Nuprl Lemma : iota-two-face-maps
∀[I:Cname List]. ∀[x,y,z:Cname]. ∀[i,j:ℕ2].
(((x:=i) o (y:=j)) o iota(z)) = (iota(z) o ((x:=i) o (y:=j))) ∈ name-morph(I;[z / I-[x; y]])
supposing (¬(x = z ∈ Cname)) ∧ (¬(y = z ∈ Cname))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
iota: iota(x),
face-map: (x:=i),
name-morph: name-morph(I;J),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
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,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
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,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
not_wf,
equal_wf,
coordinate_name_wf,
int_seg_wf,
name-comp-assoc,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
face-map_wf2,
iota_wf,
name-morph_wf,
subtype_rel_wf,
squash_wf,
true_wf,
list_wf,
list-diff2,
iff_weakening_equal,
subtype_rel_self,
iota-face-map,
name-comp_wf,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
list-diff-cons-single,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
cons_member,
member_singleton,
or_wf,
l_member_wf,
list-diff-cons,
deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
productEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
applyLambdaEquality,
lambdaFormation,
dependent_functionElimination,
instantiate,
cumulativity,
intEquality,
setElimination,
rename,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
computeAll,
addLevel,
orFunctionality,
promote_hyp,
unionElimination,
equalityElimination,
hyp_replacement
Latex:
\mforall{}[I:Cname List]. \mforall{}[x,y,z:Cname]. \mforall{}[i,j:\mBbbN{}2].
(((x:=i) o (y:=j)) o iota(z)) = (iota(z) o ((x:=i) o (y:=j))) supposing (\mneg{}(x = z)) \mwedge{} (\mneg{}(y = z))
Date html generated:
2017_10_05-AM-10_08_27
Last ObjectModification:
2017_07_28-AM-11_16_51
Theory : cubical!sets
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