Nuprl Lemma : copath-at-W
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:W(A;a.B[a])]. ∀[p:copath(a.B[a];w)]. (copath-at(w;p) ∈ W(A;a.B[a]))
Proof
Definitions occuring in Statement :
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
W: W(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
pi2: snd(t),
coW-item: coW-item(w;b),
pi1: fst(t),
coW-dom: coW-dom(a.B[a];w),
ext-eq: A ≡ B,
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-it: exposed-it,
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
coPath: coPath(a.B[a];w;n),
coPath-at: coPath-at(n;w;p),
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-item_wf,
le_wf,
not-le-2,
coPath_wf,
coW-dom_wf,
coW_wf,
W-ext,
equal_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
top_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
W_wf,
copath_wf,
W-subtype-coW
Rules used in proof :
dependent_set_memberEquality,
productEquality,
hypothesis_subsumption,
promote_hyp,
impliesFunctionality,
equalityElimination,
baseClosed,
closedConclusion,
baseApply,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_pairFormation,
unionElimination,
axiomEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
dependent_functionElimination,
universeEquality,
functionEquality,
cumulativity,
instantiate,
equalitySymmetry,
equalityTransitivity,
productElimination,
because_Cache,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:W(A;a.B[a])]. \mforall{}[p:copath(a.B[a];w)]. (copath-at(w;p) \mmember{} W(A;a.B[a]))
Date html generated:
2018_07_25-PM-01_39_03
Last ObjectModification:
2018_07_19-AM-10_18_55
Theory : co-recursion
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