Nuprl Lemma : face-map-idempotent
∀A,B:Cname List. ∀x:nameset(A). ∀g:name-morph(A-[x];B). ∀i:ℕ2.
(((x:=i) o ((x:=i) o g)) = ((x:=i) o g) ∈ name-morph(A;B))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
all: ∀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],
nameset: nameset(L),
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
coordinate_name: Cname,
int_upper: {i...},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uext: uext(g),
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
assert: ↑b,
not: ¬A,
false: False,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
face-map-comp-id,
name-comp_wf,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
nil_wf,
face-map_wf2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
false_wf,
int_seg_subtype,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
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,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
lelt_wf,
int_seg_wf,
name-morph_wf,
nameset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
natural_numberEquality,
instantiate,
cumulativity,
intEquality,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
addEquality,
independent_pairFormation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
promote_hyp,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}A,B:Cname List. \mforall{}x:nameset(A). \mforall{}g:name-morph(A-[x];B). \mforall{}i:\mBbbN{}2.
(((x:=i) o ((x:=i) o g)) = ((x:=i) o g))
Date html generated:
2017_10_05-AM-10_07_40
Last ObjectModification:
2017_07_28-AM-11_16_40
Theory : cubical!sets
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