diff --git a/Veir/Data/LLVM/Byte/Lemmas.lean b/Veir/Data/LLVM/Byte/Lemmas.lean
index 9958f42ce..60642bec6 100644
--- a/Veir/Data/LLVM/Byte/Lemmas.lean
+++ b/Veir/Data/LLVM/Byte/Lemmas.lean
@@ -1,6 +1,9 @@
 module
 
+public import Veir.Data.LLVM.Byte.Basic
+public import Veir.Data.Refinement
 import all Veir.Data.LLVM.Byte.Basic
+import all Veir.Data.Refinement
 meta import Veir.Meta.BVDecide
 
 namespace Veir.Data.LLVM.Byte
@@ -108,4 +111,228 @@ theorem shl_eq {w : Nat} (x : Byte w) (y : Int w) (nuw : Bool) :
     repeat' split
     all_goals first | rfl | simp_all | (exfalso; bv_omega)
 
+
+public section
+
+/-- A concrete integer is refined only by itself. -/
+private theorem int_eq_of_val_isRefinedBy {w : Nat} {v : BitVec w} {x : Int w}
+    (h : Int.val v ⊒ x) : x = Int.val v := by
+  cases x with
+  | poison => exact absurd h (by simp [_root_.isRefinedBy])
+  | val u =>
+    have : v = u := by simpa [_root_.isRefinedBy] using h
+    simp [this]
+
+/-- Bit-wise characterisation of byte refinement: at every bit, either the source bit is poison,
+or the value bits agree and the target bit is not poison. -/
+theorem isRefinedBy_iff {w : Nat} (x y : Byte w) :
+    x ⊒ y ↔ ∀ i, i < w →
+      (x.poison.getLsbD i = true ∨
+        (x.val.getLsbD i = y.val.getLsbD i ∧ y.poison.getLsbD i = false)) := by
+  simp only [isRefinedBy, BitVec.eq_of_getLsbD_eq_iff, BitVec.getLsbD_or,
+    BitVec.getLsbD_and, BitVec.getLsbD_xor, BitVec.getLsbD_not, BitVec.getLsbD_allOnes]
+  constructor
+  · intro h i hi
+    have hbit := h i hi
+    revert hbit
+    simp only [hi, decide_true, Bool.true_and]
+    cases x.poison.getLsbD i <;> cases x.val.getLsbD i <;> cases y.val.getLsbD i <;>
+      cases y.poison.getLsbD i <;> simp
+  · intro h i hi
+    have hbit := h i hi
+    revert hbit
+    simp only [hi, decide_true, Bool.true_and]
+    cases x.poison.getLsbD i <;> cases x.val.getLsbD i <;> cases y.val.getLsbD i <;>
+      cases y.poison.getLsbD i <;> simp
+
+/-- The all-poison byte refines every byte. -/
+theorem allPoison_isRefinedBy {w : Nat} (y : Byte w) :
+    allPoison ⊒ y := by
+  rw [isRefinedBy_iff]
+  intro i hi
+  simp [allPoison, hi]
+
+/-- Shifting left and back right is the identity exactly when the bits shifted out are zero. -/
+private theorem bv_shiftLeft_ushiftRight_eq_self_iff {w : Nat} (v : BitVec w) (n : Nat) :
+    (v <<< n) >>> n = v ↔ ∀ i, i < w → w ≤ i + n → v.getLsbD i = false := by
+  rw [BitVec.eq_of_getLsbD_eq_iff]
+  simp only [BitVec.getLsbD_ushiftRight, BitVec.getLsbD_shiftLeft, Nat.add_sub_cancel_left]
+  constructor
+  · intro h i hi hw
+    have := h i hi
+    simp only [Nat.not_lt.mpr (by omega : w ≤ n + i), decide_false, Bool.false_and] at this
+    grind
+  · intro h i hi
+    by_cases hw : n + i < w
+    · simp [hw]
+    · simp only [hw, decide_false, Bool.false_and]
+      exact (h i hi (by omega)).symm
+
+/-- Shifting right and back left is the identity exactly when the bits shifted out are zero. -/
+private theorem bv_ushiftRight_shiftLeft_eq_self_iff {w : Nat} (v : BitVec w) (n : Nat) :
+    (v >>> n) <<< n = v ↔ ∀ i, i < n → v.getLsbD i = false := by
+  rw [BitVec.eq_of_getLsbD_eq_iff]
+  simp only [BitVec.getLsbD_shiftLeft, BitVec.getLsbD_ushiftRight]
+  constructor
+  · intro h i hi
+    by_cases hiw : i < w
+    · have := h i hiw
