Nuprl Lemma : cubical-interval-filler-fills
Kan-filler(cubical-interval();cubical-interval-filler())
Proof
Definitions occuring in Statement :
cubical-interval-filler: cubical-interval-filler(),
Kan-filler: Kan-filler(X;filler),
cubical-interval: cubical-interval()
Definitions unfolded in proof :
Kan-filler: Kan-filler(X;filler),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
fills-open_box: fills-open_box(X;I;bx;cube),
fills-faces: fills-faces(X;I;bx;L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
open_box: open_box(X;I;J;x;i),
cubical-interval-filler: cubical-interval-filler(),
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
cons: [a / b],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
sq_type: SQType(T),
guard: {T},
true: True,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
ext-eq: A ≡ B,
length: ||as||,
list_ind: list_ind,
nil: [],
is-face: is-face(X;I;bx;f),
select: L[n],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
cubical-interval: cubical-interval(),
I-cube: X(I),
cube-set-restriction: f(s),
pi2: snd(t),
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
name-morph: name-morph(I;J),
sq_stable: SqStable(P),
face-name: face-name(f),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
face-direction: direction(f),
face-dimension: dimension(f),
nequal: a ≠ b ∈ T ,
isname: isname(z),
l_member: (x ∈ l),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cubical-interval-ap: u(L),
istype: istype(T),
respects-equality: respects-equality(S;T),
l_subset: l_subset(T;as;bs)
Lemmas referenced :
open_box_wf,
cubical-interval_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
length_wf,
I-face_wf,
null_wf3,
top_wf,
istype-void,
eqtt_to_assert,
assert_of_null,
list-cases,
product_subtype_list,
list_ind_nil_lemma,
list_ind_cons_lemma,
subtype_base_sq,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
nil_wf,
nameset_subtype,
hd_wf,
l_member_wf,
list-subtype,
listp_properties,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
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,
open_box-nil,
reduce_hd_cons_lemma,
decidable__equal_int,
int_seg_properties,
int_seg_subtype_special,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
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,
non_neg_length,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
face-cube_wf,
face-dimension_wf,
face-direction_wf,
I-cube_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
ob_pair_lemma,
name-morph-ext,
name-comp_wf,
name-morph_wf,
face-map_wf2,
name-morph_subtype,
list-diff-subset,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
member-list-diff,
member_singleton,
isname-nameset,
select_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
extd-nameset-nil,
get_face_wf,
pi2_wf,
pi1_wf_top,
set_subtype_base,
le_wf,
nameset_subtype_base,
lelt_wf,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
false_wf,
assert_of_le_int,
iff_weakening_equal,
istype-le,
pairwise-implies,
not_wf,
equal_wf,
face-name_wf,
nat_properties,
select_member,
cubical-interval-ap_wf,
squash_wf,
true_wf,
adjacent-compatible-iff,
subtype_rel_wf,
istype-universe,
list-diff2-sym,
eq_int_eq_false,
subtype-respects-equality,
subtype_rel_self,
istype-assert,
face-map-comp-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
natural_numberEquality,
productElimination,
isect_memberEquality_alt,
voidElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
instantiate,
cumulativity,
intEquality,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
because_Cache,
baseClosed,
setEquality,
independent_pairFormation,
addEquality,
minusEquality,
dependent_set_memberEquality_alt,
approximateComputation,
int_eqEquality,
Error :memTop,
imageElimination,
functionExtensionality,
imageMemberEquality,
independent_pairEquality,
hyp_replacement,
sqequalBase,
productEquality,
productIsType,
universeEquality
Latex:
Kan-filler(cubical-interval();cubical-interval-filler())
Date html generated:
2020_05_21-AM-10_52_36
Last ObjectModification:
2019_12_08-PM-07_05_51
Theory : cubical!sets
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