Quelle graph_deg.prf
Sprache: Lisp
(graph_degnil 3507100590"" (skosimp*)"" (lemma "finite_subset[doubleton[T]]" )"" (inst?)"" (inst -1 "edges(G!1)" )"" (assert ) (("" (hide 2) (("" (grind) nil ))))))))))))nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )nil finite_sets nil )"bool" finite_sets nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" sets nil )"bool" sets nil )"(partial_order?[set[T]])" nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil ))nil ))nil 3507100590 ("" (grind) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil graph_deg nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil )"[bool, bool -> bool]" booleans nil )nil finite_sets nil )nil graphs nil )nil graphs nil )"finite_set[doubleton[T]]" graph_degnil )"bool" sets nil )"bool" sets nil ))nil ))nil 3507100590"" (skosimp*)"" (apply-extensionality 1 :hide? t)"" (expand "emptyset" )"" (expand "incident_edges" ) (("" (ground) nil ))))))))nil )nil graph_deg nil )nil booleans nil )nil booleans nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil )"set" sets nil )"finite_set[doubleton[T]]" graph_degnil )"bool" finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )"finite_set" finite_sets nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" "incident_edges(y!1, del_edge(G!1,e!1)) = ") "1" (rewrite "card_remove[doubleton[T]]" )"1" (lift-if)"1" (ground)"1" (hide -1 2)"1" (hide -2) (("1" (grind) nil )))))))))))"2" (hide 2)"2" (apply-extensionality 1 :hide? t)"2" (iff 1)"2" (prop)"1" (expand "incident_edges" )"1" (ground)"1" (expand "remove" )"1" (expand "member" )"1" (ground)"1" (replace -1)"1" (hide -1)"1" "del_edge_lem[T]" )"1" "1" expand "member" )"1" (propax) nil )))))))))))"2" (lemma "del_edge_lem2[T]" )"2" (inst?)"2" (assert ) nil )))))))))))))))"2" (expand "incident_edges" )"2" (expand "remove" )"2" (flatten)"2" (expand "member" )"2" (ground)"2" (rewrite "del_edge_lem3" )nil ))))))))))))))))))))))))nil )"nat" graph_deg nil )nil graph_ops nil )nil graph_ops nil )nil graph_ops nil )"bool" sets nil )nil finite_sets nil )"int" integers nil )"int" integers nil )nil application-judgement "finite_set[T]" graph_deg nil )"set" sets nil )"graph[T]" graph_ops nil )"finite_set[doubleton[T]]" graph_degnil )"bool" finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )"finite_set" finite_sets nil ))nil ))nil 3507100590"" (skosimp*)"" (lemma "deg_del_edge" )"" (inst?)"" (typepred "e!1" )"" (skosimp*)"" (case "x!1=y!1" )"1" (replace -1)"1" (hide -1)"1" (inst -2 "y!2" )"1" (split -2)"1" (propax) nil )"2" (replace -1)"2" (hide -1)"2" (hide -1 -2 3)"2" (expand "dbl" )"2" "2" "2" (ground) nil )))))))))))))"3" (propax) nil )))))))))"2" (case "y!2 = y!1" )"1" (replace -1)"1" (hide -1)"1" (inst -2 "x!1" ) (("1" (assert ) nil )))))))"2" (hide -2 -4 4)"2" (replace -1)"2" (hide -1)"2" (expand "dbl" )"2" (ground) nil ))))))))))))))))))))))nil )nil graph_deg nil )NOT const-decl "[bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil application-judgement "finite_set[T]" graph_deg nil )"posint" nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" case "incident_edges(y!1, G!1) = incident_edges(y!1, del_edge(G!1, e!1))" )"1" (assert ) nil )"2" (hide 3)"2" (apply-extensionality 1 :hide? t)"2" (expand "incident_edges" )"2" (iff 1)"2" (ground)"1" (lemma "del_edge_lem3[T]" )"1" (inst?) (("1" (assert ) nil )))))"2" (lemma "del_edge_lem2[T]" )"2" (inst?)"2" (assert ) nil ))))))))))))))))))))nil )"nat" graph_deg nil )nil graph_ops nil )nil graph_ops nil )nil graph_deg nil )nil booleans nil )nil booleans nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil )nil finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )"bool" finite_sets nil )"finite_set[doubleton[T]]" graph_degnil )"graph[T]" graph_ops nil ))nil ))nil 3507100590"" (skosimp*)"" (case "e!1(y!1)" )"1" (lemma "deg_del_edge2" )"1" (inst?)"1" (assert )"1" (lemma "del_edge_lem5[T]" )"1" (inst?) (("1" (assert ) nil )))))))))))"2" (lemma "deg_del_edge3" )"2" (inst?) (("2" (assert ) nil ))))))))nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )"even_int" integersnil )"int" integers nil )nil finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil graph_ops nil )"(total_order?[real])" nil )nil graph_deg nil )nil graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (case "e!1(y!1)" )"1" (lemma "deg_del_edge2" )"1" (inst?)"1" (assert )"1" (lemma "del_edge_lem5[T]" )"1" (inst?) (("1" (assert ) nil )))))))))))"2" (lemma "deg_del_edge3" )"2" (inst?) (("2" (assert ) nil ))))))))nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )"even_int" integersnil )"int" integers nil )nil finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil graph_ops nil )"(total_order?[real])" nil )nil graph_deg nil )nil graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" (rewrite "nonempty_card[doubleton[T]]" :dir rl)"" (expand "nonempty?" )"" (expand "empty?" )"" (expand "incident_edges" )"" (skosimp*)"" (expand "member" )"" (inst?)"" (flatten)"" (assert ) nil ))))))))))))))))))))nil )"nat" graph_deg nil )"bool" sets nil )"bool" sets nil )"bool" sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )"finite_set[doubleton[T]]" graph_degnil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )"bool" finite_sets nil )nil finite_sets nil ))nil ))nil 3507100590"" (skosimp*)"" (lemma "deg_edge_exists" )"" (inst?)"" (assert )"" (skosimp*)"" (hide -1 -3)"" (typepred "G!1" )"" (inst?)"" (assert )"" (typepred "e!1" )"" (skosimp*)"" (inst-cp -2 "x!1" )"" (inst -2 "y!1" )"" (replace -1)"" "" "" expand "dbl" )"" expand "size" )"" case "subset?(add[T](x!1,singleton(y!1)),vert(G!1))" )"1" "card_subset[T]" )"1" "1" assert )"1" "1" "card_add[T]" )"1" "1" "card_singleton[T]" )"1" "1" replace -1)"1" "1" expand "singleton" )"1" replace "1" "1" assert )nil )))))))))))))))))))))))"2" rewrite "finite_add[T]" )nil )))))"2" "2" nil ))))))))))))))))))))))))))))))))))))))))nil )nil graph_deg nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"nat" graphs nil )"bool" sets nil )nil finite_sets nil )"non_empty_finite_set" finite_sets nil )"(partial_order?[set[T]])" nil )nil finite_sets nil )nil finite_sets nil )"(total_order?[real])" nil )"bool" finite_sets nil )"(singleton?)" sets nil )"bool" sets nil )"(nonempty?)" sets nil )"bool" sets nil )"bool" sets nil )"non_empty_finite_set" nil )NOT const-decl "[bool -> bool]" booleans nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" (lemma "card_empty?[doubleton[T]]" )"" (inst?)"" (iff)"" (assert )"" (hide -2)"" (expand "incident_edges" )"" (expand "empty?" )"" (expand "member" )"" (apply-extensionality 1 :hide? t)"" (expand "del_vert" )"" (inst?)"" (iff 1)"" nil ))))))))))))))))))))))))))))nil )"nat" graph_deg nil )"bool" finite_sets nil )nil finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )"finite_set[doubleton[T]]" graph_degnil )"bool" sets nil )"graph[T]" graph_ops nil )"bool" sets nil )nil finite_sets nil )nil graph_deg nil )nil booleans nil )nil booleans nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil ))nil ))nil 3507100590 ("" (subtype-tcc) nil nil )nil graph_deg nil )"boolean" notequal nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" "incident_edges(v!1, del_vert(G!1, x!1)) = ") "1" (hide -1)"1" (lemma "deg_del_edge" )"1" (inst?)"1" (inst -1 "x!1" "v!1" )"1" (prop)"1" (expand "deg" ) (("1" (assert ) nil )))))))"2" (inst?) (("2" (assert ) nil )))))))))"2" (hide 3)"2" (apply-extensionality 1 :hide? t)"1" (expand "incident_edges" )"1" (grind)"1" (hide -3)"1" (replace -2 * rl) (("1" (grind) nil )))))"2" (hide -1 -4)"2" (lemma "edge_has_2_verts[T]" )"2" (inst?)"2" (inst?) (("2" (assert ) nil )))))))))))))"2" (inst?) (("2" (assert ) nil )))))))"3" (inst?) (("3" (assert ) nil ))))))))nil )"nat" graph_deg nil )nil graphs nil )"finite_set" finite_sets nil )"bool" sets nil )"set" sets nil )"even_int" integersnil )"int" integers nil )"posint" nil )nil graph_deg nil )"graph[T]" graph_ops nil )"graph[T]" graph_ops nil )"finite_set[doubleton[T]]" graph_degnil )"bool" finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil )nil application-judgement "finite_set[T]" graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" "incident_edges(x!1, G!1) = incident_edges(x!1, del_vert(G!1, v!1))" )"" (hide 4)"" (apply-extensionality 1 :hide? t)"" (expand "incident_edges" )"" (iff 1)"" (prop)"1" (lemma "edge_in_del_vert" )"1" (inst?)"1" (assert )"1" (lemma "edge_has_2_verts" )"1" (inst?)"1" (inst?)"1" (assert ) nil )))))))))))))"2" (expand "del_vert" )"2" (flatten)"2" (propax) nil ))))))))))))))))))))nil )"nat" graph_deg nil )nil graphs nil )nil graph_ops nil )nil application-judgement "finite_set[T]" graph_deg nil )"graph[T]" graph_ops nil )"finite_set[doubleton[T]]" graph_degnil )"bool" finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )nil finite_sets nil )nil doubletons nil )"set[T]" doubletons nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "singleton?" )"" (expand "deg" )"" (rewrite "card_is_0[doubleton[T]]" )"" (apply-extensionality 1 :hide? t)"" (expand "incident_edges" )"" (expand "emptyset" )"" (flatten)"" (lemma "edge_in_card_gt_1[T]" )"" (inst?)"" (split -1)"1" (assert )"1" (expand "size" ) (("1" (assert ) nil )))))"2" (propax) nil ))))))))))))))))))))))nil )"bool" graphs nil )nil finite_sets nil )"bool" finite_sets nil )nil finite_sets nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )"finite_set[doubleton[T]]" graph_degnil )nil graph_deg nil )nil booleans nil )nil booleans nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil )"finite_set" finite_sets nil )"(strict_total_order?[real])" real_props nil )"nat" graphs nil )nil graphs nil )"set" sets nil )"nat" graph_deg nil ))nil ))nil 3507100590"" (skosimp*)"" (expand "deg" )"" (lemma "card_one[Dbl]" )"" (inst?)"" (flatten)"" (assert )"" (hide -2)"" (skosimp*)"" (inst?)"" (assert )"" "incident_edges(v!1, G!1)(x!1) = singleton(x!1)(x!1)" )"1" (hide -2)"1" (expand "incident_edges" )"1" (expand "singleton" )"1" "1" (assert ) nil )))))))))"2" (assert ) nil ))))))))))))))))))))))nil )"nat" graph_deg nil )"bool" finite_sets nil )nil finite_sets nil )nil doubletons nil )nil graphs nil )"[bool, bool -> bool]" booleans nil )nil graphs nil )"finite_set[doubleton[T]]" graph_degnil )"non_empty_finite_set" finite_sets nil )"bool" sets nil )"(singleton?)" sets nil )nil finite_sets nil )nil graph_deg nil )nil booleans nil )nil booleans nil )set type-eq-decl nil sets nil )AND const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"[T, T -> boolean]" equalities nil )"set[T]" doubletons nil )nil doubletons nil ))nil )))quality 100%
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2026-03-28
