Nuprl Lemma : C_DVALUEp-induction
∀[P:C_DVALUEp() ⟶ ℙ]
  ((∀x:Unit. P[DVp_Null(x)])
  
⇒ (∀int:ℤ. P[DVp_Int(int)])
  
⇒ (∀ptr:C_LVALUE()?. P[DVp_Pointer(ptr)])
  
⇒ (∀lower,upper:ℤ. ∀arr:{lower..upper-} ⟶ C_DVALUEp().
        ((∀u:{lower..upper-}. P[arr u]) 
⇒ P[DVp_Array(lower;upper;arr)]))
  
⇒ (∀lbls:Atom List. ∀struct:{a:Atom| (a ∈ lbls)}  ⟶ C_DVALUEp().
        ((∀u:{a:Atom| (a ∈ lbls)} . P[struct u]) 
⇒ P[DVp_Struct(lbls;struct)]))
  
⇒ {∀v:C_DVALUEp(). P[v]})
Proof
Definitions occuring in Statement : 
DVp_Struct: DVp_Struct(lbls;struct)
, 
DVp_Array: DVp_Array(lower;upper;arr)
, 
DVp_Pointer: DVp_Pointer(ptr)
, 
DVp_Int: DVp_Int(int)
, 
DVp_Null: DVp_Null(x)
, 
C_DVALUEp: C_DVALUEp()
, 
C_LVALUE: C_LVALUE()
, 
l_member: (x ∈ l)
, 
list: T List
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
guard: {T}
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
unit: Unit
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
union: left + right
, 
int: ℤ
, 
atom: Atom
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
le: A ≤ B
, 
and: P ∧ Q
, 
not: ¬A
, 
false: False
, 
ext-eq: A ≡ B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
sq_type: SQType(T)
, 
eq_atom: x =a y
, 
ifthenelse: if b then t else f fi 
, 
DVp_Null: DVp_Null(x)
, 
C_DVALUEp_size: C_DVALUEp_size(p)
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
bnot: ¬bb
, 
assert: ↑b
, 
DVp_Int: DVp_Int(int)
, 
DVp_Pointer: DVp_Pointer(ptr)
, 
DVp_Array: DVp_Array(lower;upper;arr)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
sum: Σ(f[x] | x < k)
, 
sum_aux: sum_aux(k;v;i;x.f[x])
, 
less_than': less_than'(a;b)
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
cand: A c∧ B
, 
DVp_Struct: DVp_Struct(lbls;struct)
, 
less_than: a < b
, 
squash: ↓T
, 
sq_stable: SqStable(P)
, 
l_member: (x ∈ l)
Lemmas referenced : 
and_wf, 
equal-wf-base-T, 
less_than_wf, 
ifthenelse_wf, 
DVp_Null_wf, 
DVp_Int_wf, 
DVp_Pointer_wf, 
DVp_Array_wf, 
DVp_Struct_wf, 
list_wf, 
uall_wf, 
set_wf, 
sq_stable__le, 
length_wf, 
int_seg_properties, 
list-subtype, 
l_member_wf, 
select_wf, 
length_wf_nat, 
trivial-int-eq1, 
sum-nat-less, 
int_term_value_add_lemma, 
itermAdd_wf, 
decidable__lt, 
int_seg_wf, 
lelt_wf, 
int_formula_prop_less_lemma, 
intformless_wf, 
assert_of_bnot, 
iff_weakening_uiff, 
not_wf, 
bnot_wf, 
assert_wf, 
iff_transitivity, 
bool_cases, 
add-member-int_seg1, 
false_wf, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_term_value_subtract_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermVar_wf, 
itermSubtract_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
decidable__le, 
nat_properties, 
subtract_wf, 
assert_of_le_int, 
le_int_wf, 
sum-nat, 
C_LVALUE_wf, 
unit_wf2, 
neg_assert_of_eq_atom, 
assert-bnot, 
bool_subtype_base, 
bool_cases_sqequal, 
equal_wf, 
eqff_to_assert, 
it_wf, 
unit_subtype_base, 
atom_subtype_base, 
subtype_base_sq, 
assert_of_eq_atom, 
eqtt_to_assert, 
bool_wf, 
eq_atom_wf, 
C_DVALUEp-ext, 
less_than'_wf, 
nat_wf, 
C_DVALUEp_size_wf, 
le_wf, 
isect_wf, 
C_DVALUEp_wf, 
all_wf, 
uniform-comp-nat-induction
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
sqequalRule, 
lambdaEquality, 
hypothesis, 
hypothesisEquality, 
applyEquality, 
because_Cache, 
setElimination, 
rename, 
independent_functionElimination, 
introduction, 
productElimination, 
independent_pairEquality, 
dependent_functionElimination, 
voidElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
promote_hyp, 
hypothesis_subsumption, 
tokenEquality, 
unionElimination, 
equalityElimination, 
independent_isectElimination, 
instantiate, 
cumulativity, 
atomEquality, 
dependent_pairFormation, 
inlEquality, 
inrEquality, 
dependent_set_memberEquality, 
natural_numberEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidEquality, 
independent_pairFormation, 
computeAll, 
equalityEquality, 
impliesFunctionality, 
setEquality, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
functionEquality, 
unionEquality, 
universeEquality, 
productEquality, 
addLevel, 
levelHypothesis, 
substitution
Latex:
\mforall{}[P:C\_DVALUEp()  {}\mrightarrow{}  \mBbbP{}]
    ((\mforall{}x:Unit.  P[DVp\_Null(x)])
    {}\mRightarrow{}  (\mforall{}int:\mBbbZ{}.  P[DVp\_Int(int)])
    {}\mRightarrow{}  (\mforall{}ptr:C\_LVALUE()?.  P[DVp\_Pointer(ptr)])
    {}\mRightarrow{}  (\mforall{}lower,upper:\mBbbZ{}.  \mforall{}arr:\{lower..upper\msupminus{}\}  {}\mrightarrow{}  C\_DVALUEp().
                ((\mforall{}u:\{lower..upper\msupminus{}\}.  P[arr  u])  {}\mRightarrow{}  P[DVp\_Array(lower;upper;arr)]))
    {}\mRightarrow{}  (\mforall{}lbls:Atom  List.  \mforall{}struct:\{a:Atom|  (a  \mmember{}  lbls)\}    {}\mrightarrow{}  C\_DVALUEp().
                ((\mforall{}u:\{a:Atom|  (a  \mmember{}  lbls)\}  .  P[struct  u])  {}\mRightarrow{}  P[DVp\_Struct(lbls;struct)]))
    {}\mRightarrow{}  \{\mforall{}v:C\_DVALUEp().  P[v]\})
Date html generated:
2016_05_16-AM-08_50_51
Last ObjectModification:
2016_01_17-AM-09_43_56
Theory : C-semantics
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