Nuprl Lemma : extend-face-map-same
∀[I:Cname List]. ∀[x,y:Cname]. ∀[i:ℕ2].
(x:=i)[y:=y] = (x:=i) ∈ name-morph([y / I];[y / I]-[x]) supposing (¬(y = x ∈ Cname)) ∧ (¬(y ∈ I))
Proof
Definitions occuring in Statement :
face-map: (x:=i),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
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,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
name-morph: name-morph(I;J),
face-map: (x:=i),
extend-name-morph: f[z1:=z2],
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
cand: A c∧ B,
nequal: a ≠ b ∈ T
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
istype-void,
l_member_wf,
int_seg_wf,
list_wf,
member-list-diff,
face-map_wf2,
extend-name-morph_wf,
iff_weakening_equal,
list-diff-cons-single,
true_wf,
squash_wf,
equal_wf,
int_subtype_base,
le_wf,
set_subtype_base,
list_subtype_base,
subtype_base_sq,
name-morph_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eq-cname_wf,
assert-eq-cname,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
neg_assert_of_eq_int,
nameset_wf,
decidable__equal_int,
istype-le,
nsub2_subtype_extd-nameset,
nameset_subtype_extd-nameset,
not_wf,
member_singleton,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
productIsType,
functionIsType,
equalityIstype,
inhabitedIsType,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
natural_numberEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
baseClosed,
imageMemberEquality,
because_Cache,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
lambdaEquality,
intEquality,
cumulativity,
instantiate,
rename,
setElimination,
functionExtensionality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
applyLambdaEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
promote_hyp,
Error :memTop,
dependent_set_memberEquality_alt,
dependent_set_memberEquality,
impliesFunctionality,
addLevel,
inlFormation
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y:Cname]. \mforall{}[i:\mBbbN{}2]. (x:=i)[y:=y] = (x:=i) supposing (\mneg{}(y = x)) \mwedge{} (\mneg{}(y \mmember{} I))
Date html generated:
2020_05_21-AM-10_48_56
Last ObjectModification:
2019_12_10-PM-00_09_02
Theory : cubical!sets
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