Nuprl Lemma : poset-cat-dist-add
∀[I:Cname List]. ∀[x,y,z:cat-ob(poset-cat(I))].
(poset-cat-dist(I;x;z) = (poset-cat-dist(I;x;y) + poset-cat-dist(I;y;z)) ∈ ℤ) supposing
((cat-arrow(poset-cat(I)) x y) and
(cat-arrow(poset-cat(I)) y z))
Proof
Definitions occuring in Statement :
poset-cat-dist: poset-cat-dist(I;x;y),
poset-cat: poset-cat(J),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
poset-cat-dist: poset-cat-dist(I;x;y),
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
cat-ob: cat-ob(C),
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
prop: ℙ,
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eq_int: (i =z j),
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
cat-arrow_wf,
poset-cat_wf,
cat-ob_wf,
list_wf,
coordinate_name_wf,
equal-wf-T-base,
extd-nameset_subtype_int,
nil_wf,
nameset_wf,
assert_of_le_int,
subtype_base_sq,
int_subtype_base,
extd-nameset-nil,
int_seg_wf,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
le_wf,
equal_wf,
intformless_wf,
int_formula_prop_less_lemma,
list_induction,
all_wf,
name-morph_wf,
length_wf,
filter_wf5,
l_member_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
filter_nil_lemma,
length_of_nil_lemma,
filter_cons_lemma,
cons_wf,
name-morph_subtype,
nameset_subtype,
l_subset_right_cons_trivial,
cons_member,
equal-wf-base,
int_seg_cases,
length_of_cons_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
intformimplies_wf,
int_formual_prop_imp_lemma,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
applyEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
intEquality,
setElimination,
rename,
baseClosed,
independent_pairFormation,
dependent_functionElimination,
productElimination,
independent_isectElimination,
instantiate,
cumulativity,
independent_functionElimination,
natural_numberEquality,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
productEquality,
dependent_set_memberEquality,
equalityElimination,
setEquality,
addEquality,
inlFormation,
hypothesis_subsumption,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y,z:cat-ob(poset-cat(I))].
(poset-cat-dist(I;x;z) = (poset-cat-dist(I;x;y) + poset-cat-dist(I;y;z))) supposing
((cat-arrow(poset-cat(I)) x y) and
(cat-arrow(poset-cat(I)) y z))
Date html generated:
2017_10_05-AM-10_28_23
Last ObjectModification:
2017_07_28-AM-11_23_42
Theory : cubical!sets
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