Nuprl Lemma : edge-arrows-commute_wf
∀C:SmallCategory. ∀I:Cname List. ∀L:name-morph(I;[]) ⟶ cat-ob(C). ∀E:i:nameset(I)
⟶ c:{c:name-morph(I;[])| (c i) = 0 ∈ ℕ2}
⟶ (cat-arrow(C) (L c) (L flip(c;i))).
(edge-arrows-commute(C;I;L;E) ∈ ℙ)
Proof
Definitions occuring in Statement :
edge-arrows-commute: edge-arrows-commute(C;I;L;E),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
nil: [],
list: T List,
int_seg: {i..j-},
prop: ℙ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
edge-arrows-commute: edge-arrows-commute(C;I;L;E),
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
name-morph-flip: flip(f;y),
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nameset_wf,
equal-wf-T-base,
int_seg_wf,
not_wf,
equal_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
extd-nameset-nil,
cat-arrow_wf,
name-morph-flip_wf,
cat-ob_wf,
list_wf,
small-category_wf,
cat-comp_wf,
subtype_rel-equal,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
name-morph-flips-commute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
functionEquality,
because_Cache,
setEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
setElimination,
rename,
baseClosed,
functionIsType,
universeIsType,
setIsType,
equalityIsType3,
dependent_set_memberEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_pairFormation,
productIsType,
equalityIsType1,
applyLambdaEquality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
lambdaEquality_alt,
closedConclusion
Latex:
\mforall{}C:SmallCategory. \mforall{}I:Cname List. \mforall{}L:name-morph(I;[]) {}\mrightarrow{} cat-ob(C). \mforall{}E:i:nameset(I)
{}\mrightarrow{} c:\{c:name-morph(I;[])|
(c i) = 0\}
{}\mrightarrow{} (cat-arrow(C) (L c)
(L flip(c;i))).
(edge-arrows-commute(C;I;L;E) \mmember{} \mBbbP{})
Date html generated:
2019_11_05-PM-00_32_51
Last ObjectModification:
2018_11_07-AM-11_53_14
Theory : cubical!sets
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