Nuprl Lemma : bbar-recursion_wf
∀[T:Type]. ∀[R:(T List) ⟶ 𝔹]. ∀[A:(T List) ⟶ ℙ]. ∀[b:∀s:{s:T List| ↑R[s]} . A[s]]. ∀[i:∀s:{s:T List| ¬↑R[s]}
((∀t:T. A[s @ [t]]) ⇒ A[s])].
∀[s:T List].
((∀alpha:ℕ ⟶ T. (↓∃n:ℕ. (↑R[s @ map(alpha;upto(n))]))) ⇒ (bbar-recursion(R;b;i;s) ∈ A[s]))
Proof
Definitions occuring in Statement :
bbar-recursion: bbar-recursion,
upto: upto(n),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
bbar-recursion: bbar-recursion,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
all_wf,
nat_wf,
squash_wf,
exists_wf,
assert_wf,
list_wf,
append_wf,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
upto_wf,
not_wf,
cons_wf,
nil_wf,
bool_wf,
subtype_rel_list,
top_wf,
append-nil,
decidable__assert,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
append_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
isectElimination,
thin,
functionEquality,
cumulativity,
hypothesisEquality,
lambdaEquality,
because_Cache,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
setEquality,
universeEquality,
voidElimination,
voidEquality,
barInduction,
unionElimination,
inlFormation,
inrFormation,
equalityElimination,
productElimination,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:(T List) {}\mrightarrow{} \mBbbB{}]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}]. \mforall{}[b:\mforall{}s:\{s:T List| \muparrow{}R[s]\} . A[s]].
\mforall{}[i:\mforall{}s:\{s:T List| \mneg{}\muparrow{}R[s]\} . ((\mforall{}t:T. A[s @ [t]]) {}\mRightarrow{} A[s])]. \mforall{}[s:T List].
((\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. (\muparrow{}R[s @ map(alpha;upto(n))]))) {}\mRightarrow{} (bbar-recursion(R;b;i;s) \mmember{} A[s]))
Date html generated:
2018_05_21-PM-10_17_40
Last ObjectModification:
2017_07_26-PM-06_36_26
Theory : bar!induction
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