completed proof

10 files changed, 10320 insertions, 510 deletions

diff --git a/README.md b/README.md
index 2425ab9..3f9c16d 100644
--- a/README.md
+++ b/README.md
@@ -6,6 +6,8 @@ Requires my [fork of Inpla](https://github.com/eric-marin/inpla).
 - **make** the executable
 - Copy the **inpla** executable in NNEQ/
 
+[Soundness proof](./proof.md)
+
 ## License
 
 This project is licensed under the GNU Affero General Public License v3.0 - see the [LICENSE](LICENSE) file for details.
diff --git a/examples/fashion_mnist/fashion_mnist.py b/examples/fashion_mnist/fashion_mnist.py
index 1c9dcf7..3514448 100644
--- a/examples/fashion_mnist/fashion_mnist.py
+++ b/examples/fashion_mnist/fashion_mnist.py
@@ -8,22 +8,14 @@ class FashionMNIST_MLP(nn.Module):
         super().__init__()
         self.layers = nn.Sequential(
             nn.Flatten(),
-            nn.Linear(6 * 6, hidden_dim),
+            nn.Linear(28 * 28, hidden_dim),
             nn.ReLU(),
-            nn.Linear(hidden_dim, 2)
+            nn.Linear(hidden_dim, 10),
         )
     def forward(self, x):
         return self.layers(x)
 
-transform = transforms.Compose([
-    transforms.Resize((6, 6)),
-    transforms.ToTensor(),
-])
-
-train_dataset = FashionMNIST('./', download=True, transform=transform, train=True)
-tshirts_trousers = [id for id, data in enumerate(train_dataset.targets) if data.item() == 0 or data.item() == 1]
-train_dataset = torch.utils.data.Subset(train_dataset, tshirts_trousers)
-
+train_dataset = FashionMNIST('./', download=True, transform=transforms.ToTensor(), train=True)
 trainloader = DataLoader(train_dataset, batch_size=128, shuffle=True)
 
 def train_model(name: str, dim):
@@ -32,7 +24,7 @@ def train_model(name: str, dim):
     optimizer = torch.optim.Adam(net.parameters(), lr=1e-4)
 
     print(f"Training {name}...")
-    for epoch in range(100):
+    for epoch in range(10):
         global loss
         for data in trainloader:
             inputs, targets = data
@@ -41,13 +33,13 @@ def train_model(name: str, dim):
             loss = loss_fn(outputs, targets)
             loss.backward()
             optimizer.step()
-        if (epoch + 1) % 10 == 0:
+        if (epoch + 1) % 1 == 0:
             print(f"  Epoch {epoch+1}, Loss: {loss.item():.4f}")
     return net
 
 if __name__ == "__main__":
-    torch_net_a = train_model("Network A", 6).eval()
-    torch_net_b = train_model("Network B", 12).eval()
+    torch_net_a = train_model("Network A", 28).eval()
+    torch_net_b = train_model("Network B", 56).eval()
 
-    torch.onnx.export(torch_net_a, (torch.randn(1, 1, 6, 6),), "fashion_mnist_a.onnx")
-    torch.onnx.export(torch_net_b, (torch.randn(1, 1, 6, 6),), "fashion_mnist_b.onnx")
+    torch.onnx.export(torch_net_a, (torch.randn(1, 28, 28),), "fashion_mnist_a.onnx")
+    torch.onnx.export(torch_net_b, (torch.randn(1, 28, 28),), "fashion_mnist_b.onnx")
diff --git a/examples/fashion_mnist/fashion_mnist_argmax.vnnlib b/examples/fashion_mnist/fashion_mnist_argmax.vnnlib
index 0c4412b..8d06485 100644
--- a/examples/fashion_mnist/fashion_mnist_argmax.vnnlib
+++ b/examples/fashion_mnist/fashion_mnist_argmax.vnnlib
@@ -37,10 +37,1538 @@
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+(declare-const Y_617 Real)
+(declare-const Y_618 Real)
+(declare-const Y_619 Real)
+(declare-const Y_620 Real)
+(declare-const Y_621 Real)
+(declare-const Y_622 Real)
+(declare-const Y_623 Real)
+(declare-const Y_624 Real)
+(declare-const Y_625 Real)
+(declare-const Y_626 Real)
+(declare-const Y_627 Real)
+(declare-const Y_628 Real)
+(declare-const Y_629 Real)
+(declare-const Y_630 Real)
+(declare-const Y_631 Real)
+(declare-const Y_632 Real)
+(declare-const Y_633 Real)
+(declare-const Y_634 Real)
+(declare-const Y_635 Real)
+(declare-const Y_636 Real)
+(declare-const Y_637 Real)
+(declare-const Y_638 Real)
+(declare-const Y_639 Real)
+(declare-const Y_640 Real)
+(declare-const Y_641 Real)
+(declare-const Y_642 Real)
+(declare-const Y_643 Real)
+(declare-const Y_644 Real)
+(declare-const Y_645 Real)
+(declare-const Y_646 Real)
+(declare-const Y_647 Real)
+(declare-const Y_648 Real)
+(declare-const Y_649 Real)
+(declare-const Y_650 Real)
+(declare-const Y_651 Real)
+(declare-const Y_652 Real)
+(declare-const Y_653 Real)
+(declare-const Y_654 Real)
+(declare-const Y_655 Real)
+(declare-const Y_656 Real)
+(declare-const Y_657 Real)
+(declare-const Y_658 Real)
+(declare-const Y_659 Real)
+(declare-const Y_660 Real)
+(declare-const Y_661 Real)
+(declare-const Y_662 Real)
+(declare-const Y_663 Real)
+(declare-const Y_664 Real)
+(declare-const Y_665 Real)
+(declare-const Y_666 Real)
+(declare-const Y_667 Real)
+(declare-const Y_668 Real)
+(declare-const Y_669 Real)
+(declare-const Y_670 Real)
+(declare-const Y_671 Real)
+(declare-const Y_672 Real)
+(declare-const Y_673 Real)
+(declare-const Y_674 Real)
+(declare-const Y_675 Real)
+(declare-const Y_676 Real)
+(declare-const Y_677 Real)
+(declare-const Y_678 Real)
+(declare-const Y_679 Real)
+(declare-const Y_680 Real)
+(declare-const Y_681 Real)
+(declare-const Y_682 Real)
+(declare-const Y_683 Real)
+(declare-const Y_684 Real)
+(declare-const Y_685 Real)
+(declare-const Y_686 Real)
+(declare-const Y_687 Real)
+(declare-const Y_688 Real)
+(declare-const Y_689 Real)
+(declare-const Y_690 Real)
+(declare-const Y_691 Real)
+(declare-const Y_692 Real)
+(declare-const Y_693 Real)
+(declare-const Y_694 Real)
+(declare-const Y_695 Real)
+(declare-const Y_696 Real)
+(declare-const Y_697 Real)
+(declare-const Y_698 Real)
+(declare-const Y_699 Real)
+(declare-const Y_700 Real)
+(declare-const Y_701 Real)
+(declare-const Y_702 Real)
+(declare-const Y_703 Real)
+(declare-const Y_704 Real)
+(declare-const Y_705 Real)
+(declare-const Y_706 Real)
+(declare-const Y_707 Real)
+(declare-const Y_708 Real)
+(declare-const Y_709 Real)
+(declare-const Y_710 Real)
+(declare-const Y_711 Real)
+(declare-const Y_712 Real)
+(declare-const Y_713 Real)
+(declare-const Y_714 Real)
+(declare-const Y_715 Real)
+(declare-const Y_716 Real)
+(declare-const Y_717 Real)
+(declare-const Y_718 Real)
+(declare-const Y_719 Real)
+(declare-const Y_720 Real)
+(declare-const Y_721 Real)
+(declare-const Y_722 Real)
+(declare-const Y_723 Real)
+(declare-const Y_724 Real)
+(declare-const Y_725 Real)
+(declare-const Y_726 Real)
+(declare-const Y_727 Real)
+(declare-const Y_728 Real)
+(declare-const Y_729 Real)
+(declare-const Y_730 Real)
+(declare-const Y_731 Real)
+(declare-const Y_732 Real)
+(declare-const Y_733 Real)
+(declare-const Y_734 Real)
+(declare-const Y_735 Real)
+(declare-const Y_736 Real)
+(declare-const Y_737 Real)
+(declare-const Y_738 Real)
+(declare-const Y_739 Real)
+(declare-const Y_740 Real)
+(declare-const Y_741 Real)
+(declare-const Y_742 Real)
+(declare-const Y_743 Real)
+(declare-const Y_744 Real)
+(declare-const Y_745 Real)
+(declare-const Y_746 Real)
+(declare-const Y_747 Real)
+(declare-const Y_748 Real)
+(declare-const Y_749 Real)
+(declare-const Y_750 Real)
+(declare-const Y_751 Real)
+(declare-const Y_752 Real)
+(declare-const Y_753 Real)
+(declare-const Y_754 Real)
+(declare-const Y_755 Real)
+(declare-const Y_756 Real)
+(declare-const Y_757 Real)
+(declare-const Y_758 Real)
+(declare-const Y_759 Real)
+(declare-const Y_760 Real)
+(declare-const Y_761 Real)
+(declare-const Y_762 Real)
+(declare-const Y_763 Real)
+(declare-const Y_764 Real)
+(declare-const Y_765 Real)
+(declare-const Y_766 Real)
+(declare-const Y_767 Real)
+(declare-const Y_768 Real)
+(declare-const Y_769 Real)
+(declare-const Y_770 Real)
+(declare-const Y_771 Real)
+(declare-const Y_772 Real)
+(declare-const Y_773 Real)
+(declare-const Y_774 Real)
+(declare-const Y_775 Real)
+(declare-const Y_776 Real)
+(declare-const Y_777 Real)
+(declare-const Y_778 Real)
+(declare-const Y_779 Real)
+(declare-const Y_780 Real)
+(declare-const Y_781 Real)
+(declare-const Y_782 Real)
+(declare-const Y_783 Real)
 
 ; Bounded inputs: X must be within [0, 1]
 (assert (>= X_0 0.0))
@@ -115,19 +1643,1723 @@
 (assert (<= X_34 1.0))
 (assert (>= X_35 0.0))
 (assert (<= X_35 1.0))
+(assert (>= X_36 0.0))
+(assert (<= X_36 1.0))
+(assert (>= X_37 0.0))
+(assert (<= X_37 1.0))
+(assert (>= X_38 0.0))
+(assert (<= X_38 1.0))
+(assert (>= X_39 0.0))
+(assert (<= X_39 1.0))
+(assert (>= X_40 0.0))
+(assert (<= X_40 1.0))
+(assert (>= X_41 0.0))
+(assert (<= X_41 1.0))
+(assert (>= X_42 0.0))
+(assert (<= X_42 1.0))
+(assert (>= X_43 0.0))
+(assert (<= X_43 1.0))
+(assert (>= X_44 0.0))
+(assert (<= X_44 1.0))
+(assert (>= X_45 0.0))
+(assert (<= X_45 1.0))
+(assert (>= X_46 0.0))
+(assert (<= X_46 1.0))
+(assert (>= X_47 0.0))
+(assert (<= X_47 1.0))
+(assert (>= X_48 0.0))
+(assert (<= X_48 1.0))
+(assert (>= X_49 0.0))
+(assert (<= X_49 1.0))
+(assert (>= X_50 0.0))
+(assert (<= X_50 1.0))
+(assert (>= X_51 0.0))
+(assert (<= X_51 1.0))
+(assert (>= X_52 0.0))
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+(assert (>= X_53 0.0))
+(assert (<= X_53 1.0))
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+(assert (>= X_58 0.0))
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+(assert (>= X_72 0.0))
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+(assert (>= X_589 0.0))
+(assert (<= X_589 1.0))
+(assert (>= X_590 0.0))
+(assert (<= X_590 1.0))
+(assert (>= X_591 0.0))
+(assert (<= X_591 1.0))
+(assert (>= X_592 0.0))
+(assert (<= X_592 1.0))
+(assert (>= X_593 0.0))
+(assert (<= X_593 1.0))
+(assert (>= X_594 0.0))
+(assert (<= X_594 1.0))
+(assert (>= X_595 0.0))
+(assert (<= X_595 1.0))
+(assert (>= X_596 0.0))
+(assert (<= X_596 1.0))
+(assert (>= X_597 0.0))
+(assert (<= X_597 1.0))
+(assert (>= X_598 0.0))
+(assert (<= X_598 1.0))
+(assert (>= X_599 0.0))
+(assert (<= X_599 1.0))
+(assert (>= X_600 0.0))
+(assert (<= X_600 1.0))
+(assert (>= X_601 0.0))
+(assert (<= X_601 1.0))
+(assert (>= X_602 0.0))
+(assert (<= X_602 1.0))
+(assert (>= X_603 0.0))
+(assert (<= X_603 1.0))
+(assert (>= X_604 0.0))
+(assert (<= X_604 1.0))
+(assert (>= X_605 0.0))
+(assert (<= X_605 1.0))
+(assert (>= X_606 0.0))
+(assert (<= X_606 1.0))
+(assert (>= X_607 0.0))
+(assert (<= X_607 1.0))
+(assert (>= X_608 0.0))
+(assert (<= X_608 1.0))
+(assert (>= X_609 0.0))
+(assert (<= X_609 1.0))
+(assert (>= X_610 0.0))
+(assert (<= X_610 1.0))
+(assert (>= X_611 0.0))
+(assert (<= X_611 1.0))
+(assert (>= X_612 0.0))
+(assert (<= X_612 1.0))
+(assert (>= X_613 0.0))
+(assert (<= X_613 1.0))
+(assert (>= X_614 0.0))
+(assert (<= X_614 1.0))
+(assert (>= X_615 0.0))
+(assert (<= X_615 1.0))
+(assert (>= X_616 0.0))
+(assert (<= X_616 1.0))
+(assert (>= X_617 0.0))
+(assert (<= X_617 1.0))
+(assert (>= X_618 0.0))
+(assert (<= X_618 1.0))
+(assert (>= X_619 0.0))
+(assert (<= X_619 1.0))
+(assert (>= X_620 0.0))
+(assert (<= X_620 1.0))
+(assert (>= X_621 0.0))
+(assert (<= X_621 1.0))
+(assert (>= X_622 0.0))
+(assert (<= X_622 1.0))
+(assert (>= X_623 0.0))
+(assert (<= X_623 1.0))
+(assert (>= X_624 0.0))
+(assert (<= X_624 1.0))
+(assert (>= X_625 0.0))
+(assert (<= X_625 1.0))
+(assert (>= X_626 0.0))
+(assert (<= X_626 1.0))
+(assert (>= X_627 0.0))
+(assert (<= X_627 1.0))
+(assert (>= X_628 0.0))
+(assert (<= X_628 1.0))
+(assert (>= X_629 0.0))
+(assert (<= X_629 1.0))
+(assert (>= X_630 0.0))
+(assert (<= X_630 1.0))
+(assert (>= X_631 0.0))
+(assert (<= X_631 1.0))
+(assert (>= X_632 0.0))
+(assert (<= X_632 1.0))
+(assert (>= X_633 0.0))
+(assert (<= X_633 1.0))
+(assert (>= X_634 0.0))
+(assert (<= X_634 1.0))
+(assert (>= X_635 0.0))
+(assert (<= X_635 1.0))
+(assert (>= X_636 0.0))
+(assert (<= X_636 1.0))
+(assert (>= X_637 0.0))
+(assert (<= X_637 1.0))
+(assert (>= X_638 0.0))
+(assert (<= X_638 1.0))
+(assert (>= X_639 0.0))
+(assert (<= X_639 1.0))
+(assert (>= X_640 0.0))
+(assert (<= X_640 1.0))
+(assert (>= X_641 0.0))
+(assert (<= X_641 1.0))
+(assert (>= X_642 0.0))
+(assert (<= X_642 1.0))
+(assert (>= X_643 0.0))
+(assert (<= X_643 1.0))
+(assert (>= X_644 0.0))
+(assert (<= X_644 1.0))
+(assert (>= X_645 0.0))
+(assert (<= X_645 1.0))
+(assert (>= X_646 0.0))
+(assert (<= X_646 1.0))
+(assert (>= X_647 0.0))
+(assert (<= X_647 1.0))
+(assert (>= X_648 0.0))
+(assert (<= X_648 1.0))
+(assert (>= X_649 0.0))
+(assert (<= X_649 1.0))
+(assert (>= X_650 0.0))
+(assert (<= X_650 1.0))
+(assert (>= X_651 0.0))
+(assert (<= X_651 1.0))
+(assert (>= X_652 0.0))
+(assert (<= X_652 1.0))
+(assert (>= X_653 0.0))
+(assert (<= X_653 1.0))
+(assert (>= X_654 0.0))
+(assert (<= X_654 1.0))
+(assert (>= X_655 0.0))
+(assert (<= X_655 1.0))
+(assert (>= X_656 0.0))
+(assert (<= X_656 1.0))
+(assert (>= X_657 0.0))
+(assert (<= X_657 1.0))
+(assert (>= X_658 0.0))
+(assert (<= X_658 1.0))
+(assert (>= X_659 0.0))
+(assert (<= X_659 1.0))
+(assert (>= X_660 0.0))
+(assert (<= X_660 1.0))
+(assert (>= X_661 0.0))
+(assert (<= X_661 1.0))
+(assert (>= X_662 0.0))
+(assert (<= X_662 1.0))
+(assert (>= X_663 0.0))
+(assert (<= X_663 1.0))
+(assert (>= X_664 0.0))
+(assert (<= X_664 1.0))
+(assert (>= X_665 0.0))
+(assert (<= X_665 1.0))
+(assert (>= X_666 0.0))
+(assert (<= X_666 1.0))
+(assert (>= X_667 0.0))
+(assert (<= X_667 1.0))
+(assert (>= X_668 0.0))
+(assert (<= X_668 1.0))
+(assert (>= X_669 0.0))
+(assert (<= X_669 1.0))
+(assert (>= X_670 0.0))
+(assert (<= X_670 1.0))
+(assert (>= X_671 0.0))
+(assert (<= X_671 1.0))
+(assert (>= X_672 0.0))
+(assert (<= X_672 1.0))
+(assert (>= X_673 0.0))
+(assert (<= X_673 1.0))
+(assert (>= X_674 0.0))
+(assert (<= X_674 1.0))
+(assert (>= X_675 0.0))
+(assert (<= X_675 1.0))
+(assert (>= X_676 0.0))
+(assert (<= X_676 1.0))
+(assert (>= X_677 0.0))
+(assert (<= X_677 1.0))
+(assert (>= X_678 0.0))
+(assert (<= X_678 1.0))
+(assert (>= X_679 0.0))
+(assert (<= X_679 1.0))
+(assert (>= X_680 0.0))
+(assert (<= X_680 1.0))
+(assert (>= X_681 0.0))
+(assert (<= X_681 1.0))
+(assert (>= X_682 0.0))
+(assert (<= X_682 1.0))
+(assert (>= X_683 0.0))
+(assert (<= X_683 1.0))
+(assert (>= X_684 0.0))
+(assert (<= X_684 1.0))
+(assert (>= X_685 0.0))
+(assert (<= X_685 1.0))
+(assert (>= X_686 0.0))
+(assert (<= X_686 1.0))
+(assert (>= X_687 0.0))
+(assert (<= X_687 1.0))
+(assert (>= X_688 0.0))
+(assert (<= X_688 1.0))
+(assert (>= X_689 0.0))
+(assert (<= X_689 1.0))
+(assert (>= X_690 0.0))
+(assert (<= X_690 1.0))
+(assert (>= X_691 0.0))
+(assert (<= X_691 1.0))
+(assert (>= X_692 0.0))
+(assert (<= X_692 1.0))
+(assert (>= X_693 0.0))
+(assert (<= X_693 1.0))
+(assert (>= X_694 0.0))
+(assert (<= X_694 1.0))
+(assert (>= X_695 0.0))
+(assert (<= X_695 1.0))
+(assert (>= X_696 0.0))
+(assert (<= X_696 1.0))
+(assert (>= X_697 0.0))
+(assert (<= X_697 1.0))
+(assert (>= X_698 0.0))
+(assert (<= X_698 1.0))
+(assert (>= X_699 0.0))
+(assert (<= X_699 1.0))
+(assert (>= X_700 0.0))
+(assert (<= X_700 1.0))
+(assert (>= X_701 0.0))
+(assert (<= X_701 1.0))
+(assert (>= X_702 0.0))
+(assert (<= X_702 1.0))
+(assert (>= X_703 0.0))
+(assert (<= X_703 1.0))
+(assert (>= X_704 0.0))
+(assert (<= X_704 1.0))
+(assert (>= X_705 0.0))
+(assert (<= X_705 1.0))
+(assert (>= X_706 0.0))
+(assert (<= X_706 1.0))
+(assert (>= X_707 0.0))
+(assert (<= X_707 1.0))
+(assert (>= X_708 0.0))
+(assert (<= X_708 1.0))
+(assert (>= X_709 0.0))
+(assert (<= X_709 1.0))
+(assert (>= X_710 0.0))
+(assert (<= X_710 1.0))
+(assert (>= X_711 0.0))
+(assert (<= X_711 1.0))
+(assert (>= X_712 0.0))
+(assert (<= X_712 1.0))
+(assert (>= X_713 0.0))
+(assert (<= X_713 1.0))
+(assert (>= X_714 0.0))
+(assert (<= X_714 1.0))
+(assert (>= X_715 0.0))
+(assert (<= X_715 1.0))
+(assert (>= X_716 0.0))
+(assert (<= X_716 1.0))
+(assert (>= X_717 0.0))
+(assert (<= X_717 1.0))
+(assert (>= X_718 0.0))
+(assert (<= X_718 1.0))
+(assert (>= X_719 0.0))
+(assert (<= X_719 1.0))
+(assert (>= X_720 0.0))
+(assert (<= X_720 1.0))
+(assert (>= X_721 0.0))
+(assert (<= X_721 1.0))
+(assert (>= X_722 0.0))
+(assert (<= X_722 1.0))
+(assert (>= X_723 0.0))
+(assert (<= X_723 1.0))
+(assert (>= X_724 0.0))
+(assert (<= X_724 1.0))
+(assert (>= X_725 0.0))
+(assert (<= X_725 1.0))
+(assert (>= X_726 0.0))
+(assert (<= X_726 1.0))
+(assert (>= X_727 0.0))
+(assert (<= X_727 1.0))
+(assert (>= X_728 0.0))
+(assert (<= X_728 1.0))
+(assert (>= X_729 0.0))
+(assert (<= X_729 1.0))
+(assert (>= X_730 0.0))
+(assert (<= X_730 1.0))
+(assert (>= X_731 0.0))
+(assert (<= X_731 1.0))
+(assert (>= X_732 0.0))
+(assert (<= X_732 1.0))
+(assert (>= X_733 0.0))
+(assert (<= X_733 1.0))
+(assert (>= X_734 0.0))
+(assert (<= X_734 1.0))
+(assert (>= X_735 0.0))
+(assert (<= X_735 1.0))
+(assert (>= X_736 0.0))
+(assert (<= X_736 1.0))
+(assert (>= X_737 0.0))
+(assert (<= X_737 1.0))
+(assert (>= X_738 0.0))
+(assert (<= X_738 1.0))
+(assert (>= X_739 0.0))
+(assert (<= X_739 1.0))
+(assert (>= X_740 0.0))
+(assert (<= X_740 1.0))
+(assert (>= X_741 0.0))
+(assert (<= X_741 1.0))
+(assert (>= X_742 0.0))
+(assert (<= X_742 1.0))
+(assert (>= X_743 0.0))
+(assert (<= X_743 1.0))
+(assert (>= X_744 0.0))
+(assert (<= X_744 1.0))
+(assert (>= X_745 0.0))
+(assert (<= X_745 1.0))
+(assert (>= X_746 0.0))
+(assert (<= X_746 1.0))
+(assert (>= X_747 0.0))
+(assert (<= X_747 1.0))
+(assert (>= X_748 0.0))
+(assert (<= X_748 1.0))
+(assert (>= X_749 0.0))
+(assert (<= X_749 1.0))
+(assert (>= X_750 0.0))
+(assert (<= X_750 1.0))
+(assert (>= X_751 0.0))
+(assert (<= X_751 1.0))
+(assert (>= X_752 0.0))
+(assert (<= X_752 1.0))
+(assert (>= X_753 0.0))
+(assert (<= X_753 1.0))
+(assert (>= X_754 0.0))
+(assert (<= X_754 1.0))
+(assert (>= X_755 0.0))
+(assert (<= X_755 1.0))
+(assert (>= X_756 0.0))
+(assert (<= X_756 1.0))
+(assert (>= X_757 0.0))
+(assert (<= X_757 1.0))
+(assert (>= X_758 0.0))
+(assert (<= X_758 1.0))
+(assert (>= X_759 0.0))
+(assert (<= X_759 1.0))
+(assert (>= X_760 0.0))
+(assert (<= X_760 1.0))
+(assert (>= X_761 0.0))
+(assert (<= X_761 1.0))
+(assert (>= X_762 0.0))
+(assert (<= X_762 1.0))
+(assert (>= X_763 0.0))
+(assert (<= X_763 1.0))
+(assert (>= X_764 0.0))
+(assert (<= X_764 1.0))
+(assert (>= X_765 0.0))
+(assert (<= X_765 1.0))
+(assert (>= X_766 0.0))
+(assert (<= X_766 1.0))
+(assert (>= X_767 0.0))
+(assert (<= X_767 1.0))
+(assert (>= X_768 0.0))
+(assert (<= X_768 1.0))
+(assert (>= X_769 0.0))
+(assert (<= X_769 1.0))
+(assert (>= X_770 0.0))
+(assert (<= X_770 1.0))
+(assert (>= X_771 0.0))
+(assert (<= X_771 1.0))
+(assert (>= X_772 0.0))
+(assert (<= X_772 1.0))
+(assert (>= X_773 0.0))
+(assert (<= X_773 1.0))
+(assert (>= X_774 0.0))
+(assert (<= X_774 1.0))
+(assert (>= X_775 0.0))
+(assert (<= X_775 1.0))
+(assert (>= X_776 0.0))
+(assert (<= X_776 1.0))
+(assert (>= X_777 0.0))
+(assert (<= X_777 1.0))
+(assert (>= X_778 0.0))
+(assert (<= X_778 1.0))
+(assert (>= X_779 0.0))
+(assert (<= X_779 1.0))
+(assert (>= X_780 0.0))
+(assert (<= X_780 1.0))
+(assert (>= X_781 0.0))
+(assert (<= X_781 1.0))
+(assert (>= X_782 0.0))
+(assert (<= X_782 1.0))
+(assert (>= X_783 0.0))
+(assert (<= X_783 1.0))
 
