Nuprl Lemma : face-map-comp-id
∀A,B:Cname List. ∀g:name-morph(A;B). ∀x:nameset(A). ∀i:ℕ2.
((x:=i) o g) = g ∈ name-morph(A;B) supposing (¬↑isname(g x)) ∧ ((g x) = i ∈ ℕ2)
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
name-morph: name-morph(I;J),
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
istype: istype(T),
name-comp: (f o g),
compose: f o g,
face-map: (x:=i),
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
uext: uext(g),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
istype-assert,
isname_wf,
istype-void,
not-assert-isname,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
int_seg_wf,
nameset_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
name-morphs-equal,
uext_wf,
ifthenelse_wf,
eq_int_wf,
extd-nameset_wf,
nsub2_subtype_extd-nameset,
nameset_subtype_extd-nameset,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not_assert_elim,
equal_functionality_wrt_subtype_rel2,
isname-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
hypothesis,
sqequalRule,
productIsType,
functionIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
because_Cache,
productElimination,
independent_isectElimination,
equalityIstype,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
imageElimination,
sqequalBase,
universeIsType,
functionExtensionality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination
Latex:
\mforall{}A,B:Cname List. \mforall{}g:name-morph(A;B). \mforall{}x:nameset(A). \mforall{}i:\mBbbN{}2.
((x:=i) o g) = g supposing (\mneg{}\muparrow{}isname(g x)) \mwedge{} ((g x) = i)
Date html generated:
2020_05_21-AM-10_48_28
Last ObjectModification:
2019_12_08-PM-07_06_43
Theory : cubical!sets
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