Nuprl Lemma : cube-set-restriction-face-map
∀[X:CubicalSet]. ∀[I,K:Cname List]. ∀[f:name-morph(I;K)]. ∀[s:X(I)]. ∀[c:ℕ2]. ∀[j:name-morph-domain(f;I)].
((f j:=c)(f(s)) = f((j:=c)(s)) ∈ X(K-[f j]))
Proof
Definitions occuring in Statement :
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
face-map: (x:=i),
name-morph-domain: name-morph-domain(f;I),
name-morph: name-morph(I;J),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-set: CubicalSet,
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
nameset: nameset(L),
I-cube: X(I),
top: Top,
compose: f o g,
prop: ℙ,
squash: ↓T,
cube-set-restriction: f(s),
pi2: snd(t),
name-morph-domain: name-morph-domain(f;I),
ext-eq: A ≡ B,
uimplies: b supposing a,
true: True,
face-map: (x:=i),
name-comp: (f o g),
uext: uext(g),
implies: P ⇒ Q,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
coordinate_name: Cname,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
isname: isname(z),
int_seg: {i..j-},
decidable: Dec(P),
le_int: i ≤z j,
lt_int: i <z j,
lelt: i ≤ j < k,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
name-morph_subtype_domain,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
nil_wf,
face-map_wf2,
ob_pair_lemma,
istype-void,
name-morph-domain_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
nameset_wf,
list_wf,
name-morph-domain_subtype,
name-morph_subtype_remove1,
name-morphs-equal,
name-comp_wf,
assert-isname,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
int_seg_properties,
bfalse_wf,
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,
nsub2_subtype_extd-nameset,
iff_imp_equal_bool,
le_int_wf,
btrue_wf,
iff_functionality_wrt_iff,
assert_wf,
le_wf,
iff_weakening_uiff,
assert_of_le_int,
iff_weakening_equal,
nameset_subtype_base,
member-list-diff,
intformeq_wf,
intformnot_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
set_subtype_base,
member_singleton,
l_member_wf,
name-morph_wf,
false_wf,
istype-le,
isname-nameset,
eq_int_eq_true,
subtype_rel_self,
bnot_wf,
assert_elim,
btrue_neq_bfalse,
isname_wf,
extd-nameset_subtype_int,
equal-wf-T-base,
not_wf,
istype-assert,
uiff_transitivity,
iff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
setElimination,
rename,
productElimination,
dependent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
inhabitedIsType,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
universeIsType,
axiomEquality,
isectIsTypeImplies,
natural_numberEquality,
applyLambdaEquality,
hyp_replacement,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
lambdaEquality,
functionExtensionality,
voidEquality,
isect_memberEquality,
comment,
independent_isectElimination,
lambdaFormation_alt,
equalityIsType1,
independent_functionElimination,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIsType3,
promote_hyp,
cumulativity,
intEquality,
hypothesis_subsumption,
approximateComputation,
int_eqEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
equalityIsType2,
closedConclusion,
productIsType,
baseApply,
equalityIsType4,
functionIsType,
setEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I,K:Cname List]. \mforall{}[f:name-morph(I;K)]. \mforall{}[s:X(I)]. \mforall{}[c:\mBbbN{}2].
\mforall{}[j:name-morph-domain(f;I)].
((f j:=c)(f(s)) = f((j:=c)(s)))
Date html generated:
2019_11_05-PM-00_25_50
Last ObjectModification:
2018_11_08-PM-01_15_56
Theory : cubical!sets
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