Nuprl Lemma : cu-filler-cases
∀I:Cname List. ∀J:nameset(I) List. ∀K:Cname List. ∀x:nameset(I). ∀f:name-morph(I-[x];K). ∀i:ℕ2.
∀box:open_box(c𝕌;I;J;x;i).
((J ∈ nameset(J) List)
∧ (nameset(J) ⊆r nameset(I-[x]))
∧ ((∃y:nameset(J) [((y ∈ J) ∧ (↑¬bisname(f y)) ∧ (l-first(y.¬bisname(f y);J) ~ inl y))])
∨ (((∀y∈J.¬↑¬bisname(f y)) ∧ (↑isr(l-first(y.¬bisname(f y);J))))
∧ (f[x:=fresh-cname(K)] ∈ name-morph(I;[fresh-cname(K) / K]))
∧ (nameset([x / J]) ⊆r nameset(I))
∧ (∀z:nameset([x / J]). (↑isname(f[x:=fresh-cname(K)] z)))
∧ (f[x:=fresh-cname(K)] ∈ nameset([x / J]) ⟶ nameset([fresh-cname(K) / K]))
∧ (x ∈ nameset([x / J]))
∧ (nameset([x / J]) ⊆r name-morph-domain(f[x:=fresh-cname(K)];I)))))
Proof
Definitions occuring in Statement :
cubical-universe: c𝕌,
open_box: open_box(X;I;J;x;i),
name-morph-domain: name-morph-domain(f;I),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
isname: isname(z),
fresh-cname: fresh-cname(I),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
l-first: l-first(x.f[x];L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
bnot: ¬bb,
assert: ↑b,
isr: isr(x),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
guard: {T},
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
not: ¬A,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
open_box: open_box(X;I;J;x;i),
l_subset: l_subset(T;as;bs),
false: False,
name-morph: name-morph(I;J),
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
istype: istype(T),
respects-equality: respects-equality(S;T),
uiff: uiff(P;Q),
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isr: isr(x),
sq_type: SQType(T),
bnot: ¬bb,
exists: ∃x:A. B[x],
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
extend-name-morph: f[z1:=z2],
l_member: (x ∈ l),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
sq_stable: SqStable(P),
squash: ↓T,
nat: ℕ,
l_all: (∀x∈L.P[x]),
rev_uimplies: rev_uimplies(P;Q),
ext-eq: A ≡ B
Lemmas referenced :
open_box_wf,
cubical-universe_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
name-morph_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
list_wf,
list-subtype,
l_member_wf,
member_singleton,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
member-list-diff,
l-first_wf,
bnot_wf,
isname_wf,
nameset_subtype_base,
istype-sqequal,
istype-assert,
l_all_wf2,
not_wf,
assert_wf,
isr_wf,
sq_exists_wf,
and_wf,
istype-void,
fresh-cname_wf,
extend-name-morph_wf,
fresh-cname-not-member2,
respects-equality-set,
extd-nameset_wf,
all_wf,
equal_wf,
subtype-respects-equality,
subtype_rel_set,
subtype_rel_dep_function,
nameset_subtype_cons_diff,
subtype_rel_wf,
respects-equality-set-trivial2,
assert-isname,
name-morph-domain_wf,
cons_member,
subtype_base_sq,
list-diff-subset,
nameset_subtype,
iff_weakening_uiff,
assert-bnot,
bool_wf,
bool_cases_sqequal,
bool_subtype_base,
eqff_to_assert,
assert-eq-cname,
eqtt_to_assert,
eq-cname_wf,
isname-name,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
int_seg_properties,
nat_properties,
sq_stable__le,
select_wf,
istype-le,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__assert,
length_wf,
istype-less_than,
assert_of_bnot,
name-morph-domain_subtype,
istype-inr-sqeq-inl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
independent_pairFormation,
hypothesis,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
closedConclusion,
intEquality,
equalityIsType2,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality_alt,
hyp_replacement,
productIsType,
equalityIsType4,
baseApply,
baseClosed,
applyLambdaEquality,
voidElimination,
equalityIstype,
unionElimination,
inlFormation_alt,
dependent_set_memberFormation_alt,
setIsType,
voidEquality,
inlEquality_alt,
functionEquality,
functionIsType,
functionExtensionality,
inrFormation_alt,
cumulativity,
equalityIsType1,
promote_hyp,
equalityIsType3,
dependent_pairFormation_alt,
equalityElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
imageElimination,
imageMemberEquality
Latex:
\mforall{}I:Cname List. \mforall{}J:nameset(I) List. \mforall{}K:Cname List. \mforall{}x:nameset(I). \mforall{}f:name-morph(I-[x];K). \mforall{}i:\mBbbN{}2.
\mforall{}box:open\_box(c\mBbbU{};I;J;x;i).
((J \mmember{} nameset(J) List)
\mwedge{} (nameset(J) \msubseteq{}r nameset(I-[x]))
\mwedge{} ((\mexists{}y:nameset(J) [((y \mmember{} J) \mwedge{} (\muparrow{}\mneg{}\msubb{}isname(f y)) \mwedge{} (l-first(y.\mneg{}\msubb{}isname(f y);J) \msim{} inl y))])
\mvee{} (((\mforall{}y\mmember{}J.\mneg{}\muparrow{}\mneg{}\msubb{}isname(f y)) \mwedge{} (\muparrow{}isr(l-first(y.\mneg{}\msubb{}isname(f y);J))))
\mwedge{} (f[x:=fresh-cname(K)] \mmember{} name-morph(I;[fresh-cname(K) / K]))
\mwedge{} (nameset([x / J]) \msubseteq{}r nameset(I))
\mwedge{} (\mforall{}z:nameset([x / J]). (\muparrow{}isname(f[x:=fresh-cname(K)] z)))
\mwedge{} (f[x:=fresh-cname(K)] \mmember{} nameset([x / J]) {}\mrightarrow{} nameset([fresh-cname(K) / K]))
\mwedge{} (x \mmember{} nameset([x / J]))
\mwedge{} (nameset([x / J]) \msubseteq{}r name-morph-domain(f[x:=fresh-cname(K)];I)))))
Date html generated:
2019_11_06-PM-00_54_27
Last ObjectModification:
2018_12_10-PM-03_15_12
Theory : cubical!sets
Home
Index