Nuprl Lemma : copathAgree_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[x,y:copath(a.B[a];w)]. (copathAgree(a.B[a];w;x;y) ∈ ℙ)
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
guard: {T},
uiff: uiff(P;Q),
gt: i > j
Lemmas referenced :
coPath_wf,
top_wf,
less_than_wf,
coPathAgree_wf,
coPath_subtype,
le_weakening2,
not-gt-2,
copath_wf,
coW_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
spreadEquality,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairEquality,
hypothesisEquality,
extract_by_obid,
isectElimination,
lambdaEquality,
applyEquality,
hypothesis,
setElimination,
rename,
lessCases,
independent_pairFormation,
baseClosed,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
isect_memberEquality,
because_Cache,
voidElimination,
voidEquality,
lambdaFormation,
imageElimination,
independent_functionElimination,
independent_isectElimination,
dependent_functionElimination,
axiomEquality,
instantiate,
cumulativity,
functionEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[x,y:copath(a.B[a];w)].
(copathAgree(a.B[a];w;x;y) \mmember{} \mBbbP{})
Date html generated:
2019_06_20-PM-00_56_48
Last ObjectModification:
2019_01_02-PM-01_33_51
Theory : co-recursion-2
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