Nuprl Lemma : rename-one-comp
∀I:Cname List. ∀z,z1,z2:Cname.
(rename-one-name(z;z1) o rename-one-name(z1;z2)) = rename-one-name(z;z2) ∈ name-morph([z / I];[z2 / I])
supposing (¬(z2 ∈ I)) ∧ (¬(z1 ∈ I)) ∧ (¬(z ∈ I))
Proof
Definitions occuring in Statement :
rename-one-name: rename-one-name(z1;z2),
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
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,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
rename-one-name: rename-one-name(z1;z2),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(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,
rev_implies: P ⇐ Q,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
isname: isname(z),
true: True,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
top: Top,
select: L[n],
cons: [a / b],
nat_plus: ℕ+,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_stable: SqStable(P),
ge: i ≥ j
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
rename-one-name_wf,
l_member_wf,
istype-void,
list_wf,
name-comp_wf,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
nameset_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,
equal-wf-base,
istype-le,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
length_wf,
select_wf,
nat_properties,
sq_stable__le,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
nameset_subtype_extd-nameset,
isname-nameset,
cons_member,
nameset_subtype,
l_subset_right_cons_trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
equalitySymmetry,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
inhabitedIsType,
applyEquality,
because_Cache,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
functionExtensionality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
Error :memTop,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
sqequalBase
Latex:
\mforall{}I:Cname List. \mforall{}z,z1,z2:Cname.
(rename-one-name(z;z1) o rename-one-name(z1;z2)) = rename-one-name(z;z2)
supposing (\mneg{}(z2 \mmember{} I)) \mwedge{} (\mneg{}(z1 \mmember{} I)) \mwedge{} (\mneg{}(z \mmember{} I))
Date html generated:
2020_05_21-AM-10_49_18
Last ObjectModification:
2019_12_08-PM-07_06_26
Theory : cubical!sets
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