Reorganize and extend test file for `multiset_solver`.

tests/multiset_solver.v

@@ -5,16 +5,65 @@ Section test.
   Implicit Types x y : A.
   Implicit Types X Y : gmultiset A.
 
-  Lemma test1 x y X : {[x]} ⊎ ({[y]} ⊎ X) ≠ ∅.
+  Lemma test_eq_1 x y X : {[x]} ⊎ ({[y]} ⊎ X) = {[y]} ⊎ ({[x]} ⊎ X).
   Proof. multiset_solver. Qed.
-  Lemma test2 x y X : {[x]} ⊎ ({[y]} ⊎ X) = {[y]} ⊎ ({[x]} ⊎ X).
-  Proof. multiset_solver. Qed.
-  Lemma test3 x : {[x]} ⊎ ∅ ≠@{gmultiset A} ∅.
-  Proof. multiset_solver. Qed.
-  Lemma test4 x y z X Y :
+  Lemma test_eq_2 x y z X Y :
     {[z]} ⊎ X = {[y]} ⊎ Y →
     {[x]} ⊎ ({[z]} ⊎ X) = {[y]} ⊎ ({[x]} ⊎ Y).
   Proof. multiset_solver. Qed.
-  Lemma test5 X x : X ⊎ ∅ = {[x]} → X ⊎ ∅ ≠@{gmultiset A} ∅.
+
+  Lemma test_neq_1 x y X : {[x]} ⊎ ({[y]} ⊎ X) ≠ ∅.
+  Proof. multiset_solver. Qed.
+  Lemma test_neq_2 x : {[x]} ⊎ ∅ ≠@{gmultiset A} ∅.
+  Proof. multiset_solver. Qed.
+  Lemma test_neq_3 X x : X ⊎ ∅ = {[x]} → X ⊎ ∅ ≠@{gmultiset A} ∅.
+  Proof. multiset_solver. Qed.
+  Lemma test_neq_4 x y : {[ x ]} ∖ {[ y ]} ≠@{gmultiset A} ∅ → x ≠ y.
+  Proof. multiset_solver. Qed.
+  Lemma test_neq_5 x y Y : y ∈ Y → {[ x ]} ∖ Y ≠ ∅ → x ≠ y.
+  Proof. multiset_solver. Qed.
+
+  Lemma test_multiplicity_1 x X Y :
+    2 < multiplicity x X → X ⊆ Y → 1 < multiplicity x Y.
+  Proof. multiset_solver. Qed.
+  Lemma test_multiplicity_2 x X :
+    2 < multiplicity x X → {[ x ]} ⊎ {[ x ]} ⊎ {[ x ]} ⊆ X.
+  Proof. multiset_solver. Qed.
+  Lemma test_multiplicity_3 x X :
+    multiplicity x X < 3 → {[ x ]} ⊎ {[ x ]} ⊎ {[ x ]} ⊈ X.
+  Proof. multiset_solver. Qed.
+
+  Lemma test_elem_of_1 x X : x ∈ X → {[x]} ⊆ X.
+  Proof. multiset_solver. Qed.
+  Lemma test_elem_of_2 x y X : x ≠ y → x ∈ X → y ∈ X → {[x]} ⊎ {[y]} ⊆ X.
+  Proof. multiset_solver. Qed.
+  Lemma test_elem_of_3 x y X Y : x ≠ y → x ∈ X → y ∈ Y → {[x]} ⊎ {[y]} ⊆ X ∪ Y.
+  Proof. multiset_solver. Qed.
+  Lemma test_elem_of_4 x y X Y : x ≠ y → x ∈ X → y ∈ Y → {[x]} ⊆ (X ∪ Y) ∖ {[ y ]}.
+  Proof. multiset_solver. Qed.
+
+  Lemma test_big_1 x1 x2 x3 x4 :
+    {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]} ⊆@{gmultiset A}
+      {[ x1 ]} ⊎ {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]}.
+  Proof. multiset_solver. Qed.
+  Lemma test_big_2 x1 x2 x3 x4 X :
+    2 ≤ multiplicity x4 X →
+    {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]} ⊆@{gmultiset A}
+      {[ x1 ]} ⊎ {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ X.
+  Proof. multiset_solver. Qed.
+  Lemma test_big_3 x1 x2 x3 x4 X :
+    4 ≤ multiplicity x4 X →
+    {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]} ⊎
+      {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]}
+    ⊆@{gmultiset A}
+      {[ x1 ]} ⊎ {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎
+      {[ x1 ]} ⊎ {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ X.
+  Proof. multiset_solver. Qed.
+  Lemma test_big_4 x1 x2 x3 x4 x5 x6 x7 x8 x9 :
+    {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]} ⊎
+      {[ x5 ]} ⊎ {[ x6 ]} ⊎ {[ x7 ]} ⊎ {[ x8 ]} ⊎ {[ x8 ]} ⊎ {[ x9 ]}
+    ⊆@{gmultiset A}
+      {[ x1 ]} ⊎ {[ x1 ]} ⊎ {[ x2 ]} ⊎ {[ x3 ]} ⊎ {[ x4 ]} ⊎ {[ x4 ]} ⊎
+      {[ x5 ]} ⊎ {[ x5 ]} ⊎ {[ x6 ]} ⊎ {[ x7 ]} ⊎ {[ x9 ]} ⊎ {[ x8 ]} ⊎ {[ x8 ]}.
   Proof. multiset_solver. Qed.
 End test.