 ; Violation of argmax equivalence
 (assert (or
   (and 
     (> Y_0 Y_1)
+    (> Y_0 Y_2)
+    (> Y_0 Y_3)
+    (> Y_0 Y_4)
+    (> Y_0 Y_5)
+    (> Y_0 Y_6)
+    (> Y_0 Y_7)
+    (> Y_0 Y_8)
+    (> Y_0 Y_9)
     (or
-      (> Y_3 Y_2)
+      (> Y_11 Y_10)
+      (> Y_12 Y_10)
+      (> Y_13 Y_10)
+      (> Y_14 Y_10)
+      (> Y_15 Y_10)
+      (> Y_16 Y_10)
+      (> Y_17 Y_10)
+      (> Y_18 Y_10)
+      (> Y_19 Y_10)
     )
   )
   (and 
     (> Y_1 Y_0)
+    (> Y_1 Y_2)
+    (> Y_1 Y_3)
+    (> Y_1 Y_4)
+    (> Y_1 Y_5)
+    (> Y_1 Y_6)
+    (> Y_1 Y_7)
+    (> Y_1 Y_8)
+    (> Y_1 Y_9)
     (or
-      (> Y_2 Y_3)
+      (> Y_10 Y_11)
+      (> Y_12 Y_11)
+      (> Y_13 Y_11)
+      (> Y_14 Y_11)
+      (> Y_15 Y_11)
+      (> Y_16 Y_11)
+      (> Y_17 Y_11)
+      (> Y_18 Y_11)
+      (> Y_19 Y_11)
+    )
+  )
+  (and 
+    (> Y_2 Y_0)
+    (> Y_2 Y_1)
+    (> Y_2 Y_3)
+    (> Y_2 Y_4)
+    (> Y_2 Y_5)
+    (> Y_2 Y_6)
+    (> Y_2 Y_7)
+    (> Y_2 Y_8)
+    (> Y_2 Y_9)
+    (or
+      (> Y_10 Y_12)
+      (> Y_11 Y_12)
+      (> Y_13 Y_12)
+      (> Y_14 Y_12)
+      (> Y_15 Y_12)
+      (> Y_16 Y_12)
+      (> Y_17 Y_12)
+      (> Y_18 Y_12)
+      (> Y_19 Y_12)
+    )
+  )
+  (and 
+    (> Y_3 Y_0)
+    (> Y_3 Y_1)
+    (> Y_3 Y_2)
+    (> Y_3 Y_4)
+    (> Y_3 Y_5)
+    (> Y_3 Y_6)
+    (> Y_3 Y_7)
+    (> Y_3 Y_8)
+    (> Y_3 Y_9)
+    (or
+      (> Y_10 Y_13)
+      (> Y_11 Y_13)
+      (> Y_12 Y_13)
+      (> Y_14 Y_13)
+      (> Y_15 Y_13)
+      (> Y_16 Y_13)
+      (> Y_17 Y_13)
+      (> Y_18 Y_13)
+      (> Y_19 Y_13)
+    )
+  )
+  (and 
+    (> Y_4 Y_0)
+    (> Y_4 Y_1)
+    (> Y_4 Y_2)
+    (> Y_4 Y_3)
+    (> Y_4 Y_5)
+    (> Y_4 Y_6)
+    (> Y_4 Y_7)
+    (> Y_4 Y_8)
+    (> Y_4 Y_9)
+    (or
+      (> Y_10 Y_14)
+      (> Y_11 Y_14)
+      (> Y_12 Y_14)
+      (> Y_13 Y_14)
+      (> Y_15 Y_14)
+      (> Y_16 Y_14)
+      (> Y_17 Y_14)
+      (> Y_18 Y_14)
+      (> Y_19 Y_14)
+    )
+  )
+  (and 
+    (> Y_5 Y_0)
+    (> Y_5 Y_1)
+    (> Y_5 Y_2)
+    (> Y_5 Y_3)
+    (> Y_5 Y_4)
+    (> Y_5 Y_6)
+    (> Y_5 Y_7)
+    (> Y_5 Y_8)
+    (> Y_5 Y_9)
+    (or
+      (> Y_10 Y_15)
+      (> Y_11 Y_15)
+      (> Y_12 Y_15)
+      (> Y_13 Y_15)
+      (> Y_14 Y_15)
+      (> Y_16 Y_15)
+      (> Y_17 Y_15)
+      (> Y_18 Y_15)
+      (> Y_19 Y_15)
+    )
+  )
+  (and 
+    (> Y_6 Y_0)
+    (> Y_6 Y_1)
+    (> Y_6 Y_2)
+    (> Y_6 Y_3)
+    (> Y_6 Y_4)
+    (> Y_6 Y_5)
+    (> Y_6 Y_7)
+    (> Y_6 Y_8)
+    (> Y_6 Y_9)
+    (or
+      (> Y_10 Y_16)
+      (> Y_11 Y_16)
+      (> Y_12 Y_16)
+      (> Y_13 Y_16)
+      (> Y_14 Y_16)
+      (> Y_15 Y_16)
+      (> Y_17 Y_16)
+      (> Y_18 Y_16)
+      (> Y_19 Y_16)
+    )
+  )
+  (and 
+    (> Y_7 Y_0)
+    (> Y_7 Y_1)
+    (> Y_7 Y_2)
+    (> Y_7 Y_3)
+    (> Y_7 Y_4)
+    (> Y_7 Y_5)
+    (> Y_7 Y_6)
+    (> Y_7 Y_8)
+    (> Y_7 Y_9)
+    (or
+      (> Y_10 Y_17)
+      (> Y_11 Y_17)
+      (> Y_12 Y_17)
+      (> Y_13 Y_17)
+      (> Y_14 Y_17)
+      (> Y_15 Y_17)
+      (> Y_16 Y_17)
+      (> Y_18 Y_17)
+      (> Y_19 Y_17)
+    )
+  )
+  (and 
+    (> Y_8 Y_0)
+    (> Y_8 Y_1)
+    (> Y_8 Y_2)
+    (> Y_8 Y_3)
+    (> Y_8 Y_4)
+    (> Y_8 Y_5)
+    (> Y_8 Y_6)
+    (> Y_8 Y_7)
+    (> Y_8 Y_9)
+    (or
+      (> Y_10 Y_18)
+      (> Y_11 Y_18)
+      (> Y_12 Y_18)
+      (> Y_13 Y_18)
+      (> Y_14 Y_18)
+      (> Y_15 Y_18)
+      (> Y_16 Y_18)
+      (> Y_17 Y_18)
+      (> Y_19 Y_18)
+    )
+  )
+  (and 
+    (> Y_9 Y_0)
+    (> Y_9 Y_1)
+    (> Y_9 Y_2)
+    (> Y_9 Y_3)
+    (> Y_9 Y_4)
+    (> Y_9 Y_5)
+    (> Y_9 Y_6)
+    (> Y_9 Y_7)
+    (> Y_9 Y_8)
+    (or
+      (> Y_10 Y_19)
+      (> Y_11 Y_19)
+      (> Y_12 Y_19)
+      (> Y_13 Y_19)
+      (> Y_14 Y_19)
+      (> Y_15 Y_19)
+      (> Y_16 Y_19)
+      (> Y_17 Y_19)
+      (> Y_18 Y_19)
     )
   )
 ))
diff --git a/examples/fashion_mnist/fashion_mnist_epsilon.vnnlib b/examples/fashion_mnist/fashion_mnist_epsilon.vnnlib
index 948e501..9637489 100644
--- a/examples/fashion_mnist/fashion_mnist_epsilon.vnnlib
+++ b/examples/fashion_mnist/fashion_mnist_epsilon.vnnlib
@@ -37,10 +37,1538 @@
 (declare-const X_33 Real)
 (declare-const X_34 Real)
 (declare-const X_35 Real)
+(declare-const X_36 Real)
+(declare-const X_37 Real)
+(declare-const X_38 Real)
+(declare-const X_39 Real)
+(declare-const X_40 Real)
+(declare-const X_41 Real)
+(declare-const X_42 Real)
+(declare-const X_43 Real)
+(declare-const X_44 Real)
+(declare-const X_45 Real)
+(declare-const X_46 Real)
+(declare-const X_47 Real)
+(declare-const X_48 Real)
+(declare-const X_49 Real)
+(declare-const X_50 Real)
+(declare-const X_51 Real)
+(declare-const X_52 Real)
+(declare-const X_53 Real)
+(declare-const X_54 Real)
+(declare-const X_55 Real)
+(declare-const X_56 Real)
+(declare-const X_57 Real)
+(declare-const X_58 Real)
+(declare-const X_59 Real)
+(declare-const X_60 Real)
+(declare-const X_61 Real)
+(declare-const X_62 Real)
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+(declare-const Y_755 Real)
+(declare-const Y_756 Real)
+(declare-const Y_757 Real)
+(declare-const Y_758 Real)
+(declare-const Y_759 Real)
+(declare-const Y_760 Real)
+(declare-const Y_761 Real)
+(declare-const Y_762 Real)
+(declare-const Y_763 Real)
+(declare-const Y_764 Real)
+(declare-const Y_765 Real)
+(declare-const Y_766 Real)
+(declare-const Y_767 Real)
+(declare-const Y_768 Real)
+(declare-const Y_769 Real)
+(declare-const Y_770 Real)
+(declare-const Y_771 Real)
+(declare-const Y_772 Real)
+(declare-const Y_773 Real)
+(declare-const Y_774 Real)
+(declare-const Y_775 Real)
+(declare-const Y_776 Real)
+(declare-const Y_777 Real)
+(declare-const Y_778 Real)
+(declare-const Y_779 Real)
+(declare-const Y_780 Real)
+(declare-const Y_781 Real)
+(declare-const Y_782 Real)
+(declare-const Y_783 Real)
 
