Nuprl Lemma : binary-tree_ind_wf_simple
∀[A:Type]. ∀[v:binary-tree()]. ∀[Leaf:val:ℤ ⟶ A]. ∀[Node:left:binary-tree() ⟶ right:binary-tree() ⟶ A ⟶ A ⟶ A].
(binary-tree_ind(v;
btr_Leaf(val)⇒ Leaf[val];
btr_Node(left,right)⇒ rec1,rec2.Node[left;right;rec1;rec2]) ∈ A)
Proof
Definitions occuring in Statement :
binary-tree_ind: binary-tree_ind,
binary-tree: binary-tree(),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
true: True,
guard: {T}
Lemmas referenced :
binary-tree_ind_wf,
true_wf,
binary-tree_wf,
subtype_rel_dep_function,
set_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
because_Cache,
setEquality,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality,
natural_numberEquality,
functionEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[v:binary-tree()]. \mforall{}[Leaf:val:\mBbbZ{} {}\mrightarrow{} A]. \mforall{}[Node:left:binary-tree()
{}\mrightarrow{} right:binary-tree()
{}\mrightarrow{} A
{}\mrightarrow{} A
{}\mrightarrow{} A].
(binary-tree\_ind(v;
btr\_Leaf(val){}\mRightarrow{} Leaf[val];
btr\_Node(left,right){}\mRightarrow{} rec1,rec2.Node[left;right;rec1;rec2]) \mmember{} A)
Date html generated:
2016_05_16-AM-09_07_24
Last ObjectModification:
2015_12_28-PM-06_48_28
Theory : C-semantics
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