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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