 ; Bounded inputs: X must be within [0, 1]
 (assert (>= X_0 0.0))
@@ -115,10 +1643,1514 @@
 (assert (<= X_34 1.0))
 (assert (>= X_35 0.0))
 (assert (<= X_35 1.0))
+(assert (>= X_36 0.0))
+(assert (<= X_36 1.0))
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+(assert (<= X_37 1.0))
+(assert (>= X_38 0.0))
+(assert (<= X_38 1.0))
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+(assert (>= X_40 0.0))
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+(assert (>= X_45 0.0))
+(assert (<= X_45 1.0))
+(assert (>= X_46 0.0))
+(assert (<= X_46 1.0))
+(assert (>= X_47 0.0))
+(assert (<= X_47 1.0))
+(assert (>= X_48 0.0))
+(assert (<= X_48 1.0))
+(assert (>= X_49 0.0))
+(assert (<= X_49 1.0))
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+(assert (>= X_52 0.0))
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+(assert (<= X_647 1.0))
+(assert (>= X_648 0.0))
+(assert (<= X_648 1.0))
+(assert (>= X_649 0.0))
+(assert (<= X_649 1.0))
+(assert (>= X_650 0.0))
+(assert (<= X_650 1.0))
+(assert (>= X_651 0.0))
+(assert (<= X_651 1.0))
+(assert (>= X_652 0.0))
+(assert (<= X_652 1.0))
+(assert (>= X_653 0.0))
+(assert (<= X_653 1.0))
+(assert (>= X_654 0.0))
+(assert (<= X_654 1.0))
+(assert (>= X_655 0.0))
+(assert (<= X_655 1.0))
+(assert (>= X_656 0.0))
+(assert (<= X_656 1.0))
+(assert (>= X_657 0.0))
+(assert (<= X_657 1.0))
+(assert (>= X_658 0.0))
+(assert (<= X_658 1.0))
+(assert (>= X_659 0.0))
+(assert (<= X_659 1.0))
+(assert (>= X_660 0.0))
+(assert (<= X_660 1.0))
+(assert (>= X_661 0.0))
+(assert (<= X_661 1.0))
+(assert (>= X_662 0.0))
+(assert (<= X_662 1.0))
+(assert (>= X_663 0.0))
+(assert (<= X_663 1.0))
+(assert (>= X_664 0.0))
+(assert (<= X_664 1.0))
+(assert (>= X_665 0.0))
+(assert (<= X_665 1.0))
+(assert (>= X_666 0.0))
+(assert (<= X_666 1.0))
+(assert (>= X_667 0.0))
+(assert (<= X_667 1.0))
+(assert (>= X_668 0.0))
+(assert (<= X_668 1.0))
+(assert (>= X_669 0.0))
+(assert (<= X_669 1.0))
+(assert (>= X_670 0.0))
+(assert (<= X_670 1.0))
+(assert (>= X_671 0.0))
+(assert (<= X_671 1.0))
+(assert (>= X_672 0.0))
+(assert (<= X_672 1.0))
+(assert (>= X_673 0.0))
+(assert (<= X_673 1.0))
+(assert (>= X_674 0.0))
+(assert (<= X_674 1.0))
+(assert (>= X_675 0.0))
+(assert (<= X_675 1.0))
+(assert (>= X_676 0.0))
+(assert (<= X_676 1.0))
+(assert (>= X_677 0.0))
+(assert (<= X_677 1.0))
+(assert (>= X_678 0.0))
+(assert (<= X_678 1.0))
+(assert (>= X_679 0.0))
+(assert (<= X_679 1.0))
+(assert (>= X_680 0.0))
+(assert (<= X_680 1.0))
+(assert (>= X_681 0.0))
+(assert (<= X_681 1.0))
+(assert (>= X_682 0.0))
+(assert (<= X_682 1.0))
+(assert (>= X_683 0.0))
+(assert (<= X_683 1.0))
+(assert (>= X_684 0.0))
+(assert (<= X_684 1.0))
+(assert (>= X_685 0.0))
+(assert (<= X_685 1.0))
+(assert (>= X_686 0.0))
+(assert (<= X_686 1.0))
+(assert (>= X_687 0.0))
+(assert (<= X_687 1.0))
+(assert (>= X_688 0.0))
+(assert (<= X_688 1.0))
+(assert (>= X_689 0.0))
+(assert (<= X_689 1.0))
+(assert (>= X_690 0.0))
+(assert (<= X_690 1.0))
+(assert (>= X_691 0.0))
+(assert (<= X_691 1.0))
+(assert (>= X_692 0.0))
+(assert (<= X_692 1.0))
+(assert (>= X_693 0.0))
+(assert (<= X_693 1.0))
+(assert (>= X_694 0.0))
+(assert (<= X_694 1.0))
+(assert (>= X_695 0.0))
+(assert (<= X_695 1.0))
+(assert (>= X_696 0.0))
+(assert (<= X_696 1.0))
+(assert (>= X_697 0.0))
+(assert (<= X_697 1.0))
+(assert (>= X_698 0.0))
+(assert (<= X_698 1.0))
+(assert (>= X_699 0.0))
+(assert (<= X_699 1.0))
+(assert (>= X_700 0.0))
+(assert (<= X_700 1.0))
+(assert (>= X_701 0.0))
+(assert (<= X_701 1.0))
+(assert (>= X_702 0.0))
+(assert (<= X_702 1.0))
+(assert (>= X_703 0.0))
+(assert (<= X_703 1.0))
+(assert (>= X_704 0.0))
+(assert (<= X_704 1.0))
+(assert (>= X_705 0.0))
+(assert (<= X_705 1.0))
+(assert (>= X_706 0.0))
+(assert (<= X_706 1.0))
+(assert (>= X_707 0.0))
+(assert (<= X_707 1.0))
+(assert (>= X_708 0.0))
+(assert (<= X_708 1.0))
+(assert (>= X_709 0.0))
+(assert (<= X_709 1.0))
+(assert (>= X_710 0.0))
+(assert (<= X_710 1.0))
+(assert (>= X_711 0.0))
+(assert (<= X_711 1.0))
+(assert (>= X_712 0.0))
+(assert (<= X_712 1.0))
+(assert (>= X_713 0.0))
+(assert (<= X_713 1.0))
+(assert (>= X_714 0.0))
+(assert (<= X_714 1.0))
+(assert (>= X_715 0.0))
+(assert (<= X_715 1.0))
+(assert (>= X_716 0.0))
+(assert (<= X_716 1.0))
+(assert (>= X_717 0.0))
+(assert (<= X_717 1.0))
+(assert (>= X_718 0.0))
+(assert (<= X_718 1.0))
+(assert (>= X_719 0.0))
+(assert (<= X_719 1.0))
+(assert (>= X_720 0.0))
+(assert (<= X_720 1.0))
+(assert (>= X_721 0.0))
+(assert (<= X_721 1.0))
+(assert (>= X_722 0.0))
+(assert (<= X_722 1.0))
+(assert (>= X_723 0.0))
+(assert (<= X_723 1.0))
+(assert (>= X_724 0.0))
+(assert (<= X_724 1.0))
+(assert (>= X_725 0.0))
+(assert (<= X_725 1.0))
+(assert (>= X_726 0.0))
+(assert (<= X_726 1.0))
+(assert (>= X_727 0.0))
+(assert (<= X_727 1.0))
+(assert (>= X_728 0.0))
+(assert (<= X_728 1.0))
+(assert (>= X_729 0.0))
+(assert (<= X_729 1.0))
+(assert (>= X_730 0.0))
+(assert (<= X_730 1.0))
+(assert (>= X_731 0.0))
+(assert (<= X_731 1.0))
+(assert (>= X_732 0.0))
+(assert (<= X_732 1.0))
+(assert (>= X_733 0.0))
+(assert (<= X_733 1.0))
+(assert (>= X_734 0.0))
+(assert (<= X_734 1.0))
+(assert (>= X_735 0.0))
+(assert (<= X_735 1.0))
+(assert (>= X_736 0.0))
+(assert (<= X_736 1.0))
+(assert (>= X_737 0.0))
+(assert (<= X_737 1.0))
+(assert (>= X_738 0.0))
+(assert (<= X_738 1.0))
+(assert (>= X_739 0.0))
+(assert (<= X_739 1.0))
+(assert (>= X_740 0.0))
+(assert (<= X_740 1.0))
+(assert (>= X_741 0.0))
+(assert (<= X_741 1.0))
+(assert (>= X_742 0.0))
+(assert (<= X_742 1.0))
+(assert (>= X_743 0.0))
+(assert (<= X_743 1.0))
+(assert (>= X_744 0.0))
+(assert (<= X_744 1.0))
+(assert (>= X_745 0.0))
+(assert (<= X_745 1.0))
+(assert (>= X_746 0.0))
+(assert (<= X_746 1.0))
+(assert (>= X_747 0.0))
+(assert (<= X_747 1.0))
+(assert (>= X_748 0.0))
+(assert (<= X_748 1.0))
+(assert (>= X_749 0.0))
+(assert (<= X_749 1.0))
+(assert (>= X_750 0.0))
+(assert (<= X_750 1.0))
+(assert (>= X_751 0.0))
+(assert (<= X_751 1.0))
+(assert (>= X_752 0.0))
+(assert (<= X_752 1.0))
+(assert (>= X_753 0.0))
+(assert (<= X_753 1.0))
+(assert (>= X_754 0.0))
+(assert (<= X_754 1.0))
+(assert (>= X_755 0.0))
+(assert (<= X_755 1.0))
+(assert (>= X_756 0.0))
+(assert (<= X_756 1.0))
+(assert (>= X_757 0.0))
+(assert (<= X_757 1.0))
+(assert (>= X_758 0.0))
+(assert (<= X_758 1.0))
+(assert (>= X_759 0.0))
+(assert (<= X_759 1.0))
+(assert (>= X_760 0.0))
+(assert (<= X_760 1.0))
+(assert (>= X_761 0.0))
+(assert (<= X_761 1.0))
+(assert (>= X_762 0.0))
+(assert (<= X_762 1.0))
+(assert (>= X_763 0.0))
+(assert (<= X_763 1.0))
+(assert (>= X_764 0.0))
+(assert (<= X_764 1.0))
+(assert (>= X_765 0.0))
+(assert (<= X_765 1.0))
+(assert (>= X_766 0.0))
+(assert (<= X_766 1.0))
+(assert (>= X_767 0.0))
+(assert (<= X_767 1.0))
+(assert (>= X_768 0.0))
+(assert (<= X_768 1.0))
+(assert (>= X_769 0.0))
+(assert (<= X_769 1.0))
+(assert (>= X_770 0.0))
+(assert (<= X_770 1.0))
+(assert (>= X_771 0.0))
+(assert (<= X_771 1.0))
+(assert (>= X_772 0.0))
+(assert (<= X_772 1.0))
+(assert (>= X_773 0.0))
+(assert (<= X_773 1.0))
+(assert (>= X_774 0.0))
+(assert (<= X_774 1.0))
+(assert (>= X_775 0.0))
+(assert (<= X_775 1.0))
+(assert (>= X_776 0.0))
+(assert (<= X_776 1.0))
+(assert (>= X_777 0.0))
+(assert (<= X_777 1.0))
+(assert (>= X_778 0.0))
+(assert (<= X_778 1.0))
+(assert (>= X_779 0.0))
+(assert (<= X_779 1.0))
+(assert (>= X_780 0.0))
+(assert (<= X_780 1.0))
+(assert (>= X_781 0.0))
+(assert (<= X_781 1.0))
+(assert (>= X_782 0.0))
+(assert (<= X_782 1.0))
+(assert (>= X_783 0.0))
+(assert (<= X_783 1.0))
 
