Nuprl Lemma : C_TYPE_ind_wf
∀[A:Type]. ∀[R:A ⟶ C_TYPE() ⟶ ℙ]. ∀[v:C_TYPE()]. ∀[Void:{x:A| R[x;C_Void()]} ]. ∀[Int:{x:A| R[x;C_Int()]} ].
∀[Struct:fields:((Atom × C_TYPE()) List)
         ⟶ (∀u∈fields.let u1,u2 = u 
                       in {x:A| R[x;u2]} )
         ⟶ {x:A| R[x;C_Struct(fields)]} ]. ∀[Array:length:ℕ
                                                   ⟶ elems:C_TYPE()
                                                   ⟶ {x:A| R[x;elems]} 
                                                   ⟶ {x:A| R[x;C_Array(length;elems)]} ].
∀[Pointer:to:C_TYPE() ⟶ {x:A| R[x;to]}  ⟶ {x:A| R[x;C_Pointer(to)]} ].
  (C_TYPE_ind(v
   Void=>Void
   Int=>Int
   Struct(fields)=>rec1.Struct[fields;rec1]
   Array(length,elems)=>rec2.Array[length;elems;rec2]
   Pointer(to)=>rec3.Pointer[to;rec3]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement : 
C_TYPE_ind: C_TYPE_ind, 
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
, 
so_apply: x[s1;s2;s3]
, 
so_apply: x[s1;s2]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
spread: spread def, 
product: x:A × B[x]
, 
atom: Atom
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
C_TYPE_ind: C_TYPE_ind, 
so_apply: x[s1;s2]
, 
so_apply: x[s1;s2;s3]
, 
C_TYPE-definition, 
C_TYPE-induction, 
uniform-comp-nat-induction, 
C_TYPE-ext, 
eq_atom: x =a y
, 
bool_cases_sqequal, 
eqff_to_assert, 
any: any x
, 
btrue: tt
, 
bfalse: ff
, 
it: ⋅
, 
top: Top
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
has-value: (a)↓
, 
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
, 
so_apply: x[s1;s2;s3;s4]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
strict4: strict4(F)
, 
and: P ∧ Q
, 
prop: ℙ
, 
guard: {T}
, 
or: P ∨ Q
, 
squash: ↓T
, 
so_lambda: λ2x y.t[x; y]
, 
subtype_rel: A ⊆r B
Lemmas referenced : 
C_TYPE-definition, 
C_TYPE-induction, 
uniform-comp-nat-induction, 
C_TYPE-ext, 
bool_cases_sqequal, 
eqff_to_assert, 
guard_wf, 
all_wf, 
C_Pointer_wf, 
C_Array_wf, 
nat_wf, 
C_Struct_wf, 
set_wf, 
l_member_wf, 
l_all_wf2, 
list_wf, 
C_Int_wf, 
C_Void_wf, 
C_TYPE_wf, 
lifting-strict-spread, 
base_wf, 
lifting-strict-atom_eq, 
is-exception_wf, 
has-value_wf_base, 
top_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalRule, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
thin, 
lemma_by_obid, 
hypothesis, 
lambdaFormation, 
because_Cache, 
sqequalSqle, 
divergentSqle, 
callbyvalueDecide, 
sqequalHypSubstitution, 
unionEquality, 
unionElimination, 
sqleReflexivity, 
equalityEquality, 
equalityTransitivity, 
equalitySymmetry, 
hypothesisEquality, 
dependent_functionElimination, 
independent_functionElimination, 
decideExceptionCases, 
axiomSqleEquality, 
exceptionSqequal, 
baseApply, 
closedConclusion, 
baseClosed, 
isectElimination, 
independent_isectElimination, 
independent_pairFormation, 
inrFormation, 
imageMemberEquality, 
imageElimination, 
inlFormation, 
callbyvalueApply, 
applyExceptionCases, 
instantiate, 
extract_by_obid, 
applyEquality, 
lambdaEquality, 
isectEquality, 
functionEquality, 
cumulativity, 
universeEquality, 
setEquality, 
setElimination, 
rename, 
productEquality, 
atomEquality, 
productElimination, 
independent_pairEquality, 
dependent_set_memberEquality, 
axiomEquality, 
spreadEquality
Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  C\_TYPE()  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[v:C\_TYPE()].  \mforall{}[Void:\{x:A|  R[x;C\_Void()]\}  ].
\mforall{}[Int:\{x:A|  R[x;C\_Int()]\}  ].  \mforall{}[Struct:fields:((Atom  \mtimes{}  C\_TYPE())  List)
                                                                          {}\mrightarrow{}  (\mforall{}u\mmember{}fields.let  u1,u2  =  u 
                                                                                                      in  \{x:A|  R[x;u2]\}  )
                                                                          {}\mrightarrow{}  \{x:A|  R[x;C\_Struct(fields)]\}  ].
\mforall{}[Array:length:\mBbbN{}  {}\mrightarrow{}  elems:C\_TYPE()  {}\mrightarrow{}  \{x:A|  R[x;elems]\}    {}\mrightarrow{}  \{x:A|  R[x;C\_Array(length;elems)]\}  ].
\mforall{}[Pointer:to:C\_TYPE()  {}\mrightarrow{}  \{x:A|  R[x;to]\}    {}\mrightarrow{}  \{x:A|  R[x;C\_Pointer(to)]\}  ].
    (C\_TYPE\_ind(v
      Void=>Void
      Int=>Int
      Struct(fields)=>rec1.Struct[fields;rec1]
      Array(length,elems)=>rec2.Array[length;elems;rec2]
      Pointer(to)=>rec3.Pointer[to;rec3])  \mmember{}  \{x:A|  R[x;v]\}  )
Date html generated:
2016_05_16-AM-08_45_22
Last ObjectModification:
2016_01_17-AM-09_44_21
Theory : C-semantics
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