Nuprl Lemma : C_TYPE-induction3
∀[P:C_TYPE() ⟶ ℙ]
(P[C_Void()]
⇒ P[C_Int()]
⇒ (∀fields:(Atom × C_TYPE()) List. ((∀ct∈map(λp.(snd(p));fields).P[ct]) ⇒ 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)]))
⇒ {∀x:C_TYPE(). P[x]})
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]),
map: map(f;as),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
atom: Atom
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
pi2: snd(t),
subtype_rel: A ⊆r B,
l_all: (∀x∈L.P[x]),
nat: ℕ,
le: A ≤ B
Lemmas referenced :
lelt_wf,
less_than_wf,
nat_wf,
equal_wf,
and_wf,
length_wf_nat,
map-length,
top_wf,
subtype_rel_list,
select-map,
C_Void_wf,
C_Int_wf,
C_Struct_wf,
l_member_wf,
map_wf,
l_all_wf2,
list_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,
int_seg_properties,
select_wf,
C_TYPE_wf,
length_wf,
int_seg_wf,
all_wf,
C_TYPE-induction2
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
natural_numberEquality,
productEquality,
atomEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
universeEquality,
cumulativity,
dependent_set_memberEquality,
substitution
Latex:
\mforall{}[P:C\_TYPE() {}\mrightarrow{} \mBbbP{}]
(P[C\_Void()]
{}\mRightarrow{} P[C\_Int()]
{}\mRightarrow{} (\mforall{}fields:(Atom \mtimes{} C\_TYPE()) List. ((\mforall{}ct\mmember{}map(\mlambda{}p.(snd(p));fields).P[ct]) {}\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{}x:C\_TYPE(). P[x]\})
Date html generated:
2016_05_16-AM-08_46_07
Last ObjectModification:
2016_01_17-AM-09_43_12
Theory : C-semantics
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