt,
Error :assert-bnot,
decidable__int_equal,
decidable__all_int_seg,
subtype_rel_dep_function,
decidable_wf,
le-add-cancel,
add_functionality_wrt_le,
add-swap,
zero-add,
minus-add,
condition-implies-le,
not-le-2,
decidable__le,
seq-adjoin_wf,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-commutes,
minus-one-mul,
add-associates,
not-gt-2,
add-is-int-iff,
subtract_nat_wf,
subtract_wf,
istype-assert,
assert_of_bnot,
iff_weakening_uiff,
not_wf,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
iff_weakening_equal,
istype-less_than,
istype-universe,
true_wf,
squash_wf,
equal_wf,
istype-void,
istype-top,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
subtype_rel_self,
subtype_rel_function,
istype-false,
int_seg_subtype_nat,
seq-append_wf,
istype-le,
sq_stable__le,
add_nat_wf,
exists_wf,
sq_exists_wf,
all_wf,
isect_wf,
istype-nat,
project-seq_wf,
less_than_wf,
and_wf,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
pi1_wf,
equal-wf-base,
int_seg_wf,
nat_wf,
basic_bar_induction
Rules used in proof :
functionExtensionality,
dependent_pairEquality_alt,
functionExtensionality_alt,
hyp_replacement,
dependent_set_memberFormation_alt,
inrFormation_alt,
inlFormation_alt,
multiplyEquality,
setIsType,
isectIsType,
axiomEquality,
sqequalBase,
minusEquality,
baseApply,
cumulativity,
promote_hyp,
dependent_pairFormation_alt,
universeEquality,
instantiate,
voidElimination,
isectIsTypeImplies,
isect_memberEquality_alt,
axiomSqEquality,
lessCases,
equalityElimination,
unionElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
independent_functionElimination,
addEquality,
dependent_set_memberEquality_alt,
closedConclusion,
productIsType,
universeIsType,
functionIsType,
productElimination,
because_Cache,
independent_isectElimination,
intEquality,
setElimination,
natural_numberEquality,
functionEquality,
isectEquality,
applyEquality,
productEquality,
isectElimination,
extract_by_obid,
rename,
inhabitedIsType,
functionIsTypeImplies,
baseClosed,
imageMemberEquality,
hypothesis,
imageElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality_alt,
sqequalHypSubstitution,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[T:\mBbbN{} {}\mrightarrow{} Type]
(\mforall{}i:\mBbbN{}. Bounded(T[i]))
{}\mRightarrow{} (\mforall{}[X:n:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{}n {}\mrightarrow{} T[i]) {}\mrightarrow{} \mBbbP{}]
(\mforall{}n:\mBbbN{}. \mforall{}s:i:\mBbbN{}n {}\mrightarrow{} T[i]. Dec(X[n;s])) {}\mRightarrow{} (\mexists{}k:\mBbbN{} [(\mforall{}f:i:\mBbbN{} {}\mrightarrow{} T[i]. \mexists{}n:\mBbbN{}k. X[n;f])])
supposing \mforall{}f:i:\mBbbN{} {}\mrightarrow{} T[i]. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f]))
supposing \mforall{}i:\mBbbN{}. T[i]
Date html generated:
2019_10_15-AM-10_20_16
Last ObjectModification:
2019_10_07-PM-04_40_21
Theory : bar-induction
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