Nuprl Lemma : coW-wfdd_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. (coW-wfdd(a.B[a];w) ∈ ℙ)
Proof
Definitions occuring in Statement :
coW-wfdd: coW-wfdd(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
top: Top,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
coW-wfdd: coW-wfdd(a.B[a];w),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW_wf,
not_wf,
exists_wf,
squash_wf,
copathAgree_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
copath-length_wf,
equal_wf,
copath_wf,
nat_wf,
all_wf
Rules used in proof :
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
voidEquality,
isect_memberEquality,
minusEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
addEquality,
dependent_set_memberEquality,
rename,
setElimination,
intEquality,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesisEquality,
cumulativity,
hypothesis,
functionEquality,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. (coW-wfdd(a.B[a];w) \mmember{} \mBbbP{})
Date html generated:
2018_07_25-PM-01_41_54
Last ObjectModification:
2018_07_23-PM-03_19_11
Theory : co-recursion
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