Nuprl Lemma : pW-sup_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. ∀[par:P]. ∀[a:A[par]].
∀[f:b:B[par;a] ⟶ (pW C[par;a;b])].
(pW-sup(a;f) ∈ pW par)
Proof
Definitions occuring in Statement :
pW-sup: pW-sup(a;f),
param-W: pW,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pW-sup: pW-sup(a;f),
subtype_rel: A ⊆r B,
param-W: pW,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pcw-path: Path,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
exists: ∃x:A. B[x],
ext-family: F ≡ G,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pcw-step-agree: StepAgree(s;p1;w),
spreadn: spread3,
ext-eq: A ≡ B,
pi1: fst(t),
pi2: snd(t),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
true: True,
pcw-steprel: StepRel(s1;s2),
label: ...$L... t,
istype: istype(T),
pcw-partial: pcw-partial(path;n),
pcw-pp-barred: Barred(pp),
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
less_than: a < b
Lemmas referenced :
param-co-W_wf,
param-W_wf,
istype-universe,
pcw-path_wf,
pcw-step-agree_wf,
istype-false,
le_wf,
squash_wf,
exists_wf,
nat_wf,
pcw-pp-barred_wf,
pcw-partial_wf,
param-co-W-ext,
ext-eq_inversion,
subtype_rel_weakening,
equal_functionality_wrt_subtype_rel2,
subtype_rel-equal,
pcw-path-shift,
pcw-steprel_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-void,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
equal_wf,
true_wf,
subtype_rel_dep_function,
iff_weakening_equal,
subtype_rel_self,
subtype_rel_wf,
decidable__lt,
not-lt-2,
less-iff-le,
general_arith_equation1,
less_than_wf,
assert_wf,
bfalse_wf,
btrue_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
Error :universeIsType,
applyEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
because_Cache,
Error :isect_memberEquality_alt,
universeEquality,
Error :dependent_pairEquality_alt,
Error :functionExtensionality_alt,
setElimination,
rename,
Error :setIsType,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
Error :lambdaFormation_alt,
dependent_functionElimination,
productEquality,
functionEquality,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
hypothesis_subsumption,
Error :equalityIsType1,
independent_functionElimination,
unionElimination,
applyLambdaEquality,
addEquality,
voidElimination,
minusEquality,
hyp_replacement,
Error :productIsType,
promote_hyp,
instantiate,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P]. \mforall{}[par:P].
\mforall{}[a:A[par]]. \mforall{}[f:b:B[par;a] {}\mrightarrow{} (pW C[par;a;b])].
(pW-sup(a;f) \mmember{} pW par)
Date html generated:
2019_06_20-PM-00_35_45
Last ObjectModification:
2018_10_06-AM-11_20_33
Theory : co-recursion
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