Nuprl Lemma : W-bars_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[w:co-W(A;a.B[a])]. ∀p:ℕ ⟶ a:A ⟶ (B[a]?). (W-bars(w;p) ∈ ℙ)
Proof
Definitions occuring in Statement :
W-bars: W-bars(w;p),
co-W: co-W(A;a.B[a]),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
W-bars: W-bars(w;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,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
squash_wf,
exists_wf,
nat_wf,
assert_wf,
isr_wf,
co-W_wf,
unit_wf2,
W-select_wf,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
upto_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
cumulativity,
hypothesisEquality,
applyEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
functionEquality,
unionEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:co-W(A;a.B[a])]. \mforall{}p:\mBbbN{} {}\mrightarrow{} a:A {}\mrightarrow{} (B[a]?). (W-bars(w;p) \mmember{} \mBbbP{})
Date html generated:
2016_05_15-PM-10_06_42
Last ObjectModification:
2015_12_27-PM-05_50_29
Theory : bar!induction
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