Nuprl Lemma : copath-hd_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:copath(a.B[a];w)].
copath-hd(p) ∈ coW-dom(a.B[a];w) supposing 0 < copath-length(p)
Proof
Definitions occuring in Statement :
copath-hd: copath-hd(p),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
copath-hd: copath-hd(p),
copath: copath(a.B[a];w),
pi1: fst(t),
pi2: snd(t),
copath-length: copath-length(p),
coPath: coPath(a.B[a];w;n),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
false: False,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
less_than_wf,
copath-length_wf,
nat_wf,
copath_wf,
coW_wf,
eq_int_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
isect_memberEquality,
because_Cache,
instantiate,
cumulativity,
functionEquality,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
intEquality,
baseClosed,
spreadEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
impliesFunctionality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:copath(a.B[a];w)].
copath-hd(p) \mmember{} coW-dom(a.B[a];w) supposing 0 < copath-length(p)
Date html generated:
2018_07_25-PM-01_39_41
Last ObjectModification:
2018_06_01-AM-11_27_14
Theory : co-recursion
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