chore: remove lean4checker from release repos (#13121)

Lean4checker has been merged into this repository and is no longer a
standalone repo.

script/release_repos.yml

@@ -14,13 +14,6 @@ repositories:
     bump-branch: true
     dependencies: []
 
-  - name: lean4checker
-    url: https://github.com/leanprover/lean4checker
-    toolchain-tag: true
-    stable-branch: true
-    branch: master
-    dependencies: []
-
   - name: quote4
     url: https://github.com/leanprover-community/quote4
     toolchain-tag: true

src/Init/Control/ExceptCps.lean

@@ -37,7 +37,7 @@ set_option linter.unusedVariables false in  -- `s` unused
 Use a monadic action that may throw an exception by providing explicit success and failure
 continuations.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def runK {ε α : Type u} (x : ExceptCpsT ε m α) (s : ε) (ok : α → m β) (error : ε → m β) : m β :=
   x _ ok error
 
@@ -83,6 +83,8 @@ of `True`.
 -/
 instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
+@[simp] theorem throw_bind [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : (throw e >>= f : ExceptCpsT ε m β) = throw e := rfl
+
 @[simp] theorem run_pure [Monad m] : run (pure x : ExceptCpsT ε m α) = pure (Except.ok x) := rfl
 
 @[simp] theorem run_lift {α ε : Type u} [Monad m] (x : m α) : run (ExceptCpsT.lift x : ExceptCpsT ε m α) = (x >>= fun a => pure (Except.ok a) : m (Except ε α)) := rfl
@@ -91,7 +93,20 @@ instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
 @[simp] theorem run_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT ε m β) : run (ExceptCpsT.lift x >>= f : ExceptCpsT ε m β) = x >>= fun a => run (f a) := rfl
 
-@[simp] theorem run_bind_throw [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : run (throw e >>= f : ExceptCpsT ε m β) = run (throw e) := rfl
+@[deprecated throw_bind (since := "2026-03-13")]
+theorem run_bind_throw [Monad m] (e : ε) (f : α → ExceptCpsT ε m β) : run (throw e >>= f : ExceptCpsT ε m β) = run (throw e) := rfl
+
+@[simp] theorem runK_pure :
+    runK (pure x : ExceptCpsT ε m α) s ok error = ok x := rfl
+
+@[simp] theorem runK_lift {α ε : Type u} [Monad m] (x : m α) (s : ε) (ok : α → m β) (error : ε → m β) :
+    runK (ExceptCpsT.lift x : ExceptCpsT ε m α) s ok error = x >>= ok := rfl
+
+@[simp] theorem runK_throw [Monad m] :
+    runK (throw e : ExceptCpsT ε m β) s ok error = error e := rfl
+
+@[simp] theorem runK_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT ε m β) :
+    runK (ExceptCpsT.lift x >>= f : ExceptCpsT ε m β) s ok error = x >>= fun a => runK (f a) s ok error := rfl
 
 @[simp] theorem runCatch_pure [Monad m] : runCatch (pure x : ExceptCpsT α m α) = pure x := rfl
 
@@ -102,6 +117,7 @@ instance : MonadAttach (ExceptCpsT ε m) := .trivial
 
 @[simp] theorem runCatch_bind_lift [Monad m] (x : m α) (f : α → ExceptCpsT β m β) : runCatch (ExceptCpsT.lift x >>= f : ExceptCpsT β m β) = x >>= fun a => runCatch (f a) := rfl
 
-@[simp] theorem runCatch_bind_throw [Monad m] (e : β) (f : α → ExceptCpsT β m β) : runCatch (throw e >>= f : ExceptCpsT β m β) = pure e := rfl
+@[deprecated throw_bind (since := "2026-03-13")]
+theorem runCatch_bind_throw [Monad m] (e : β) (f : α → ExceptCpsT β m β) : runCatch (throw e >>= f : ExceptCpsT β m β) = pure e := rfl
 
 end ExceptCpsT

src/Init/Data/String/Lemmas/FindPos.lean

@@ -201,6 +201,10 @@ theorem Pos.prev_eq_iff {s : Slice} {p q : s.Pos} {h} :
 theorem Pos.prev_lt {s : Slice} {p : s.Pos} {h} : p.prev h < p := by
   simp
 
