Nuprl Lemma : binary-tree_ind_wf
∀[A:Type]. ∀[R:A ⟶ binary-tree() ⟶ ℙ]. ∀[v:binary-tree()]. ∀[Leaf:val:ℤ ⟶ {x:A| R[x;btr_Leaf(val)]} ].
∀[Node:left:binary-tree()
       ⟶ right:binary-tree()
       ⟶ {x:A| R[x;left]} 
       ⟶ {x:A| R[x;right]} 
       ⟶ {x:A| R[x;btr_Node(left;right)]} ].
  (binary-tree_ind(v;
                   btr_Leaf(val)
⇒ Leaf[val];
                   btr_Node(left,right)
⇒ rec1,rec2.Node[left;right;rec1;rec2])  ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement : 
binary-tree_ind: binary-tree_ind, 
btr_Node: btr_Node(left;right)
, 
btr_Leaf: btr_Leaf(val)
, 
binary-tree: binary-tree()
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2;s3;s4]
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
int: ℤ
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
binary-tree_ind: binary-tree_ind, 
so_apply: x[s1;s2;s3;s4]
, 
so_apply: x[s]
, 
so_apply: x[s1;s2]
, 
binary-tree-definition, 
binary-tree-induction, 
uniform-comp-nat-induction, 
binary-tree-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_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 : 
binary-tree-definition, 
binary-tree-induction, 
uniform-comp-nat-induction, 
binary-tree-ext, 
bool_cases_sqequal, 
eqff_to_assert, 
set_wf, 
all_wf, 
guard_wf, 
btr_Node_wf, 
btr_Leaf_wf, 
binary-tree_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, 
functionEquality, 
cumulativity, 
universeEquality, 
intEquality, 
setEquality, 
setElimination, 
rename, 
dependent_set_memberEquality, 
axiomEquality
Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  binary-tree()  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[v:binary-tree()].  \mforall{}[Leaf:val:\mBbbZ{}  {}\mrightarrow{}  \{x:A| 
                                                                                                                                                            R[x;btr\_Leaf(val)]\}  ].
\mforall{}[Node:left:binary-tree()
              {}\mrightarrow{}  right:binary-tree()
              {}\mrightarrow{}  \{x:A|  R[x;left]\} 
              {}\mrightarrow{}  \{x:A|  R[x;right]\} 
              {}\mrightarrow{}  \{x:A|  R[x;btr\_Node(left;right)]\}  ].
    (binary-tree\_ind(v;
                                      btr\_Leaf(val){}\mRightarrow{}  Leaf[val];
                                      btr\_Node(left,right){}\mRightarrow{}  rec1,rec2.Node[left;right;rec1;rec2])    \mmember{}  \{x:A|  R[x;v]\}  )
Date html generated:
2016_05_16-AM-09_07_18
Last ObjectModification:
2016_01_17-AM-09_42_45
Theory : C-semantics
Home
Index