Nuprl Lemma : MultiTree-definition
∀[T,A:Type]. ∀[R:A ⟶ MultiTree(T) ⟶ ℙ].
((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)} ⟶ MultiTree(T).
((∀u:{a:Atom| (a ∈ labels)} . {x:A| R[x;children u]} ) ⇒ {x:A| R[x;MTree_Node(labels;children)]} ))
⇒ (∀val:T. {x:A| R[x;MTree_Leaf(val)]} )
⇒ {∀v:MultiTree(T). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
MTree_Leaf: MTree_Leaf(val),
MTree_Node: MTree_Node(labels;children),
MultiTree: MultiTree(T),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
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]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
atom: Atom,
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 :
MultiTree-induction,
set_wf,
MultiTree_wf,
all_wf,
MTree_Leaf_wf,
list_wf,
less_than_wf,
length_wf,
l_member_wf,
MTree_Node_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,
universeEquality,
setEquality,
atomEquality,
natural_numberEquality,
setElimination,
rename,
functionEquality,
cumulativity,
dependent_set_memberEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[R:A {}\mrightarrow{} MultiTree(T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}labels:\{L:Atom List| 0 < ||L||\} . \mforall{}children:\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTree(T).
((\mforall{}u:\{a:Atom| (a \mmember{} labels)\} . \{x:A| R[x;children u]\} ) {}\mRightarrow{} \{x:A| R[x;MTree\_Node(labels;children\000C)]\} ))
{}\mRightarrow{} (\mforall{}val:T. \{x:A| R[x;MTree\_Leaf(val)]\} )
{}\mRightarrow{} \{\mforall{}v:MultiTree(T). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_16-AM-08_53_38
Last ObjectModification:
2015_12_28-PM-06_54_04
Theory : C-semantics
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