Goal:
  
  G1 : t
  G2 : t
  H : invariants G1
  ext := EdgeSet.diff (E G2) (E G1) : EdgeSet.t
  bs := border_set G1 ext : VSet.t
  H0 : invariants G2
  H1 : acyclic_no_loop G1
  H2 : acyclic_no_loop G2
  h : forall x y : VSet.elt,
      VSet.In x bs -> VSet.In y bs -> (lsp G2 x y <= lsp G1 x y)%nbar
  _s_ : VSet.t
  _s'_ : VSet.t
  x : V.t
  y : V.t
  z : V.t
  _d_ : DisjointAdd x _s_ _s'_
  e : EdgeOf G2 x y
  q : SPath G2 _s_ y z
  ih : VSet.In z (V G1) ->
       (Some (sweight q) <=
        lsp G2 y (first_in G1 G2 q) + lsp G1 (first_in G1 G2 q) z)%nbar
  hz : VSet.In z (V G1)
  Ee : ~ EdgeSet.In (x, e.π1, y) (E G1)
  H3 : (Some e.π1 <= lsp G2 x y)%nbar
  ============================
   (Some (sweight (spath_step G2 _s_ _s'_ x y z _d_ e q)) <=
    lsp G2 x
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x 
         (VSet.mem x (V G1)) z _s_ y _d_ e q) +
    lsp G1
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x 
         (VSet.mem x (V G1)) z _s_ y _d_ e q) z)%nbar


DEBUG:Going to execute:
<coq-core.plugins.ssreflect::ssrcase@0> $1 $2

Goals:
  
  G1 : t
  G2 : t
  H : invariants G1
  ext := EdgeSet.diff (E G2) (E G1) : EdgeSet.t
  bs := border_set G1 ext : VSet.t
  H0 : invariants G2
  H1 : acyclic_no_loop G1
  H2 : acyclic_no_loop G2
  h : forall x y : VSet.elt,
      VSet.In x bs -> VSet.In y bs -> (lsp G2 x y <= lsp G1 x y)%nbar
  _s_ : VSet.t
  _s'_ : VSet.t
  x : V.t
  y : V.t
  z : V.t
  _d_ : DisjointAdd x _s_ _s'_
  e : EdgeOf G2 x y
  q : SPath G2 _s_ y z
  ih : VSet.In z (V G1) ->
       (Some (sweight q) <=
        lsp G2 y (first_in G1 G2 q) + lsp G1 (first_in G1 G2 q) z)%nbar
  hz : VSet.In z (V G1)
  Ee : ~ EdgeSet.In (x, e.π1, y) (E G1)
  H3 : (Some e.π1 <= lsp G2 x y)%nbar
  Ex : VSet.mem x (V G1) = true
  ============================
   (Some (sweight (spath_step G2 _s_ _s'_ x y z _d_ e q)) <=
    lsp G2 x
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x true z _s_ y _d_ e q) +
    lsp G1
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x true z _s_ y _d_ e q) z)%nbar

  
  G1 : t
  G2 : t
  H : invariants G1
  ext := EdgeSet.diff (E G2) (E G1) : EdgeSet.t
  bs := border_set G1 ext : VSet.t
  H0 : invariants G2
  H1 : acyclic_no_loop G1
  H2 : acyclic_no_loop G2
  h : forall x y : VSet.elt,
      VSet.In x bs -> VSet.In y bs -> (lsp G2 x y <= lsp G1 x y)%nbar
  _s_ : VSet.t
  _s'_ : VSet.t
  x : V.t
  y : V.t
  z : V.t
  _d_ : DisjointAdd x _s_ _s'_
  e : EdgeOf G2 x y
  q : SPath G2 _s_ y z
  ih : VSet.In z (V G1) ->
       (Some (sweight q) <=
        lsp G2 y (first_in G1 G2 q) + lsp G1 (first_in G1 G2 q) z)%nbar
  hz : VSet.In z (V G1)
  Ee : ~ EdgeSet.In (x, e.π1, y) (E G1)
  H3 : (Some e.π1 <= lsp G2 x y)%nbar
  Ex : VSet.mem x (V G1) = false
  ============================
   (Some (sweight (spath_step G2 _s_ _s'_ x y z _d_ e q)) <=
    lsp G2 x
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x false z _s_ y _d_ e q) +
    lsp G1
      (first_in_clause_2 G2 (@first_in G1 G2) _s'_ x false z _s_ y _d_ e q) z)%nbar


DEBUG:Going to execute:
simp first_in
DEBUG LTAC Evaluated term: (VSet.mem ?e ?t)