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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