Nuprl Lemma : alpha-rename-equivalent
∀[opr:Type]
∀t:term(opr). ∀f:{v:varname()| (v ∈ all-vars(t))} ⟶ varname().
alpha-eq-terms(opr;alpha-rename(f;t);t)
supposing ((∀x:{v:varname()| (v ∈ all-vars(t))} . ((f x ∈ free-vars(t)) ⇒ ((f x) = x ∈ varname())))
∧ (∀x:{v:varname()| (v ∈ all-vars(t))} . (((f x) = nullvar() ∈ varname()) ⇒ (x = nullvar() ∈ varname()))))
∧ Inj({v:varname()| (v ∈ all-vars(t))} ;varname();f)
Proof
Definitions occuring in Statement :
alpha-rename: alpha-rename(f;t),
all-vars: all-vars(t),
free-vars: free-vars(t),
alpha-eq-terms: alpha-eq-terms(opr;a;b),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
l_member: (x ∈ l),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
istype: istype(T),
guard: {T},
not: ¬A,
false: False,
bound-term: bound-term(opr),
pi2: snd(t),
alpha-rename: alpha-rename(f;t),
alpha-eq-terms: alpha-eq-terms(opr;a;b),
free-vars: free-vars(t),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
all-vars: all-vars(t),
varterm: varterm(v),
cons: [a / b],
alpha-rename-aux: alpha-rename-aux(f;bnds;t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
alpha-aux: alpha-aux(opr;vs;ws;a;b),
sq_stable: SqStable(P),
squash: ↓T,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
inject: Inj(A;B;f),
label: ...$L... t,
true: True,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
select: L[n],
cand: A c∧ B,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
same-binding: same-binding(vs;ws;v;w),
map: map(f;as),
list_ind: list_ind,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_uimplies: rev_uimplies(P;Q),
band: p ∧b q,
free-vars-aux: free-vars-aux(bnds;t),
nil: [],
colength: colength(L),
less_than: a < b,
mkterm: mkterm(opr;bts),
pi1: fst(t),
has-value: (a)↓,
lelt: i ≤ j < k,
int_seg: {i..j-},
reverse: rev(as),
respects-equality: respects-equality(S;T)
Lemmas referenced :
list-subtype,
varname_wf,
map_wf,
l_member_wf,
subtype_rel_dep_function,
append_wf,
all-vars_wf,
subtype_rel_sets_simple,
member_append,
list_wf,
term_wf,
term-induction,
all_wf,
isect_wf,
free-vars-aux_wf,
equal_wf,
equal-wf-T-base,
inject_wf,
alpha-aux_wf,
alpha-rename-aux_wf,
nullvar_wf,
varterm_wf,
subtype_rel_list,
not_wf,
istype-void,
mkterm_wf,
bound-term_wf,
nil_wf,
list_ind_nil_lemma,
map_nil_lemma,
istype-universe,
deq-member_wf,
var-deq_wf,
eqtt_to_assert,
assert-deq-member,
sq_stable_from_decidable,
assert_wf,
same-binding_wf,
decidable__assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
cons_wf,
member_wf,
iff_weakening_equal,
list_induction,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
istype-assert,
istype-le,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
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,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
eq_var_wf,
assert-eq_var,
iff_weakening_uiff,
map_cons_lemma,
spread_cons_lemma,
iff_transitivity,
bnot_wf,
bool_cases,
band_wf,
cons_member,
bfalse_wf,
assert_of_band,
assert_of_bnot,
member_map,
subtype_rel_sets,
l_member-settype,
map-length,
ge_wf,
assert_witness,
list-cases,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
set_subtype_base,
int_subtype_base,
le_weakening2,
non_neg_length,
colength_wf_list,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
istype-nat,
member-all-vars-mkterm,
pi1_wf,
pi2_wf,
member-rev-append,
rev-append_wf,
eager-map_wf,
product-value-type,
value-type-has-value,
list-value-type,
eager-map-is-map,
int_seg_wf,
alpha-aux-mkterm,
map_length,
squash_wf,
true_wf,
subtype_rel_self,
top_wf,
int_seg_properties,
select-map,
map-reverse,
map_append_sq,
rev-append-property,
select_member,
sq_stable__equal,
sq_stable__all,
member-l-union-list,
respects-equality-set-trivial2,
respects-equality-list,
member-map,
inject-subtype,
respects-equality-trivial,
respects-equality-sets,
strong-subtype-iff-respects-equality,
less_than_wf,
length-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
setEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
setIsType,
universeIsType,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
inlFormation_alt,
functionIsType,
functionEquality,
setElimination,
rename,
productEquality,
dependent_set_memberEquality_alt,
baseClosed,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
voidElimination,
inhabitedIsType,
isectIsType,
Error :memTop,
instantiate,
universeEquality,
unionElimination,
equalityElimination,
imageMemberEquality,
imageElimination,
dependent_pairFormation_alt,
promote_hyp,
cumulativity,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
independent_pairFormation,
approximateComputation,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
inrFormation_alt,
isect_memberEquality_alt,
intWeakElimination,
functionIsTypeImplies,
sqequalBase,
hypothesis_subsumption,
addEquality,
intEquality,
independent_pairEquality,
unionIsType,
dependent_pairEquality_alt,
callbyvalueReduce,
axiomEquality,
spreadEquality
Latex:
\mforall{}[opr:Type]
\mforall{}t:term(opr). \mforall{}f:\{v:varname()| (v \mmember{} all-vars(t))\} {}\mrightarrow{} varname().
alpha-eq-terms(opr;alpha-rename(f;t);t)
supposing ((\mforall{}x:\{v:varname()| (v \mmember{} all-vars(t))\} . ((f x \mmember{} free-vars(t)) {}\mRightarrow{} ((f x) = x)))
\mwedge{} (\mforall{}x:\{v:varname()| (v \mmember{} all-vars(t))\} . (((f x) = nullvar()) {}\mRightarrow{} (x = nullvar()))))
\mwedge{} Inj(\{v:varname()| (v \mmember{} all-vars(t))\} ;varname();f)
Date html generated:
2020_05_19-PM-09_56_58
Last ObjectModification:
2020_03_09-PM-04_09_30
Theory : terms
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