Nuprl Lemma : maximal-copath_wf
∀[A:𝕌']. ∀B:A ⟶ Type. ∀w:coW(A;a.B[a]). (maximal-copath(a.B[a];w) ∈ Type)
Proof
Definitions occuring in Statement :
maximal-copath: maximal-copath(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
true: True,
top: Top,
subtract: n - m,
squash: ↓T,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
maximal-copath: maximal-copath(a.B[a];w),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW_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,
decidable__le,
copathAgree_wf,
subtract_wf,
false_wf,
int_seg_subtype_nat,
copath-length_wf,
equal_wf,
int_seg_wf,
all_wf,
copath_wf,
nat_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
universeEquality,
instantiate,
minusEquality,
voidEquality,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
productElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
independent_isectElimination,
intEquality,
rename,
setElimination,
natural_numberEquality,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
functionEquality,
setEquality,
sqequalRule,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}w:coW(A;a.B[a]). (maximal-copath(a.B[a];w) \mmember{} Type)
Date html generated:
2018_07_25-PM-01_49_12
Last ObjectModification:
2018_07_25-AM-10_05_42
Theory : co-recursion
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