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Step * of Lemma causal_order_or

∀[T:Type]
  ∀L:T List
    ∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ]. ∀[P1,P2,P3:ℕ||L|| ⟶ ℙ].
      (Trans(ℕ||L||)(R _1 _2)
      ⇒ causal_order(L;R;P1;P2)
      ⇒ causal_order(L;R;P1;P3)
      ⇒ causal_order(L;R;P1;λi.((P2 i) ∨ (P3 i))))
BY
{ (((Unfold `causal_order` 0 THEN Reduce 0 THEN Auto THEN ((D (-1)) THENL [FwdThruHyp 8 [(-1)]; FwdThruHyp 9 [(-1)]]))
    THENA Auto
    )
   THEN ExRepD
   THEN InstConcl [j]⋅
   THEN Auto) }

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Latex:

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Latex:
\mforall{}[T:Type]
    \mforall{}L:T  List
        \mforall{}[R:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P1,P2,P3:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].
            (Trans(\mBbbN{}||L||)(R  $_{1}$  $_{2}$)
            {}\mRightarrow{}  causal\_order(L;R;P1;P2)
            {}\mRightarrow{}  causal\_order(L;R;P1;P3)
            {}\mRightarrow{}  causal\_order(L;R;P1;\mlambda{}i.((P2  i)  \mvee{}  (P3  i))))


By

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Latex:
(((Unfold  `causal\_order`  0
      THEN  Reduce  0
      THEN  Auto
      THEN  ((D  (-1))  THENL  [FwdThruHyp  8  [(-1)];  FwdThruHyp  9  [(-1)]]))
    THENA  Auto
    )
  THEN  ExRepD
  THEN  InstConcl  [j]\mcdot{}
  THEN  Auto)

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