Nuprl Lemma : copath-last_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:copath(a.B[a];w)].
copath-last(w;p) ∈ w':coW(A;a.B[a]) × coW-dom(a.B[a];w') supposing 0 < copath-length(p)
Proof
Definitions occuring in Statement :
copath-last: copath-last(w;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],
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
copath: copath(a.B[a];w),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
copath-last: copath-last(w;p),
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW_wf,
le_reflexive,
le-add-cancel-alt,
not-le-2,
le-add-cancel2,
not-equal-2,
not-lt-2,
decidable__lt,
length-copath-tl,
copath-tl_wf,
coW-item_wf,
nat_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
coW-dom_wf,
copath-hd_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_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,
copath_wf,
copath-length_wf,
le_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties
Rules used in proof :
universeEquality,
functionEquality,
independent_pairEquality,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
dependent_pairEquality,
equalityElimination,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_pairFormation,
unionElimination,
productElimination,
because_Cache,
applyEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
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])]. \mforall{}[p:copath(a.B[a];w)].
copath-last(w;p) \mmember{} w':coW(A;a.B[a]) \mtimes{} coW-dom(a.B[a];w') supposing 0 < copath-length(p)
Date html generated:
2018_07_25-PM-01_40_35
Last ObjectModification:
2018_07_24-AM-09_42_16
Theory : co-recursion
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