### Nuprl Lemma : transitive-reflexive-closure-base-case

`∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].  ∀x,y:A.  ((x R y) `` (x R^* y))`

Proof

Definitions occuring in Statement :  transitive-reflexive-closure: `R^*` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` transitive-reflexive-closure: `R^*` infix_ap: `x f y` guard: `{T}` or: `P ∨ Q` member: `t ∈ T` prop: `ℙ` rel_implies: `R1 => R2`
Lemmas referenced :  equal_wf transitive-closure-contains
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule cut hypothesis inrFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality applyEquality functionExtensionality functionEquality universeEquality dependent_functionElimination independent_functionElimination

Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].    \mforall{}x,y:A.    ((x  R  y)  {}\mRightarrow{}  (x  R\^{}*  y))

Date html generated: 2017_01_19-PM-02_17_47
Last ObjectModification: 2017_01_14-PM-05_04_26

Theory : relations2

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