Nuprl Lemma : poset_functor_extend-face-map
∀[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)))].
∀[y:nameset(I)]. ∀[a:ℕ2]. ∀[c1,c2:name-morph(I-[y];[])].
poset_functor_extend(C;I;L;E;((y:=a) o c1);((y:=a) o c2))
= poset_functor_extend(C;I-[y];L o (λf.((y:=a) o f));λz,f. (E z ((y:=a) o f));c1;c2)
∈ (cat-arrow(C) (L ((y:=a) o c1)) (L ((y:=a) o c2)))
supposing ∀x:nameset(I-[y]). ((c1 x) ≤ (c2 x))
Proof
Definitions occuring in Statement :
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
name-morph-flip: flip(f;y),
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
compose: f o g,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
uimplies: b supposing a,
nameset: nameset(L),
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
coordinate_name: Cname,
int_upper: {i...},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: 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,
int_seg: {i..j-},
decidable: Dec(P),
uext: uext(g),
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True
Lemmas referenced :
poset_functor_extend-face-map1,
int_seg_wf,
nameset_wf,
set_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
equal-wf-T-base,
extd-nameset-nil,
cat-arrow_wf,
name-morph-flip_wf,
cat-ob_wf,
list_wf,
small-category_wf,
all_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
le_wf,
name-comp_wf,
face-map_wf2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
int_seg_properties,
false_wf,
int_seg_subtype,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
member-list-diff,
intformeq_wf,
intformnot_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
member_singleton,
l_member_wf,
not_wf,
squash_wf,
true_wf,
uext-ap-name,
extd-nameset_subtype_int,
iff_weakening_equal,
face-map-idempotent,
face-map-comp-trivial,
member_wf,
poset_functor_extend_wf,
nameset_subtype,
list-diff-subset,
subtype_rel-equal,
name-morph-flip-face-map1,
subtype_rel_self
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
functionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
because_Cache,
baseClosed,
lambdaFormation,
functionExtensionality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
independent_pairFormation,
hypothesis_subsumption,
addEquality,
int_eqEquality,
voidEquality,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addLevel,
impliesFunctionality,
productEquality,
universeEquality,
hyp_replacement,
andLevelFunctionality,
impliesLevelFunctionality,
equalityUniverse,
levelHypothesis,
setEquality
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)))].
\mforall{}[y:nameset(I)]. \mforall{}[a:\mBbbN{}2]. \mforall{}[c1,c2:name-morph(I-[y];[])].
poset\_functor\_extend(C;I;L;E;((y:=a) o c1);((y:=a) o c2))
= poset\_functor\_extend(C;I-[y];L o (\mlambda{}f.((y:=a) o f));\mlambda{}z,f. (E z ((y:=a) o f));c1;c2)
supposing \mforall{}x:nameset(I-[y]). ((c1 x) \mleq{} (c2 x))
Date html generated:
2017_10_05-AM-10_30_45
Last ObjectModification:
2017_07_28-AM-11_24_28
Theory : cubical!sets
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