Nuprl Lemma : MultiTree_ind_wf
∀[T,A:Type]. ∀[R:A ⟶ MultiTree(T) ⟶ ℙ]. ∀[v:MultiTree(T)]. ∀[Node:labels:{L:Atom List| 0 < ||L||} 
                                                                    ⟶ children:({a:Atom| (a ∈ labels)}  ⟶ MultiTree(T)\000C)
                                                                    ⟶ (u:{a:Atom| (a ∈ labels)}  ⟶ {x:A| R[x;children \000Cu]} )
                                                                    ⟶ {x:A| R[x;MTree_Node(labels;children)]} ].
∀[Leaf:val:T ⟶ {x:A| R[x;MTree_Leaf(val)]} ].
  (MultiTree_ind(v;
                 MTree_Node(labels,children)
⇒ rec1.Node[labels;children;rec1];
                 MTree_Leaf(val)
⇒ Leaf[val])  ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement : 
MultiTree_ind: MultiTree_ind, 
MTree_Leaf: MTree_Leaf(val)
, 
MTree_Node: MTree_Node(labels;children)
, 
MultiTree: MultiTree(T)
, 
l_member: (x ∈ l)
, 
length: ||as||
, 
list: T List
, 
less_than: a < b
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2;s3]
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
atom: Atom
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
MultiTree_ind: MultiTree_ind, 
so_apply: x[s]
, 
so_apply: x[s1;s2;s3]
, 
so_apply: x[s1;s2]
, 
MultiTree-definition, 
MultiTree-induction, 
uniform-comp-nat-induction, 
MultiTree-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]
, 
uimplies: b supposing a
, 
strict4: strict4(F)
, 
and: P ∧ Q
, 
prop: ℙ
, 
guard: {T}
, 
or: P ∨ Q
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
Lemmas referenced : 
MultiTree-definition, 
MultiTree-induction, 
uniform-comp-nat-induction, 
MultiTree-ext, 
bool_cases_sqequal, 
eqff_to_assert, 
set_wf, 
all_wf, 
MTree_Leaf_wf, 
MTree_Node_wf, 
l_member_wf, 
length_wf, 
less_than_wf, 
list_wf, 
MultiTree_wf, 
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, 
instantiate, 
extract_by_obid, 
applyEquality, 
lambdaEquality, 
isectEquality, 
universeEquality, 
functionEquality, 
cumulativity, 
setEquality, 
atomEquality, 
natural_numberEquality, 
setElimination, 
rename, 
dependent_set_memberEquality, 
axiomEquality
Latex:
\mforall{}[T,A:Type].  \mforall{}[R:A  {}\mrightarrow{}  MultiTree(T)  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[v:MultiTree(T)].
\mforall{}[Node:labels:\{L:Atom  List|  0  <  ||L||\} 
              {}\mrightarrow{}  children:(\{a:Atom|  (a  \mmember{}  labels)\}    {}\mrightarrow{}  MultiTree(T))
              {}\mrightarrow{}  (u:\{a:Atom|  (a  \mmember{}  labels)\}    {}\mrightarrow{}  \{x:A|  R[x;children  u]\}  )
              {}\mrightarrow{}  \{x:A|  R[x;MTree\_Node(labels;children)]\}  ].  \mforall{}[Leaf:val:T  {}\mrightarrow{}  \{x:A|  R[x;MTree\_Leaf(val)]\}  ].
    (MultiTree\_ind(v;
                                  MTree\_Node(labels,children){}\mRightarrow{}  rec1.Node[labels;children;rec1];
                                  MTree\_Leaf(val){}\mRightarrow{}  Leaf[val])    \mmember{}  \{x:A|  R[x;v]\}  )
Date html generated:
2016_05_16-AM-08_53_46
Last ObjectModification:
2016_01_17-AM-09_42_34
Theory : C-semantics
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