Nuprl Lemma : name-morph-extend-face-map
∀[I,K:Cname List]. ∀[i:ℕ2]. ∀[f:name-morph(I;K)].
(((fresh-cname(I):=i) o f) = ((f)+ o (fresh-cname(K):=i)) ∈ name-morph(I+;K))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph-extend: (f)+,
name-morph: name-morph(I;J),
add-fresh-cname: I+,
fresh-cname: fresh-cname(I),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
uimplies: b supposing a,
name-morph-extend: (f)+,
name-comp: (f o g),
has-value: (a)↓,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq-cname: eq-cname(x;y),
compose: f o g,
nameset: nameset(L),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
coordinate_name: Cname,
int_upper: {i...},
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
face-map: (x:=i),
uext: uext(g),
isname: isname(z),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
nequal: a ≠ b ∈ T ,
top: Top,
add-fresh-cname: I+
Lemmas referenced :
name-morphs-equal,
add-fresh-cname_wf,
name-comp_wf,
face-map_wf_fresh-cname,
name-morph-extend_wf,
value-type-has-value,
coordinate_name_wf,
not_wf,
l_member_wf,
set-value-type,
coordinate_name-value-type,
fresh-cname_wf,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf,
nameset_wf,
name-morph_wf,
int_seg_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
eq_int_wf,
assert_of_eq_int,
bool_wf,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
false_wf,
assert_of_le_int,
iff_weakening_equal,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-void,
isname-name,
nsub2_subtype_extd-nameset,
assert_elim,
bnot_wf,
eq_int_eq_true,
btrue_neq_bfalse,
neg_assert_of_eq_int,
btrue_wf,
not_assert_elim,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
istype-true,
cons_member,
assert-isname,
nameset_subtype_extd-nameset,
isname_wf,
extd-nameset_subtype_int,
equal-wf-T-base,
uiff_transitivity,
assert_of_bnot,
fresh-cname-not-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_isectElimination,
callbyvalueReduce,
setEquality,
universeIsType,
because_Cache,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
natural_numberEquality,
cumulativity,
intEquality,
imageElimination,
independent_pairFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
Error :memTop,
dependent_set_memberEquality_alt,
productIsType,
universeEquality
Latex:
\mforall{}[I,K:Cname List]. \mforall{}[i:\mBbbN{}2]. \mforall{}[f:name-morph(I;K)].
(((fresh-cname(I):=i) o f) = ((f)+ o (fresh-cname(K):=i)))
Date html generated:
2020_05_21-AM-10_48_22
Last ObjectModification:
2019_12_10-PM-00_28_26
Theory : cubical!sets
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