+      simp only [hiw, decide_true, Bool.true_and, hi, decide_true, Bool.not_true,
+        Bool.false_and] at this
+      exact this.symm
+    · exact BitVec.getLsbD_of_ge v i (by omega)
+  · intro h i hi
+    by_cases hin : i < n
+    · simp only [hi, decide_true, Bool.true_and, hin, decide_true, Bool.not_true, Bool.false_and]
+      exact (h i hin).symm
+    · simp only [hi, decide_true, Bool.true_and, hin, decide_false, Bool.not_false,
+        Bool.true_and]
+      congr 1
+      omega
+
+/-- Shifting both the value and the poison mask left preserves byte refinement. -/
+theorem shiftLeft_isRefinedBy {w : Nat} {x y : Byte w} (h : x ⊒ y)
+    (s : BitVec w) (hx : (x.val <<< s) &&& (x.poison <<< s) = 0)
+    (hy : (y.val <<< s) &&& (y.poison <<< s) = 0) :
+    (⟨x.val <<< s, x.poison <<< s, hx⟩ : Byte w) ⊒
+      ⟨y.val <<< s, y.poison <<< s, hy⟩ := by
+  rw [isRefinedBy_iff] at h ⊢
+  intro i hi
+  simp only [BitVec.shiftLeft_eq', BitVec.getLsbD_shiftLeft, hi, decide_true, Bool.true_and]
+  by_cases hlt : i < s.toNat
+  · simp [hlt]
+  · simp only [hlt, decide_false, Bool.not_false, Bool.true_and]
+    exact h (i - s.toNat) (by omega)
+
+/-- Shifting both the value and the poison mask right preserves byte refinement. -/
+theorem ushiftRight_isRefinedBy {w : Nat} {x y : Byte w} (h : x ⊒ y)
+    (s : BitVec w) (hx : (x.val >>> s) &&& (x.poison >>> s) = 0)
+    (hy : (y.val >>> s) &&& (y.poison >>> s) = 0) :
+    (⟨x.val >>> s, x.poison >>> s, hx⟩ : Byte w) ⊒
+      ⟨y.val >>> s, y.poison >>> s, hy⟩ := by
+  rw [isRefinedBy_iff] at h ⊢
+  intro i hi
+  simp only [BitVec.ushiftRight_eq', BitVec.getLsbD_ushiftRight]
+  by_cases hlt : s.toNat + i < w
+  · exact h (s.toNat + i) hlt
+  · right
+    exact ⟨by rw [BitVec.getLsbD_of_ge _ _ (by omega), BitVec.getLsbD_of_ge _ _ (by omega)],
+      BitVec.getLsbD_of_ge _ _ (by omega)⟩
+
+/-- If the source byte loses no bit when shifted left (the `nuw` check of `shl`), then
+neither does a byte refining it: the bits shifted out are concrete zeros in the source, hence also
+in the target. -/
+theorem shl_noWrap_of_isRefinedBy {w : Nat} {x y : Byte w} (h : x ⊒ y)
+    (s : BitVec w) (hv : (x.val <<< s) >>> s = x.val)
+    (hp : (x.poison <<< s) >>> s = x.poison) :
+    (y.val <<< s) >>> s = y.val ∧ (y.poison <<< s) >>> s = y.poison := by
+  rw [isRefinedBy_iff] at h
+  rw [BitVec.shiftLeft_eq', BitVec.ushiftRight_eq'] at hv hp
+  rw [bv_shiftLeft_ushiftRight_eq_self_iff] at hv hp
+  constructor <;>
+    · rw [BitVec.shiftLeft_eq', BitVec.ushiftRight_eq', bv_shiftLeft_ushiftRight_eq_self_iff]
+      intro i hi hw
+      rcases h i hi with hpoison | ⟨hval, hypoison⟩
+      · simp [hp i hi hw] at hpoison
+      · first
+          | (rw [← hval]; exact hv i hi hw)
+          | exact hypoison
+
+/-- If the source byte loses no bit when shifted right (the `exact` check of `lshr`),
+then neither does a byte refining it. -/
+theorem lshr_exact_of_isRefinedBy {w : Nat} {x y : Byte w} (h : x ⊒ y)
+    (s : BitVec w) (hv : (x.val >>> s) <<< s = x.val)
+    (hp : (x.poison >>> s) <<< s = x.poison) :
+    (y.val >>> s) <<< s = y.val ∧ (y.poison >>> s) <<< s = y.poison := by
+  rw [isRefinedBy_iff] at h
+  rw [BitVec.shiftLeft_eq', BitVec.ushiftRight_eq'] at hv hp
+  rw [bv_ushiftRight_shiftLeft_eq_self_iff] at hv hp
+  constructor <;>
+    · rw [BitVec.shiftLeft_eq', BitVec.ushiftRight_eq', bv_ushiftRight_shiftLeft_eq_self_iff]
