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
Lemmas :
top_wf,
has-value_wf_base,
lifting-strict-atom_eq,
base_wf,
btr_Leaf_wf,
binary-tree_wf,
btr_Node_wf,
all_wf,
set_wf,
subtype_rel-equal
\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:
2015_07_17-AM-07_52_24
Last ObjectModification:
2015_01_29-PM-04_39_12
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