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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
copath: copath(a.B[a];w),
copath-at: copath-at(w;p),
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
coPath: coPath(a.B[a];w;n),
coPath-at: coPath-at(n;w;p),
not: ¬A,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
coW-item: coW-item(w;b),
pi1: fst(t),
coW-dom: coW-dom(a.B[a];w),
ext-eq: A ≡ B,
decidable: Dec(P),
or: P ∨ Q,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
W-subtype-coW,
copath_wf,
W_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
subtract-1-ge-0,
istype-top,
eq_int_wf,
equal-wf-base,
bool_wf,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
false_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-universe,
W-ext,
coW-dom_wf,
coPath_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
coW-item_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :universeIsType,
hypothesis,
because_Cache,
productElimination,
instantiate,
cumulativity,
Error :functionIsType,
universeEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
intEquality,
Error :equalityIsType4,
unionElimination,
equalityElimination,
independent_pairFormation,
Error :equalityIsType1,
hypothesis_subsumption,
promote_hyp,
functionExtensionality,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
addEquality,
Error :isect_memberEquality_alt,
minusEquality
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:
2019_06_20-PM-00_56_28
Last ObjectModification:
2019_01_02-PM-01_33_16
Theory : co-recursion-2
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