proof that liquidating with maxRepaid is enough to make the account healthy

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 .certora_internal
+.*.cache

rocq/maxRepaidHealthy.v

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+From Stdlib Require Import ZArith Lia Psatz.
+
+Open Scope Z_scope.
+
+Definition WAD : Z := 1000000000000000000.
+Definition ORACLE_PRICE_SCALE : Z := 1000000000000000000000000000000000000.
+
+Definition ceil_div (numerator denominator : Z) : Z :=
+  (numerator + denominator - 1) / denominator.
+
+(**
+If:
+- maxRepaid is computed as:
+  ceil((debt - maxDebt) * WAD^2 / (WAD^2 - lif * lltv))
+- seizedAssets is computed as:
+  floor(floor(maxRepaid * lif / WAD) * ORACLE_PRICE_SCALE / price)
+- maxDebt is computed from the current collateral as:
+  otherCollatContribution + floor(floor(collat * price / ORACLE_PRICE_SCALE) * lltv / WAD)
+
+then after liquidating maxRepaid, the borrower is healthy:
+  newDebt <= newMaxDebt
+
+This proof is entirely over integer arithmetic.
+It does not use real-number approximations.
+The assumptions that the collateral seized does not exceed the current collateral does not narrow the generality of the proof: in that case the transaction would revert independently of the RCF mechanism, so that mechanism does not restrict the possibility to go back to health.
+*)
+
+Definition max_repaid_liquidation_leaves_healthy_statement : Prop :=
+  forall debt maxDebt otherCollatContribution collat seizedAssets price lltv lif maxRepaid,
+    0 < price ->
+    0 <= debt -> 0 <= otherCollatContribution ->
+    0 <= collat -> 0 <= seizedAssets -> seizedAssets <= collat ->
+    0 <= lltv -> 0 <= lif ->
+    lif * lltv < WAD * WAD ->
+    maxDebt =
+      otherCollatContribution
+      + (collat * price / ORACLE_PRICE_SCALE) * lltv / WAD ->
+    seizedAssets =
+      (maxRepaid * lif / WAD * ORACLE_PRICE_SCALE) / price ->
+    maxRepaid =
+      ceil_div ((debt - maxDebt) * (WAD * WAD))
+        (WAD * WAD - lif * lltv) ->
+    maxDebt <= debt ->
+    let newDebt := debt - maxRepaid in
+    let newMaxDebt :=
+      otherCollatContribution
+      + ((collat - seizedAssets) * price / ORACLE_PRICE_SCALE) * lltv / WAD
+    in
+    newDebt <= newMaxDebt.
+
+(* -------------------------------------------------------------------------- *)
+(* Generic integer-division lemmas                                             *)
+(* -------------------------------------------------------------------------- *)
+
+Lemma WAD_pos : 0 < WAD.
+Proof. unfold WAD; lia. Qed.
+
+Lemma ORACLE_PRICE_SCALE_pos : 0 < ORACLE_PRICE_SCALE.
+Proof. unfold ORACLE_PRICE_SCALE; lia. Qed.
+
+Lemma div_upper_strict :
+  forall numerator denominator,
+    0 < denominator -> 0 <= numerator ->
+    numerator < denominator * (numerator / denominator + 1).
+Proof.
+  intros numerator denominator Hden Hnum.
+  pose proof (Z.div_mod numerator denominator) as Hdivmod.
+  specialize (Hdivmod ltac:(lia)).
+  pose proof (Z.mod_pos_bound numerator denominator Hden) as Hmod.
+  nia.
+Qed.
+
+Lemma floor_mul_le :
+  forall numerator denominator,
+    0 < denominator -> 0 <= numerator ->
+    (numerator / denominator) * denominator <= numerator.
+Proof.
+  intros numerator denominator Hden Hnum.
+  pose proof (Z.div_mod numerator denominator) as Hdivmod.
+  specialize (Hdivmod ltac:(lia)).
