Nuprl Lemma : extended-face-map
∀I:Cname List. ∀x1,x2:nameset(I). ∀i:ℕ2. ∀y1,y2:Cname.
(x2:=i)[y1:=y2] = ((x2:=i) o rename-one-name(y1;y2)) ∈ name-morph([y1 / I-[x1]];[y2 / I-[x1; x2]])
supposing (¬(y2 ∈ I-[x1; x2])) ∧ (¬(y1 ∈ I))
Proof
Definitions occuring in Statement :
rename-one-name: rename-one-name(z1;z2),
name-comp: (f o g),
face-map: (x:=i),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nameset: nameset(L),
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
false: False,
true: True,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
name-morph: name-morph(I;J),
cand: A c∧ B,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
so_apply: x[s],
sq_type: SQType(T),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
rename-one-name: rename-one-name(z1;z2),
face-map: (x:=i),
name-comp: (f o g),
extend-name-morph: f[z1:=z2],
compose: f o g,
uext: uext(g),
isname: isname(z),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
l_member: (x ∈ l),
nat: ℕ,
less_than': less_than'(a;b),
top: Top,
select: L[n],
cons: [a / b],
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
sq_stable: SqStable(P),
ge: i ≥ j
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
extend-name-morph_wf,
l_member_wf,
istype-void,
int_seg_wf,
nameset_wf,
list_wf,
face-map_wf2,
name-morph_wf,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
list-diff2,
iff_weakening_equal,
subtype_rel_self,
name-comp_wf,
rename-one-name_wf,
member-list-diff,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
equal_wf,
list-diff-cons-single,
cons_member,
member_singleton,
list-diff-cons,
deq-member_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
assert_wf,
false_wf,
iff_weakening_uiff,
assert_of_le_int,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
eq-cname_wf,
assert-eq-cname,
equal-wf-T-base,
nsub2_subtype_extd-nameset,
isname-nameset,
nameset_subtype_extd-nameset,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
length_wf,
select_wf,
nat_properties,
sq_stable__le,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
nameset_subtype,
l_subset_right_cons_trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
inhabitedIsType,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
cumulativity,
intEquality,
equalityIstype,
hyp_replacement,
dependent_set_memberEquality_alt,
applyLambdaEquality,
voidElimination,
unionElimination,
inlFormation_alt,
inrFormation_alt,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
functionExtensionality,
approximateComputation,
int_eqEquality,
Error :memTop,
isect_memberEquality_alt,
pointwiseFunctionality,
baseApply,
closedConclusion,
sqequalBase,
unionIsType
Latex:
\mforall{}I:Cname List. \mforall{}x1,x2:nameset(I). \mforall{}i:\mBbbN{}2. \mforall{}y1,y2:Cname.
(x2:=i)[y1:=y2] = ((x2:=i) o rename-one-name(y1;y2)) supposing (\mneg{}(y2 \mmember{} I-[x1; x2])) \mwedge{} (\mneg{}(y1 \mmember{} I))
Date html generated:
2020_05_21-AM-10_49_28
Last ObjectModification:
2019_12_10-PM-00_44_07
Theory : cubical!sets
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