Nuprl Lemma : coPath-extend_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[n:ℕ]. ∀[w:coW(A;a.B[a])]. ∀[p:coPath(a.B[a];w;n)]. ∀[t:coW-dom(a.B[a];coPath-at(n;w;p))].
(coPath-extend(n;p;t) ∈ coPath(a.B[a];w;n + 1))
Proof
Definitions occuring in Statement :
coPath-extend: coPath-extend(n;p;t),
coPath-at: coPath-at(n;w;p),
coPath: coPath(a.B[a];w;n),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
exists: ∃x:A. B[x],
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
top: Top,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
coPath-at: coPath-at(n;w;p),
coPath: coPath(a.B[a];w;n),
coPath-extend: coPath-extend(n;p;t),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
uiff_transitivity,
zero-mul,
add-mul-special,
not-equal-2,
not-le-2,
coPath_subtype,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
not_wf,
bnot_wf,
le_weakening,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
nat_wf,
le_weakening2,
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,
subtract_wf,
decidable__le,
le_wf,
false_wf,
top_wf,
coW-item_wf,
it_wf,
coW_wf,
coPath_wf,
coPath-at_wf,
coW-dom_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties
Rules used in proof :
impliesFunctionality,
productEquality,
multiplyEquality,
promote_hyp,
dependent_pairFormation,
equalityElimination,
baseClosed,
closedConclusion,
baseApply,
universeEquality,
functionEquality,
minusEquality,
intEquality,
voidEquality,
addEquality,
productElimination,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairEquality,
instantiate,
because_Cache,
functionExtensionality,
applyEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:coPath(a.B[a];w;n)].
\mforall{}[t:coW-dom(a.B[a];coPath-at(n;w;p))].
(coPath-extend(n;p;t) \mmember{} coPath(a.B[a];w;n + 1))
Date html generated:
2018_07_25-PM-01_38_42
Last ObjectModification:
2018_07_18-PM-07_07_06
Theory : co-recursion
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