Nuprl Lemma : alpha-rename-aux_wf
∀[opr:Type]. ∀[t:term(opr)]. ∀[bnds:varname() List].
∀f:{v:varname()| (v ∈ bnds @ all-vars(t))} ⟶ varname()
alpha-rename-aux(f;bnds;t) ∈ term(opr)
supposing ∀x:{v:varname()| (v ∈ bnds @ all-vars(t))}
(((f x) = nullvar() ∈ varname()) ⇒ (x = nullvar() ∈ varname()))
Proof
Definitions occuring in Statement :
alpha-rename-aux: alpha-rename-aux(f;bnds;t),
all-vars: all-vars(t),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
coterm-fun: coterm-fun(opr;T),
all-vars: all-vars(t),
alpha-rename-aux: alpha-rename-aux(f;bnds;t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
has-value: (a)↓,
bound-term: bound-term(opr),
istype: istype(T),
mkterm: mkterm(opr;bts),
cand: A c∧ B,
pi1: fst(t),
pi2: snd(t),
reverse: rev(as),
squash: ↓T,
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
term-ext,
subtype_rel_weakening,
term_wf,
coterm-fun_wf,
ext-eq_inversion,
nullvar_wf,
l_member_wf,
varname_wf,
append_wf,
all-vars_wf,
list_wf,
term-size_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
istype-universe,
deq-member_wf,
var-deq_wf,
eqtt_to_assert,
assert-deq-member,
varterm_wf,
member_append,
cons_wf,
nil_wf,
value-type-has-value,
bound-term_wf,
list-value-type,
mkterm_wf,
list-subtype,
eager-map_wf,
product-value-type,
map_wf,
subtype_rel_dep_function,
member-all-vars-mkterm,
term-size-positive,
term_size_mkterm_lemma,
rev-append_wf,
rev-append-property,
reverse_wf,
member-reverse,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
summand-le-lsum,
pi2_wf,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
functionIsType,
equalityIstype,
setIsType,
addEquality,
universeEquality,
equalityElimination,
inlFormation_alt,
baseClosed,
sqequalBase,
callbyvalueReduce,
productEquality,
independent_pairEquality,
setEquality,
inrFormation_alt,
unionIsType,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[opr:Type]. \mforall{}[t:term(opr)]. \mforall{}[bnds:varname() List].
\mforall{}f:\{v:varname()| (v \mmember{} bnds @ all-vars(t))\} {}\mrightarrow{} varname()
alpha-rename-aux(f;bnds;t) \mmember{} term(opr)
supposing \mforall{}x:\{v:varname()| (v \mmember{} bnds @ all-vars(t))\} . (((f x) = nullvar()) {}\mRightarrow{} (x = nullvar()))
Date html generated:
2020_05_19-PM-09_56_39
Last ObjectModification:
2020_03_09-PM-04_09_26
Theory : terms
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