Nuprl Lemma : coPath-at_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[n:ℕ]. ∀[w:coW(A;a.B[a])]. ∀[p:coPath(a.B[a];w;n)]. (coPath-at(n;w;p) ∈ coW(A;a.B[a]))
Proof
Definitions occuring in Statement :
coPath-at: coPath-at(n;w;p),
coPath: coPath(a.B[a];w;n),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
coPath-at: coPath-at(n;w;p),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
coPath: coPath(a.B[a];w;n),
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
coPath_wf,
istype-universe,
coW_wf,
istype-false,
le_wf,
subtract-1-ge-0,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
le_weakening2,
nat_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
coW-item_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
instantiate,
cumulativity,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType2,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
Error :equalityIsType1,
Error :functionIsType,
universeEquality,
intEquality,
Error :equalityIsType4
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)].
(coPath-at(n;w;p) \mmember{} coW(A;a.B[a]))
Date html generated:
2019_06_20-PM-00_56_17
Last ObjectModification:
2019_01_02-PM-01_33_06
Theory : co-recursion-2
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