+@[simp]
+theorem Pos.prev_le {s : Slice} {p : s.Pos} {h} : p.prev h ≤ p :=
+  Std.le_of_lt (by simp)
+
 @[simp]
 theorem Pos.prev_ne_endPos {s : Slice} {p : s.Pos} {h} : p.prev h ≠ s.endPos :=
   ne_endPos_of_lt prev_lt
@@ -211,6 +215,29 @@ theorem Pos.prevn_le {s : Slice} {p : s.Pos} {n : Nat} : p.prevn n ≤ p := by
   | case2 p n h ih => exact Std.le_of_lt (by simpa using ih)
   | case3 => simp
 
+theorem Pos.ofSliceTo_prev {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    Pos.ofSliceTo (p.prev h) = (Pos.ofSliceTo p).prev (by simpa [← Pos.ofSliceTo_inj] using h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceTo_lt_ofSliceTo_iff, Pos.le_ofSliceTo_iff]
+  simp [Pos.lt_ofSliceTo_iff]
+
+theorem Pos.prev_ofSliceTo {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    (Pos.ofSliceTo p).prev h = Pos.ofSliceTo (p.prev (by simpa [← Pos.ofSliceTo_inj])) := by
+  simp [ofSliceTo_prev]
+
+theorem Pos.ofSliceFrom_prev {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    Pos.ofSliceFrom (p.prev h) = (Pos.ofSliceFrom p).prev (by exact ofSliceFrom_ne_startPos h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceFrom_lt_ofSliceFrom_iff,  Pos.le_ofSliceFrom_iff]
+  simp [Pos.lt_ofSliceFrom_iff]
+
+theorem Pos.ofSlice_prev {s : Slice} {p₀ p₁ : s.Pos} {h}
+    {p : (s.slice p₀ p₁ h).Pos} {h'} :
+    Pos.ofSlice (p.prev h') = (Pos.ofSlice p).prev (by exact ofSlice_ne_startPos h') := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [ofSlice_lt_ofSlice_iff,  le_ofSlice_iff]
+  simpa +contextual [← ofSlice_lt_ofSlice_iff] using fun q hq => Std.le_of_lt (Std.lt_of_lt_of_le hq ofSlice_le)
+
 @[simp]
 theorem Pos.prev_next {s : Slice} {p : s.Pos} {h} : (p.next h).prev (by simp) = p :=
   prev_eq_iff.2 (by simp)
@@ -439,6 +466,10 @@ theorem Pos.prev_eq_iff {s : String} {p q : s.Pos} {h} :
 theorem Pos.prev_lt {s : String} {p : s.Pos} {h} : p.prev h < p := by
   simp
 
+@[simp]
+theorem Pos.prev_le {s : String} {p : s.Pos} {h} : p.prev h ≤ p :=
+  Std.le_of_lt (by simp)
+
 @[simp]
 theorem Pos.prev_ne_endPos {s : String} {p : s.Pos} {h} : p.prev h ≠ s.endPos :=
   ne_endPos_of_lt prev_lt
@@ -463,6 +494,29 @@ theorem Pos.prevn_le {s : String} {p : s.Pos} {n : Nat} :
     p.prevn n ≤ p := by
   simpa [Pos.le_iff, ← offset_toSlice] using Slice.Pos.prevn_le
 
+theorem Pos.ofSliceTo_prev {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    Pos.ofSliceTo (p.prev h) = (Pos.ofSliceTo p).prev (by simpa [← Pos.ofSliceTo_inj] using h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceTo_lt_ofSliceTo_iff, Pos.le_ofSliceTo_iff]
+  simp [Pos.lt_ofSliceTo_iff]
+
+theorem Pos.prev_ofSliceTo {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
+    (Pos.ofSliceTo p).prev h = Pos.ofSliceTo (p.prev (by simpa [← Pos.ofSliceTo_inj])) := by
+  simp [ofSliceTo_prev]
+
+theorem Pos.ofSliceFrom_prev {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    Pos.ofSliceFrom (p.prev h) = (Pos.ofSliceFrom p).prev (by exact ofSliceFrom_ne_startPos h) := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [Pos.ofSliceFrom_lt_ofSliceFrom_iff, Pos.le_ofSliceFrom_iff]
+  simp [Pos.lt_ofSliceFrom_iff]
+
+theorem Pos.ofSlice_prev {s : String} {p₀ p₁ : s.Pos} {h}
+    {p : (s.slice p₀ p₁ h).Pos} {h'} :
+    Pos.ofSlice (p.prev h') = (Pos.ofSlice p).prev (by exact ofSlice_ne_startPos h') := by
+  rw [eq_comm, Pos.prev_eq_iff]
+  simp only [ofSlice_lt_ofSlice_iff, le_ofSlice_iff]
+  simpa +contextual [← ofSlice_lt_ofSlice_iff] using fun q hq => Std.le_of_lt (Std.lt_of_lt_of_le hq ofSlice_le)
+
 @[simp]
 theorem Pos.prev_next {s : String} {p : s.Pos} {h} : (p.next h).prev (by simp) = p :=
   prev_eq_iff.2 (by simp)

src/Init/Data/String/Lemmas/IsEmpty.lean

@@ -204,7 +204,7 @@ theorem Slice.copy_sliceTo_startPos {s : Slice} : (s.sliceTo s.startPos).copy =
   simp
 