+  pose proof (Z.mod_pos_bound numerator denominator Hden) as Hmod.
+  nia.
+Qed.
+
+Lemma ceil_div_mul_ge :
+  forall numerator denominator,
+    0 < denominator -> 0 <= numerator ->
+    numerator <= ceil_div numerator denominator * denominator.
+Proof.
+  intros numerator denominator Hden Hnum.
+  unfold ceil_div.
+  pose proof (Z.div_mod (numerator + denominator - 1) denominator) as Hdivmod.
+  specialize (Hdivmod ltac:(lia)).
+  pose proof (Z.mod_pos_bound (numerator + denominator - 1) denominator Hden) as Hmod.
+  nia.
+Qed.
+
+Lemma ceil_div_le_of_mul_ge :
+  forall numerator denominator bound,
+    0 < denominator -> 0 <= numerator -> numerator <= bound * denominator ->
+    ceil_div numerator denominator <= bound.
+Proof.
+  intros numerator denominator bound Hden Hnum Hbound.
+  unfold ceil_div.
+  assert ((numerator + denominator - 1) / denominator < bound + 1) as Hlt.
+  { apply Z.div_lt_upper_bound; nia. }
+  lia.
+Qed.
+
+Lemma ceil_div_mono :
+  forall left right denominator,
+    0 < denominator -> left <= right ->
+    ceil_div left denominator <= ceil_div right denominator.
+Proof.
+  intros left right denominator Hden Hle.
+  unfold ceil_div.
+  apply Z.div_le_mono; lia.
+Qed.
+
+Lemma floor_drop_div_le_ceil :
+  forall base drop denominator,
+    0 < denominator -> 0 <= base -> 0 <= drop ->
+    (base + drop) / denominator - base / denominator <= ceil_div drop denominator.
+Proof.
+  intros base drop denominator Hden Hbase Hdrop.
+  assert (Hbase_lt : base < denominator * (base / denominator + 1)).
+  { apply div_upper_strict; lia. }
+  assert (Hdrop_le : drop <= ceil_div drop denominator * denominator).
+  { apply ceil_div_mul_ge; lia. }
+  assert ((base + drop) / denominator < base / denominator + ceil_div drop denominator + 1) as Hlt.
+  { apply Z.div_lt_upper_bound; nia. }
+  lia.
+Qed.
+
+(* -------------------------------------------------------------------------- *)
+(* Midnight-specific rounding lemmas                                           *)
+(* -------------------------------------------------------------------------- *)
+
+Lemma seized_value_drop_le_maxSeizedValue :
+  forall maxRepaid lif price seizedAssets,
+    0 < price ->
+    0 <= maxRepaid -> 0 <= lif ->
+    seizedAssets = (maxRepaid * lif / WAD * ORACLE_PRICE_SCALE) / price ->
+    ceil_div (seizedAssets * price) ORACLE_PRICE_SCALE <= maxRepaid * lif / WAD.
+Proof.
+  intros maxRepaid lif price seizedAssets Hprice HmaxRepaid Hlif Hseized.
+  subst seizedAssets.
+  assert (HWAD : 0 < WAD) by apply WAD_pos.
+  assert (Hscale : 0 < ORACLE_PRICE_SCALE) by apply ORACLE_PRICE_SCALE_pos.
+  assert (HmaxSeizedValue_nonneg : 0 <= maxRepaid * lif / WAD).
+  { apply Z.div_pos; nia. }
+  apply ceil_div_le_of_mul_ge.
+  - exact Hscale.
+  - apply Z.mul_nonneg_nonneg; [apply Z.div_pos |]; nia.
+  - pose proof (floor_mul_le (maxRepaid * lif / WAD * ORACLE_PRICE_SCALE) price Hprice) as Hfloor.
+    specialize (Hfloor ltac:(nia)).
+    nia.
+Qed.