 ; Violation of epsilon equivalence (epsilon = 0.1)
 (define-fun absolute ((x Real)) Real (if (>= x 0) x (- x)))
 (assert (or
-	(> (absolute (- Y_0 Y_2)) 0.1)
-	(> (absolute (- Y_1 Y_3)) 0.1)
+	(> (absolute (- Y_0 Y_10)) 0.1)
+	(> (absolute (- Y_1 Y_11)) 0.1)
+	(> (absolute (- Y_2 Y_12)) 0.1)
+	(> (absolute (- Y_3 Y_13)) 0.1)
+	(> (absolute (- Y_4 Y_14)) 0.1)
+	(> (absolute (- Y_5 Y_15)) 0.1)
+	(> (absolute (- Y_6 Y_16)) 0.1)
+	(> (absolute (- Y_7 Y_17)) 0.1)
+	(> (absolute (- Y_8 Y_18)) 0.1)
+	(> (absolute (- Y_9 Y_19)) 0.1)
 ))
diff --git a/examples/fashion_mnist/fashion_mnist_strict.vnnlib b/examples/fashion_mnist/fashion_mnist_strict.vnnlib
index 8242c34..c8b3f8e 100644
--- a/examples/fashion_mnist/fashion_mnist_strict.vnnlib
+++ b/examples/fashion_mnist/fashion_mnist_strict.vnnlib
@@ -37,10 +37,1538 @@
 (declare-const X_33 Real)
 (declare-const X_34 Real)
 (declare-const X_35 Real)
+(declare-const X_36 Real)
+(declare-const X_37 Real)
+(declare-const X_38 Real)
+(declare-const X_39 Real)
+(declare-const X_40 Real)
+(declare-const X_41 Real)
+(declare-const X_42 Real)
+(declare-const X_43 Real)
+(declare-const X_44 Real)
+(declare-const X_45 Real)
+(declare-const X_46 Real)
+(declare-const X_47 Real)
+(declare-const X_48 Real)
+(declare-const X_49 Real)
+(declare-const X_50 Real)
+(declare-const X_51 Real)
+(declare-const X_52 Real)
+(declare-const X_53 Real)
+(declare-const X_54 Real)
+(declare-const X_55 Real)
+(declare-const X_56 Real)
+(declare-const X_57 Real)
+(declare-const X_58 Real)
+(declare-const X_59 Real)
+(declare-const X_60 Real)
+(declare-const X_61 Real)
+(declare-const X_62 Real)
+(declare-const X_63 Real)
+(declare-const X_64 Real)
+(declare-const X_65 Real)
+(declare-const X_66 Real)
+(declare-const X_67 Real)
+(declare-const X_68 Real)
+(declare-const X_69 Real)
+(declare-const X_70 Real)
+(declare-const X_71 Real)
+(declare-const X_72 Real)
+(declare-const X_73 Real)
+(declare-const X_74 Real)
+(declare-const X_75 Real)
+(declare-const X_76 Real)
+(declare-const X_77 Real)
+(declare-const X_78 Real)
+(declare-const X_79 Real)
+(declare-const X_80 Real)
+(declare-const X_81 Real)
+(declare-const X_82 Real)
+(declare-const X_83 Real)
+(declare-const X_84 Real)
+(declare-const X_85 Real)
+(declare-const X_86 Real)
+(declare-const X_87 Real)
+(declare-const X_88 Real)
+(declare-const X_89 Real)
+(declare-const X_90 Real)
+(declare-const X_91 Real)
+(declare-const X_92 Real)
+(declare-const X_93 Real)
+(declare-const X_94 Real)
+(declare-const X_95 Real)
+(declare-const X_96 Real)
+(declare-const X_97 Real)
+(declare-const X_98 Real)
+(declare-const X_99 Real)
+(declare-const X_100 Real)
+(declare-const X_101 Real)
+(declare-const X_102 Real)
+(declare-const X_103 Real)
+(declare-const X_104 Real)
+(declare-const X_105 Real)
+(declare-const X_106 Real)
+(declare-const X_107 Real)
+(declare-const X_108 Real)
+(declare-const X_109 Real)
+(declare-const X_110 Real)
+(declare-const X_111 Real)
+(declare-const X_112 Real)
+(declare-const X_113 Real)
+(declare-const X_114 Real)
+(declare-const X_115 Real)
+(declare-const X_116 Real)
+(declare-const X_117 Real)
+(declare-const X_118 Real)
+(declare-const X_119 Real)
+(declare-const X_120 Real)
+(declare-const X_121 Real)
+(declare-const X_122 Real)
+(declare-const X_123 Real)
+(declare-const X_124 Real)
+(declare-const X_125 Real)
+(declare-const X_126 Real)
+(declare-const X_127 Real)
+(declare-const X_128 Real)
+(declare-const X_129 Real)
+(declare-const X_130 Real)
+(declare-const X_131 Real)
+(declare-const X_132 Real)
+(declare-const X_133 Real)
+(declare-const X_134 Real)
+(declare-const X_135 Real)
+(declare-const X_136 Real)
+(declare-const X_137 Real)
+(declare-const X_138 Real)
+(declare-const X_139 Real)
+(declare-const X_140 Real)
+(declare-const X_141 Real)
+(declare-const X_142 Real)
+(declare-const X_143 Real)
+(declare-const X_144 Real)
+(declare-const X_145 Real)
+(declare-const X_146 Real)
+(declare-const X_147 Real)
+(declare-const X_148 Real)
+(declare-const X_149 Real)
+(declare-const X_150 Real)
+(declare-const X_151 Real)
+(declare-const X_152 Real)
+(declare-const X_153 Real)
+(declare-const X_154 Real)
+(declare-const X_155 Real)
+(declare-const X_156 Real)
+(declare-const X_157 Real)
+(declare-const X_158 Real)
+(declare-const X_159 Real)
+(declare-const X_160 Real)
+(declare-const X_161 Real)
+(declare-const X_162 Real)
+(declare-const X_163 Real)
+(declare-const X_164 Real)
+(declare-const X_165 Real)
+(declare-const X_166 Real)
+(declare-const X_167 Real)
+(declare-const X_168 Real)
+(declare-const X_169 Real)
+(declare-const X_170 Real)
+(declare-const X_171 Real)
+(declare-const X_172 Real)
+(declare-const X_173 Real)
+(declare-const X_174 Real)
+(declare-const X_175 Real)
+(declare-const X_176 Real)
+(declare-const X_177 Real)
+(declare-const X_178 Real)
+(declare-const X_179 Real)
+(declare-const X_180 Real)
+(declare-const X_181 Real)
+(declare-const X_182 Real)
+(declare-const X_183 Real)
+(declare-const X_184 Real)
+(declare-const X_185 Real)
+(declare-const X_186 Real)
+(declare-const X_187 Real)
+(declare-const X_188 Real)
+(declare-const X_189 Real)
+(declare-const X_190 Real)
+(declare-const X_191 Real)
+(declare-const X_192 Real)
+(declare-const X_193 Real)
+(declare-const X_194 Real)
+(declare-const X_195 Real)
+(declare-const X_196 Real)
+(declare-const X_197 Real)
+(declare-const X_198 Real)
+(declare-const X_199 Real)
+(declare-const X_200 Real)
+(declare-const X_201 Real)
+(declare-const X_202 Real)
+(declare-const X_203 Real)
+(declare-const X_204 Real)
+(declare-const X_205 Real)
+(declare-const X_206 Real)
+(declare-const X_207 Real)
+(declare-const X_208 Real)
+(declare-const X_209 Real)
+(declare-const X_210 Real)
+(declare-const X_211 Real)
+(declare-const X_212 Real)
+(declare-const X_213 Real)
+(declare-const X_214 Real)
+(declare-const X_215 Real)
+(declare-const X_216 Real)
+(declare-const X_217 Real)
+(declare-const X_218 Real)
+(declare-const X_219 Real)
+(declare-const X_220 Real)
+(declare-const X_221 Real)
+(declare-const X_222 Real)
+(declare-const X_223 Real)
+(declare-const X_224 Real)
+(declare-const X_225 Real)
+(declare-const X_226 Real)
+(declare-const X_227 Real)
+(declare-const X_228 Real)
+(declare-const X_229 Real)
+(declare-const X_230 Real)
+(declare-const X_231 Real)
+(declare-const X_232 Real)
+(declare-const X_233 Real)
+(declare-const X_234 Real)
+(declare-const X_235 Real)
+(declare-const X_236 Real)
+(declare-const X_237 Real)
+(declare-const X_238 Real)
+(declare-const X_239 Real)
+(declare-const X_240 Real)
+(declare-const X_241 Real)
+(declare-const X_242 Real)
+(declare-const X_243 Real)
+(declare-const X_244 Real)
+(declare-const X_245 Real)
+(declare-const X_246 Real)
+(declare-const X_247 Real)
+(declare-const X_248 Real)
+(declare-const X_249 Real)
+(declare-const X_250 Real)
+(declare-const X_251 Real)
+(declare-const X_252 Real)
+(declare-const X_253 Real)
+(declare-const X_254 Real)
+(declare-const X_255 Real)
+(declare-const X_256 Real)
+(declare-const X_257 Real)
+(declare-const X_258 Real)
+(declare-const X_259 Real)
+(declare-const X_260 Real)
+(declare-const X_261 Real)
+(declare-const X_262 Real)
+(declare-const X_263 Real)
+(declare-const X_264 Real)
+(declare-const X_265 Real)
+(declare-const X_266 Real)
+(declare-const X_267 Real)
+(declare-const X_268 Real)
+(declare-const X_269 Real)
+(declare-const X_270 Real)
+(declare-const X_271 Real)
+(declare-const X_272 Real)
+(declare-const X_273 Real)
+(declare-const X_274 Real)
+(declare-const X_275 Real)
+(declare-const X_276 Real)
+(declare-const X_277 Real)
+(declare-const X_278 Real)
+(declare-const X_279 Real)
+(declare-const X_280 Real)
+(declare-const X_281 Real)
+(declare-const X_282 Real)
+(declare-const X_283 Real)
+(declare-const X_284 Real)
+(declare-const X_285 Real)
+(declare-const X_286 Real)
+(declare-const X_287 Real)
+(declare-const X_288 Real)
+(declare-const X_289 Real)
+(declare-const X_290 Real)
+(declare-const X_291 Real)
+(declare-const X_292 Real)
+(declare-const X_293 Real)
+(declare-const X_294 Real)
+(declare-const X_295 Real)
+(declare-const X_296 Real)
+(declare-const X_297 Real)
+(declare-const X_298 Real)
+(declare-const X_299 Real)
+(declare-const X_300 Real)
+(declare-const X_301 Real)
+(declare-const X_302 Real)
+(declare-const X_303 Real)
+(declare-const X_304 Real)
+(declare-const X_305 Real)
+(declare-const X_306 Real)
+(declare-const X_307 Real)
+(declare-const X_308 Real)
+(declare-const X_309 Real)
+(declare-const X_310 Real)
+(declare-const X_311 Real)
+(declare-const X_312 Real)
+(declare-const X_313 Real)
+(declare-const X_314 Real)
+(declare-const X_315 Real)
+(declare-const X_316 Real)
+(declare-const X_317 Real)
+(declare-const X_318 Real)
+(declare-const X_319 Real)
+(declare-const X_320 Real)
+(declare-const X_321 Real)
+(declare-const X_322 Real)
+(declare-const X_323 Real)
+(declare-const X_324 Real)
+(declare-const X_325 Real)
+(declare-const X_326 Real)
+(declare-const X_327 Real)
+(declare-const X_328 Real)
+(declare-const X_329 Real)
+(declare-const X_330 Real)
+(declare-const X_331 Real)
+(declare-const X_332 Real)
+(declare-const X_333 Real)
+(declare-const X_334 Real)
+(declare-const X_335 Real)
+(declare-const X_336 Real)
+(declare-const X_337 Real)
+(declare-const X_338 Real)
+(declare-const X_339 Real)
+(declare-const X_340 Real)
+(declare-const X_341 Real)
+(declare-const X_342 Real)
+(declare-const X_343 Real)
+(declare-const X_344 Real)
+(declare-const X_345 Real)
+(declare-const X_346 Real)
+(declare-const X_347 Real)
+(declare-const X_348 Real)
+(declare-const X_349 Real)
+(declare-const X_350 Real)
+(declare-const X_351 Real)
+(declare-const X_352 Real)
+(declare-const X_353 Real)
+(declare-const X_354 Real)
+(declare-const X_355 Real)
+(declare-const X_356 Real)
+(declare-const X_357 Real)
+(declare-const X_358 Real)
+(declare-const X_359 Real)
+(declare-const X_360 Real)
+(declare-const X_361 Real)
+(declare-const X_362 Real)
+(declare-const X_363 Real)
+(declare-const X_364 Real)
+(declare-const X_365 Real)
+(declare-const X_366 Real)
+(declare-const X_367 Real)
+(declare-const X_368 Real)
+(declare-const X_369 Real)
+(declare-const X_370 Real)
+(declare-const X_371 Real)
+(declare-const X_372 Real)
+(declare-const X_373 Real)
+(declare-const X_374 Real)
+(declare-const X_375 Real)
+(declare-const X_376 Real)
+(declare-const X_377 Real)
+(declare-const X_378 Real)
+(declare-const X_379 Real)
+(declare-const X_380 Real)
+(declare-const X_381 Real)
+(declare-const X_382 Real)
+(declare-const X_383 Real)
+(declare-const X_384 Real)
+(declare-const X_385 Real)
+(declare-const X_386 Real)
+(declare-const X_387 Real)
+(declare-const X_388 Real)
+(declare-const X_389 Real)
+(declare-const X_390 Real)
+(declare-const X_391 Real)
+(declare-const X_392 Real)
+(declare-const X_393 Real)
+(declare-const X_394 Real)
+(declare-const X_395 Real)
+(declare-const X_396 Real)
+(declare-const X_397 Real)
+(declare-const X_398 Real)
+(declare-const X_399 Real)
+(declare-const X_400 Real)
+(declare-const X_401 Real)
+(declare-const X_402 Real)
+(declare-const X_403 Real)
+(declare-const X_404 Real)
+(declare-const X_405 Real)
+(declare-const X_406 Real)
+(declare-const X_407 Real)
+(declare-const X_408 Real)
+(declare-const X_409 Real)
+(declare-const X_410 Real)
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+(declare-const Y_364 Real)
+(declare-const Y_365 Real)
+(declare-const Y_366 Real)
+(declare-const Y_367 Real)
+(declare-const Y_368 Real)
+(declare-const Y_369 Real)
+(declare-const Y_370 Real)
+(declare-const Y_371 Real)
+(declare-const Y_372 Real)
+(declare-const Y_373 Real)
+(declare-const Y_374 Real)
+(declare-const Y_375 Real)
+(declare-const Y_376 Real)
+(declare-const Y_377 Real)
+(declare-const Y_378 Real)
+(declare-const Y_379 Real)
+(declare-const Y_380 Real)
+(declare-const Y_381 Real)
+(declare-const Y_382 Real)
+(declare-const Y_383 Real)
+(declare-const Y_384 Real)
+(declare-const Y_385 Real)
+(declare-const Y_386 Real)
+(declare-const Y_387 Real)
+(declare-const Y_388 Real)
+(declare-const Y_389 Real)
+(declare-const Y_390 Real)
+(declare-const Y_391 Real)
+(declare-const Y_392 Real)
+(declare-const Y_393 Real)
+(declare-const Y_394 Real)
+(declare-const Y_395 Real)
+(declare-const Y_396 Real)
+(declare-const Y_397 Real)
+(declare-const Y_398 Real)
+(declare-const Y_399 Real)
+(declare-const Y_400 Real)
+(declare-const Y_401 Real)
+(declare-const Y_402 Real)
+(declare-const Y_403 Real)
+(declare-const Y_404 Real)
+(declare-const Y_405 Real)
+(declare-const Y_406 Real)
+(declare-const Y_407 Real)
+(declare-const Y_408 Real)
+(declare-const Y_409 Real)
+(declare-const Y_410 Real)
+(declare-const Y_411 Real)
+(declare-const Y_412 Real)
+(declare-const Y_413 Real)
+(declare-const Y_414 Real)
+(declare-const Y_415 Real)
+(declare-const Y_416 Real)
+(declare-const Y_417 Real)
+(declare-const Y_418 Real)
+(declare-const Y_419 Real)
+(declare-const Y_420 Real)
+(declare-const Y_421 Real)
+(declare-const Y_422 Real)
+(declare-const Y_423 Real)
+(declare-const Y_424 Real)
+(declare-const Y_425 Real)
+(declare-const Y_426 Real)
+(declare-const Y_427 Real)
+(declare-const Y_428 Real)
+(declare-const Y_429 Real)
+(declare-const Y_430 Real)
+(declare-const Y_431 Real)
+(declare-const Y_432 Real)
+(declare-const Y_433 Real)
+(declare-const Y_434 Real)
+(declare-const Y_435 Real)
+(declare-const Y_436 Real)
+(declare-const Y_437 Real)
+(declare-const Y_438 Real)
+(declare-const Y_439 Real)
+(declare-const Y_440 Real)
+(declare-const Y_441 Real)
+(declare-const Y_442 Real)
+(declare-const Y_443 Real)
+(declare-const Y_444 Real)
+(declare-const Y_445 Real)
+(declare-const Y_446 Real)
+(declare-const Y_447 Real)
+(declare-const Y_448 Real)
+(declare-const Y_449 Real)
+(declare-const Y_450 Real)
+(declare-const Y_451 Real)
+(declare-const Y_452 Real)
+(declare-const Y_453 Real)
+(declare-const Y_454 Real)
+(declare-const Y_455 Real)
+(declare-const Y_456 Real)
+(declare-const Y_457 Real)
+(declare-const Y_458 Real)
+(declare-const Y_459 Real)
+(declare-const Y_460 Real)
+(declare-const Y_461 Real)
+(declare-const Y_462 Real)
+(declare-const Y_463 Real)
+(declare-const Y_464 Real)
+(declare-const Y_465 Real)
+(declare-const Y_466 Real)
+(declare-const Y_467 Real)
+(declare-const Y_468 Real)
+(declare-const Y_469 Real)
+(declare-const Y_470 Real)
+(declare-const Y_471 Real)
+(declare-const Y_472 Real)
+(declare-const Y_473 Real)
+(declare-const Y_474 Real)
+(declare-const Y_475 Real)
+(declare-const Y_476 Real)
+(declare-const Y_477 Real)
+(declare-const Y_478 Real)
+(declare-const Y_479 Real)
+(declare-const Y_480 Real)
+(declare-const Y_481 Real)
+(declare-const Y_482 Real)
+(declare-const Y_483 Real)
+(declare-const Y_484 Real)
+(declare-const Y_485 Real)
+(declare-const Y_486 Real)
+(declare-const Y_487 Real)
+(declare-const Y_488 Real)
+(declare-const Y_489 Real)
+(declare-const Y_490 Real)
+(declare-const Y_491 Real)
+(declare-const Y_492 Real)
+(declare-const Y_493 Real)
+(declare-const Y_494 Real)
+(declare-const Y_495 Real)
+(declare-const Y_496 Real)
+(declare-const Y_497 Real)
+(declare-const Y_498 Real)
+(declare-const Y_499 Real)
+(declare-const Y_500 Real)
+(declare-const Y_501 Real)
+(declare-const Y_502 Real)
+(declare-const Y_503 Real)
+(declare-const Y_504 Real)
+(declare-const Y_505 Real)
+(declare-const Y_506 Real)
+(declare-const Y_507 Real)
+(declare-const Y_508 Real)
+(declare-const Y_509 Real)
+(declare-const Y_510 Real)
+(declare-const Y_511 Real)
+(declare-const Y_512 Real)
+(declare-const Y_513 Real)
+(declare-const Y_514 Real)
+(declare-const Y_515 Real)
+(declare-const Y_516 Real)
+(declare-const Y_517 Real)
+(declare-const Y_518 Real)
+(declare-const Y_519 Real)
+(declare-const Y_520 Real)
+(declare-const Y_521 Real)
+(declare-const Y_522 Real)
+(declare-const Y_523 Real)
+(declare-const Y_524 Real)
+(declare-const Y_525 Real)
+(declare-const Y_526 Real)
+(declare-const Y_527 Real)
+(declare-const Y_528 Real)
+(declare-const Y_529 Real)
+(declare-const Y_530 Real)
+(declare-const Y_531 Real)
+(declare-const Y_532 Real)
+(declare-const Y_533 Real)
+(declare-const Y_534 Real)
+(declare-const Y_535 Real)
+(declare-const Y_536 Real)
+(declare-const Y_537 Real)
+(declare-const Y_538 Real)
+(declare-const Y_539 Real)
+(declare-const Y_540 Real)
+(declare-const Y_541 Real)
+(declare-const Y_542 Real)
+(declare-const Y_543 Real)
+(declare-const Y_544 Real)
+(declare-const Y_545 Real)
+(declare-const Y_546 Real)
+(declare-const Y_547 Real)
+(declare-const Y_548 Real)
+(declare-const Y_549 Real)
+(declare-const Y_550 Real)
+(declare-const Y_551 Real)
+(declare-const Y_552 Real)
+(declare-const Y_553 Real)
+(declare-const Y_554 Real)
+(declare-const Y_555 Real)
+(declare-const Y_556 Real)
+(declare-const Y_557 Real)
+(declare-const Y_558 Real)
+(declare-const Y_559 Real)
+(declare-const Y_560 Real)
+(declare-const Y_561 Real)
+(declare-const Y_562 Real)
+(declare-const Y_563 Real)
+(declare-const Y_564 Real)
+(declare-const Y_565 Real)
+(declare-const Y_566 Real)
+(declare-const Y_567 Real)
+(declare-const Y_568 Real)
+(declare-const Y_569 Real)
+(declare-const Y_570 Real)
+(declare-const Y_571 Real)
+(declare-const Y_572 Real)
+(declare-const Y_573 Real)
+(declare-const Y_574 Real)
+(declare-const Y_575 Real)
+(declare-const Y_576 Real)
+(declare-const Y_577 Real)
+(declare-const Y_578 Real)
+(declare-const Y_579 Real)
+(declare-const Y_580 Real)
+(declare-const Y_581 Real)
+(declare-const Y_582 Real)
+(declare-const Y_583 Real)
+(declare-const Y_584 Real)
+(declare-const Y_585 Real)
+(declare-const Y_586 Real)
+(declare-const Y_587 Real)
+(declare-const Y_588 Real)
+(declare-const Y_589 Real)
+(declare-const Y_590 Real)
+(declare-const Y_591 Real)
+(declare-const Y_592 Real)
+(declare-const Y_593 Real)
+(declare-const Y_594 Real)
+(declare-const Y_595 Real)
+(declare-const Y_596 Real)
+(declare-const Y_597 Real)
+(declare-const Y_598 Real)
+(declare-const Y_599 Real)
+(declare-const Y_600 Real)
+(declare-const Y_601 Real)
+(declare-const Y_602 Real)
+(declare-const Y_603 Real)
+(declare-const Y_604 Real)
+(declare-const Y_605 Real)
+(declare-const Y_606 Real)
+(declare-const Y_607 Real)
+(declare-const Y_608 Real)
+(declare-const Y_609 Real)
+(declare-const Y_610 Real)
+(declare-const Y_611 Real)
+(declare-const Y_612 Real)
+(declare-const Y_613 Real)
+(declare-const Y_614 Real)
+(declare-const Y_615 Real)
+(declare-const Y_616 Real)
+(declare-const Y_617 Real)
+(declare-const Y_618 Real)
+(declare-const Y_619 Real)
+(declare-const Y_620 Real)
+(declare-const Y_621 Real)
+(declare-const Y_622 Real)
+(declare-const Y_623 Real)
+(declare-const Y_624 Real)
+(declare-const Y_625 Real)
+(declare-const Y_626 Real)
+(declare-const Y_627 Real)
+(declare-const Y_628 Real)
+(declare-const Y_629 Real)
+(declare-const Y_630 Real)
+(declare-const Y_631 Real)
+(declare-const Y_632 Real)
+(declare-const Y_633 Real)
+(declare-const Y_634 Real)
+(declare-const Y_635 Real)
+(declare-const Y_636 Real)
+(declare-const Y_637 Real)
+(declare-const Y_638 Real)
+(declare-const Y_639 Real)
+(declare-const Y_640 Real)
+(declare-const Y_641 Real)
+(declare-const Y_642 Real)
+(declare-const Y_643 Real)
+(declare-const Y_644 Real)
+(declare-const Y_645 Real)
+(declare-const Y_646 Real)
+(declare-const Y_647 Real)
+(declare-const Y_648 Real)
+(declare-const Y_649 Real)
+(declare-const Y_650 Real)
+(declare-const Y_651 Real)
+(declare-const Y_652 Real)
+(declare-const Y_653 Real)
+(declare-const Y_654 Real)
+(declare-const Y_655 Real)
+(declare-const Y_656 Real)
+(declare-const Y_657 Real)
+(declare-const Y_658 Real)
+(declare-const Y_659 Real)
+(declare-const Y_660 Real)
+(declare-const Y_661 Real)
+(declare-const Y_662 Real)
+(declare-const Y_663 Real)
+(declare-const Y_664 Real)
+(declare-const Y_665 Real)
+(declare-const Y_666 Real)
+(declare-const Y_667 Real)
+(declare-const Y_668 Real)
+(declare-const Y_669 Real)
+(declare-const Y_670 Real)
+(declare-const Y_671 Real)
+(declare-const Y_672 Real)
+(declare-const Y_673 Real)
+(declare-const Y_674 Real)
+(declare-const Y_675 Real)
+(declare-const Y_676 Real)
+(declare-const Y_677 Real)
+(declare-const Y_678 Real)
+(declare-const Y_679 Real)
+(declare-const Y_680 Real)
+(declare-const Y_681 Real)
+(declare-const Y_682 Real)
+(declare-const Y_683 Real)
+(declare-const Y_684 Real)
+(declare-const Y_685 Real)
+(declare-const Y_686 Real)
+(declare-const Y_687 Real)
+(declare-const Y_688 Real)
+(declare-const Y_689 Real)
+(declare-const Y_690 Real)
+(declare-const Y_691 Real)
+(declare-const Y_692 Real)
+(declare-const Y_693 Real)
+(declare-const Y_694 Real)
+(declare-const Y_695 Real)
+(declare-const Y_696 Real)
+(declare-const Y_697 Real)
+(declare-const Y_698 Real)
+(declare-const Y_699 Real)
+(declare-const Y_700 Real)
+(declare-const Y_701 Real)
+(declare-const Y_702 Real)
+(declare-const Y_703 Real)
+(declare-const Y_704 Real)
+(declare-const Y_705 Real)
+(declare-const Y_706 Real)
+(declare-const Y_707 Real)
+(declare-const Y_708 Real)
+(declare-const Y_709 Real)
+(declare-const Y_710 Real)
+(declare-const Y_711 Real)
+(declare-const Y_712 Real)
+(declare-const Y_713 Real)
+(declare-const Y_714 Real)
+(declare-const Y_715 Real)
+(declare-const Y_716 Real)
+(declare-const Y_717 Real)
+(declare-const Y_718 Real)
+(declare-const Y_719 Real)
+(declare-const Y_720 Real)
+(declare-const Y_721 Real)
+(declare-const Y_722 Real)
+(declare-const Y_723 Real)
+(declare-const Y_724 Real)
+(declare-const Y_725 Real)
+(declare-const Y_726 Real)
+(declare-const Y_727 Real)
+(declare-const Y_728 Real)
+(declare-const Y_729 Real)
+(declare-const Y_730 Real)
+(declare-const Y_731 Real)
+(declare-const Y_732 Real)
+(declare-const Y_733 Real)
+(declare-const Y_734 Real)
+(declare-const Y_735 Real)
+(declare-const Y_736 Real)
+(declare-const Y_737 Real)
+(declare-const Y_738 Real)
+(declare-const Y_739 Real)
+(declare-const Y_740 Real)
+(declare-const Y_741 Real)
+(declare-const Y_742 Real)
+(declare-const Y_743 Real)
+(declare-const Y_744 Real)
+(declare-const Y_745 Real)
+(declare-const Y_746 Real)
+(declare-const Y_747 Real)
+(declare-const Y_748 Real)
+(declare-const Y_749 Real)
+(declare-const Y_750 Real)
+(declare-const Y_751 Real)
+(declare-const Y_752 Real)
+(declare-const Y_753 Real)
+(declare-const Y_754 Real)
+(declare-const Y_755 Real)
+(declare-const Y_756 Real)
+(declare-const Y_757 Real)
+(declare-const Y_758 Real)
+(declare-const Y_759 Real)
+(declare-const Y_760 Real)
+(declare-const Y_761 Real)
+(declare-const Y_762 Real)
+(declare-const Y_763 Real)
+(declare-const Y_764 Real)
+(declare-const Y_765 Real)
+(declare-const Y_766 Real)
+(declare-const Y_767 Real)
+(declare-const Y_768 Real)
+(declare-const Y_769 Real)
+(declare-const Y_770 Real)
+(declare-const Y_771 Real)
+(declare-const Y_772 Real)
+(declare-const Y_773 Real)
+(declare-const Y_774 Real)
+(declare-const Y_775 Real)
+(declare-const Y_776 Real)
+(declare-const Y_777 Real)
+(declare-const Y_778 Real)
+(declare-const Y_779 Real)
+(declare-const Y_780 Real)
+(declare-const Y_781 Real)
+(declare-const Y_782 Real)
+(declare-const Y_783 Real)
 
