diff --git a/content/blog/05_spa_agda_semantics/index.md b/content/blog/05_spa_agda_semantics/index.md
index 3ce4778..944fa17 100644
--- a/content/blog/05_spa_agda_semantics/index.md
+++ b/content/blog/05_spa_agda_semantics/index.md
@@ -285,6 +285,9 @@ the exact rules that were used in computing a particular value. The variables
 `x`, `y`, and `z` are pre-defined for your convenience.
 
 {{< bergamot_widget id="expr-widget" query="" prompt="eval(extend(extend(extend(empty, x, 17), y, 42), z, 0), TERM, ?v)" modes="Expression:Expression" >}}
+hidden section "" {
+  Eq @ eq(?x, ?x) <-;
+}
 section "" {
   EvalNum @ eval(?rho, lit(?n), ?n) <- int(?n);
   EvalVar @ eval(?rho, var(?x), ?v) <- inenv(?x, ?v, ?rho);
@@ -295,7 +298,7 @@ section "" {
 }
 hidden section "" {
   EnvTake @ inenv(?x, ?v, extend(?rho, ?x, ?v)) <-;
-  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(symeq(?x, ?y));
+  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(eq(?x, ?y));
 }
 {{< /bergamot_widget >}}
 
@@ -349,6 +352,9 @@ a Bergamot instance, this time with simple statements. Try out `noop` or some
 sort of variable assignment, like `x = x + 1`.
 
 {{< bergamot_widget id="basic-stmt-widget" query="" prompt="stepbasic(extend(extend(extend(empty, x, 17), y, 42), z, 0), TERM, ?env)" modes="Basic Statement:Basic Statement" >}}
+hidden section "" {
+  Eq @ eq(?x, ?x) <-;
+}
 hidden section "" {
   EvalNum @ eval(?rho, lit(?n), ?n) <- int(?n);
   EvalVar @ eval(?rho, var(?x), ?v) <- inenv(?x, ?v, ?rho);
@@ -363,7 +369,7 @@ section "" {
 }
 hidden section "" {
   EnvTake @ inenv(?x, ?v, extend(?rho, ?x, ?v)) <-;
-  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(symeq(?x, ?y));
+  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(eq(?x, ?y));
 }
 {{< /bergamot_widget >}}
 
@@ -420,6 +426,9 @@ The rule for evaluating a conditional with a "false" branch is very similar.
 {{< /latex >}}
 
 {{< bergamot_widget id="stmt-widget" query="" prompt="step(extend(extend(extend(empty, x, 17), y, 42), z, 0), TERM, ?env)" modes="Statement:Statement" >}}
+hidden section "" {
+  Eq @ eq(?x, ?x) <-;
+}
 hidden section "" {
   EvalNum @ eval(?rho, lit(?n), ?n) <- int(?n);
   EvalVar @ eval(?rho, var(?x), ?v) <- inenv(?x, ?v, ?rho);
@@ -440,14 +449,14 @@ hidden section "" {
   StepLiftBasic @ step(?rho_1, ?s, ?rho_2) <- stepbasic(?rho_1, ?s, ?rho_2);
 }
 section "" {
-  StepIfTrue @ step(?rho_1, if(?e, ?s_1, ?s_2), ?rho_2) <- eval(?rho_1, ?e, ?v), not(numeq(?v, 0)), step(?rho_1, ?s_1, ?rho_2);
-  StepIfFalse @ step(?rho_1, if(?e, ?s_1, ?s_2), ?rho_2) <- eval(?rho_1, ?e, ?v), numeq(?v, 0), step(?rho_1, ?s_2, ?rho_2);
-  StepWhileTrue @ step(?rho_1, while(?e, ?s), ?rho_3) <- eval(?rho_1, ?e, ?v), not(numeq(?v, 0)), step(?rho_1, ?s, ?rho_2), step(?rho_2, while(?e, ?s), ?rho_3);
-  StepWhileFalse @ step(?rho_1, while(?e, ?s), ?rho_1) <- eval(?rho_1, ?e, ?v), numeq(?v, 0);
+  StepIfTrue @ step(?rho_1, if(?e, ?s_1, ?s_2), ?rho_2) <- eval(?rho_1, ?e, ?v), not(eq(?v, 0)), step(?rho_1, ?s_1, ?rho_2);
+  StepIfFalse @ step(?rho_1, if(?e, ?s_1, ?s_2), ?rho_2) <- eval(?rho_1, ?e, ?v), eq(?v, 0), step(?rho_1, ?s_2, ?rho_2);
+  StepWhileTrue @ step(?rho_1, while(?e, ?s), ?rho_3) <- eval(?rho_1, ?e, ?v), not(eq(?v, 0)), step(?rho_1, ?s, ?rho_2), step(?rho_2, while(?e, ?s), ?rho_3);
+  StepWhileFalse @ step(?rho_1, while(?e, ?s), ?rho_1) <- eval(?rho_1, ?e, ?v), eq(?v, 0);
   StepSeq @ step(?rho_1, seq(?s_1, ?s_2), ?rho_3) <- step(?rho_1, ?s_1, ?rho_2), step(?rho_2, ?s_2, ?rho_3);
 }
 hidden section "" {
   EnvTake @ inenv(?x, ?v, extend(?rho, ?x, ?v)) <-;
-  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(symeq(?x, ?y));
+  EnvSkip @ inenv(?x, ?v_1, extend(?rho, ?y, ?v_2)) <- inenv(?x, ?v_1, ?rho), not(eq(?x, ?y));
 }
 {{< /bergamot_widget >}}