Nuprl Lemma : poset-cat-arrow-cases
∀I:Cname List. ∀c1,c2:cat-ob(poset-cat(I)). ∀f:cat-arrow(poset-cat(I)) c1 c2.
((c1 = c2 ∈ cat-ob(poset-cat(I)))
∨ (∃y:{y:nameset(I)| (c1 y) = 0 ∈ ℕ2} . (c2 = flip(c1;y) ∈ cat-ob(poset-cat(I))))
∨ (∃c3:cat-ob(poset-cat(I))
∃g:cat-arrow(poset-cat(I)) c1 c3
∃h:cat-arrow(poset-cat(I)) c3 c2. ((¬(c1 = c3 ∈ cat-ob(poset-cat(I)))) ∧ (¬(c2 = c3 ∈ cat-ob(poset-cat(I)))))))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
respects-equality: respects-equality(S;T),
implies: P ⇒ Q,
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
name-morph: name-morph(I;J),
and: P ∧ Q,
not: ¬A,
false: False,
decidable: Dec(P),
cand: A c∧ B,
true: True,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
prop: ℙ,
subtract: n - m,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
guard: {T},
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
rev_uimplies: rev_uimplies(P;Q),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-it: exposed-it,
nameset: nameset(L),
pi2: snd(t),
cat-arrow: cat-arrow(C),
name-morph-flip: flip(f;y)
Lemmas referenced :
poset-cat-ob-cases,
nameset_wf,
int_seg_wf,
extd-nameset-respects-equality,
nil_wf,
coordinate_name_wf,
name-morph-flip_wf,
subtype_rel_self,
name-morph_wf,
cat-ob_wf,
poset-cat_wf,
cat-arrow_wf,
istype-void,
poset-cat-arrow-not-equal,
decidable__equal-poset-cat-ob,
extd-nameset_wf,
istype-assert,
isname_wf,
list_wf,
int_term_value_var_lemma,
itermVar_wf,
l_member_wf,
istype-le,
extd-nameset_subtype_int,
equal_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
decidable__equal-coordinate_name,
sq_stable__l_member,
sq_stable__le,
lelt_wf,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
subtype_base_sq,
assert_of_le_int,
subtract_wf,
le_int_wf,
assert_witness,
assert-eq-cname,
eqtt_to_assert,
eq-cname_wf,
extd-nameset-nil,
equal-wf-T-base,
member_wf,
not_wf,
bnot_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
intformand_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
unionElimination,
inlFormation_alt,
sqequalRule,
unionIsType,
productIsType,
setIsType,
universeIsType,
isectElimination,
equalityIstype,
natural_numberEquality,
applyEquality,
because_Cache,
baseClosed,
independent_functionElimination,
sqequalBase,
equalitySymmetry,
inhabitedIsType,
setElimination,
rename,
functionIsType,
equalityTransitivity,
voidElimination,
inrFormation_alt,
productElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
independent_pairFormation,
axiomEquality,
closedConclusion,
isect_memberEquality_alt,
int_eqEquality,
Error :memTop,
approximateComputation,
imageElimination,
imageMemberEquality,
applyLambdaEquality,
intEquality,
cumulativity,
instantiate,
promote_hyp,
independent_isectElimination,
equalityElimination,
hyp_replacement
Latex:
\mforall{}I:Cname List. \mforall{}c1,c2:cat-ob(poset-cat(I)). \mforall{}f:cat-arrow(poset-cat(I)) c1 c2.
((c1 = c2)
\mvee{} (\mexists{}y:\{y:nameset(I)| (c1 y) = 0\} . (c2 = flip(c1;y)))
\mvee{} (\mexists{}c3:cat-ob(poset-cat(I))
\mexists{}g:cat-arrow(poset-cat(I)) c1 c3
\mexists{}h:cat-arrow(poset-cat(I)) c3 c2. ((\mneg{}(c1 = c3)) \mwedge{} (\mneg{}(c2 = c3)))))
Date html generated:
2020_05_21-AM-10_52_47
Last ObjectModification:
2020_02_07-PM-08_16_58
Theory : cubical!sets
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