Nuprl Lemma : C_TYPE-induction
∀[P:C_TYPE() ⟶ ℙ]
(P[C_Void()]
⇒ P[C_Int()]
⇒ (∀fields:(Atom × C_TYPE()) List. ((∀u∈fields.let u1,u2 = u in P[u2]) ⇒ P[C_Struct(fields)]))
⇒ (∀length:ℕ. ∀elems:C_TYPE(). (P[elems] ⇒ P[C_Array(length;elems)]))
⇒ (∀to:C_TYPE(). (P[to] ⇒ P[C_Pointer(to)]))
⇒ {∀v:C_TYPE(). P[v]})
Proof
Definitions occuring in Statement :
C_Pointer: C_Pointer(to),
C_Array: C_Array(length;elems),
C_Struct: C_Struct(fields),
C_Int: C_Int(),
C_Void: C_Void(),
C_TYPE: C_TYPE(),
l_all: (∀x∈L.P[x]),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
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 ,
C_Void: C_Void(),
C_TYPE_size: C_TYPE_size(p),
select: L[n],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
C_Int: C_Int(),
C_Struct: C_Struct(fields),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
pi2: snd(t),
C_Array: C_Array(length;elems),
C_Pointer: C_Pointer(to)
Lemmas referenced :
C_Void_wf,
C_Int_wf,
C_Struct_wf,
l_member_wf,
l_all_wf2,
list_wf,
C_Array_wf,
C_Pointer_wf,
uall_wf,
lelt_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
sum-nat-less,
int_term_value_add_lemma,
itermAdd_wf,
int_seg_wf,
pi2_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
sum-nat,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
base_wf,
stuck-spread,
it_wf,
unit_subtype_base,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
C_TYPE-ext,
less_than'_wf,
nat_wf,
C_TYPE_size_wf,
le_wf,
isect_wf,
C_TYPE_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,
baseClosed,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
productEquality,
natural_numberEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
imageElimination,
equalityEquality,
setEquality,
dependent_set_memberEquality,
functionEquality,
spreadEquality,
universeEquality
Latex:
\mforall{}[P:C\_TYPE() {}\mrightarrow{} \mBbbP{}]
(P[C\_Void()]
{}\mRightarrow{} P[C\_Int()]
{}\mRightarrow{} (\mforall{}fields:(Atom \mtimes{} C\_TYPE()) List. ((\mforall{}u\mmember{}fields.let u1,u2 = u in P[u2]) {}\mRightarrow{} P[C\_Struct(fields)]))
{}\mRightarrow{} (\mforall{}length:\mBbbN{}. \mforall{}elems:C\_TYPE(). (P[elems] {}\mRightarrow{} P[C\_Array(length;elems)]))
{}\mRightarrow{} (\mforall{}to:C\_TYPE(). (P[to] {}\mRightarrow{} P[C\_Pointer(to)]))
{}\mRightarrow{} \{\mforall{}v:C\_TYPE(). P[v]\})
Date html generated:
2016_05_16-AM-08_45_15
Last ObjectModification:
2016_01_17-AM-09_44_02
Theory : C-semantics
Home
Index