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Step * of Lemma run-event-cases

∀[M:Type ⟶ Type]
  ∀S0:System(P.M[P]). ∀r:pRunType(P.M[P]). ∀e1,e2:runEvents(r).
    (((run-event-local-pred(r;e2) = run-event-local-pred(r;e1) ∈ (runEvents(r)?))
       ∧ (run-event-interval(r;e1;e2) = [e2] ∈ (runEvents(r) List)))
       ∨ (∃e:runEvents(r)
           (run-event-step(e) < run-event-step(e2)
           ∧ (run-event-step(e1) ≤ run-event-step(e))
           ∧ ((run-event-loc(e1) = run-event-loc(e) ∈ Id) ∧ (run-event-local-pred(r;e2) = (inl e) ∈ (runEvents(r)?)))
           ∧ (run-event-interval(r;e1;e2) = (run-event-interval(r;e1;e) @ [e2]) ∈ (runEvents(r) List))))) supposing 
       ((run-event-step(e1) ≤ run-event-step(e2)) and 
       (run-event-loc(e1) = run-event-loc(e2) ∈ Id))
BY
{ Auto }

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1. [M] : Type ⟶ Type
2. S0 : System(P.M[P])@i
3. r : pRunType(P.M[P])@i
4. e1 : runEvents(r)@i
5. e2 : runEvents(r)@i
6. run-event-loc(e1) = run-event-loc(e2) ∈ Id
7. run-event-step(e1) ≤ run-event-step(e2)
⊢ ((run-event-local-pred(r;e2) = run-event-local-pred(r;e1) ∈ (runEvents(r)?))
∧ (run-event-interval(r;e1;e2) = [e2] ∈ (runEvents(r) List)))
∨ (∃e:runEvents(r)
    (run-event-step(e) < run-event-step(e2)
    ∧ (run-event-step(e1) ≤ run-event-step(e))
    ∧ ((run-event-loc(e1) = run-event-loc(e) ∈ Id) ∧ (run-event-local-pred(r;e2) = (inl e) ∈ (runEvents(r)?)))
    ∧ (run-event-interval(r;e1;e2) = (run-event-interval(r;e1;e) @ [e2]) ∈ (runEvents(r) List))))

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Latex:

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Latex:
\mforall{}[M:Type  {}\mrightarrow{}  Type]
    \mforall{}S0:System(P.M[P]).  \mforall{}r:pRunType(P.M[P]).  \mforall{}e1,e2:runEvents(r).
        (((run-event-local-pred(r;e2)  =  run-event-local-pred(r;e1))
              \mwedge{}  (run-event-interval(r;e1;e2)  =  [e2]))
              \mvee{}  (\mexists{}e:runEvents(r)
                      (run-event-step(e)  <  run-event-step(e2)
                      \mwedge{}  (run-event-step(e1)  \mleq{}  run-event-step(e))
                      \mwedge{}  ((run-event-loc(e1)  =  run-event-loc(e))  \mwedge{}  (run-event-local-pred(r;e2)  =  (inl  e)))
                      \mwedge{}  (run-event-interval(r;e1;e2)  =  (run-event-interval(r;e1;e)  @  [e2])))))  supposing 
              ((run-event-step(e1)  \mleq{}  run-event-step(e2))  and 
              (run-event-loc(e1)  =  run-event-loc(e2)))


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Latex:
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