 ; Bounded inputs: X must be within [0, 1]
 (assert (>= X_0 0.0))
@@ -115,9 +1643,1513 @@
 (assert (<= X_34 1.0))
 (assert (>= X_35 0.0))
 (assert (<= X_35 1.0))
+(assert (>= X_36 0.0))
+(assert (<= X_36 1.0))
+(assert (>= X_37 0.0))
+(assert (<= X_37 1.0))
+(assert (>= X_38 0.0))
+(assert (<= X_38 1.0))
+(assert (>= X_39 0.0))
+(assert (<= X_39 1.0))
+(assert (>= X_40 0.0))
+(assert (<= X_40 1.0))
+(assert (>= X_41 0.0))
+(assert (<= X_41 1.0))
+(assert (>= X_42 0.0))
+(assert (<= X_42 1.0))
+(assert (>= X_43 0.0))
+(assert (<= X_43 1.0))
+(assert (>= X_44 0.0))
+(assert (<= X_44 1.0))
+(assert (>= X_45 0.0))
+(assert (<= X_45 1.0))
+(assert (>= X_46 0.0))
+(assert (<= X_46 1.0))
+(assert (>= X_47 0.0))
+(assert (<= X_47 1.0))
+(assert (>= X_48 0.0))
+(assert (<= X_48 1.0))
+(assert (>= X_49 0.0))
+(assert (<= X_49 1.0))
+(assert (>= X_50 0.0))
+(assert (<= X_50 1.0))
+(assert (>= X_51 0.0))
+(assert (<= X_51 1.0))
+(assert (>= X_52 0.0))
+(assert (<= X_52 1.0))
+(assert (>= X_53 0.0))
+(assert (<= X_53 1.0))
+(assert (>= X_54 0.0))
+(assert (<= X_54 1.0))
+(assert (>= X_55 0.0))
+(assert (<= X_55 1.0))
+(assert (>= X_56 0.0))
+(assert (<= X_56 1.0))
+(assert (>= X_57 0.0))
+(assert (<= X_57 1.0))
+(assert (>= X_58 0.0))
+(assert (<= X_58 1.0))
+(assert (>= X_59 0.0))
+(assert (<= X_59 1.0))
+(assert (>= X_60 0.0))
+(assert (<= X_60 1.0))
+(assert (>= X_61 0.0))
+(assert (<= X_61 1.0))
+(assert (>= X_62 0.0))
+(assert (<= X_62 1.0))
+(assert (>= X_63 0.0))
+(assert (<= X_63 1.0))
+(assert (>= X_64 0.0))
+(assert (<= X_64 1.0))
+(assert (>= X_65 0.0))
+(assert (<= X_65 1.0))
+(assert (>= X_66 0.0))
+(assert (<= X_66 1.0))
+(assert (>= X_67 0.0))
+(assert (<= X_67 1.0))
+(assert (>= X_68 0.0))
+(assert (<= X_68 1.0))
+(assert (>= X_69 0.0))
+(assert (<= X_69 1.0))
+(assert (>= X_70 0.0))
+(assert (<= X_70 1.0))
+(assert (>= X_71 0.0))
+(assert (<= X_71 1.0))
+(assert (>= X_72 0.0))
+(assert (<= X_72 1.0))
+(assert (>= X_73 0.0))
+(assert (<= X_73 1.0))
+(assert (>= X_74 0.0))
+(assert (<= X_74 1.0))
+(assert (>= X_75 0.0))
+(assert (<= X_75 1.0))
+(assert (>= X_76 0.0))
+(assert (<= X_76 1.0))
+(assert (>= X_77 0.0))
+(assert (<= X_77 1.0))
+(assert (>= X_78 0.0))
+(assert (<= X_78 1.0))
+(assert (>= X_79 0.0))
+(assert (<= X_79 1.0))
+(assert (>= X_80 0.0))
+(assert (<= X_80 1.0))
+(assert (>= X_81 0.0))
+(assert (<= X_81 1.0))
+(assert (>= X_82 0.0))
+(assert (<= X_82 1.0))
+(assert (>= X_83 0.0))
+(assert (<= X_83 1.0))
+(assert (>= X_84 0.0))
+(assert (<= X_84 1.0))
+(assert (>= X_85 0.0))
+(assert (<= X_85 1.0))
+(assert (>= X_86 0.0))
+(assert (<= X_86 1.0))
+(assert (>= X_87 0.0))
+(assert (<= X_87 1.0))
+(assert (>= X_88 0.0))
+(assert (<= X_88 1.0))
+(assert (>= X_89 0.0))
+(assert (<= X_89 1.0))
+(assert (>= X_90 0.0))
+(assert (<= X_90 1.0))
+(assert (>= X_91 0.0))
+(assert (<= X_91 1.0))
+(assert (>= X_92 0.0))
+(assert (<= X_92 1.0))
+(assert (>= X_93 0.0))
+(assert (<= X_93 1.0))
+(assert (>= X_94 0.0))
+(assert (<= X_94 1.0))
+(assert (>= X_95 0.0))
+(assert (<= X_95 1.0))
+(assert (>= X_96 0.0))
+(assert (<= X_96 1.0))
+(assert (>= X_97 0.0))
+(assert (<= X_97 1.0))
+(assert (>= X_98 0.0))
+(assert (<= X_98 1.0))
+(assert (>= X_99 0.0))
+(assert (<= X_99 1.0))
+(assert (>= X_100 0.0))
+(assert (<= X_100 1.0))
+(assert (>= X_101 0.0))
+(assert (<= X_101 1.0))
+(assert (>= X_102 0.0))
+(assert (<= X_102 1.0))
+(assert (>= X_103 0.0))
+(assert (<= X_103 1.0))
+(assert (>= X_104 0.0))
+(assert (<= X_104 1.0))
+(assert (>= X_105 0.0))
+(assert (<= X_105 1.0))
+(assert (>= X_106 0.0))
+(assert (<= X_106 1.0))
+(assert (>= X_107 0.0))
+(assert (<= X_107 1.0))
+(assert (>= X_108 0.0))
+(assert (<= X_108 1.0))
+(assert (>= X_109 0.0))
+(assert (<= X_109 1.0))
+(assert (>= X_110 0.0))
+(assert (<= X_110 1.0))
+(assert (>= X_111 0.0))
+(assert (<= X_111 1.0))
+(assert (>= X_112 0.0))
+(assert (<= X_112 1.0))
+(assert (>= X_113 0.0))
+(assert (<= X_113 1.0))
+(assert (>= X_114 0.0))
+(assert (<= X_114 1.0))
+(assert (>= X_115 0.0))
+(assert (<= X_115 1.0))
+(assert (>= X_116 0.0))
+(assert (<= X_116 1.0))
+(assert (>= X_117 0.0))
+(assert (<= X_117 1.0))
+(assert (>= X_118 0.0))
+(assert (<= X_118 1.0))
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+(assert (<= X_119 1.0))
+(assert (>= X_120 0.0))
+(assert (<= X_120 1.0))
+(assert (>= X_121 0.0))
+(assert (<= X_121 1.0))
+(assert (>= X_122 0.0))
+(assert (<= X_122 1.0))
+(assert (>= X_123 0.0))
+(assert (<= X_123 1.0))
+(assert (>= X_124 0.0))
+(assert (<= X_124 1.0))
+(assert (>= X_125 0.0))
+(assert (<= X_125 1.0))
+(assert (>= X_126 0.0))
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+(assert (>= X_127 0.0))
+(assert (<= X_127 1.0))
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+(assert (>= X_129 0.0))
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+(assert (>= X_130 0.0))
+(assert (<= X_130 1.0))
+(assert (>= X_131 0.0))
+(assert (<= X_131 1.0))
+(assert (>= X_132 0.0))
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+(assert (<= X_133 1.0))
+(assert (>= X_134 0.0))
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+(assert (<= X_135 1.0))
+(assert (>= X_136 0.0))
+(assert (<= X_136 1.0))
+(assert (>= X_137 0.0))
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+(assert (>= X_138 0.0))
+(assert (<= X_138 1.0))
+(assert (>= X_139 0.0))
+(assert (<= X_139 1.0))
+(assert (>= X_140 0.0))
+(assert (<= X_140 1.0))
+(assert (>= X_141 0.0))
+(assert (<= X_141 1.0))
+(assert (>= X_142 0.0))
+(assert (<= X_142 1.0))
+(assert (>= X_143 0.0))
+(assert (<= X_143 1.0))
+(assert (>= X_144 0.0))
+(assert (<= X_144 1.0))
+(assert (>= X_145 0.0))
+(assert (<= X_145 1.0))
+(assert (>= X_146 0.0))
+(assert (<= X_146 1.0))
+(assert (>= X_147 0.0))
+(assert (<= X_147 1.0))
+(assert (>= X_148 0.0))
+(assert (<= X_148 1.0))
+(assert (>= X_149 0.0))
+(assert (<= X_149 1.0))
+(assert (>= X_150 0.0))
+(assert (<= X_150 1.0))
+(assert (>= X_151 0.0))
+(assert (<= X_151 1.0))
+(assert (>= X_152 0.0))
+(assert (<= X_152 1.0))
+(assert (>= X_153 0.0))
+(assert (<= X_153 1.0))
+(assert (>= X_154 0.0))
+(assert (<= X_154 1.0))
+(assert (>= X_155 0.0))
+(assert (<= X_155 1.0))
+(assert (>= X_156 0.0))
+(assert (<= X_156 1.0))
+(assert (>= X_157 0.0))
+(assert (<= X_157 1.0))
+(assert (>= X_158 0.0))
+(assert (<= X_158 1.0))
+(assert (>= X_159 0.0))
+(assert (<= X_159 1.0))
+(assert (>= X_160 0.0))
+(assert (<= X_160 1.0))
+(assert (>= X_161 0.0))
+(assert (<= X_161 1.0))
+(assert (>= X_162 0.0))
+(assert (<= X_162 1.0))
+(assert (>= X_163 0.0))
+(assert (<= X_163 1.0))
+(assert (>= X_164 0.0))
+(assert (<= X_164 1.0))
+(assert (>= X_165 0.0))
+(assert (<= X_165 1.0))
+(assert (>= X_166 0.0))
+(assert (<= X_166 1.0))
+(assert (>= X_167 0.0))
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+(assert (>= X_782 0.0))
+(assert (<= X_782 1.0))
+(assert (>= X_783 0.0))
+(assert (<= X_783 1.0))
 
 ; Violation of strict equivalence
 (assert (or
-	(not (= Y_0 Y_2))
-	(not (= Y_1 Y_3))
+	(not (= Y_0 Y_10))
+	(not (= Y_1 Y_11))
+	(not (= Y_2 Y_12))
+	(not (= Y_3 Y_13))
+	(not (= Y_4 Y_14))
+	(not (= Y_5 Y_15))
+	(not (= Y_6 Y_16))
+	(not (= Y_7 Y_17))
+	(not (= Y_8 Y_18))
+	(not (= Y_9 Y_19))
 ))
diff --git a/examples/verify_fashion_mnist.py b/examples/verify_fashion_mnist.py
index 4de0d1e..91ca1ae 100644
--- a/examples/verify_fashion_mnist.py
+++ b/examples/verify_fashion_mnist.py
@@ -1,18 +1,3 @@
-# Copyright (C) 2026 Eric Marin
-#
-# This program is free software: you can redistribute it and/or modify
-# it under the terms of the GNU Affero General Public License as
-# published by the Free Software Foundation, either version 3 of the
-# License, or (at your option) any later version.
-#
-# This program is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-# GNU Affero General Public License for more details.
-#
-# You should have received a copy of the GNU Affero General Public License
-# along with this program.  If not, see <https://www.gnu.org/licenses/>.
-
 import sys, os
 sys.path.append(os.path.dirname(os.path.dirname(os.path.abspath(__file__))))
 
diff --git a/examples/verify_xor.py b/examples/verify_xor.py
index f1de0cb..a9f5e23 100644
--- a/examples/verify_xor.py
+++ b/examples/verify_xor.py
@@ -1,18 +1,3 @@
-# Copyright (C) 2026 Eric Marin
-#
-# This program is free software: you can redistribute it and/or modify
-# it under the terms of the GNU Affero General Public License as
-# published by the Free Software Foundation, either version 3 of the
-# License, or (at your option) any later version.
-#
-# This program is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-# GNU Affero General Public License for more details.
-#
-# You should have received a copy of the GNU Affero General Public License
-# along with this program.  If not, see <https://www.gnu.org/licenses/>.
-
 import sys, os
 sys.path.append(os.path.dirname(os.path.dirname(os.path.abspath(__file__))))
 
diff --git a/nneq.py b/nneq.py
index b919b17..cab8322 100644
--- a/nneq.py
+++ b/nneq.py
@@ -1,17 +1,15 @@
 # Copyright (C) 2026 Eric Marin
 #
-# This program is free software: you can redistribute it and/or modify
-# it under the terms of the GNU Affero General Public License as
-# published by the Free Software Foundation, either version 3 of the
-# License, or (at your option) any later version.
+# This program is free software: you can redistribute it and/or modify it under the terms of the
+# GNU Affero General Public License as published by the Free Software Foundation, either version 3
+# of the License, or (at your option) any later version.
 #
-# This program is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-# GNU Affero General Public License for more details.
+# This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without
+# even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+# Affero General Public License for more details.
 #
-# You should have received a copy of the GNU Affero General Public License
-# along with this program.  If not, see <https://www.gnu.org/licenses/>.
+# You should have received a copy of the GNU Affero General Public License along with this program.
+# If not, see <https://www.gnu.org/licenses/>. 
 
