Nuprl Lemma : alpha-aux-mkterm
∀[opr:Type]
  ∀a,b:opr. ∀as,bs:bound-term(opr) List. ∀vs,ws:varname() List.
    (alpha-aux(opr;vs;ws;mkterm(a;as);mkterm(b;bs))
    
⇐⇒ (a = b ∈ opr)
        ∧ (||as|| = ||bs|| ∈ ℤ)
        ∧ (∀i:ℕ||as||
             (alpha-aux(opr;rev(fst(as[i])) + vs;rev(fst(bs[i])) + ws;snd(as[i]);snd(bs[i]))
             ∧ (||fst(as[i])|| = ||fst(bs[i])|| ∈ ℤ))))
Proof
Definitions occuring in Statement : 
alpha-aux: alpha-aux(opr;vs;ws;a;b)
, 
bound-term: bound-term(opr)
, 
mkterm: mkterm(opr;bts)
, 
varname: varname()
, 
select: L[n]
, 
length: ||as||
, 
rev-append: rev(as) + bs
, 
list: T List
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
natural_number: $n
, 
int: ℤ
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
bound-term: bound-term(opr)
, 
prop: ℙ
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
le: A ≤ B
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
select: L[n]
, 
nil: []
, 
it: ⋅
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
alpha-aux: alpha-aux(opr;vs;ws;a;b)
, 
mkterm: mkterm(opr;bts)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
cons: [a / b]
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
less_than': less_than'(a;b)
, 
nat_plus: ℕ+
, 
uiff: uiff(P;Q)
, 
less_than: a < b
, 
squash: ↓T
, 
label: ...$L... t
, 
true: True
, 
subtract: n - m
Lemmas referenced : 
list_induction, 
bound-term_wf, 
all_wf, 
list_wf, 
varname_wf, 
iff_wf, 
alpha-aux_wf, 
mkterm_wf, 
equal_wf, 
equal-wf-base, 
int_seg_wf, 
rev-append_wf, 
length_wf_nat, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
length_wf, 
length_of_nil_lemma, 
stuck-spread, 
istype-base, 
nil_wf, 
term_wf, 
int_seg_properties, 
full-omega-unsat, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf, 
length_of_cons_lemma, 
non_neg_length, 
decidable__le, 
intformnot_wf, 
int_formula_prop_not_lemma, 
istype-void, 
intformeq_wf, 
itermAdd_wf, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
cons_wf, 
spread_cons_lemma, 
decidable__equal_int, 
subtype_base_sq, 
select_wf, 
decidable__lt, 
istype-false, 
add_nat_plus, 
istype-less_than, 
nat_plus_properties, 
add-is-int-iff, 
false_wf, 
istype-le, 
subtype_rel_self, 
istype-universe, 
select-cons-tl, 
subtract_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
select_cons_tl_sq2, 
int_seg_subtype_nat, 
add-associates, 
add-swap, 
add-commutes, 
zero-add, 
squash_wf, 
true_wf, 
iff_weakening_equal, 
add-subtract-cancel
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
cut, 
thin, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
sqequalRule, 
lambdaEquality_alt, 
because_Cache, 
productEquality, 
closedConclusion, 
natural_numberEquality, 
equalityTransitivity, 
equalitySymmetry, 
inhabitedIsType, 
productElimination, 
equalityIstype, 
dependent_functionElimination, 
independent_functionElimination, 
applyEquality, 
setElimination, 
rename, 
independent_isectElimination, 
universeIsType, 
baseClosed, 
Error :memTop, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
independent_pairFormation, 
voidElimination, 
productIsType, 
sqequalBase, 
functionIsType, 
unionElimination, 
addEquality, 
intEquality, 
independent_pairEquality, 
instantiate, 
cumulativity, 
dependent_set_memberEquality_alt, 
applyLambdaEquality, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
imageElimination, 
universeEquality, 
imageMemberEquality
Latex:
\mforall{}[opr:Type]
    \mforall{}a,b:opr.  \mforall{}as,bs:bound-term(opr)  List.  \mforall{}vs,ws:varname()  List.
        (alpha-aux(opr;vs;ws;mkterm(a;as);mkterm(b;bs))
        \mLeftarrow{}{}\mRightarrow{}  (a  =  b)
                \mwedge{}  (||as||  =  ||bs||)
                \mwedge{}  (\mforall{}i:\mBbbN{}||as||
                          (alpha-aux(opr;rev(fst(as[i]))  +  vs;rev(fst(bs[i]))  +  ws;snd(as[i]);snd(bs[i]))
                          \mwedge{}  (||fst(as[i])||  =  ||fst(bs[i])||))))
Date html generated:
2020_05_19-PM-09_55_28
Last ObjectModification:
2020_03_09-PM-04_08_55
Theory : terms
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