another example for eHoare

Author: namasikanam

examples/ehoare/random_boolean_matrix.ec

@@ -0,0 +1,248 @@
+require import AllCore Array Real RealExp List.
+(*---*) import RField.
+require import Distr DBool Xreal.
+(*---*) import Biased.
+require import StdOrder.
+(*---*) import RealOrder.
+
+lemma mul0z (x : int) : 0 * x = 0.
+proof. by auto. qed.
+
+lemma neg0: -0 = 0.
+proof. by auto. qed.
+
+lemma xle_rle:
+  forall (x y : real),
+    0%r <= x <= y => x%xr <= y%xr.
+proof.
+move => x y [??] /=.
+by rewrite !to_pos_pos //; exact (ler_trans x).
+qed.
+
+lemma Ep_dbiased (p : real) (f : bool -> xreal) :
+  0%r <= p <= 1%r => Ep (dbiased p) f = p ** f true + (1%r - p) ** f false.
+proof.
+move => ?.
+rewrite (Ep_fin [true; false]) //.
++ by case.
+by rewrite /BXA.big /predT /= !dbiased1E /= !clamp_id //.
+qed.
+
+(* uniformly sampling a 2-d boolean array of size n x m *)
+module M = {
+  proc sample (n : int, m : int, a : bool array) : (bool array) = {
+    var i, j : int;
+    var b : bool;
+    i <- 0;
+    while (i < n) {
+      j <- 0;
+      while (j < m) {
+        b <$ dbiased 0.5;
+        a.[i * m + j] <- b;
+        j <- j + 1;
+      }
+      i <- i + 1;
+    }
+    return a;
+  }
+}.
+
+op outer_shape_pred (i n m : int) (a a' : bool array) =
+    0 <= i <= n
+ /\ 0 <= m
+ /\ size a = n * m
+ /\ size a = size a'.
+
+op shape_pred (i j n m : int) (a a' : bool array) =
+    0 <= i < n
+ /\ 0 <= j <= m
+ /\ size a = n * m
+ /\ size a = size a'.
+
+op row_eq_upto (i m : int) (a1 a2 : bool array) =
+  forall (i' j' : int),
+     0 <= i' < i
+  => 0 <= j' < m
+  => a1.[i' * m + j'] = a2.[i' * m + j'].
+
+op cell_eq_upto (i j m : int) (a1 a2 : bool array) =
+  forall (j' : int),
+     0 <= j' < j
+  => a1.[i * m + j'] = a2.[i * m + j'].
+
+lemma row_eq_upto_increase (i m : int) (a1 a2 : bool array):
+      0 <= i
+   => (row_eq_upto i m a1 a2 /\ cell_eq_upto i m m a1 a2
+   <=> row_eq_upto (i + 1) m a1 a2).
+proof.
+move => ? @/row_eq_upto @/cell_eq_upto; split.
+- move => ? i' j' *.
+  by case: (i' < i) => /#.
+- move => H; split.
+  - move => i' j' ??.
+    have ?: 0 <= i' < i + 1 by smt().
+    by have := H i' j' _ _ => //.
+  - by have := H i => /#.
+qed.
+
+lemma cell_eq_upto_false (i j' j m : int) (a1 a2 : bool array) :
+     0 <= j' < j
+  => a1.[i * m + j'] <> a2.[i * m + j']
+  => cell_eq_upto i j m a1 a2 = false.
+proof. by smt(). qed.
+
+lemma cell_eq_upto_split (i j m : int) (a1 a2 : bool array) :
+     0 <= j < m
+  => (cell_eq_upto i (j + 1) m a1 a2
+         <=> (cell_eq_upto i j m a1 a2
+              /\ a1.[i * m + j] = a2.[i * m + j])
+     ).
+proof.
+move => ? @/cell_eq_upto; split.
+- move => H; split.
+  - move => j' ?.
+    have ?: 0 <= j' < j + 1 by smt().
+    have := H j' _ => //.
+  - by smt().
+- move => ? j' ?.
+  by case (j' < j) => /#.
+qed.
+
+lemma row_eq_upto_unrelated_set (i m x : int) (v : bool) (a1 a2 : bool array):
+     i * m <= x < size a1
+  => (row_eq_upto i m a1 a2 <=> row_eq_upto i m a1.[x <- v] a2).
+proof.
+move => ?.
+rewrite /row_eq_upto; split.
+- move => ? i' j' ??.
+  rewrite get_set 1:/#.
+  have -> /=: !(i' * m + j' = x) by smt().
+  by smt().
+- move => ? i' j' ??.
+  by rewrite (_: a1.[i' * m + j'] = a1.[x <- v].[i' * m + j']) 1:get_set /#.
+qed.
+
+lemma cell_eq_upto_unrelated_set (i j m x : int) (v : bool) (a1 a2 : bool array) :
+     0 <= i /\ 0 <= j < m /\ i * m + j <= x < size a1
+  => (cell_eq_upto i j m a1 a2 <=> cell_eq_upto i j m a1.[x <- v] a2).
+proof.
+move => [#] ?????.
+rewrite /cell_eq_upto; split.
+- move => ? j' ?.
+  rewrite get_set 1:/#.
+  have -> /=: !(i * m + j' = x) by smt().
+  by smt().
+- move => ? j' ?.
+  by rewrite (_: a1.[i * m + j'] = a1.[x <- v].[i * m + j']) 1:get_set /#.
+qed.
+
+(* The probability of every possible boolean matrix of size n x m is no more than 2 ^ -(n * m) *)
+lemma L:
+  forall (a0 : bool array),