 import z3
 import re
diff --git a/proof.md b/proof.md
new file mode 100644
index 0000000..7fb6413
--- /dev/null
+++ b/proof.md
@@ -0,0 +1,532 @@
+# Soundness Proof
+## Mathematical Definitions
+![Linear(x, q, r) ~ out => out = q*x + r](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;out\Rightarrow&space;out=q*x&plus;r&space;)
+![Concrete(k) ~ out => out = ](https://latex.codecogs.com/svg.image?Concrete(k)\sim&space;out\Rightarrow&space;out=k&space;)
+![Add(out, b) ~ a => out = a + b](https://latex.codecogs.com/svg.image?Add(out,b)\sim&space;a\Rightarrow&space;out=a&plus;b&space;)
+![AddCheckLinear(out, x, q, r) ~ b => out = q*x + (r + b)](https://latex.codecogs.com/svg.image?AddCheckLinear(out,x,q,r)\sim&space;b\Rightarrow&space;out=q*x&plus;(r&plus;b))
+![AddCheckConcrete(out, k) ~ b => out = k + b](https://latex.codecogs.com/svg.image?AddCheckConcrete(out,k)\sim&space;b\Rightarrow&space;out=k&plus;b&space;)
+![Mul(out, b) ~ a => out = a * b](https://latex.codecogs.com/svg.image?Mul(out,b)\sim&space;a\Rightarrow&space;out=a*b&space;)
+![MulCheckLinear(out, x, q, r) ~ b => out = q*b*x + r*b](https://latex.codecogs.com/svg.image?MulCheckLinear(out,x,q,r)\sim&space;b\Rightarrow&space;out=q*b*x&plus;r*b&space;)
+![MulCheckConcrete(out, k) ~ b => out = k*b](https://latex.codecogs.com/svg.image?MulCheckConcrete(out,k)\sim&space;b\Rightarrow&space;out=k*b&space;)
+![ReLU(out) ~ x => out = IF (x > 0) THEN x ELSE 0](https://latex.codecogs.com/svg.image?ReLU(out)\sim&space;x\Rightarrow&space;out=IF\;(x>0)\;THEN\;x\;ELSE\;0&space;)
+![Materialize(out) ~ x => out = x](https://latex.codecogs.com/svg.image?Materialize(out)\sim&space;x\Rightarrow&space;out=x&space;)
+
+## Soundness of Translation
+### ReLU
+ONNX ReLU node is defined as:
+![Y = X if X > 0 else 0](https://latex.codecogs.com/svg.image?Y=X\;if\;X>0\;else\;0&space;)
+
+The translation defines the interactions:
+![x_i ~ ReLU(y_i)](https://latex.codecogs.com/svg.image?x_i\sim&space;ReLU(y_i))
+
+By definition this interaction is equal to:
+![y_i = IF (x_i > 0) THEN x_i ELSE 0](https://latex.codecogs.com/svg.image?y_i=IF\;(x_i>0)\;THEN\;x_i\;ELSE\;0&space;)
+
+### Gemm
+ONNX Gemm node is defined as:
+![Y = alpha * A * B + beta * C](https://latex.codecogs.com/svg.image?Y=\alpha\cdot&space;A\cdot&space;B&plus;\beta\cdot&space;C&space;)
+
+The translation defines the interactions:
+![a_i ~ Mul(v_i, Concrete(alpha * b_i))](https://latex.codecogs.com/svg.image?a_i\sim&space;Mul(v_i,Concrete(\alpha*b_i)))
+![Add(...(Add(y_i, v_1), ...), v_n) ~ Concrete(beta * c_i)](https://latex.codecogs.com/svg.image?Add(...(Add(y_i,v_1),...),v_n)\sim&space;Concrete(\beta*c_i))
+
+By definition this interaction is equal to:
+![v_i = alpha * a_i * b_i](https://latex.codecogs.com/svg.image?v_i=\alpha*a_i*b_i&space;)
+![y_i = v_1 + v_2 + ... + v_n + beta * c_i](https://latex.codecogs.com/svg.image?y_i=v_1&plus;v_2&plus;...&plus;v_n&plus;\beta*c_i&space;)
+
+By grouping the operations we get:
+![Y = alpha * A * B + beta * C](https://latex.codecogs.com/svg.image?Y=\alpha\cdot&space;A\cdot&space;B&plus;\beta\cdot&space;C&space;)
+
+### Flatten
+Just identity mapping.
+
+### MatMul
+Equal to Gemm with ![alpha=1](https://latex.codecogs.com/svg.image?\inline&space;\alpha=1) and ![beta=0](https://latex.codecogs.com/svg.image?\inline&space;\beta=0).
+
+### Reshape
+Just identity mapping.
+
+## Soundness of Interaction Rules
+### Materialize
+The Materialize agent transforms a Linear agent into a tree of explicit mathematical operations
+that are used as final representation for the solver.
+In the Python module the terms are defined as:
+```python
+def TermAdd(a, b):
+    return a + b  
+def TermMul(a, b):
+    return a * b
+def TermReLU(x):
+    return z3.If(x > 0, x, 0)
+```
+
+#### Linear >< Materialize
+![Linear(x, q, r) >< Materialize(out) => (1), (2), (3), (4), (5)](https://latex.codecogs.com/svg.image?\inline&space;&space;Linear(x,q,r)><Materialize(out)\Rightarrow&space;(1),(2),(3),(4),(5))
+
+LHS:
+![Linear(x, q, r) ~ wire](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire)
+![Materialize(out) ~ wire](https://latex.codecogs.com/svg.image?Materialize(out)\sim&space;wire)
+![q*x + r = wire](https://latex.codecogs.com/svg.image?q*x&plus;r=wire)
+![out = wire](https://latex.codecogs.com/svg.image?out=wire)
+![out = q*x + r](https://latex.codecogs.com/svg.image?out=q*x&plus;r)
+
+##### Case 1:
+![q = 0 => out ~ Concrete(r), x ~ Eraser](https://latex.codecogs.com/svg.image?q=0\Rightarrow&space;out\sim&space;Concrete(r),x\sim&space;Eraser)
+
+RHS:
+![out = r](https://latex.codecogs.com/svg.image?out=r)
+
+EQUIVALENCE:
+![0*x + r = r => r = r](https://latex.codecogs.com/svg.image?0*x&plus;r=r\Rightarrow&space;r=r)
+
+##### Case 2:
+![q = 1, r = 0 => out ~ x](https://latex.codecogs.com/svg.image?q=1,r=0\Rightarrow&space;out~x)
+
+RHS:
+![x = out](https://latex.codecogs.com/svg.image?x=out)
+![out = x](https://latex.codecogs.com/svg.image?out=x)
+
+EQUIVALENCE:
+![1*x + 0 = x => x = x](https://latex.codecogs.com/svg.image?1*x&plus;0=x\Rightarrow&space;x=x)
+
+##### Case 3:
+![q = 1 => out ~ TermAdd(x, Concrete(r))](https://latex.codecogs.com/svg.image?q=1\Rightarrow&space;out\sim&space;TermAdd(x,Concrete(r)))
+
+RHS:
+![out = x + r](https://latex.codecogs.com/svg.image?out=x&plus;r)
+
+EQUIVALENCE:
+![1*x + r = x + r => x + r = x + r](https://latex.codecogs.com/svg.image?1*x&plus;r=x&plus;r\Rightarrow&space;x&plus;r=x&plus;r)
+
+##### Case 4:
+![r = 0 => out ~ TermMul(Concrete(q), x)](https://latex.codecogs.com/svg.image?r=0\Rightarrow&space;out\sim&space;TermMul(Concrete(q),x))
+
+RHS:
+![out = q*x](https://latex.codecogs.com/svg.image?out=q*x)
+
+EQUIVALENCE:
+![q*x + 0 = q*x => q*x = q*x](https://latex.codecogs.com/svg.image?q*x&plus;0=q*x\Rightarrow&space;q*x=q*x)
+
+##### Case 5:
+![otherwise => out ~ TermAdd(TermMul(Concrete(q), x), Concrete(r))](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;out\sim&space;TermAdd(TermMul(Concrete(q),x),Concrete(r)))
+
+RHS:
+![out = q*x + r](https://latex.codecogs.com/svg.image?out=q*x&plus;r)
+
+EQUIVALENCE:
+![q*x + r = q*x + r](https://latex.codecogs.com/svg.image?q*x&plus;r=q*x&plus;r)
+
+#### Concrete >< Materialize
+![Concrete(k) >< Materialize(out) => out ~ Concrete(k)](https://latex.codecogs.com/svg.image?Concrete(k)><Materialize(out)\Rightarrow&space;out\sim&space;Concrete(k))
+
+LHS:
+![Concrete(k) ~ wire](https://latex.codecogs.com/svg.image?Concrete(k)\sim&space;wire)
+![Materialize(out) ~ wire](https://latex.codecogs.com/svg.image?Materialize(out)\sim&space;wire)
+![k = wire](https://latex.codecogs.com/svg.image?k=wire)
+![out = wire](https://latex.codecogs.com/svg.image?out=wire)
+![out = k](https://latex.codecogs.com/svg.image?out=k)
+
+RHS:
+![out = k](https://latex.codecogs.com/svg.image?out=k)
+
+EQUIVALENCE:
+![k = k](https://latex.codecogs.com/svg.image?k=k)
+
+### Add
+#### Linear >< Add
+![Linear(x, q, r) >< Add(out, b) => b ~ AddCheckLinear(out, x, q, r)](https://latex.codecogs.com/svg.image?Linear(x,q,r)><Add(out,b)\Rightarrow&space;b\sim&space;AddCheckLinear(out,x,q,r))
+
+LHS:
+![Linear(x, q, r) ~ wire](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire)
+![Add(out, b) ~ wire](https://latex.codecogs.com/svg.image?Add(out,b)\sim&space;wire)
+![q*x + r = wire](https://latex.codecogs.com/svg.image?q*x&plus;r=wire)
+![out = wire + b](https://latex.codecogs.com/svg.image?out=wire&plus;b)
+![out = q*x + r + b](https://latex.codecogs.com/svg.image?out=q*x&plus;r&plus;b)
+
+RHS:
+![out = q*x + (r + b)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;b))
+
+EQUIVALENCE:
+![q*x + r + b = q*x + (r + b) => q*x + (r + b) = q*x + (r + b)](https://latex.codecogs.com/svg.image?q*x&plus;r&plus;b=q*x&plus;(r&plus;b)\Rightarrow&space;q*x&plus;(r&plus;b)=q*x&plus;(r&plus;b))
+
+#### Concrete >< Add
+![Concrete(k) >< Add(out, b) => (1), (2)](https://latex.codecogs.com/svg.image?Concrete(k)><Add(out,b)\Rightarrow&space;(1),(2))
+
+LHS:
+![Concrete(k) ~ wire](https://latex.codecogs.com/svg.image?Concrete(k)\sim&space;wire)
+![Add(out, b) ~ wire](https://latex.codecogs.com/svg.image?Add(out,b)\sim&space;wire)
+![k = wire](https://latex.codecogs.com/svg.image?k=wire)
+![out = wire + b](https://latex.codecogs.com/svg.image?out=wire&plus;b)
+![out = k + b](https://latex.codecogs.com/svg.image?out=k&plus;b)
+
+##### Case 1:
+![k = 0 => out ~ b](https://latex.codecogs.com/svg.image?k=0\Rightarrow&space;out\sim&space;b)
+
+RHS:
+![out = b](https://latex.codecogs.com/svg.image?out=b)
+
+EQUIVALENCE:
+![0 + b = b => b = b](https://latex.codecogs.com/svg.image?0&plus;b=b\Rightarrow&space;b=b)
+
+##### Case 2:
+![otherwise => b ~ AddCheckConcrete(out, k)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;b\sim&space;AddCheckConcrete(out,k))
+
+RHS:
+![out = k + b](https://latex.codecogs.com/svg.image?out=k&plus;b)
+
+EQUIVALENCE:
+![k + b = k + b](https://latex.codecogs.com/svg.image?k&plus;b=k&plus;b)
+
+#### Linear >< AddCheckLinear
+![Linear(y, s, t) >< AddCheckLinear(out, x, q, r) => (1), (2), (3), (4)](https://latex.codecogs.com/svg.image?Linear(y,s,t)><AddCheckLinear(out,x,q,r)\Rightarrow&space;(1),(2),(3),(4))
+
+LHS:
+![Linear(y, s, t) ~ wire](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire)
+![AddCheckLinear(out, x, q, r) ~ wire](https://latex.codecogs.com/svg.image?AddCheckLinear(out,x,q,r)\sim&space;wire)
+![s*y + t = wire](https://latex.codecogs.com/svg.image?s*y&plus;t=wire)
+![out = q*x + (r + wire)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;wire))
+![out = q*x + (r + s*y + t)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;s*y&plus;t))
+
+##### Case 1:
+![q,r,s,t = 0 => out ~ Concrete(0), x ~ Eraser, y ~ Eraser](https://latex.codecogs.com/svg.image?q,r,s,t=0\Rightarrow&space;out\sim&space;Concrete(0),x\sim&space;Eraser,y\sim&space;Eraser)
+
+RHS:
+![out = 0](https://latex.codecogs.com/svg.image?out=0)
+
+EQUIVALENCE:
+![0*x + (0 + 0*y + 0) = 0 => 0 = 0](https://latex.codecogs.com/svg.image?0*x&plus;(0&plus;0*y&plus;0)=0\Rightarrow&space;0=0)
+
+##### Case 2:
+![s,t = 0 => out ~ Linear(x, q, r), y ~ Eraser](https://latex.codecogs.com/svg.image?s,t=0\Rightarrow&space;out\sim&space;Linear(x,q,r),y\sim&space;Eraser)
+
+RHS:
+![out = q*x + r](https://latex.codecogs.com/svg.image?out=q*x&plus;r)
+
+EQUIVALENCE:
+![q*x + (r + 0*y + 0) = q*x + r => q*x + r = q*x + r](https://latex.codecogs.com/svg.image?q*x&plus;(r&plus;0*y&plus;0)=q*x&plus;r\Rightarrow&space;q*x&plus;r=q*x&plus;r)
+
+##### Case 3:
+![q, r = 0 => out ~ Linear(y, s, t), x ~ Eraser](https://latex.codecogs.com/svg.image?q,r=0\Rightarrow&space;out\sim&space;Linear(y,s,t),x\sim&space;Eraser)
+
+RHS:
+![out = s*y + t](https://latex.codecogs.com/svg.image?out=s*y&plus;t)
+
+EQUIVALENCE:
+![0*x + (0 + s*y + t) = s*y + t => s*y + t = s*y + t](https://latex.codecogs.com/svg.image?0*x&plus;(0&plus;s*y&plus;t)=s*y&plus;t\Rightarrow&space;s*y&plus;t=s*y&plus;t)
+
+##### Case 4:
+![otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermAdd(out_x, out_y), 1, 0)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;Linear(x,q,r)\sim&space;Materialize(out_x),Linear(y,s,t)\sim&space;Materialize(out_y),out\sim&space;Linear(TermAdd(out_x,out_y),1,0))
+
+RHS:
+![Linear(x, q, r) ~ wire_1](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire_1)
+![Materialize(out_x) ~ wire_1](https://latex.codecogs.com/svg.image?Materialize(out_x)\sim&space;wire_1)
+![q*x + r = wire_1](https://latex.codecogs.com/svg.image?q*x&plus;r=wire_1)
+![out_x = wire_1](https://latex.codecogs.com/svg.image?out_x=wire_1)
+![Linear(y, s, t) ~ wire_2](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire_2)
+![Materialize(out_y) ~ wire_2](https://latex.codecogs.com/svg.image?Materialize(out_y)\sim&space;wire_2)
+![s*y + t = wire_2](https://latex.codecogs.com/svg.image?s*y&plus;t=wire_2)
+![out_y = wire_2](https://latex.codecogs.com/svg.image?out_y=wire_2)
+![out = 1*TermAdd(out_x, out_y) + 0](https://latex.codecogs.com/svg.image?out=1*TermAdd(out_x,out_y)&plus;0)
+Because `TermAdd(a, b)` is defined as `a+b`:
+![out = 1*(q*x + r + s*y + t) + 0](https://latex.codecogs.com/svg.image?out=1*(q*x&plus;r&plus;s*y&plus;t)&plus;0)
+
+EQUIVALENCE:
+![q*x + (r + s*y + t) = 1*(q*x + r + s*y + t) + 0 => q*x + r + s*y + t = q*x + r + s*y + t](https://latex.codecogs.com/svg.image?q*x&plus;(r&plus;s*y&plus;t)=1*(q*x&plus;r&plus;s*y&plus;t)&plus;0\Rightarrow&space;q*x&plus;r&plus;s*y&plus;t=q*x&plus;r&plus;s*y&plus;t)
+
+#### Concrete >< AddCheckLinear
+![Concrete(j) >< AddCheckLinear(out, x, q, r) => out ~ Linear(x, q, r + j)](https://latex.codecogs.com/svg.image?Concrete(j)><AddCheckLinear(out,x,q,r)\Rightarrow&space;out\sim&space;Linear(x,q,r&plus;j))
+
+LHS:
+![Concrete(j) ~ wire](https://latex.codecogs.com/svg.image?Concrete(j)\sim&space;wire)
+![AddCheckLinear(out, x, q, r) ~ wire](https://latex.codecogs.com/svg.image?AddCheckLinear(out,x,q,r)\sim&space;wire)
+![j = wire](https://latex.codecogs.com/svg.image?j=wire)
+![out = q*x + (r + wire)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;wire))
+![out = q*x + (r + j)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;j))
+
+RHS:
+![out = q*x + (r + j)](https://latex.codecogs.com/svg.image?out=q*x&plus;(r&plus;j))
+
+EQUIVALENCE:
+![q*x + (r + j) = q*x + (r + j)](https://latex.codecogs.com/svg.image?q*x&plus;(r&plus;j)=q*x&plus;(r&plus;j))
+
+#### Linear >< AddCheckConcrete
+![Linear(y, s, t) >< AddCheckConcrete(out, k) => out ~ Linear(y, s, t + k)](https://latex.codecogs.com/svg.image?Linear(y,s,t)><AddCheckConcrete(out,k)\Rightarrow&space;out\sim&space;Linear(y,s,t&plus;k))
+
+LHS:
+![Linear(y, s, t) ~ wire](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire)
+![AddCheckConcrete(out, k) ~ wire](https://latex.codecogs.com/svg.image?AddCheckConcrete(out,k)\sim&space;wire)
+![s*y + t = wire](https://latex.codecogs.com/svg.image?s*y&plus;t=wire)
+![out = k + wire](https://latex.codecogs.com/svg.image?out=k&plus;wire)
+![out = k + s*y + t](https://latex.codecogs.com/svg.image?out=k&plus;s*y&plus;t)
+
+RHS:
+![out = s*y + (t + k)](https://latex.codecogs.com/svg.image?out=s*y&plus;(t&plus;k))
+
+EQUIVALENCE:
+![k + s*y + t = s*y + (t + k) => s*y + (t + k) = s*y + (t + k)](https://latex.codecogs.com/svg.image?k&plus;s*y&plus;t=s*y&plus;(t&plus;k)\Rightarrow&space;s*y&plus;(t&plus;k)=s*y&plus;(t&plus;k))
+
+#### Concrete >< AddCheckConcrete
+![Concrete(j) >< AddCheckConcrete(out, k) => (1), (2)](https://latex.codecogs.com/svg.image?Concrete(j)><AddCheckConcrete(out,k)\Rightarrow&space;(1),(2))
+
+LHS:
+![Concrete(j) ~ wire](https://latex.codecogs.com/svg.image?Concrete(j)\sim&space;wire)
+![AddCheckConcrete(out, k) ~ wire](https://latex.codecogs.com/svg.image?AddCheckConcrete(out,k)\sim&space;wire)
+![j = wire](https://latex.codecogs.com/svg.image?j=wire)
+![out = k + wire](https://latex.codecogs.com/svg.image?out=k&plus;wire)
+![out = k + j](https://latex.codecogs.com/svg.image?out=k&plus;j)
+
+##### Case 1:
+![j = 0 => out ~ Concrete(k)](https://latex.codecogs.com/svg.image?j=0\Rightarrow&space;out\sim&space;Concrete(k))
+
