Nuprl Lemma : iota'-comp
∀[I,J:Cname List]. ∀[f:name-morph(I;J)]. ((iota'(I) o (f)+) = (f o iota'(J)) ∈ name-morph(I;J+))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
iota': iota'(I),
name-morph-extend: (f)+,
name-morph: name-morph(I;J),
add-fresh-cname: I+,
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
uimplies: b supposing a,
name-morph-extend: (f)+,
iota': iota'(I),
name-comp: (f o g),
iota: iota(x),
uext: uext(g),
compose: f o g,
has-value: (a)↓,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
isname: isname(z),
implies: P ⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
all: ∀x:A. B[x],
eq-cname: eq-cname(x;y),
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
add-fresh-cname: I+
Lemmas referenced :
name-morphs-equal,
add-fresh-cname_wf,
name-comp_wf,
iota'_wf,
name-morph-extend_wf,
value-type-has-value,
coordinate_name_wf,
not_wf,
l_member_wf,
set-value-type,
coordinate_name-value-type,
fresh-cname_wf,
nameset_wf,
name-morph_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
le_int_wf,
btrue_wf,
iff_functionality_wrt_iff,
assert_wf,
le_wf,
true_wf,
iff_weakening_uiff,
assert_of_le_int,
iff_weakening_equal,
set_subtype_base,
istype-int,
int_subtype_base,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
equal_wf,
isname_wf,
extd-nameset_subtype,
cons_wf,
l_subset_right_cons_trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
sqequalRule,
independent_isectElimination,
callbyvalueReduce,
setEquality,
universeIsType,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
instantiate,
cumulativity,
natural_numberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
dependent_functionElimination,
intEquality,
closedConclusion,
voidElimination,
equalityIsType4,
equalityIsType3,
equalityIsType1,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp
Latex:
\mforall{}[I,J:Cname List]. \mforall{}[f:name-morph(I;J)]. ((iota'(I) o (f)+) = (f o iota'(J)))
Date html generated:
2019_11_05-PM-00_24_53
Last ObjectModification:
2018_11_08-PM-00_15_40
Theory : cubical!sets
Home
Index