+
+Lemma max_debt_contribution_drop_bound :
+  forall collat seizedAssets price lltv maxRepaid lif,
+    0 < price ->
+    0 <= collat -> 0 <= seizedAssets -> seizedAssets <= collat ->
+    0 <= lltv -> 0 <= maxRepaid -> 0 <= lif ->
+    seizedAssets = (maxRepaid * lif / WAD * ORACLE_PRICE_SCALE) / price ->
+    let currentCollatValue := collat * price / ORACLE_PRICE_SCALE in
+    let newCollatValue := (collat - seizedAssets) * price / ORACLE_PRICE_SCALE in
+    currentCollatValue * lltv / WAD - newCollatValue * lltv / WAD
+      <= ceil_div (maxRepaid * lif * lltv) (WAD * WAD).
+Proof.
+  intros collat seizedAssets price lltv maxRepaid lif
+    Hprice Hcollat HseizedNonneg HseizedLe Hlltv HmaxRepaid Hlif HseizedEq
+    currentCollatValue newCollatValue.
+  subst currentCollatValue newCollatValue.
+  assert (HWAD : 0 < WAD) by apply WAD_pos.
+  assert (Hscale : 0 < ORACLE_PRICE_SCALE) by apply ORACLE_PRICE_SCALE_pos.
+
+  set (collatValueDrop :=
+    collat * price / ORACLE_PRICE_SCALE - (collat - seizedAssets) * price / ORACLE_PRICE_SCALE).
+  set (maxSeizedValue := maxRepaid * lif / WAD).
+  set (maxDebtDropBound := ceil_div (maxRepaid * lif * lltv) (WAD * WAD)).
+
+  assert (HcollatValueDrop_le : collatValueDrop <= maxSeizedValue).
+  {
+    unfold collatValueDrop, maxSeizedValue.
+    replace (collat * price)
+      with ((collat - seizedAssets) * price + seizedAssets * price) by nia.
+    eapply Z.le_trans.
+    - apply floor_drop_div_le_ceil; nia.
+    - eapply seized_value_drop_le_maxSeizedValue; eauto; nia.
+  }
+
+  assert (Hold_ge_new_value :
+    (collat - seizedAssets) * price / ORACLE_PRICE_SCALE <= collat * price / ORACLE_PRICE_SCALE).
+  {
+    apply Z.div_le_mono; nia.
+  }
+
+  assert (HmaxDebtDrop_by_valueDrop :
+    collat * price / ORACLE_PRICE_SCALE * lltv / WAD
+      - (collat - seizedAssets) * price / ORACLE_PRICE_SCALE * lltv / WAD
+      <= ceil_div (collatValueDrop * lltv) WAD).
+  {
+    unfold collatValueDrop.
+    replace (collat * price / ORACLE_PRICE_SCALE * lltv)
+      with (((collat - seizedAssets) * price / ORACLE_PRICE_SCALE) * lltv
+            + (collat * price / ORACLE_PRICE_SCALE - (collat - seizedAssets) * price / ORACLE_PRICE_SCALE) * lltv) by nia.
+    eapply floor_drop_div_le_ceil.
+    - exact HWAD.
+    - apply Z.mul_nonneg_nonneg; try nia.
+      apply Z.div_pos; nia.
+    - apply Z.mul_nonneg_nonneg; nia.
+  }
+
+  assert (Hceil_valueDrop_le_maxSeizedValue :
+    ceil_div (collatValueDrop * lltv) WAD <= ceil_div (maxSeizedValue * lltv) WAD).
+  {
+    apply ceil_div_mono; nia.
+  }
+
+  assert (HmaxSeizedValue_floor : maxSeizedValue * WAD <= maxRepaid * lif).
+  {
+    unfold maxSeizedValue.
+    apply floor_mul_le; nia.
+  }
+
+  assert (HmaxDebtDropBound_mul : maxRepaid * lif * lltv <= maxDebtDropBound * (WAD * WAD)).
+  {
+    unfold maxDebtDropBound.
+    apply ceil_div_mul_ge; nia.