+RHS:
+![out = k](https://latex.codecogs.com/svg.image?out=k)
+
+EQUIVALENCE:
+![k + 0 = k => k = k](https://latex.codecogs.com/svg.image?k&plus;0=k\Rightarrow&space;k=k)
+
+##### Case 2:
+![otherwise => out ~ Concrete(k + j)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;out\sim&space;Concrete(k&plus;j))
+
+RHS:
+![out = k + j](https://latex.codecogs.com/svg.image?out=k&plus;j)
+
+EQUIVALENCE:
+![k + j = k + j](https://latex.codecogs.com/svg.image?k&plus;j=k&plus;j)
+
+### Mul
+#### Linear >< Mul
+![Linear(x, q, r) >< Mul(out, b) => b ~ MulCheckLinear(out, x, q, r)](https://latex.codecogs.com/svg.image?Linear(x,q,r)><Mul(out,b)\Rightarrow&space;b\sim&space;MulCheckLinear(out,x,q,r))
+
+LHS:
+![Linear(x, q, r) ~ wire](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire)
+![Mul(out, b) ~ wire](https://latex.codecogs.com/svg.image?Mul(out,b)\sim&space;wire)
+![q*x + r = wire](https://latex.codecogs.com/svg.image?q*x&plus;r=wire)
+![out = wire * b](https://latex.codecogs.com/svg.image?out=wire*b)
+![out = (q*x + r) * b](https://latex.codecogs.com/svg.image?out=(q*x&plus;r)*b)
+
+RHS:
+![out = q*b*x + r*b](https://latex.codecogs.com/svg.image?out=q*b*x&plus;r*b)
+
+EQUIVALENCE:
+![(q*x + r) * b = q*b*x + r*b => q*b*x + r*b = q*b*x + r*b](https://latex.codecogs.com/svg.image?(q*x&plus;r)*b=q*b*x&plus;r*b\Rightarrow&space;q*b*x&plus;r*b=q*b*x&plus;r*b)
+
+#### Concrete >< Mul
+![Concrete(k) >< Mul(out, b) => (1), (2), (3)](https://latex.codecogs.com/svg.image?Concrete(k)><Mul(out,b)\Rightarrow&space;(1),(2),(3))
+
+LHS:
+![Concrete(k) ~ wire](https://latex.codecogs.com/svg.image?Concrete(k)\sim&space;wire)
+![Mul(out, b) ~ wire](https://latex.codecogs.com/svg.image?Mul(out,b)\sim&space;wire)
+![k = wire](https://latex.codecogs.com/svg.image?k=wire)
+![out = wire * b](https://latex.codecogs.com/svg.image?out=wire*b)
+![out = k * b](https://latex.codecogs.com/svg.image?out=k*b)
+
+##### Case 1:
+![k = 0 => out ~ Concrete(0), b ~ Eraser](https://latex.codecogs.com/svg.image?k=0\Rightarrow&space;out\sim&space;Concrete(0),b\sim&space;Eraser)
+
+RHS:
+![out = 0](https://latex.codecogs.com/svg.image?out=0)
+
+EQUIVALENCE:
+![0 * b = 0 => 0 = 0](https://latex.codecogs.com/svg.image?0*b=0\Rightarrow&space;0=0)
+
+##### Case 2:
+![k = 1 => out ~ b](https://latex.codecogs.com/svg.image?k=1\Rightarrow&space;out\sim&space;b)
+
+RHS:
+![out = b](https://latex.codecogs.com/svg.image?out=b)
+
+EQUIVALENCE:
+![1 * b = b => b = b](https://latex.codecogs.com/svg.image?1*b=b\Rightarrow&space;b=b)
+
+##### Case 3:
+![otherwise => b ~ MulCheckConcrete(out, k)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;b\sim&space;MulCheckConcrete(out,k))
+
+RHS:
+![out = k * b](https://latex.codecogs.com/svg.image?out=k*b)
+
+EQUIVALENCE:
+![k * b = k * b](https://latex.codecogs.com/svg.image?k*b=k*b)
+
+#### Linear >< MulCheckLinear
+![Linear(y, s, t) >< MulCheckLinear(out, x, q, r) => (1), (2)](https://latex.codecogs.com/svg.image?Linear(y,s,t)><MulCheckLinear(out,x,q,r)\Rightarrow&space;(1),(2))
+
+LHS:
+![Linear(y, s, t) ~ wire](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire)
+![MulCheckLinear(out, x, q, r) ~ wire](https://latex.codecogs.com/svg.image?MulCheckLinear(out,x,q,r)\sim&space;wire)
+![s*y + t = wire](https://latex.codecogs.com/svg.image?s*y&plus;t=wire)
+![out = q*wire*x + r*wire](https://latex.codecogs.com/svg.image?out=q*wire*x&plus;r*wire)
+![out = q*(s*y + t)*x + r*(s*y + t)](https://latex.codecogs.com/svg.image?out=q*(s*y&plus;t)*x&plus;r*(s*y&plus;t))
+
+##### Case 1:
+![(q,r = 0) or (s,t = 0) => x ~ Eraser, y ~ Eraser, out ~ Concrete(0)](https://latex.codecogs.com/svg.image?(q,r=0)\lor(s,t=0)\Rightarrow&space;x\sim&space;Eraser,y\sim&space;Eraser,out\sim&space;Concrete(0))
+
+RHS:
+![out = 0](https://latex.codecogs.com/svg.image?out=0)
+
+EQUIVALENCE:
+![0*(s*y + t)*x + 0*(s*y + t) = 0 => 0 = 0](https://latex.codecogs.com/svg.image?0*(s*y&plus;t)*x&plus;0*(s*y&plus;t)=0\Rightarrow&space;0=0)
+![or](https://latex.codecogs.com/svg.image?\lor)
+![q*(0*y + 0)*x + r*(0*y + 0) = 0 => 0 = 0](https://latex.codecogs.com/svg.image?q*(0*y&plus;0)*x&plus;r*(0*y&plus;0)=0\Rightarrow&space;0=0)
+
+##### Case 2:
+![otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermMul(out_x, out_y), 1, 0)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;Linear(x,q,r)\sim&space;Materialize(out_x),Linear(y,s,t)\sim&space;Materialize(out_y),out\sim&space;Linear(TermMul(out_x,out_y),1,0))
+
+RHS:
+![Linear(x, q, r) ~ wire_1](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire_1)
+![Materialize(out_x) ~ wire_1](https://latex.codecogs.com/svg.image?Materialize(out_x)\sim&space;wire_1)
+![q*x + r = wire_1](https://latex.codecogs.com/svg.image?q*x&plus;r=wire_1)
+![out_x = wire_1](https://latex.codecogs.com/svg.image?out_x=wire_1)
+![Linear(y, s, t) ~ wire_2](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire_2)
+![Materialize(out_y) ~ wire_2](https://latex.codecogs.com/svg.image?Materialize(out_y)\sim&space;wire_2)
+![s*y + t = wire_2](https://latex.codecogs.com/svg.image?s*y&plus;t=wire_2)
+![out_y = wire_2](https://latex.codecogs.com/svg.image?out_y=wire_2)
+![out = 1*TermMul(out_x, out_y) + 0](https://latex.codecogs.com/svg.image?out=1*TermMul(out_x,out_y)&plus;0)
+Because `TermMul(a, b)` is defined as `a*b`:
+![out = 1*(q*x + r)*(s*y + t) + 0](https://latex.codecogs.com/svg.image?out=1*(q*x&plus;r)*(s*y&plus;t)&plus;0)
+
+EQUIVALENCE:
+![q*(s*y + t)*x + r*(s*y + t) = 1*(q*x + r)*(s*y + t) => 
+q*(s*y + t)*x + r*(s*y + t) = (q*x + r)*(s*y + t) => 
+q*(s*y + t)*x + r*(s*y + t) = q*(s*y + t)*x + r*(s*y + t)](https://latex.codecogs.com/svg.image?q*(s*y&plus;t)*x&plus;r*(s*y&plus;t)=1*(q*x&plus;r)*(s*y&plus;t)\Rightarrow&space;q*(s*y&plus;t)*x&plus;r*(s*y&plus;t)=(q*x&plus;r)*(s*y&plus;t)\Rightarrow&space;q*(s*y&plus;t)*x&plus;r*(s*y&plus;t)=q*(s*y&plus;t)*x&plus;r*(s*y&plus;t))
+
+
+#### Concrete >< MulCheckLinear
+![Concrete(j) >< MulCheckLinear(out, x, q, r) => out ~ Linear(x, q * j, r * j)](https://latex.codecogs.com/svg.image?Concrete(j)><MulCheckLinear(out,x,q,r)\Rightarrow&space;out\sim&space;Linear(x,q*j,r*j))
+
+LHS:
+![Concrete(j) ~ wire](https://latex.codecogs.com/svg.image?Concrete(j)\sim&space;wire)
+![MulCheckLinear(out, x, q, r) ~ wire](https://latex.codecogs.com/svg.image?MulCheckLinear(out,x,q,r)\sim&space;wire)
+![j = wire](https://latex.codecogs.com/svg.image?j=wire)
+![out = q*wire*x + r*wire](https://latex.codecogs.com/svg.image?out=q*wire*x&plus;r*wire)
+![out = q*j*x + r*j](https://latex.codecogs.com/svg.image?out=q*j*x&plus;r*j)
+
+RHS:
+![out = q*j*x + r*j](https://latex.codecogs.com/svg.image?out=q*j*x&plus;r*j)
+
+EQUIVALENCE:
+![q*j*x + r*j = q*j*x + r*j](https://latex.codecogs.com/svg.image?q*j*x&plus;r*j=q*j*x&plus;r*j)
+
+#### Linear >< MulCheckConcrete
+![Linear(y, s, t) >< MulCheckConcrete(out, k) => out ~ Linear(y, s * k, t * k)](https://latex.codecogs.com/svg.image?Linear(y,s,t)><MulCheckConcrete(out,k)\Rightarrow&space;out\sim&space;Linear(y,s*k,t*k))
+
+LHS:
+![Linear(y, s, t) ~ wire](https://latex.codecogs.com/svg.image?Linear(y,s,t)\sim&space;wire)
+![MulCheckConcrete(out, k) ~ wire](https://latex.codecogs.com/svg.image?MulCheckConcrete(out,k)\sim&space;wire)
+![s*y + t = wire](https://latex.codecogs.com/svg.image?s*y&plus;t=wire)
+![out = k * wire](https://latex.codecogs.com/svg.image?out=k*wire)
+![out = k * (s*y + t)](https://latex.codecogs.com/svg.image?out=k*(s*y&plus;t))
+
+RHS:
+![out = s*k*y + t*k](https://latex.codecogs.com/svg.image?out=s*k*y&plus;t*k)
+
+EQUIVALENCE:
+![k * (s*y + t) = s*k*y + t*k => s*k*y + t*k = s*k*y + t*k](https://latex.codecogs.com/svg.image?k*(s*y&plus;t)=s*k*y&plus;t*k\Rightarrow&space;s*k*y&plus;t*k=s*k*y&plus;t*k)
+
+
+#### Concrete >< MulCheckConcrete
+![Concrete(j) >< MulCheckConcrete(out, k) => (1), (2), (3)](https://latex.codecogs.com/svg.image?Concrete(j)><MulCheckConcrete(out,k)\Rightarrow&space;(1),(2),(3))
+
+LHS:
+![Concrete(j) ~ wire](https://latex.codecogs.com/svg.image?Concrete(j)\sim&space;wire)
+![MulCheckConcrete(out, k) ~ wire](https://latex.codecogs.com/svg.image?MulCheckConcrete(out,k)\sim&space;wire)
+![j = wire](https://latex.codecogs.com/svg.image?j=wire)
+![out = k * wire](https://latex.codecogs.com/svg.image?out=k*wire)
+![out = k * j](https://latex.codecogs.com/svg.image?out=k*j)
+
+##### Case 1:
+![j = 0 => out ~ Concrete(0)](https://latex.codecogs.com/svg.image?j=0\Rightarrow&space;out\sim&space;Concrete(0))
+
+RHS:
+![out = 0](https://latex.codecogs.com/svg.image?out=0)
+
+EQUIVALENCE:
+![k * 0 = 0 => 0 = 0](https://latex.codecogs.com/svg.image?k*0=0\Rightarrow&space;0=0)
+
+##### Case 2:
+![j = 1 => out ~ Concrete(k)](https://latex.codecogs.com/svg.image?j=1\Rightarrow&space;out\sim&space;Concrete(k))
+
+RHS:
+![out = k](https://latex.codecogs.com/svg.image?out=k)
+
+EQUIVALENCE:
+![k * 1 = k => k = k](https://latex.codecogs.com/svg.image?k*1=k\Rightarrow&space;k=k)
+
+##### Case 3:
+![otherwise => out ~ Concrete(k * j)](https://latex.codecogs.com/svg.image?otherwise\Rightarrow&space;out\sim&space;Concrete(k*j))
+
+RHS:
+![out = k * j](https://latex.codecogs.com/svg.image?out=k*j)
+
+EQUIVALENCE:
+![k * j = k * j](https://latex.codecogs.com/svg.image?k*j=k*j)
+
+### ReLU
+#### Linear >< ReLU
+![Linear(x, q, r) >< ReLU(out) => Linear(x, q, r) ~ Materialize(out_x), out ~ Linear(TermReLU(out_x), 1, 0)](https://latex.codecogs.com/svg.image?Linear(x,q,r)><ReLU(out)\Rightarrow&space;Linear(x,q,r)\sim&space;Materialize(out_x),out\sim&space;Linear(TermReLU(out_x),1,0))
+
+LHS:
+![Linear(x, q, r) ~ wire](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire)
+![ReLU(out) ~ wire](https://latex.codecogs.com/svg.image?ReLU(out)\sim&space;wire)
+![q*x + r = wire](https://latex.codecogs.com/svg.image?q*x&plus;r=wire)
+![out = IF wire > 0 THEN wire ELSE 0](https://latex.codecogs.com/svg.image?out=IF\;wire>0\;THEN\;wire\;ELSE\;0)
+![out = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0](https://latex.codecogs.com/svg.image?out=IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0)
+
+RHS:
+![Linear(x, q, r) ~ wire](https://latex.codecogs.com/svg.image?Linear(x,q,r)\sim&space;wire)
+![Materialize(out_x) ~ wire](https://latex.codecogs.com/svg.image?Materialize(out_x)\sim&space;wire)
+![q*x + r = wire](https://latex.codecogs.com/svg.image?q*x&plus;r=wire)
+![out_x = wire](https://latex.codecogs.com/svg.image?out_x=wire)
+![out = 1*TermReLU(out_x) + 0](https://latex.codecogs.com/svg.image?out=1*TermReLU(out_x)&plus;0)
+Because `TermReLU(x)` is defined as `z3.If(x > 0, x, 0)`:
+![out = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0](https://latex.codecogs.com/svg.image?out=1*(IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0)&plus;0)
+
+EQUIVALENCE:
+![IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0 =>
+IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0](https://latex.codecogs.com/svg.image?IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0=1*(IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0)&plus;0\Rightarrow&space;IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0=IF\;(q*x&plus;r)>0\;THEN\;(q*x&plus;r)\;ELSE\;0)
+
+
+#### Concrete >< ReLU
+![Concrete(k) >< ReLU(out) => (1), (2)](https://latex.codecogs.com/svg.image?Concrete(k)><ReLU(out)\Rightarrow&space;(1),(2))
+
+LHS:
+![Concrete(k) ~ wire](https://latex.codecogs.com/svg.image?Concrete(k)\sim&space;wire)
+![ReLU(out) ~ wire](https://latex.codecogs.com/svg.image?ReLU(out)\sim&space;wire)
+![k = wire](https://latex.codecogs.com/svg.image?k=wire)
+![out = IF wire > 0 THEN wire ELSE 0](https://latex.codecogs.com/svg.image?out=IF\;wire>0\;THEN\;wire\;ELSE\;0)
+![out = IF k > 0 THEN k ELSE 0](https://latex.codecogs.com/svg.image?out=IF\;k>0\;THEN\;k\;ELSE\;0)
+
+##### Case 1:
+![k > 0 => out ~ Concrete(k)](https://latex.codecogs.com/svg.image?k>0\Rightarrow&space;out\sim&space;Concrete(k))
+
+RHS:
+![out = k](https://latex.codecogs.com/svg.image?out=k)
+
+EQUIVALENCE:
+![IF true THEN k ELSE 0 = k => k = k](https://latex.codecogs.com/svg.image?IF\;true\;THEN\;k\;ELSE\;0=k\Rightarrow&space;k=k)
+
+##### Case 2:
+![k <= 0 => out ~ Concrete(0)](https://latex.codecogs.com/svg.image?k\leq&space;0\Rightarrow&space;out\sim&space;Concrete(0))
+
+RHS:
+![out = 0](https://latex.codecogs.com/svg.image?out=0)
+
+EQUIVALENCE:
+![IF false THEN k ELSE 0 = 0 => 0 = 0](https://latex.codecogs.com/svg.image?IF\;false\;THEN\;k\;ELSE\;0=0\Rightarrow&space;0=0)
+
+## Soundness of Reduction 
+Let ![IN_0](https://latex.codecogs.com/svg.image?\mathrm{IN_0}) be the Interaction Net translated from a Neural Network ![NN](https://latex.codecogs.com/svg.image?\inline&space;\mathrm{NN}). Let ![IN_n](https://latex.codecogs.com/svg.image?\inline&space;\mathrm{IN_n}) be the state of the net
+after ![n](https://latex.codecogs.com/svg.image?\inline&space;n) reduction steps. Then ![forall n in N, [IN_n] = [NN]](https://latex.codecogs.com/svg.image?\inline&space;\forall&space;n\in\mathbb{N},[\mathrm{IN_n}]=[\mathrm{NN}]).
+
+### Proof by Induction
+- Base Case (![n = 0](https://latex.codecogs.com/svg.image?\inline&space;n=0)): By the [Soundness of Translation](#soundness-of-translation), the initial net ![IN_0](https://latex.codecogs.com/svg.image?\mathrm{IN_0}) is constructed such that
+its semantics ![[IN_0]](https://latex.codecogs.com/svg.image?\inline&space;[\mathrm{IN_0}]) exactly match the mathematical definition of the ONNX nodes in ![NN](https://latex.codecogs.com/svg.image?\inline&space;\mathrm{NN}).
+- Induction Step (![n -> n + 1](https://latex.codecogs.com/svg.image?\inline&space;n\to&space;n&plus;1)): Assume ![[IN_n] = [NN]](https://latex.codecogs.com/svg.image?\inline&space;[\mathrm{IN_n}]=[\mathrm{NN}]). If ![IN_n](https://latex.codecogs.com/svg.image?\inline&space;\mathrm{IN_n}) is in normal form, the proof is complete.
+Otherwise, there exists an active pair ![A](https://latex.codecogs.com/svg.image?\inline&space;A) that reduces ![IN_n to IN_{n+1}](https://latex.codecogs.com/svg.image?\inline&space;\mathrm{IN_n}\Rightarrow&space;\mathrm{IN_{n&plus;1}}).
+By the [Soundness of Interaction Rules](#soundness-of-interaction-rules), the mathematical definition is preserved after any reduction step,
+it follows that ![[IN_{n+1}] = [IN_n]](https://latex.codecogs.com/svg.image?\inline&space;[\mathrm{IN_{n&plus;1}}]=[\mathrm{IN_n}]). By the inductive hypothesis, ![[IN_{n+1}] = [NN]](https://latex.codecogs.com/svg.image?\inline&space;[\mathrm{IN_{n&plus;1}}]=[\mathrm{NN}]).
+
+  By the principle of mathematical induction, the Interaction Net remains semantically equivalent to the original 
+  Neural Network at every step of the reduction process.
+
+Since Interaction Nets are confluent, the reduced mathematical expression is unique regardless
+of order in which rules are applied.
diff --git a/proof.norg b/proof.norg
index 8ae8490..21f1fcd 100644
--- a/proof.norg
+++ b/proof.norg
@@ -4,63 +4,64 @@ description:
 authors: ericmarin
 categories: research
 created: 2026-03-16T11:34:52
-updated: 2026-03-25T08:18:04
+updated: 2026-03-28T17:37:48
 version: 1.1.1
 @end
 