 @[simp]
-theorem Slice.copy_sliceFrom_startPos {s : Slice} : (s.sliceFrom s.endPos).copy = "" := by
+theorem Slice.copy_sliceFrom_endPos {s : Slice} : (s.sliceFrom s.endPos).copy = "" := by
   simp
 
 end CopyEqEmpty

src/Init/Data/String/Lemmas/Order.lean

@@ -11,6 +11,7 @@ import Init.Data.String.OrderInstances
 import Init.Data.String.Lemmas.Basic
 import Init.Data.Order.Lemmas
 import Init.Omega
+import Init.ByCases
 
 public section
 
@@ -70,7 +71,7 @@ theorem Pos.le_startPos {s : String} (p : s.Pos) : p ≤ s.startPos ↔ p = s.st
   ⟨fun h => Std.le_antisymm h (startPos_le _), by simp +contextual⟩
 
 @[simp]
-theorem Pos.startPos_lt_iff {s : String} {p : s.Pos} : s.startPos < p ↔ p ≠ s.startPos := by
+theorem Pos.startPos_lt_iff {s : String} (p : s.Pos) : s.startPos < p ↔ p ≠ s.startPos := by
   simp [← le_startPos, Std.not_le]
 
 @[simp]
@@ -235,6 +236,10 @@ theorem Slice.Pos.ofSliceFrom_next {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom
     Pos.next_le_iff_lt, true_and]
   simp [Pos.ofSliceFrom_lt_iff]
 
+theorem Slice.Pos.next_ofSliceFrom {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    (Pos.ofSliceFrom p).next h = Pos.ofSliceFrom (p.next (by simpa [← Pos.ofSliceFrom_inj])) := by
+  simp [ofSliceFrom_next]
+
 theorem Pos.ofSliceFrom_next {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
     Pos.ofSliceFrom (p.next h) = (Pos.ofSliceFrom p).next (by simpa [← Pos.ofSliceFrom_inj] using h) := by
   rw [eq_comm, Pos.next_eq_iff]
@@ -242,6 +247,10 @@ theorem Pos.ofSliceFrom_next {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀)
     Slice.Pos.next_le_iff_lt, true_and]
   simp [Pos.ofSliceFrom_lt_iff]
 
+theorem Pos.next_ofSliceFrom {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos} {h} :
+    (Pos.ofSliceFrom p).next h = Pos.ofSliceFrom (p.next (by simpa [← Pos.ofSliceFrom_inj])) := by
+  simp [Pos.ofSliceFrom_next]
+
 theorem Slice.Pos.le_ofSliceTo_iff {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {q : s.Pos} :
     q ≤ Pos.ofSliceTo p ↔ ∃ h, Slice.Pos.sliceTo p₀ q h ≤ p := by
   refine ⟨fun h => ⟨Slice.Pos.le_trans h Pos.ofSliceTo_le, ?_⟩, fun ⟨h, h'⟩ => ?_⟩
@@ -359,11 +368,21 @@ theorem Slice.Pos.ofSliceTo_ne_endPos {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₀))
   simpa [← lt_endPos_iff, ← ofSliceTo_lt_ofSliceTo_iff] using h
 
+theorem Slice.Pos.ofSliceFrom_ne_startPos {s : Slice} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos}
+    (h : p ≠ (s.sliceFrom p₀).startPos) : Pos.ofSliceFrom p ≠ s.startPos := by
+  refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
+  simpa [← startPos_lt_iff, ← ofSliceFrom_lt_ofSliceFrom_iff] using h
+
 theorem Pos.ofSliceTo_ne_endPos {s : String} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos}
     (h : p ≠ (s.sliceTo p₀).endPos) : Pos.ofSliceTo p ≠ s.endPos := by
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₀))
   simpa [← Slice.Pos.lt_endPos_iff, ← ofSliceTo_lt_ofSliceTo_iff] using h
 
