Nuprl Lemma : apply_gen_wf2
∀[B:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[q:ℕm + 1]. ∀[A:ℕn ⟶ Type]. ∀[f:funtype(n - q;λx.(A (x + q));B)].
∀[lst:k:{q..n-} ⟶ (A k)].
  (apply_gen(m;lst) q f ∈ funtype(n - m;λx.(A (x + m));B))
Proof
Definitions occuring in Statement : 
apply_gen: apply_gen(n;lst)
, 
funtype: funtype(n;A;T)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
apply: f a
, 
lambda: λx.A[x]
, 
function: x:A ⟶ B[x]
, 
subtract: n - m
, 
add: n + m
, 
natural_number: $n
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
exists: ∃x:A. B[x]
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
ge: i ≥ j 
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
top: Top
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
sq_type: SQType(T)
, 
guard: {T}
, 
le: A ≤ B
, 
uiff: uiff(P;Q)
, 
less_than: a < b
, 
apply_gen: apply_gen(n;lst)
, 
squash: ↓T
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
nequal: a ≠ b ∈ T 
, 
bfalse: ff
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
funtype: funtype(n;A;T)
, 
subtract: n - m
Lemmas referenced : 
int_seg_properties, 
subtract_wf, 
nat_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermSubtract_wf, 
itermVar_wf, 
intformless_wf, 
itermAdd_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_subtract_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_term_value_add_lemma, 
int_formula_prop_wf, 
le_wf, 
decidable__equal_int, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
equal-wf-base-T, 
int_subtype_base, 
subtype_base_sq, 
funtype_wf, 
int_seg_wf, 
add-member-int_seg2, 
lelt_wf, 
add-member-int_seg1, 
decidable__lt, 
nat_wf, 
ge_wf, 
less_than_wf, 
bool_wf, 
bool_subtype_base, 
equal_wf, 
squash_wf, 
true_wf, 
eq_int_eq_true, 
btrue_wf, 
iff_weakening_equal, 
eq_int_eq_false, 
equal-wf-base, 
bfalse_wf, 
subtype_rel_dep_function, 
int_seg_subtype, 
subtype_rel_self, 
primrec-unroll, 
eq_int_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
equal-wf-T-base, 
subtype_rel-equal, 
minus-add, 
minus-minus, 
add-associates, 
minus-one-mul, 
add-swap, 
add-mul-special, 
add-commutes, 
mul-distributes-right, 
two-mul, 
zero-add, 
one-mul, 
zero-mul, 
itermMultiply_wf, 
int_term_value_mul_lemma, 
primrec_wf, 
add-zero, 
bool_cases, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isectElimination, 
thin, 
natural_numberEquality, 
addEquality, 
setElimination, 
rename, 
because_Cache, 
hypothesis, 
hypothesisEquality, 
dependent_pairFormation, 
productElimination, 
dependent_set_memberEquality, 
dependent_functionElimination, 
unionElimination, 
independent_isectElimination, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
applyEquality, 
instantiate, 
cumulativity, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
lambdaFormation, 
functionExtensionality, 
functionEquality, 
universeEquality, 
isect_memberFormation, 
axiomEquality, 
intWeakElimination, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
baseApply, 
closedConclusion, 
multiplyEquality, 
minusEquality, 
impliesFunctionality
Latex:
\mforall{}[B:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[m:\mBbbN{}n  +  1].  \mforall{}[q:\mBbbN{}m  +  1].  \mforall{}[A:\mBbbN{}n  {}\mrightarrow{}  Type].  \mforall{}[f:funtype(n  -  q;\mlambda{}x.(A  (x  +  q));B)].
\mforall{}[lst:k:\{q..n\msupminus{}\}  {}\mrightarrow{}  (A  k)].
    (apply\_gen(m;lst)  q  f  \mmember{}  funtype(n  -  m;\mlambda{}x.(A  (x  +  m));B))
Date html generated:
2017_10_01-AM-09_03_32
Last ObjectModification:
2017_07_26-PM-04_44_23
Theory : bags
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