### Nuprl Lemma : tree-definition

`∀[E,A:Type]. ∀[R:A ⟶ tree(E) ⟶ ℙ].`
`  ((∀value:E. {x:A| R[x;tree_leaf(value)]} )`
`  `` (∀left,right:tree(E).  ({x:A| R[x;left]}  `` {x:A| R[x;right]}  `` {x:A| R[x;tree_node(left;right)]} ))`
`  `` {∀v:tree(E). {x:A| R[x;v]} })`

Proof

Definitions occuring in Statement :  tree_node: `tree_node(left;right)` tree_leaf: `tree_leaf(value)` tree: `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 :  tree-induction set_wf tree_wf all_wf tree_node_wf 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{}[E,A:Type].  \mforall{}[R:A  {}\mrightarrow{}  tree(E)  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}value:E.  \{x:A|  R[x;tree\_leaf(value)]\}  )
{}\mRightarrow{}  (\mforall{}left,right:tree(E).
(\{x:A|  R[x;left]\}    {}\mRightarrow{}  \{x:A|  R[x;right]\}    {}\mRightarrow{}  \{x:A|  R[x;tree\_node(left;right)]\}  ))
{}\mRightarrow{}  \{\mforall{}v:tree(E).  \{x:A|  R[x;v]\}  \})

Date html generated: 2016_05_15-PM-01_49_52
Last ObjectModification: 2015_12_27-AM-00_12_40

Theory : tree_1

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