Nuprl Lemma : MMTree-definition
∀[T,A:Type]. ∀[R:A ⟶ MMTree(T) ⟶ ℙ].
((∀val:T. {x:A| R[x;MMTree_Leaf(val)]} )
⇒ (∀forest:MMTree(T) List List. ((∀u∈forest.(∀u1∈u.{x:A| R[x;u1]} )) ⇒ {x:A| R[x;MMTree_Node(forest)]} ))
⇒ {∀v:MMTree(T). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
MMTree_Node: MMTree_Node(forest),
MMTree_Leaf: MMTree_Leaf(val),
MMTree: MMTree(T),
l_all: (∀x∈L.P[x]),
list: T List,
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 :
MMTree-induction,
set_wf,
MMTree_wf,
all_wf,
list_wf,
l_all_wf2,
l_member_wf,
MMTree_Node_wf,
MMTree_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,
setElimination,
rename,
setEquality,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[R:A {}\mrightarrow{} MMTree(T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}val:T. \{x:A| R[x;MMTree\_Leaf(val)]\} )
{}\mRightarrow{} (\mforall{}forest:MMTree(T) List List
((\mforall{}u\mmember{}forest.(\mforall{}u1\mmember{}u.\{x:A| R[x;u1]\} )) {}\mRightarrow{} \{x:A| R[x;MMTree\_Node(forest)]\} ))
{}\mRightarrow{} \{\mforall{}v:MMTree(T). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_16-AM-08_55_37
Last ObjectModification:
2015_12_28-PM-06_53_30
Theory : C-semantics
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