Nuprl Lemma : copath-cons_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[b:coW-dom(a.B[a];w)]. ∀[p:copath(a.B[a];coW-item(w;b))].
(copath-cons(b;p) ∈ copath(a.B[a];w))
Proof
Definitions occuring in Statement :
copath-cons: copath-cons(b;x),
copath: copath(a.B[a];w),
coW-item: coW-item(w;b),
coW-dom: coW-dom(a.B[a];w),
coW: coW(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,
copath: copath(a.B[a];w),
copath-cons: copath-cons(b;x),
coPath: coPath(a.B[a];w;n),
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
int_upper: {i...}
Lemmas referenced :
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
set_subtype_base,
int_subtype_base,
coPath_wf,
subtract_wf,
not-equal-2,
minus-minus,
coW-item_wf,
top_wf,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
coW-dom_wf,
copath_wf,
coW_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
extract_by_obid,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
because_Cache,
minusEquality,
equalityElimination,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
lambdaEquality,
functionExtensionality,
hypothesis_subsumption,
productEquality,
intEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[b:coW-dom(a.B[a];w)].
\mforall{}[p:copath(a.B[a];coW-item(w;b))].
(copath-cons(b;p) \mmember{} copath(a.B[a];w))
Date html generated:
2018_07_25-PM-01_39_57
Last ObjectModification:
2018_06_01-AM-10_01_56
Theory : co-recursion
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