Nuprl Lemma : Kan-interval_wf
Kan-interval() ∈ KanCubicalSet
Proof
Definitions occuring in Statement :
Kan-interval: Kan-interval(),
Kan-cubical-set: KanCubicalSet,
member: t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
Kan-cubical-set: KanCubicalSet,
Kan-interval: Kan-interval(),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
and: P ∧ Q,
cand: A c∧ B,
uniform-Kan-filler: uniform-Kan-filler(X;filler),
all: ∀x:A. B[x],
implies: P ⇒ Q,
name-morph: name-morph(I;J),
prop: ℙ,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
nat: ℕ,
squash: ↓T,
sq_stable: SqStable(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
open_box: open_box(X;I;J;x;i),
less_than: a < b,
I-face: I-face(X;I),
pi1: fst(t),
cubical-interval: cubical-interval(),
cubical-interval-filler: cubical-interval-filler(),
I-cube: X(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
cons: [a / b],
l_exists: (∃x∈L. P[x]),
open_box_image: open_box_image(X;I;K;f;bx),
cube-set-restriction: f(s),
pi2: snd(t),
face-image: face-image(X;I;K;f;face),
spreadn: spread3,
face-cube: cube(f),
le: A ≤ B,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
listp: A List+,
name-comp: (f o g),
compose: f o g,
uext: uext(g),
face-name: face-name(f),
cubical-interval-ap: u(L)
Lemmas referenced :
cubical-interval_wf,
cubical-interval-filler_wf,
list_wf,
coordinate_name_wf,
nameset_wf,
int_seg_wf,
open_box_wf,
subtype_rel_list,
I-cube_wf,
cubical-interval-filler-fills,
istype-assert,
isname_wf,
l_member_wf,
name-morph_wf,
Kan-filler_wf,
uniform-Kan-filler_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
select_wf,
sq_stable__le,
nat_properties,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
void_wf,
assert-isname,
list-subtype,
map_wf,
cons_wf,
name-morph-domain_wf,
assert_wf,
subtype_rel_functionality_wrt_iff,
ext-eq_weakening,
name-morph-domain_subtype,
cons_member,
length_wf,
I-face_wf,
sq_stable__assert,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
ob_pair_lemma,
null-map,
null_wf3,
top_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
iff_weakening_uiff,
equal-wf-T-base,
list-cases,
product_subtype_list,
length_of_nil_lemma,
reduce_hd_cons_lemma,
map_cons_lemma,
name-comp_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
name-morph_subtype,
nameset_subtype,
list-diff-subset,
map_nil_lemma,
extd-nameset_wf,
extd-nameset_subtype_base,
hd_wf,
listp_properties,
length_of_cons_lemma,
length_wf_nat,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-less_than,
hd_member,
not-ge-2,
add-swap,
le-add-cancel2,
extd-nameset-nil,
get_face_image,
isname-nameset,
lelt_wf,
get_face_wf,
equal_wf,
cube-set-restriction_wf,
face-cube_wf,
face-dimension_wf,
cubical-interval-ap_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
functionIsType,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
independent_pairFormation,
lambdaFormation_alt,
because_Cache,
productElimination,
productIsType,
instantiate,
cumulativity,
intEquality,
closedConclusion,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
functionExtensionality,
functionExtensionality_alt,
equalityIsType1,
setEquality,
equalityElimination,
equalityIsType3,
promote_hyp,
hypothesis_subsumption,
addEquality,
minusEquality,
hyp_replacement
Latex:
Kan-interval() \mmember{} KanCubicalSet
Date html generated:
2019_11_05-PM-00_31_40
Last ObjectModification:
2018_11_08-PM-01_16_07
Theory : cubical!sets
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