-* Mathematical Definitions
-	- Linear(x, q, r) ~ out => out = q*x + r
-	- Concrete(k) ~ out => out = k
-	- Add(out, b) ~ a => out = a + b
-	- AddCheckLinear(out, x, q, r) ~ b => out = q*x + (r + b)
-	- AddCheckConcrete(out, k) ~ b => out = k + b
-	- Mul(out, b) ~ a => out = a * b
-	- MulCheckLinear(out, x, q, r) ~ b => out = q*b*x + r*b
-	- MulCheckConcrete(out, k) ~ b => out = k*b
-	- ReLU(out) ~ x => out = IF (x > 0) THEN x ELSE 0
-	- Materialize(out) ~ x => out = x
-
-* Proof for translation from Pytorch representation to Interaction Net graph
-** ReLU
-	 ONNX ReLU node is defined as:
-	 /Y = X if X > 0 else 0/
-
-	 The translation defines the interactions:
-	 /x_i ~ ReLU(y_i)/
-
-	 By definition this interaction is equal to:
-	 /y_i = IF (x_i >0) THEN x_i ELSE 0/
-
-** Gemm
-	 ONNX Gemm node is defined as:
-	 Y = alpha * A * B + beta * C
-
-	 The translation defines the interactions:
-	 /a_i ~ Mul(v_i, Concrete(alpha * b_i))/
-	 /Add(...(Add(y_i, v_1), ...), v_n) ~ Concrete(beta * c_i)/
-
-	 By definition this interaction is equal to:
-	 /v_i = alpha * a_i * b_i/
-	 /y_i = v_1 + v_2 + ... + v_n + beta * c_i/
-
-	 By grouping the operations we get:
-	 /Y = alpha * A * B + beta * C/
-
-** Flatten
-	 Just identity mapping.
-
-** MatMul
-	 Equal to Gemm with alpha=1 and beta=0.
-
-** Reshape
-	 Just identity mapping.
-
-* Proof for the Interaction Rules
-** Materialize
-	 The Materialize agent transforms a Linear agent into a tree of explicit mathematical operations
-	 that are used as final representation for the solver.
-	 In the Python module the terms are defined as:
-	 @code python
+* Soundness Proof
+** Mathematical Definitions
+	 - $Linear(x, q, r) ~ out => out = q*x + r$
+	 - $Concrete(k) ~ out => out = k$
+	 - $Add(out, b) ~ a => out = a + b$
+	 - $AddCheckLinear(out, x, q, r) ~ b => out = q*x + (r + b)$
+	 - $AddCheckConcrete(out, k) ~ b => out = k + b$
+	 - $Mul(out, b) ~ a => out = a * b$
+	 - $MulCheckLinear(out, x, q, r) ~ b => out = q*b*x + r*b$
+	 - $MulCheckConcrete(out, k) ~ b => out = k*b$
+	 - $ReLU(out) ~ x => out = IF (x > 0) THEN x ELSE 0$
+	 - $Materialize(out) ~ x => out = x$
+
+** Soundness of Translation
+*** ReLU
+		ONNX ReLU node is defined as:
+		$Y = X if X > 0 else 0$
+
+		The translation defines the interactions:
+		$x_i ~ ReLU(y_i)$
+
+		By definition this interaction is equal to:
+		$y_i = IF (x_i > 0) THEN x_i ELSE 0$
+
+*** Gemm
+		ONNX Gemm node is defined as:
+		$Y = alpha * A * B + beta * C$
+
+		The translation defines the interactions:
+		$a_i ~ Mul(v_i, Concrete(alpha * b_i))$
+		$Add(...(Add(y_i, v_1), ...), v_n) ~ Concrete(beta * c_i)$
+
+		By definition this interaction is equal to:
+		$v_i = alpha * a_i * b_i$
+		$y_i = v_1 + v_2 + ... + v_n + beta * c_i$
+
+		By grouping the operations we get:
+		$Y = alpha * A * B + beta * C$
+
+*** Flatten
+		Just identity mapping.
+
+*** MatMul
+		Equal to Gemm with $alpha=1$ and $beta=0$.
+
+*** Reshape
+		Just identity mapping.
+
+** Soundness of Interaction Rules
+*** Materialize
+		The Materialize agent transforms a Linear agent into a tree of explicit mathematical operations
+		that are used as final representation for the solver.
+		In the Python module the terms are defined as:
+		@code python
 		def TermAdd(a, b):
 			return a + b  
 		def TermMul(a, b):
@@ -68,396 +69,415 @@ version: 1.1.1
 		def TermReLU(x):
 			return z3.If(x > 0, x, 0)
 	@end
-*** Linear(x, q, r) >< Materialize(out) => (1), (2), (3), (4), (5)
-		LHS:
-		Linear(x, q, r) ~ wire
-		Materialize(out) ~ wire
-		q*x + r = wire
-		out = wire
-		out = q*x + r
-
-		$$ Case 1: q = 0 => out ~ Concrete(r), x ~ Eraser
-		RHS:
-		out = r
-
-		EQUIVALENCE:
-		0*x + r = r => r = r
-		$$
-
-		$$ Case 2: q = 1, r = 0 => out ~ x
-		RHS:
-		x = out
-		out = x
-
-		EQUIVALENCE:
-		1*x + 0 = x => x = x
-		$$
-
-		$$ Case 3: q = 1 => out ~ TermAdd(x, Concrete(r))
-		RHS:
-		out = x + r
-
-		EQUIVALENCE:
-		1*x + r = x + r => x + r = x + r
-		$$
-
-		$$ Case 4: r = 0 => out ~ TermMul(Concrete(q), x)
-		RHS:
-		out = q*x
-
-		EQUIVALENCE:
-		q*x + 0 = q*x => q*x = q*x
-		$$
-
-		$$ Case 5: otherwise => out ~ TermAdd(TermMul(Concrete(q), x), Concrete(r))
-		RHS:
-		out = q*x + r
-
-		EQUIVALENCE:
-		q*x + r = q*x + r
-		$$
-
-*** Concrete(k) >< Materialize(out) => out ~ Concrete(k)
-		LHS:
-		Concrete(k) ~ wire
-		Materialize(out) ~ wire
-		k = wire
-		out = wire
-		out = k
-
-		RHS:
-		out = k
-
-		EQUIVALENCE:
-		k = k
-
-** Add
-*** Linear(x, q, r) >< Add(out, b) => b ~ AddCheckLinear(out, x, q, r)
-		LHS:
-		Linear(x, q, r) ~ wire
-		Add(out, b) ~ wire
-		q*x + r = wire
-		out = wire + b
-		out = q*x + r + b
-
-		RHS:
-		out = q*x + (r + b)
-
-		EQUIVALENCE:
-		q*x + r + b = q*x + (r + b) => q*x + (r + b) = q*x + (r + b)
-
-*** Concrete(k) >< Add(out, b) => (1), (2)
-		LHS:
-		Concrete(k) ~ wire
-		Add(out, b) ~ wire
-		k = wire
-		out = wire + b
-		out = k + b
-
-		$$ Case 1: k = 0 => out ~ b
-		RHS:
-		out = b
-
-		EQUIVALENCE:
-		0 + b = b => b = b
-		$$
-
-		$$ Case 2: otherwise => b ~ AddCheckConcrete(out, k)
-		RHS:
-		out = k + b
-
-		EQUIVALENCE:
-		k + b = k + b
-		$$
-
-*** Linear(y, s, t) >< AddCheckLinear(out, x, q, r) => (1), (2), (3), (4)
-		LHS:
-		Linear(y, s, t) ~ wire
-		AddCheckLinear(out, x, q, r) ~ wire
-		s*y + t = wire
-		out = q*x + (r + wire)
-		out = q*x + (r + s*y + t)
-
-		$$ Case 1: q,r,s,t = 0 => out ~ Concrete(0), x ~ Eraser, y ~ Eraser
-		RHS:
-		out = 0
-
-		EQUIVALENCE:
-		0*x + (0 + 0*y + 0) = 0 => 0 = 0
-		$$
-
-		$$ Case 2: s,t = 0 => out ~ Linear(x, q, r), y ~ Eraser
-		RHS:
-		out = q*x + r
-
-		EQUIVALENCE:
-		q*x + (r + 0*y + 0) = q*x + r => q*x + r = q*x + r
-		$$
-
-		$$ Case 3: q, r = 0 => out ~ Linear(y, s, t), x ~ Eraser
-		RHS:
-		out = s*y + t
-
-		EQUIVALENCE:
-		0*x + (0 + s*y + t) = s*y + t => s*y + t = s*y + t
-		$$
-
-		$$ Case 4: otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermAdd(out_x, out_y), 1, 0)
-		RHS:
-		Linear(x, q, r) ~ wire1
-		Materialize(out_x) ~ wire1
-		q*x + r = wire1
-		out_x = wire1
-		Linear(y, s, t) ~ wire2
-		Materialize(out_y) ~ wire2
-		s*y + t = wire2
-		out_y = wire2
-		out = 1*TermAdd(out_x, out_y) + 0
-		Because TermAdd(a, b) is defined as "a+b":
-		out = 1*(q*x + r + s*y + t) + 0
-
-		EQUIVALENCE:
-		q*x + (r + s*y + t) = 1*(q*x + r + s*y + t) + 0 => q*x + r + s*y + t = q*x + r + s*y + t
-		$$
-
-*** Concrete(j) >< AddCheckLinear(out, x, q, r) => out ~ Linear(x, q, r + j)
-		LHS:
-		Concrete(j) ~ wire
-		AddCheckLinear(out, x, q, r) ~ wire
-		j = wire
-		out = q*x + (r + wire)
-		out = q*x + (r + j)
-
-		RHS:
-		out = q*x + (r + j)
-
-		EQUIVALENCE:
-		q*x + (r + j) = q*x + (r + j)
-
-*** Linear(y, s, t) >< AddCheckConcrete(out, k) => out ~ Linear(y, s, t + k)
-		LHS:
-		Linear(y, s, t) ~ wire
-		AddCheckConcrete(out, k) ~ wire
-		s*y + t = wire
-		out = k + wire
-		out = k + s*y + t
-
-		RHS:
-		out = s*y + (t + k)
-
-		EQUIVALENCE:
-		k + s*y + t = s*y + (t + k) => s*y + (t + k) = s*y + (t + k)
-
-*** Concrete(j) >< AddCheckConcrete(out, k) => (1), (2)
-		LHS:
-		Concrete(j) ~ wire
-		AddCheckConcrete(out, k) ~ wire
-		j = wire
-		out = k + wire
-		out = k + j
-
-		$$ Case 1: j = 0 => out ~ Concrete(k)
-		RHS:
-		out = k
-
-		EQUIVALENCE:
-		k + 0 = k => k = k
-		$$
-
-		$$ Case 2: otherwise => out ~ Concrete(k + j)
-		RHS:
-		out = k + j
-
-		EQUIVALENCE:
-		k + j = k + j
-		$$
-
-** Mul
-*** Linear(x, q, r) >< Mul(out, b) => b ~ MulCheckLinear(out, x, q, r)
-		LHS:
-		Linear(x, q, r) ~ wire
-		Mul(out, b) ~ wire
-		q*x + r = wire
-		out = wire * b
-		out = (q*x + r) * b
-
-		RHS:
-		out = q*b*x + r*b
-
-		EQUIVALENCE:
-		(q*x + r) * b = q*b*x + r*b => q*b*x + r*b = q*b*x + r*b
-
-*** Concrete(k) >< Mul(out, b) => (1), (2), (3)
-		LHS:
-		Concrete(k) ~ wire
-		Mul(out, b) ~ wire
-		k = wire
-		out = wire * b
-		out = k * b
-
-		$$ Case 1: k = 0 => out ~ Concrete(0), b ~ Eraser
-		RHS:
-		out = 0
-
-		EQUIVALENCE:
-		0 * b = 0 => 0 = 0
-		$$
-
-		$$ Case 2: k = 1 => out ~ b
-		RHS:
-		out = b
-
-		EQUIVALENCE:
-		1 * b = b => b = b
-		$$
-
-		$$ Case 3: otherwise => b ~ MulCheckConcrete(out, k)
-		RHS:
-		out = k * b
-
-		EQUIVALENCE:
-		k * b = k * b
-		$$
-
-*** Linear(y, s, t) >< MulCheckLinear(out, x, q, r) => (1), (2)
-		LHS:
-		Linear(y, s, t) ~ wire
-		MulCheckLinear(out, x, q, r) ~ wire
-		s*y + t = wire
-		out = q*wire*x + r*wire
-		out = q\*(s*y + t)\*x + r*(s*y + t)
-
-		$$ Case 1: (q,r = 0) or (s,t = 0) => x ~ Eraser, y ~ Eraser, out ~ Concrete(0)
-		RHS:
-		out = 0
-
-		EQUIVALENCE:
-		0\*(s*y + t)\*x + 0*(s*y + t) = 0 => 0 = 0
-		or
-		q\*(0*y + 0)\*x + r*(0*y + 0) = 0 => 0 = 0
-		$$
-
-		$$ Case 2: otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermMul(out_x, out_y), 1, 0)
-		RHS:
-		Linear(x, q, r) ~ wire1
-		Materialize(out_x) ~ wire1
-		q*x + r = wire1
-		out_x = wire1
-		Linear(y, s, t) ~ wire2
-		Materialize(out_y) ~ wire2
-		s*y + t = wire2
-		out_y = wire2
-		out = 1*TermMul(out_x, out_y) + 0
-		Because TermMul(a, b) is defined as "a*b":
-		out = 1*(q*x + r)*(s*y + t) + 0
-
-		EQUIVALENCE:
-		q*(s*y + t)\*x + r*(s*y + t) = 1*(q*x + r)\*(s*y + t) => 
-		q\*(s*y + t)\*x + r*(s*y + t) = (q*x + r)\*(s*y + t) => 
-		q\*(s*y + t)\*x + r*(s*y + t) = q\*(s*y + t)\*x + r*(s*y + t)
-		$$
-
-*** Concrete(j) >< MulCheckLinear(out, x, q, r) => out ~ Linear(x, q * j, r * j)
-		LHS:
-		Concrete(j) ~ wire
-		MulCheckLinear(out, x, q, r) ~ wire
-		j = wire
-		out = q*wire*x + r*wire
-		out = q*j*x + r*j
-
-		RHS:
-		out = q*j*x + r*j
-
-		EQUIVALENCE:
-		q*j*x + r*j = q*j*x + r*j
-
-*** Linear(y, s, t) >< MulCheckConcrete(out, k) => out ~ Linear(y, s * k, t * k)
-		LHS:
-		Linear(y, s, t) ~ wire
-		MulCheckConcrete(out, k) ~ wire
-		s*y + t = wire
-		out = k * wire
-		out = k * (s*y + t)
-
-		RHS:
-		out = s*k*y + t*k
-
-		EQUIVALENCE:
-		k * (s*y + t) = s*k*y + t*k => s*k*y + t*k = s*k*y + t*k
-
-*** Concrete(j) >< MulCheckConcrete(out, k) => (1), (2), (3)
-		LHS:
-		Concrete(j) ~ wire
-		MulCheckConcrete(out, k) ~ wire
-		j = wire
-		out = k * wire
-		out = k * j
-
-		$$ Case 1: j = 0 => out ~ Concrete(0)
-		RHS:
-		out = 0
-
-		EQUIVALENCE:
-		k * 0 = 0 => 0 = 0
-		$$
-
-		$$ Case 2: j = 1 => out ~ Concrete(k)
-		RHS:
-		out = k
-
-		EQUIVALENCE:
-		k * 1 = k => k = k
-		$$
-
-		$$ Case 3: otherwise => out ~ Concrete(k * j)
-		RHS:
-		out = k * j
-
-		EQUIVALENCE:
-		k * j = k * j
-
-** ReLU
-*** Linear(x, q, r) >< ReLU(out) => Linear(x, q, r) ~ Materialize(out_x), out ~ Linear(TermReLU(out_x), 1, 0)
-		LHS:
-		Linear(x, q, r) ~ wire
-		ReLU(out) ~ wire
-		q*x + r = wire
-		out = IF wire > 0 THEN wire ELSE 0
-		out = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0
-
-		RHS:
-		Linear(x, q, r) ~ wire
-		Materialize(out_x) ~ wire
-		q*x + r = wire
-		out_x = wire
-		out = 1*TermReLU(out_x) + 0
-		Because TermReLU(x) is defined as "z3.If(x > 0, x, 0)":
-		out = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0
-
-		EQUIVALENCE:
-		IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0 =>
-		IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0
-
-*** Concrete(k) >< ReLU(out) => (1), (2)
-		LHS:
-		Concrete(k) ~ wire
-		ReLU(out) ~ wire
-		k = wire
-		out = IF wire > 0 THEN wire ELSE 0
-		out = IF k > 0 THEN k ELSE 0
-
-		$$ Case 1: k > 0 => out ~ Concrete(k)
-		RHS:
-		out = k
-
-		EQUIVALENCE:
-		IF true THEN k ELSE 0 = k => k = k
-		$$
-
-		$$ Case 2: k <= 0 => out ~ Concrete(0)
-		RHS:
-		out = 0
-
-		EQUIVALENCE:
-		IF false THEN k ELSE 0 = 0 => 0 = 0
-		$$
+**** $Linear(x, q, r) >< Materialize(out) => (1), (2), (3), (4), (5)$
+		 LHS:
+		 $Linear(x, q, r) ~ wire$
+		 $Materialize(out) ~ wire$
+		 $q*x + r = wire$
+		 $out = wire$
+		 $out = q*x + r$
+
+		 $$ Case 1: $q = 0 => out ~ Concrete(r), x ~ Eraser$
+		 RHS:
+		 $out = r$
+
+		 EQUIVALENCE:
+		 $0*x + r = r => r = r$
+		 $$
+
+		 $$ Case 2: $q = 1, r = 0 => out ~ x$
+		 RHS:
+		 $x = out$
+		 $out = x$
+
+		 EQUIVALENCE:
+		 $1*x + 0 = x => x = x$
+		 $$
+
+		 $$ Case 3: $q = 1 => out ~ TermAdd(x, Concrete(r))$
+		 RHS:
+		 $out = x + r$
+
+		 EQUIVALENCE:
+		 $1*x + r = x + r => x + r = x + r$
+		 $$
+
+		 $$ Case 4: $r = 0 => out ~ TermMul(Concrete(q), x)$
+		 RHS:
+		 $out = q*x$
+
+		 EQUIVALENCE:
+		 $q*x + 0 = q*x => q*x = q*x$
+		 $$
+
+		 $$ Case 5: $otherwise => out ~ TermAdd(TermMul(Concrete(q), x), Concrete(r))$
+		 RHS:
+		 $out = q*x + r$
+
+		 EQUIVALENCE:
+		 $q*x + r = q*x + r$
+		 $$
+
+**** $Concrete(k) >< Materialize(out) => out ~ Concrete(k)$
+		 LHS:
+		 $Concrete(k) ~ wire$
+		 $Materialize(out) ~ wire$
+		 $k = wire$
+		 $out = wire$
+		 $out = k$
+
+		 RHS:
+		 $out = k$
+
+		 EQUIVALENCE:
+		 $k = k$
+
+*** Add
+**** $Linear(x, q, r) >< Add(out, b) => b ~ AddCheckLinear(out, x, q, r)$
+		 LHS:
+		 $Linear(x, q, r) ~ wire$
+		 $Add(out, b) ~ wire$
+		 $q*x + r = wire$
+		 $out = wire + b$
+		 $out = q*x + r + b$
+
+		 RHS:
+		 $out = q*x + (r + b)$
+
+		 EQUIVALENCE:
+		 $q*x + r + b = q*x + (r + b) => q*x + (r + b) = q*x + (r + b)$
+
+**** $Concrete(k) >< Add(out, b) => (1), (2)$
+		 LHS:
+		 $Concrete(k) ~ wire$
+		 $Add(out, b) ~ wire$
+		 $k = wire$
+		 $out = wire + b$
+		 $out = k + b$
+
+		 $$ Case 1: $k = 0 => out ~ b$
+		 RHS:
+		 $out = b$
+
+		 EQUIVALENCE:
+		 $0 + b = b => b = b$
+		 $$
+
+		 $$ Case 2: $otherwise => b ~ AddCheckConcrete(out, k)$
+		 RHS:
+		 $out = k + b$
+
+		 EQUIVALENCE:
+		 $k + b = k + b$
+		 $$
+
+**** $Linear(y, s, t) >< AddCheckLinear(out, x, q, r) => (1), (2), (3), (4)$
+		 LHS:
+		 $Linear(y, s, t) ~ wire$
+		 $AddCheckLinear(out, x, q, r) ~ wire$
+		 $s*y + t = wire$
+		 $out = q*x + (r + wire)$
+		 $out = q*x + (r + s*y + t)$
+
+		 $$ Case 1: $q,r,s,t = 0 => out ~ Concrete(0), x ~ Eraser, y ~ Eraser$
+		 RHS:
+		 $out = 0$
+
+		 EQUIVALENCE:
+		 $0*x + (0 + 0*y + 0) = 0 => 0 = 0$
+		 $$
+
+		 $$ Case 2: $s,t = 0 => out ~ Linear(x, q, r), y ~ Eraser$
+		 RHS:
+		 $out = q*x + r$
+
+		 EQUIVALENCE:
+		 $q*x + (r + 0*y + 0) = q*x + r => q*x + r = q*x + r$
+		 $$
+
+		 $$ Case 3: $q, r = 0 => out ~ Linear(y, s, t), x ~ Eraser$
+		 RHS:
+		 $out = s*y + t$
+
+		 EQUIVALENCE:
+		 $0*x + (0 + s*y + t) = s*y + t => s*y + t = s*y + t$
+		 $$
+
+		 $$ Case 4: $otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermAdd(out_x, out_y), 1, 0)$
+		 RHS:
+		 $Linear(x, q, r) ~ wire1$
+		 $Materialize(out_x) ~ wire1$
+		 $q*x + r = wire1$
+		 $out_x = wire1$
+		 $Linear(y, s, t) ~ wire2$
+		 $Materialize(out_y) ~ wire2$
+		 $s*y + t = wire2$
+		 $out_y = wire2$
+		 $out = 1*TermAdd(out_x, out_y) + 0$
+		 Because $TermAdd(a, b)$ is defined as "a+b$:
+		 $out = 1*(q*x + r + s*y + t) + 0$
+
+		 EQUIVALENCE:
+		 $q*x + (r + s*y + t) = 1*(q*x + r + s*y + t) + 0 => q*x + r + s*y + t = q*x + r + s*y + t
+		 $$
+
+**** $Concrete(j) >< AddCheckLinear(out, x, q, r) => out ~ Linear(x, q, r + j)$
+		 LHS:
+		 $Concrete(j) ~ wire$
+		 $AddCheckLinear(out, x, q, r) ~ wire$
+		 $j = wire$
+		 $out = q*x + (r + wire)$
+		 $out = q*x + (r + j)$
+
+		 RHS:
+		 $out = q*x + (r + j)$
+
+		 EQUIVALENCE:
+		 $q*x + (r + j) = q*x + (r + j)$
+
+**** $Linear(y, s, t) >< AddCheckConcrete(out, k) => out ~ Linear(y, s, t + k)$
+		 LHS:
+		 $Linear(y, s, t) ~ wire$
+		 $AddCheckConcrete(out, k) ~ wire$
+		 $s*y + t = wire$
+		 $out = k + wire$
+		 $out = k + s*y + t$
+
+		 RHS:
+		 $out = s*y + (t + k)$
+
+		 EQUIVALENCE:
+		 $k + s*y + t = s*y + (t + k) => s*y + (t + k) = s*y + (t + k)$
+
+**** $Concrete(j) >< AddCheckConcrete(out, k) => (1), (2)$
+		 LHS:
+		 $Concrete(j) ~ wire$
+		 $AddCheckConcrete(out, k) ~ wire$
+		 $j = wire$
+		 $out = k + wire$
+		 $out = k + j$
+
+		 $$ Case 1: $j = 0 => out ~ Concrete(k)$
+		 RHS:
+		 $out = k$
+
+		 EQUIVALENCE:
+		 $k + 0 = k => k = k$
+		 $$
+
+		 $$ Case 2: $otherwise => out ~ Concrete(k + j)$
+		 RHS:
+		 $out = k + j$
+
+		 EQUIVALENCE:
+		 $k + j = k + j$
+		 $$
+
+*** Mul
+**** $Linear(x, q, r) >< Mul(out, b) => b ~ MulCheckLinear(out, x, q, r)$
+		 LHS:
+		 $Linear(x, q, r) ~ wire$
+		 $Mul(out, b) ~ wire$
+		 $q*x + r = wire$
+		 $out = wire * b$
+		 $out = (q*x + r) * b$
+
+		 RHS:
+		 $out = q*b*x + r*b$
+
+		 EQUIVALENCE:
+		 $(q*x + r) * b = q*b*x + r*b => q*b*x + r*b = q*b*x + r*b$
+
+**** $Concrete(k) >< Mul(out, b) => (1), (2), (3)$
+		 LHS:
+		 $Concrete(k) ~ wire$
+		 $Mul(out, b) ~ wire$
+		 $k = wire$
+		 $out = wire * b$
+		 $out = k * b$
+
+		 $$ Case 1: $k = 0 => out ~ Concrete(0), b ~ Eraser$
+		 RHS:
+		 $out = 0$
+
+		 EQUIVALENCE:
+		 $0 * b = 0 => 0 = 0$
+		 $$
+
+		 $$ Case 2: $k = 1 => out ~ b$
+		 RHS:
+		 $out = b$
+
+		 EQUIVALENCE:
+		 $1 * b = b => b = b$
+		 $$
+
+		 $$ Case 3: $otherwise => b ~ MulCheckConcrete(out, k)$
+		 RHS:
+		 $out = k * b$
+
+		 EQUIVALENCE:
+		 $k * b = k * b$
+		 $$
+
+**** $Linear(y, s, t) >< MulCheckLinear(out, x, q, r) => (1), (2)$
+		 LHS:
+		 $Linear(y, s, t) ~ wire$
+		 $MulCheckLinear(out, x, q, r) ~ wire$
+		 $s*y + t = wire$
+		 $out = q*wire*x + r*wire$
+		 $out = q*(s*y + t)*x + r*(s*y + t)$
+
+		 $$ Case 1: $(q,r = 0) or (s,t = 0) => x ~ Eraser, y ~ Eraser, out ~ Concrete(0)$
+		 RHS:
+		 $out = 0$
+
+		 EQUIVALENCE:
+		 $0*(s*y + t)*x + 0*(s*y + t) = 0 => 0 = 0$
+		 or
+		 $q*(0*y + 0)*x + r*(0*y + 0) = 0 => 0 = 0$
+		 $$
+
+		 $$ Case 2: $otherwise => Linear(x, q, r) ~ Materialize(out_x), Linear(y, s, t) ~ Materialize(out_y), out ~ Linear(TermMul(out_x, out_y), 1, 0)$
+		 RHS:
+		 $Linear(x, q, r) ~ wire1$
+		 $Materialize(out_x) ~ wire1$
+		 $q*x + r = wire1$
+		 $out_x = wire1$
+		 $Linear(y, s, t) ~ wire2$
+		 $Materialize(out_y) ~ wire2$
+		 $s*y + t = wire2$
+		 $out_y = wire2$
+		 $out = 1*TermMul(out_x, out_y) + 0$
+		 Because $TermMul(a, b)$ is defined as $a*b$:
+		 $out = 1*(q*x + r)*(s*y + t) + 0$
+
+		 EQUIVALENCE:
+		 $q*(s*y + t)*x + r*(s*y + t) = 1*(q*x + r)*(s*y + t) => 
+		 q*(s*y + t)*x + r*(s*y + t) = (q*x + r)*(s*y + t) => 
+		 q*(s*y + t)*x + r*(s*y + t) = q*(s*y + t)*x + r*(s*y + t)$
+		 $$
+
+**** $Concrete(j) >< MulCheckLinear(out, x, q, r) => out ~ Linear(x, q * j, r * j)$
+		 LHS:
+		 $Concrete(j) ~ wire$
+		 $MulCheckLinear(out, x, q, r) ~ wire$
+		 $j = wire$
+		 $out = q*wire*x + r*wire$
+		 $out = q*j*x + r*j$
+
+		 RHS:
+		 $out = q*j*x + r*j$
+
+		 EQUIVALENCE:
+		 $q*j*x + r*j = q*j*x + r*j$
+
+**** $Linear(y, s, t) >< MulCheckConcrete(out, k) => out ~ Linear(y, s * k, t * k)$
+		 LHS:
+		 $Linear(y, s, t) ~ wire$
+		 $MulCheckConcrete(out, k) ~ wire$
+		 $s*y + t = wire$
+		 $out = k * wire$
+		 $out = k * (s*y + t)$
+
+		 RHS:
+		 $out = s*k*y + t*k$
+
+		 EQUIVALENCE:
+		 $k * (s*y + t) = s*k*y + t*k => s*k*y + t*k = s*k*y + t*k$
+
+**** $Concrete(j) >< MulCheckConcrete(out, k) => (1), (2), (3)$
+		 LHS:
+		 $Concrete(j) ~ wire$
+		 $MulCheckConcrete(out, k) ~ wire$
+		 $j = wire$
+		 $out = k * wire$
+		 $out = k * j$
+
+		 $$ Case 1: $j = 0 => out ~ Concrete(0)$
+		 RHS:
+		 $out = 0$
+
+		 EQUIVALENCE:
+		 $k * 0 = 0 => 0 = 0$
+		 $$
+
+		 $$ Case 2: $j = 1 => out ~ Concrete(k)$
+		 RHS:
+		 $out = k$
+
+		 EQUIVALENCE:
+		 $k * 1 = k => k = k$
+		 $$
+
+		 $$ Case 3: $otherwise => out ~ Concrete(k * j)$
+		 RHS:
+		 $out = k * j$
+
+		 EQUIVALENCE:
+		 $k * j = k * j$
+
+*** ReLU
+**** $Linear(x, q, r) >< ReLU(out) => Linear(x, q, r) ~ Materialize(out_x), out ~ Linear(TermReLU(out_x), 1, 0)$
+		 LHS:
+		 $Linear(x, q, r) ~ wire$
+		 $ReLU(out) ~ wire$
+		 $q*x + r = wire$
+		 $out = IF wire > 0 THEN wire ELSE 0$
+		 $out = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0$
+
+		 RHS:
+		 $Linear(x, q, r) ~ wire$
+		 $Materialize(out_x) ~ wire$
+		 $q*x + r = wire$
+		 $out_x = wire$
+		 $out = 1*TermReLU(out_x) + 0$
+		 Because $TermReLU(x)$ is defined as $z3.If(x > 0, x, 0)$:
+		 $out = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0$
+
+		 EQUIVALENCE:
+		 $IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = 1*(IF (q*x + r) > 0 THEN (q*x + r) ELSE 0) + 0 =>
+		 IF (q*x + r) > 0 THEN (q*x + r) ELSE 0 = IF (q*x + r) > 0 THEN (q*x + r) ELSE 0$
+
+**** $Concrete(k) >< ReLU(out) => (1), (2)$
+		 LHS:
+		 $Concrete(k) ~ wire$
+		 $ReLU(out) ~ wire$
+		 $k = wire$
+		 $out = IF wire > 0 THEN wire ELSE 0$
+		 $out = IF k > 0 THEN k ELSE 0$
+
+		 $$ Case 1: $k > 0 => out ~ Concrete(k)$
+		 RHS:
+		 $out = k$
+
+		 EQUIVALENCE:
+		 $IF true THEN k ELSE 0 = k => k = k$
+		 $$
+
+		 $$ Case 2: $k <= 0 => out ~ Concrete(0)$
+		 RHS:
+		 $out = 0$
+
+		 EQUIVALENCE:
+		 $IF false THEN k ELSE 0 = 0 => 0 = 0$
+		 $$
+
+** Soundness of Reduction 
+	 Let $IN_0$ be the Interaction Net translated from a Neural Network $NN$. Let $IN_n$ be the state of the net
+	 after $n$ reduction steps. Then $\forall n \in N$, $[IN_n] = [NN]$.
+
+*** Proof by Induction
+		- Base Case ($n = 0$): By the {* Soundness of Translation}, the initial net $IN_0$ is constructed such that
+			its semantics $[IN_0]$ exactly match the mathematical definition of the ONNX nodes in $NN$.
+		- Induction Step ($n -> n + 1$): Assume $[IN_n] = [NN]$. If $IN_n$ is in normal form, the proof is complete.
+			Otherwise, there exists an active pair $A$ that reduces $IN_n$ to $IN_{n+1}$.
+			By the {* Soundness of Interaction Rules}, the mathematical definition is preserved after any reduction step,
+		it follows that $[IN_{n+1}] = [IN_n]$. By the inductive hypothesis, $[IN_{n+1}] = [NN]$.
+		---
+
+	 By the principle of mathematical induction, the Interaction Net remains semantically equivalent to the original 
+	 Neural Network at every step of the reduction process.
+
+	 Since Interaction Nets are confluent, the reduced mathematical expression is unique regardless
+	 of order in which rules are applied.