Nuprl Lemma : MultiTree-definition
∀[T,A:Type]. ∀[R:A ⟶ MultiTree(T) ⟶ ℙ].
  ((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)}  ⟶ MultiTree(T).
      ((∀u:{a:Atom| (a ∈ labels)} . {x:A| R[x;children u]} ) 
⇒ {x:A| R[x;MTree_Node(labels;children)]} ))
  
⇒ (∀val:T. {x:A| R[x;MTree_Leaf(val)]} )
  
⇒ {∀v:MultiTree(T). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement : 
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
, 
guard: {T}
, 
so_apply: x[s1;s2]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
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
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s1;s2]
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s]
, 
prop: ℙ
, 
all: ∀x:A. B[x]
Lemmas referenced : 
MultiTree-induction, 
set_wf, 
MultiTree_wf, 
all_wf, 
MTree_Leaf_wf, 
list_wf, 
less_than_wf, 
length_wf, 
l_member_wf, 
MTree_Node_wf
Rules used in proof : 
cut, 
lemma_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
lambdaFormation, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
because_Cache, 
independent_functionElimination, 
universeEquality, 
setEquality, 
atomEquality, 
natural_numberEquality, 
setElimination, 
rename, 
functionEquality, 
cumulativity, 
dependent_set_memberEquality
Latex:
\mforall{}[T,A:Type].  \mforall{}[R:A  {}\mrightarrow{}  MultiTree(T)  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}labels:\{L:Atom  List|  0  <  ||L||\}  .  \mforall{}children:\{a:Atom|  (a  \mmember{}  labels)\}    {}\mrightarrow{}  MultiTree(T).
            ((\mforall{}u:\{a:Atom|  (a  \mmember{}  labels)\}  .  \{x:A|  R[x;children  u]\}  )  {}\mRightarrow{}  \{x:A|  R[x;MTree\_Node(labels;children\000C)]\}  ))
    {}\mRightarrow{}  (\mforall{}val:T.  \{x:A|  R[x;MTree\_Leaf(val)]\}  )
    {}\mRightarrow{}  \{\mforall{}v:MultiTree(T).  \{x:A|  R[x;v]\}  \})
Date html generated:
2016_05_16-AM-08_53_38
Last ObjectModification:
2015_12_28-PM-06_54_04
Theory : C-semantics
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