still be -> is still

adamant-pwn · adamant-pwn · commit 888b75bf5c50 · 2023-12-13T14:10:34.000+01:00
diff --git a/src/graph/hungarian-algorithm.md b/src/graph/hungarian-algorithm.md
@@ -107,7 +107,7 @@ Note that conditions are not mutually exclusive.
 
 ??? info "Proof"
 
-    We will show that, after recalculation, $u[i]+v[j]\leq A[i][j]$ for all $i,j$. For all the elements of A with $i\in Z_1$ and $j\in Z_2$, the sum $u[i]+v[j]$ does not change, so the inequality remains true. For all the elements with $i\notin Z_1$ and $j\in Z_2$, the sum $u[i]+v[j]$ decreases by $\Delta$, so the inequality still be true. For the other elements whose $i\in Z_1$ and $j\notin Z_2$, the sum increases, but the inequality is still preserved, since the value $\Delta$ is, by definition, the maximum increase that does not change the inequality.
+    We will show that, after recalculation, $u[i]+v[j]\leq A[i][j]$ for all $i,j$. For all the elements of A with $i\in Z_1$ and $j\in Z_2$, the sum $u[i]+v[j]$ does not change, so the inequality remains true. For all the elements with $i\notin Z_1$ and $j\in Z_2$, the sum $u[i]+v[j]$ decreases by $\Delta$, so the inequality is still true. For the other elements whose $i\in Z_1$ and $j\notin Z_2$, the sum increases, but the inequality is still preserved, since the value $\Delta$ is, by definition, the maximum increase that does not change the inequality.
 
 !!! info ""
 
@@ -123,7 +123,7 @@ Note that conditions are not mutually exclusive.
 
 ??? info "Proof"
 
-    First, note that any vertex that was reachable before recalculation, still be reachable. Indeed, if some vertex is reachable, then there is some path from reachable vertices to it, starting from the unsaturated vertex of the left part; since for edges of the form $(i,j),\ i\in Z_1,\ j\in Z_2$ the sum $u[i]+v[j]$ does not change, this entire path will be preserved after changing the potential.
+    First, note that any vertex that was reachable before recalculation, is still reachable. Indeed, if some vertex is reachable, then there is some path from reachable vertices to it, starting from the unsaturated vertex of the left part; since for edges of the form $(i,j),\ i\in Z_1,\ j\in Z_2$ the sum $u[i]+v[j]$ does not change, this entire path will be preserved after changing the potential.
     Secondly, we show that after a recalculation, at least one new vertex will be reachable. This follows by the definition of $\Delta$: the edge $(i,j)$ which $\Delta$ refers to will become rigid, so vertex $j$ will be reachable from vertex $i$.
 
 Due to the last lemma, **no more than $n$ potential recalculations can occur** before an augmenting path is found and the matching cardinality of $M$ is increased.
