Nuprl Lemma : face-map-comp2
∀A,B:Cname List. ∀g:name-morph(A;B). ∀x,y:nameset(A). ∀i,j:ℕ2.
(g o ((g x:=i) o (g y:=j))) = (((x:=i) o (y:=j)) o g) ∈ name-morph(A;B-[g x; g y])
supposing ((↑isname(g x)) ∧ (↑isname(g y))) ∧ (¬(x = y ∈ Cname))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
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,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
not: ¬A,
implies: P ⇒ Q,
nameset: nameset(L),
false: False,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
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,
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
top: Top,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isname: isname(z),
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
l_member: (x ∈ l),
respects-equality: respects-equality(S;T),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
assert-isname,
istype-assert,
isname_wf,
coordinate_name_wf,
istype-void,
int_seg_wf,
nameset_wf,
name-morph_wf,
name-morph-ext,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
name-comp_wf,
face-map_wf2,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
list_wf,
list-diff2,
iff_weakening_equal,
subtype_rel_self,
name-morph_subtype_remove1,
member-list-diff,
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,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
int_formula_prop_wf,
istype-le,
member_singleton,
l_member_wf,
eq_int_wf,
intformless_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
assert_wf,
le_wf,
false_wf,
iff_weakening_uiff,
assert_of_le_int,
extd-nameset_subtype_int,
isname-nameset,
nsub2_subtype_extd-nameset,
nameset_subtype_base,
extd-nameset-respects-equality,
respects-equality-set-trivial2,
cons_member,
nameset_subtype_extd-nameset,
not-assert-isname
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
sqequalRule,
productIsType,
functionIsType,
equalityIstype,
universeIsType,
inhabitedIsType,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
applyLambdaEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
cumulativity,
intEquality,
unionElimination,
voidElimination,
dependent_set_memberEquality_alt,
promote_hyp,
equalityElimination,
isect_memberEquality_alt,
inlFormation_alt,
inrFormation_alt,
unionIsType
Latex:
\mforall{}A,B:Cname List. \mforall{}g:name-morph(A;B). \mforall{}x,y:nameset(A). \mforall{}i,j:\mBbbN{}2.
(g o ((g x:=i) o (g y:=j))) = (((x:=i) o (y:=j)) o g)
supposing ((\muparrow{}isname(g x)) \mwedge{} (\muparrow{}isname(g y))) \mwedge{} (\mneg{}(x = y))
Date html generated:
2020_05_21-AM-10_48_43
Last ObjectModification:
2019_12_10-PM-00_28_30
Theory : cubical!sets
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