Nuprl Lemma : fan-realizer_wf
fan-realizer ∈ ∀[X:(𝔹 List) ⟶ ℙ]. (tbar(𝔹;X) ⇒ Decidable(X) ⇒ (∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. (X map(f;upto(n)))))
Proof
Definitions occuring in Statement :
fan-realizer: fan-realizer,
tbar: tbar(T;X),
dec-predicate: Decidable(X),
upto: upto(n),
map: map(f;as),
list: T List,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
fan-theorem,
simple-fan-theorem,
simple_fan_theorem,
basic_bar_induction,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
false: False,
seq-normalize: seq-normalize(n;s),
fan-realizer: fan-realizer,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
pi1: fst(t),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
fan-theorem,
lifting-strict-less,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
istype-universe,
exception-not-value,
is-exception_wf,
strictness-apply,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
bottom-sqle,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
bottom_diverge,
exception-not-bottom,
dec-predicate_wf,
list_wf,
tbar_wf,
nat_wf,
set-value-type,
le_wf,
istype-int,
istype-nat,
int_seg_wf,
map_wf,
upto_wf,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
simple-fan-theorem,
simple_fan_theorem,
basic_bar_induction
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
instantiate,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
introduction,
isectElimination,
thin,
baseClosed,
Error :memTop,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueAdd,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
because_Cache,
universeIsType,
addExceptionCases,
exceptionSqequal,
inlFormation_alt,
imageMemberEquality,
imageElimination,
sqleReflexivity,
independent_functionElimination,
voidElimination,
isect_memberEquality_alt,
lambdaEquality_alt,
sqequalSqle,
divergentSqle,
callbyvalueLess,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
natural_numberEquality,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
cumulativity,
lessExceptionCases,
axiomSqleEquality,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueExceptionCases,
functionIsType,
universeEquality,
applyEquality,
isectIsType,
productIsType,
setElimination,
rename,
productEquality,
functionEquality
Latex:
fan-realizer \mmember{} \mforall{}[X:(\mBbbB{} List) {}\mrightarrow{} \mBbbP{}]
(tbar(\mBbbB{};X) {}\mRightarrow{} Decidable(X) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}k. (X map(f;upto(n)))))
Date html generated:
2020_05_20-AM-09_07_38
Last ObjectModification:
2020_01_10-PM-03_32_46
Theory : bar!induction
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