Nuprl Lemma : bar-recursion_wf
∀[T:Type]. ∀[R,A:(T List) ⟶ ℙ]. ∀[d:∀s:T List. Dec(R[s])]. ∀[b:∀s:T List. (R[s] ⇒ A[s])]. ∀[i:∀s:T List
((∀t:T. A[s @ [t]])
⇒ A[s])].
∀[s:T List].
((∀alpha:ℕ ⟶ T. (↓∃n:ℕ. R[s @ map(alpha;upto(n))])) ⇒ (bar-recursion(d;b;i;s) ∈ A[s]))
Proof
Definitions occuring in Statement :
bar-recursion: bar-recursion,
upto: upto(n),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
member: t ∈ T,
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),
bar-recursion: bar-recursion,
or: P ∨ Q
Lemmas referenced :
all_wf,
nat_wf,
squash_wf,
exists_wf,
append_wf,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
upto_wf,
list_wf,
cons_wf,
nil_wf,
decidable_wf,
subtype_rel_list,
top_wf,
append-nil,
equal_wf,
append_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
functionEquality,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality,
voidElimination,
voidEquality,
barInduction,
unionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R,A:(T List) {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\mforall{}s:T List. Dec(R[s])]. \mforall{}[b:\mforall{}s:T List. (R[s] {}\mRightarrow{} A[s])].
\mforall{}[i:\mforall{}s:T List. ((\mforall{}t:T. A[s @ [t]]) {}\mRightarrow{} A[s])]. \mforall{}[s:T List].
((\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. R[s @ map(alpha;upto(n))])) {}\mRightarrow{} (bar-recursion(d;b;i;s) \mmember{} A[s]))
Date html generated:
2016_05_15-PM-10_05_00
Last ObjectModification:
2015_12_27-PM-05_51_05
Theory : bar!induction
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