Nuprl Lemma : face-maps-comp-property
∀L:(Cname × ℕ2) List
∀[I:Cname List]
∀y:nameset(map(λp.(fst(p));L) @ I)
(((↑isname(face-maps-comp(L) y)) ⇒ ((¬(y ∈ map(λp.(fst(p));L))) ∧ ((face-maps-comp(L) y) = y ∈ nameset(I))))
∧ ((¬↑isname(face-maps-comp(L) y))
⇒ ((y ∈ map(λp.(fst(p));L)) ∧ ((face-maps-comp(L) y) = outl(apply-alist(CnameDeq;L;y)) ∈ ℕ2))))
Proof
Definitions occuring in Statement :
face-maps-comp: face-maps-comp(L),
isname: isname(z),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
map: map(f;as),
append: as @ bs,
list: T List,
int_seg: {i..j-},
outl: outl(x),
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
pi1: fst(t),
name-morph: name-morph(I;J),
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
nameset: nameset(L),
uimplies: b supposing a,
uiff: uiff(P;Q),
or: P ∨ Q,
iff: P ⇐⇒ Q,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
top: Top,
face-maps-comp: face-maps-comp(L),
rev_uimplies: rev_uimplies(P;Q),
isname: isname(z),
int_upper: {i...},
coordinate_name: Cname,
pi2: snd(t),
decidable: Dec(P),
sq_type: SQType(T),
compose: f o g,
name-comp: (f o g),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
int_seg: {i..j-},
true: True,
squash: ↓T,
face-map: (x:=i),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
bnot: ¬bb,
uext: uext(g),
lt_int: i <z j,
le_int: i ≤z j,
outl: outl(x),
respects-equality: respects-equality(S;T)
Lemmas referenced :
list_wf,
coordinate_name_wf,
int_seg_wf,
list_induction,
map_wf,
append_wf,
nameset_wf,
face-maps-comp_wf,
isname_wf,
assert_wf,
equal_wf,
l_member_wf,
not_wf,
assert-isname,
member_append,
not-assert-isname,
cname_deq_wf,
isl-apply-alist,
outl_wf,
nameset_subtype_extd-nameset,
istype-assert,
btrue_neq_bfalse,
nil_wf,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
list_ind_nil_lemma,
map_nil_lemma,
istype-void,
reduce_nil_lemma,
assert_of_le_int,
map_cons_lemma,
reduce_cons_lemma,
apply_alist_cons_lemma,
list_ind_cons_lemma,
cons_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
decidable__equal-coordinate_name,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
int_seg_properties,
bnot_wf,
iff_weakening_equal,
subtype_rel_self,
eq_int_eq_true,
istype-universe,
true_wf,
squash_wf,
bool_wf,
equal-wf-base,
eq_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-le,
assert-bnot,
bool_cases_sqequal,
bool_subtype_base,
lelt_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
le_int_wf,
int_seg_cases,
int_seg_subtype_special,
decidable__equal_int,
safe-assert-deq,
eqof_wf,
cons_member,
neg_assert_of_eq_int,
equal-wf-T-base,
false_wf,
iff_functionality_wrt_iff,
bfalse_wf,
iff_imp_equal_bool,
respects-equality-set,
respects-equality-set-trivial2,
apply-alist_wf,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesis,
natural_numberEquality,
sqequalRule,
independent_functionElimination,
lambdaEquality_alt,
isectEquality,
functionEquality,
productIsType,
hypothesisEquality,
productElimination,
because_Cache,
closedConclusion,
dependent_functionElimination,
equalityIsType1,
inhabitedIsType,
rename,
setElimination,
equalitySymmetry,
equalityTransitivity,
applyEquality,
independent_isectElimination,
unionElimination,
voidElimination,
dependent_set_memberEquality_alt,
functionIsType,
axiomEquality,
functionIsTypeImplies,
independent_pairEquality,
independent_pairFormation,
isect_memberFormation_alt,
isect_memberEquality_alt,
intEquality,
cumulativity,
instantiate,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
applyLambdaEquality,
equalityIsType4,
imageMemberEquality,
universeEquality,
imageElimination,
baseClosed,
baseApply,
equalityElimination,
promote_hyp,
hypothesis_subsumption,
inlFormation_alt,
equalityIstype,
inrFormation_alt,
equalityIsType3,
unionIsType,
isectIsType
Latex:
\mforall{}L:(Cname \mtimes{} \mBbbN{}2) List
\mforall{}[I:Cname List]
\mforall{}y:nameset(map(\mlambda{}p.(fst(p));L) @ I)
(((\muparrow{}isname(face-maps-comp(L) y))
{}\mRightarrow{} ((\mneg{}(y \mmember{} map(\mlambda{}p.(fst(p));L))) \mwedge{} ((face-maps-comp(L) y) = y)))
\mwedge{} ((\mneg{}\muparrow{}isname(face-maps-comp(L) y))
{}\mRightarrow{} ((y \mmember{} map(\mlambda{}p.(fst(p));L)) \mwedge{} ((face-maps-comp(L) y) = outl(apply-alist(CnameDeq;L;y))))))
Date html generated:
2019_11_05-PM-00_25_15
Last ObjectModification:
2018_12_11-PM-11_41_33
Theory : cubical!sets
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