Nuprl Lemma : binary-tree-definition
∀[A:Type]. ∀[R:A ⟶ binary-tree() ⟶ ℙ].
  ((∀val:ℤ. {x:A| R[x;btr_Leaf(val)]} )
  
⇒ (∀left,right:binary-tree().  ({x:A| R[x;left]}  
⇒ {x:A| R[x;right]}  
⇒ {x:A| R[x;btr_Node(left;right)]} ))
  
⇒ {∀v:binary-tree(). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement : 
btr_Node: btr_Node(left;right)
, 
btr_Leaf: btr_Leaf(val)
, 
binary-tree: binary-tree()
, 
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]} 
, 
function: x:A ⟶ B[x]
, 
int: ℤ
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
member: t ∈ T
, 
so_apply: x[s1;s2]
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s]
, 
prop: ℙ
Lemmas referenced : 
binary-tree-induction, 
set_wf, 
binary-tree_wf, 
all_wf, 
btr_Node_wf, 
btr_Leaf_wf
Rules used in proof : 
cut, 
lemma_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
sqequalRule, 
lambdaEquality, 
hypothesisEquality, 
applyEquality, 
because_Cache, 
independent_functionElimination, 
functionEquality, 
universeEquality, 
intEquality, 
cumulativity
Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  binary-tree()  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}val:\mBbbZ{}.  \{x:A|  R[x;btr\_Leaf(val)]\}  )
    {}\mRightarrow{}  (\mforall{}left,right:binary-tree().
                (\{x:A|  R[x;left]\}    {}\mRightarrow{}  \{x:A|  R[x;right]\}    {}\mRightarrow{}  \{x:A|  R[x;btr\_Node(left;right)]\}  ))
    {}\mRightarrow{}  \{\mforall{}v:binary-tree().  \{x:A|  R[x;v]\}  \})
Date html generated:
2016_05_16-AM-09_07_08
Last ObjectModification:
2015_12_28-PM-06_48_51
Theory : C-semantics
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