+      intro i hi
+      by_cases hiw : i < w
+      · rcases h i hiw with hpoison | ⟨hval, hypoison⟩
+        · simp [hp i hi] at hpoison
+        · first
+            | (rw [← hval]; exact hv i hi)
+            | exact hypoison
+      · exact BitVec.getLsbD_of_ge _ _ (by omega)
+
+/-- `shl` is monotone in both of its operands. -/
+theorem shl_mono {w : Nat} (x y : Byte w) (a b : Int w) (nuw : Bool)
+    (hxy : x ⊒ y) (hab : a ⊒ b) :
+    shl x a nuw ⊒ shl y b nuw := by
+  cases a with
+  | poison =>
+    rw [shl_eq]
+    simp only [Int.isPoison_of_poison, Bool.true_or, if_true]
+    exact allPoison_isRefinedBy _
+  | val s =>
+    rw [int_eq_of_val_isRefinedBy hab, shl_eq, shl_eq]
+    simp only [Int.isPoison_of_val, Int.getValueD_val, Bool.false_or]
+    split
+    · exact allPoison_isRefinedBy _
+    · by_cases hnuw : nuw = true
+      · subst hnuw
+        split
+        · exact allPoison_isRefinedBy _
+        · next hv =>
+          split
+          · exact allPoison_isRefinedBy _
+          · next hp =>
+            have hv' : (x.val <<< s) >>> s = x.val := by simpa using hv
+            have hp' : (x.poison <<< s) >>> s = x.poison := by simpa using hp
+            obtain ⟨hvy, hpy⟩ := shl_noWrap_of_isRefinedBy hxy s hv' hp'
+            rw [if_neg (by simpa using hvy), if_neg (by simpa using hpy)]
+            exact shiftLeft_isRefinedBy hxy s _ _
+      · simp only [Bool.not_eq_true] at hnuw
+        subst hnuw
+        simp only [Bool.false_eq_true, false_and, if_false]
+        exact shiftLeft_isRefinedBy hxy s _ _
+
+/-- `lshr` is monotone in both of its operands. -/
+theorem lshr_mono {w : Nat} (x y : Byte w) (a b : Int w) (exact : Bool)
+    (hxy : x ⊒ y) (hab : a ⊒ b) :
+    lshr x a exact ⊒ lshr y b exact := by
+  cases a with
+  | poison =>
+    simp only [lshr, Int.isPoison_of_poison, Bool.true_or, if_true]
+    exact allPoison_isRefinedBy _
+  | val s =>
+    rw [int_eq_of_val_isRefinedBy hab]
+    simp only [lshr, Int.isPoison_of_val, Int.getValueD_val, Bool.false_or]
+    split
+    · exact allPoison_isRefinedBy _
+    · by_cases hexact : exact = true
+      · subst hexact
+        split
+        · exact allPoison_isRefinedBy _
+        · next hv =>
+          split
+          · exact allPoison_isRefinedBy _
+          · next hp =>
+            have hv' : (x.val >>> s) <<< s = x.val := by simpa using hv
+            have hp' : (x.poison >>> s) <<< s = x.poison := by simpa using hp
+            obtain ⟨hvy, hpy⟩ := lshr_exact_of_isRefinedBy hxy s hv' hp'
+            rw [if_neg (by simpa using hvy), if_neg (by simpa using hpy)]
+            exact ushiftRight_isRefinedBy hxy s _ _
+      · simp only [Bool.not_eq_true] at hexact
+        subst hexact
+        simp only [Bool.false_eq_true, false_and, if_false]
+        exact ushiftRight_isRefinedBy hxy s _ _
+
+/-- `trunc` is monotone. -/
+theorem trunc_mono {w w' : Nat} (x y : Byte w) (h : x ⊒ y) :
+    trunc x w' ⊒ trunc y w' := by
+  rw [isRefinedBy_iff] at h ⊢
+  intro i hi
+  simp only [trunc, BitVec.getLsbD_setWidth, hi, decide_true, Bool.true_and]
+  by_cases hiw : i < w
+  · exact h i hiw
+  · right
+    exact ⟨by rw [BitVec.getLsbD_of_ge _ _ (by omega), BitVec.getLsbD_of_ge _ _ (by omega)],
+      BitVec.getLsbD_of_ge _ _ (by omega)⟩
+
+end
+
 end Veir.Data.LLVM.Byte