Nuprl Lemma : face-maps-commute
∀I:Cname List. ∀x:nameset(I). ∀i:ℕ2. ∀y:nameset(I). ∀j:ℕ2.
((¬(x = y ∈ Cname)) ⇒ (((x:=i) o (y:=j)) = ((y:=j) o (x:=i)) ∈ name-morph(I;I-[x; y])))
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],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nameset: nameset(L),
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
true: True,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
name-morph: name-morph(I;J),
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
coordinate_name: Cname,
int_upper: {i...},
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],
sq_type: SQType(T),
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
top: Top,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
isname: isname(z),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
face-map_wf2,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
nil_wf,
istype-void,
int_seg_wf,
nameset_wf,
name-morph_wf,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
list-diff2,
iff_weakening_equal,
name-morphs-equal,
name-comp_wf,
list_wf,
list-diff2-sym,
eq_int_wf,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
intformnot_wf,
int_formula_prop_not_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
istype-le,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bool_wf,
le_int_wf,
assert_of_le_int,
iff_weakening_uiff,
assert_wf,
le_wf,
nsub2_subtype_extd-nameset,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
l_member_wf,
member-list-diff,
member_singleton,
cons_member,
set_subtype_base,
nameset_subtype_extd-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
setElimination,
rename,
applyEquality,
sqequalRule,
functionIsType,
equalityIstype,
universeIsType,
inhabitedIsType,
natural_numberEquality,
because_Cache,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionExtensionality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
cumulativity,
intEquality,
unionElimination,
dependent_set_memberEquality_alt,
equalityElimination,
isect_memberEquality_alt,
promote_hyp,
inlFormation_alt,
applyLambdaEquality,
inrFormation_alt,
unionIsType
Latex:
\mforall{}I:Cname List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2. \mforall{}y:nameset(I). \mforall{}j:\mBbbN{}2.
((\mneg{}(x = y)) {}\mRightarrow{} (((x:=i) o (y:=j)) = ((y:=j) o (x:=i))))
Date html generated:
2020_05_21-AM-10_48_50
Last ObjectModification:
2019_12_08-PM-07_06_37
Theory : cubical!sets
Home
Index