Nuprl Lemma : extend-name-morph-face-map
∀I,K:Cname List. ∀f:name-morph(I;K). ∀z,x:Cname. ∀i:ℕ2.
(f[z:=x] o (x:=i)) = ((z:=i) o f) ∈ name-morph([z / I];K) supposing (¬(x ∈ K)) ∧ (¬(z ∈ I))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
coordinate_name: Cname,
l_member: (x ∈ l),
cons: [a / b],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
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,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
face-map: (x:=i),
name-comp: (f o g),
extend-name-morph: f[z1:=z2],
compose: f o g,
uext: uext(g),
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
squash: ↓T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
top: Top,
le: A ≤ B,
less_than: a < b,
decidable: Dec(P),
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
isname: isname(z),
true: True
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
name-comp_wf,
extend-name-morph_wf,
face-map_wf,
l_member_wf,
istype-void,
int_seg_wf,
name-morph_wf,
nameset_wf,
eq-cname_wf,
eq_int_wf,
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,
int_formula_prop_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
decidable__equal_int,
istype-le,
assert-eq-cname,
bool_wf,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
set_subtype_base,
le_wf,
iff_imp_equal_bool,
le_int_wf,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
assert_of_le_int,
iff_weakening_equal,
istype-true,
bfalse_wf,
false_wf,
intformle_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
nsub2_subtype_extd-nameset,
isname-nameset,
cons_member,
isname_wf,
assert-isname,
extd-nameset_subtype_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
natural_numberEquality,
inhabitedIsType,
because_Cache,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
functionExtensionality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
voidElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
isect_memberEquality_alt,
cumulativity,
intEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}z,x:Cname. \mforall{}i:\mBbbN{}2.
(f[z:=x] o (x:=i)) = ((z:=i) o f) supposing (\mneg{}(x \mmember{} K)) \mwedge{} (\mneg{}(z \mmember{} I))
Date html generated:
2020_05_21-AM-10_49_02
Last ObjectModification:
2019_12_08-PM-07_06_29
Theory : cubical!sets
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