Nuprl Lemma : bs_tree-definition
∀[E,A:Type]. ∀[R:A ⟶ bs_tree(E) ⟶ ℙ].
({x:A| R[x;bst_null()]}
⇒ (∀value:E. {x:A| R[x;bst_leaf(value)]} )
⇒ (∀left:bs_tree(E). ∀value:E. ∀right:bs_tree(E).
({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;bst_node(left;value;right)]} ))
⇒ {∀v:bs_tree(E). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
bst_node: bst_node(left;value;right),
bst_leaf: bst_leaf(value),
bst_null: bst_null(),
bs_tree: bs_tree(E),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x]
Lemmas referenced :
bst_null_wf,
bst_leaf_wf,
bst_node_wf,
all_wf,
bs_tree_wf,
set_wf,
bs_tree-induction
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
sqequalRule,
lambdaEquality,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
independent_functionElimination,
functionEquality,
setEquality,
setElimination,
rename,
universeEquality
Latex:
\mforall{}[E,A:Type]. \mforall{}[R:A {}\mrightarrow{} bs\_tree(E) {}\mrightarrow{} \mBbbP{}].
(\{x:A| R[x;bst\_null()]\}
{}\mRightarrow{} (\mforall{}value:E. \{x:A| R[x;bst\_leaf(value)]\} )
{}\mRightarrow{} (\mforall{}left:bs\_tree(E). \mforall{}value:E. \mforall{}right:bs\_tree(E).
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;bst\_node(left;value;right)]\} ))
{}\mRightarrow{} \{\mforall{}v:bs\_tree(E). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_15-PM-01_50_57
Last ObjectModification:
2016_04_07-PM-02_26_29
Theory : tree_1
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