+  }
+
+  assert (HmaxSeizedValue_to_maxDebtDropBound : maxSeizedValue * lltv <= maxDebtDropBound * WAD).
+  {
+    nia.
+  }
+
+  assert (Hceil_maxSeizedValue_le_maxDebtDropBound :
+    ceil_div (maxSeizedValue * lltv) WAD <= maxDebtDropBound).
+  {
+    apply ceil_div_le_of_mul_ge; nia.
+  }
+
+  lia.
+Qed.
+
+(* -------------------------------------------------------------------------- *)
+(* Final theorem                                                               *)
+(* -------------------------------------------------------------------------- *)
+
+Theorem max_repaid_liquidation_leaves_healthy :
+  max_repaid_liquidation_leaves_healthy_statement.
+Proof.
+  unfold max_repaid_liquidation_leaves_healthy_statement.
+  intros debt maxDebt otherCollatContribution collat seizedAssets price lltv lif maxRepaid
+    Hprice Hdebt Hother Hcollat HseizedNonneg HseizedLe Hlltv Hlif
+    Hden HmaxDebtEq HseizedEq HmaxRepaidEq HgapNonneg.
+  assert (HWAD : 0 < WAD) by apply WAD_pos.
+  assert (Hscale : 0 < ORACLE_PRICE_SCALE) by apply ORACLE_PRICE_SCALE_pos.
+
+  assert (HmaxDebt : 0 <= maxDebt).
+  {
+    rewrite HmaxDebtEq.
+    assert (HcollatValue_nonneg : 0 <= collat * price / ORACLE_PRICE_SCALE).
+    { apply Z.div_pos; nia. }
+    assert (HcollatContribution_nonneg :
+      0 <= (collat * price / ORACLE_PRICE_SCALE) * lltv / WAD).
+    { apply Z.div_pos; nia. }
+    nia.
+  }
+
+  set (gap := debt - maxDebt).
+  set (maxDebtDropBound := ceil_div (maxRepaid * lif * lltv) (WAD * WAD)).
+
+  assert (Hgap_nonneg : 0 <= gap) by (unfold gap; lia).
+  assert (Hden_pos : 0 < WAD * WAD - lif * lltv) by nia.
+
+  assert (HmaxRepaid : 0 <= maxRepaid).
+  {
+    rewrite HmaxRepaidEq.
+    unfold ceil_div.
+    apply Z.div_pos; nia.
+  }
+
+  assert (HmaxRepaid_def_le :
+    gap * (WAD * WAD) <= maxRepaid * (WAD * WAD - lif * lltv)).
+  {
+    rewrite HmaxRepaidEq.
+    unfold gap.
+    apply ceil_div_mul_ge; nia.
+  }
+
+  assert (Hextra_mul :
+    (maxRepaid - gap) * (WAD * WAD) >= maxRepaid * lif * lltv).
+  {
+    nia.
+  }
+
+  assert (HmaxDebtDropBound_le_extra : maxDebtDropBound <= maxRepaid - gap).
+  {
+    unfold maxDebtDropBound.
+    apply ceil_div_le_of_mul_ge; nia.
+  }
+
+  assert (HmaxDebtDrop_le_bound : maxDebt -
+      (otherCollatContribution + ((collat - seizedAssets) * price / ORACLE_PRICE_SCALE) * lltv / WAD)
+      <= maxDebtDropBound).
+  {
+    rewrite HmaxDebtEq.
+    replace (otherCollatContribution + collat * price / ORACLE_PRICE_SCALE * lltv / WAD -
+      (otherCollatContribution + (collat - seizedAssets) * price / ORACLE_PRICE_SCALE * lltv / WAD))
+      with (collat * price / ORACLE_PRICE_SCALE * lltv / WAD -
+        (collat - seizedAssets) * price / ORACLE_PRICE_SCALE * lltv / WAD) by lia.
+    unfold maxDebtDropBound.
+    eapply max_debt_contribution_drop_bound; eauto; nia.
+  }
+
+  unfold gap in *.
+  lia.
+Qed.