Nuprl Lemma : l_tree-definition
∀[L,T,A:Type]. ∀[R:A ⟶ l_tree(L;T) ⟶ ℙ].
((∀val:L. {x:A| R[x;l_tree_leaf(val)]} )
⇒ (∀val:T. ∀left_subtree,right_subtree:l_tree(L;T).
({x:A| R[x;left_subtree]}
⇒ {x:A| R[x;right_subtree]}
⇒ {x:A| R[x;l_tree_node(val;left_subtree;right_subtree)]} ))
⇒ {∀v:l_tree(L;T). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
l_tree_node: l_tree_node(val;left_subtree;right_subtree),
l_tree_leaf: l_tree_leaf(val),
l_tree: l_tree(L;T),
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 :
l_tree-induction,
set_wf,
l_tree_wf,
all_wf,
l_tree_node_wf,
l_tree_leaf_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
independent_functionElimination,
cumulativity,
functionEquality,
setEquality,
setElimination,
rename,
universeEquality
Latex:
\mforall{}[L,T,A:Type]. \mforall{}[R:A {}\mrightarrow{} l\_tree(L;T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}val:L. \{x:A| R[x;l\_tree\_leaf(val)]\} )
{}\mRightarrow{} (\mforall{}val:T. \mforall{}left$_{subtree}$,right$_{subtree}$:l\_tree(L;T\000C).
(\{x:A| R[x;left$_{subtree}$]\}
{}\mRightarrow{} \{x:A| R[x;right$_{subtree}$]\}
{}\mRightarrow{} \{x:A| R[x;l\_tree\_node(val;left$_{subtree}$;right$_{subtre\000Ce}$)]\} ))
{}\mRightarrow{} \{\mforall{}v:l\_tree(L;T). \{x:A| R[x;v]\} \})
Date html generated:
2018_05_22-PM-09_39_15
Last ObjectModification:
2015_12_28-PM-06_41_47
Theory : labeled!trees
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