+    ehoare [M.sample :
+            (0 <= arg.`1
+          /\ 0 <= arg.`2
+          /\ size arg.`3 = arg.`1 * arg.`2
+          /\ size arg.`3 = size a0)
+        `|` (1%r / (2%r ^ (n * m)))%xr ==> (res = a0)%xr].
+proof.
+move => a0.
+proc.
+while ((0 <= i <= n
+     /\ 0 <= m
+     /\ size a = n * m
+     /\ size a0 = size a)
+   `|`  (2%r ^ ((-(n - i) * m)%r))%xr
+       * (row_eq_upto i m a a0)%xr).
+(* !cond => inv => pos_f <= inv_f *)
++ move => &hr.
+  rewrite xle_cxr_r => *. (* Bug in EC: the precedent is lost when rewriting xle_cxr_r twice. *)
+  rewrite xle_cxr_r => *.
+  have ->: n{hr} - i{hr} = 0 by smt().
+  rewrite mul0z neg0 rpow0 mul1m_simpl.
+  apply xle_rle; split => * ;1: by smt().
+  exact le_b2r.
+(* {cond /\ inv | inv_f} c {inv | inv_f} *)
++ wp.
+  while (( 0 <= i < n
+        /\ 0 <= j <= m
+        /\ size a = n * m
+        /\ size a = size a0)
+          `|` (2%r ^ ((-((n - i) * m - j))%r))%xr
+            * (row_eq_upto i m a a0 /\ cell_eq_upto i j m a a0)%xr).
+  (* !cond => inv => pos_f <= inv_f *)
+  + move => &hr />.
+    rewrite xle_cxr_r => *.
+    rewrite xle_cxr_l => *.
+    + by smt().
+    + rewrite (_: - (n{hr} - (i{hr} + 1)) * m{hr}
+                = - ((n{hr} - i{hr}) * m{hr} - j{hr})) //= 1:/#.
+      rewrite (_: j{hr} = m{hr}) 1:/#.
+      rewrite -row_eq_upto_increase 1:/#.
+      rewrite ler_eqVlt; left; reflexivity.
+  (* {cond /\ inv | inv_f} c {inv | inv_f} *)
+  + wp; skip.
+    move => &hr.
+    move => />.
+    rewrite xle_cxr_r => [#] 5? Hsize *.
+    rewrite Ep_dbiased /= 1:/#.
+    rewrite (_: ((0 <= i{hr} < n{hr}) <=> true)) 1:/#.
+    rewrite (_: ((0 <= j{hr} + 1 <= m{hr}) <=> true)) 1:/#.
+    rewrite !size_set.
+    rewrite (_: ((size a{hr} = n{hr} * m{hr}) <=> true)) 1:/#.
+    rewrite (_: (size a{hr} = size a0) <=> true) /= 1:/#.
+    rewrite !to_pos_pos 1,2,3,4,5:#smt:(rpow_gt0 b2r_ge0).
+    rewrite !cell_eq_upto_split 1,2:/#.
+    rewrite !get_set //=.
+    - split; 1: by smt().
+      move => ?.
+      by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+    - split; 1: by smt().
+      move => ?.
+      by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+    case (a0.[i{hr} * m{hr} + j{hr}]) => Hcase /=.
+    + rewrite -row_eq_upto_unrelated_set.
+      - split; 1: by smt().
+        move => ?.
+        by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+      rewrite -cell_eq_upto_unrelated_set.
+      - do! split; 1,2,3: by smt().
+        move => ?.
+        by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+      rewrite -{2} (rpow1 2%r) //.
+      rewrite -rpowN // -mulrA.
+      rewrite (mulrC
+                (b2r
+                    (row_eq_upto i{hr} m{hr} a{hr} a0 /\
+                        cell_eq_upto i{hr} j{hr} m{hr} a{hr} a0))
+                (2%r ^ - 1%r)).
+      by rewrite mulrA -rpowD // /#.
+    + rewrite /= -row_eq_upto_unrelated_set.
+      - split; 1: by smt().
+        move => ?.
+        by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+      rewrite -cell_eq_upto_unrelated_set.
+      - do! split; 1,2,3: by smt().
+        move => ?.
+        by apply (IntOrder.ltr_le_trans ((n{hr} - 1) * m{hr} + m{hr})) => /#.
+      rewrite -{2} (rpow1 2%r) //.
+      rewrite -rpowN // -mulrA.
+      rewrite (mulrC
+                (b2r
+                    (row_eq_upto i{hr} m{hr} a{hr} a0 /\
+                        cell_eq_upto i{hr} j{hr} m{hr} a{hr} a0))
+                (2%r ^ - 1%r)).
+      by rewrite mulrA -rpowD // /#.
+  (* pre => inv *)
+  + wp; skip.
+    move => &hr />.
+    rewrite xle_cxr_r => *.
+    rewrite xle_cxr_l 1:/#.
+    rewrite (_: cell_eq_upto i{hr} 0 m{hr} a{hr} a0 = true) //= 1:/#.
+wp.
+skip => /> &hr.
+rewrite xle_cxr_r => *.
+rewrite xle_cxr_l 1:/#.
+rewrite fromintN rpowN //= rpow_int //=.
+by rewrite (_: row_eq_upto 0 m{hr} a{hr} a0 = true) //= 1:/#.
+qed.