Nuprl Lemma : simple_more_general_fan_theorem-ext
∀[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 :
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
decidable__false,
decidable__implies,
decidable__not,
decidable__exists_int_seg,
any: any x,
decidable__int_equal,
decidable__all_int_seg,
basic_bar_induction,
simple_more_general_fan_theorem,
prop: ℙ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
squash: ↓T,
true: True,
top: Top,
less_than': less_than'(a;b),
less_than: a < b,
uimplies: b supposing a,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
all: ∀x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
has-value: (a)↓,
genFAN: genFAN(max;d),
ifthenelse: if b then t else f fi ,
seq-normalize: seq-normalize(n;s),
bottom: ⊥,
pi2: snd(t),
project-seq: project-seq(s),
member: t ∈ T
Lemmas referenced :
lifting-strict-int_eq,
lifting-strict-callbyvalue,
strict4-decide,
lifting-strict-decide,
int-value-type,
value-type-has-value,
exception-not-value,
istype-assert,
istype-less_than,
assert_of_bnot,
iff_weakening_uiff,
less_than_wf,
not_wf,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
bottom-sqle,
strictness-apply,
istype-void,
istype-top,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
is-exception_wf,
has-value_wf_base,
exception-not-bottom,
bottom_diverge,
simple_more_general_fan_theorem,
decidable__false,
decidable__implies,
decidable__not,
decidable__exists_int_seg,
decidable__int_equal,
decidable__all_int_seg,
basic_bar_induction
Rules used in proof :
intEquality,
lessExceptionCases,
universeIsType,
functionIsType,
cumulativity,
dependent_functionElimination,
promote_hyp,
equalityIstype,
dependent_pairFormation_alt,
imageElimination,
imageMemberEquality,
natural_numberEquality,
independent_pairFormation,
isectIsTypeImplies,
isect_memberEquality_alt,
axiomSqEquality,
isect_memberFormation_alt,
lessCases,
independent_isectElimination,
equalityElimination,
unionElimination,
lambdaFormation_alt,
inhabitedIsType,
productElimination,
callbyvalueLess,
because_Cache,
isectElimination,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
sqleReflexivity,
exceptionSqequal,
axiomSqleEquality,
callbyvalueExceptionCases,
voidElimination,
independent_functionElimination,
callbyvalueReduce,
callbyvalueCallbyvalue,
divergentSqle,
sqequalSqle,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
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_19
Last ObjectModification:
2019_10_07-PM-04_40_56
Theory : bar-induction
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