+theorem Pos.ofSliceFrom_ne_startPos {s : String} {p₀ : s.Pos} {p : (s.sliceFrom p₀).Pos}
+    (h : p ≠ (s.sliceFrom p₀).startPos) : Pos.ofSliceFrom p ≠ s.startPos := by
+  refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
+  simpa [← Slice.Pos.startPos_lt_iff, ← ofSliceFrom_lt_ofSliceFrom_iff] using h
+
 theorem Slice.Pos.ofSliceTo_next {s : Slice} {p₀ : s.Pos} {p : (s.sliceTo p₀).Pos} {h} :
     Pos.ofSliceTo (p.next h) = (Pos.ofSliceTo p).next (ofSliceTo_ne_endPos h) := by
   rw [eq_comm, Pos.next_eq_iff]
@@ -406,16 +425,110 @@ theorem Pos.slice_le_slice_iff {s : String} {p₀ p₁ : s.Pos} {q r : s.Pos}
   simp [Slice.Pos.le_iff, Pos.le_iff, Pos.Raw.le_iff] at h₁ h₁' ⊢
   omega
 
+theorem Slice.Pos.le_ofSlice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q ≤ Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Slice.Pos.slice q p₀ p₁ h₀ h₁ ≤ p := by
+  refine ⟨fun h => ⟨Std.le_trans h ofSlice_le, fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Slice.Pos.slice_ofSlice (pos := p), slice_le_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Slice.Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_le_ofSlice_iff] using h h₀
+    · exact Std.le_of_lt (Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice)
+
+theorem Slice.Pos.ofSlice_lt_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p < q ↔ ∀ h₁, ∃ h₀, p < Slice.Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_le, le_ofSlice_iff]
+
+theorem Slice.Pos.lt_ofSlice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q < Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Slice.Pos.slice q p₀ p₁ h₀ h₁ < p := by
+  refine ⟨fun h => ⟨Std.le_of_lt (Std.lt_of_lt_of_le h ofSlice_le), fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Slice.Pos.slice_ofSlice (pos := p), slice_lt_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Slice.Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_lt_ofSlice_iff] using h h₀
+    · exact Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice
+
+theorem Slice.Pos.ofSlice_le_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p ≤ q ↔ ∀ h₁, ∃ h₀, p ≤ Slice.Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_lt, lt_ofSlice_iff]
+
+theorem Pos.le_ofSlice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q ≤ Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Pos.slice q p₀ p₁ h₀ h₁ ≤ p := by
+  refine ⟨fun h => ⟨Std.le_trans h ofSlice_le, fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Pos.slice_ofSlice (pos := p), slice_le_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_le_ofSlice_iff] using h h₀
+    · exact Std.le_of_lt (Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice)
+
+theorem Pos.ofSlice_lt_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p < q ↔ ∀ h₁, ∃ h₀, p < Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_le, le_ofSlice_iff]
+
+theorem Pos.lt_ofSlice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    q < Pos.ofSlice p ↔ ∃ h₁, ∀ h₀, Pos.slice q p₀ p₁ h₀ h₁ < p := by
+  refine ⟨fun h => ⟨Std.le_of_lt (Std.lt_of_lt_of_le h ofSlice_le), fun h' => ?_⟩, fun ⟨h₁, h⟩ => ?_⟩
+  · simp only [← Pos.slice_ofSlice (pos := p), slice_lt_slice_iff]
+    simpa
+  · by_cases h₀ : p₀ ≤ q
+    · simpa only [← Pos.ofSlice_slice (h₁ := h₀) (h₂ := h₁), ofSlice_lt_ofSlice_iff] using h h₀
+    · exact Std.lt_of_lt_of_le (Std.not_le.1 h₀) le_ofSlice
+
+theorem Pos.ofSlice_le_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} :
+    Pos.ofSlice p ≤ q ↔ ∀ h₁, ∃ h₀, p ≤ Pos.slice q p₀ p₁ h₀ h₁ := by
+  simp [← Std.not_lt, lt_ofSlice_iff]
+
+theorem Slice.Pos.slice_le_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Slice.Pos.slice q p₀ p₁ h₀ h₁ ≤ p ↔ q ≤ Pos.ofSlice p := by
+  simp [le_ofSlice_iff, h₀, h₁]
+
+theorem Slice.Pos.lt_slice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p < Slice.Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p < q := by
+  simp [ofSlice_lt_iff, h₀, h₁]
+
+theorem Slice.Pos.slice_lt_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Slice.Pos.slice q p₀ p₁ h₀ h₁ < p ↔ q < Pos.ofSlice p := by
+  simp [lt_ofSlice_iff, h₀, h₁]
+
+theorem Slice.Pos.le_slice_iff {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p ≤ Slice.Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p ≤ q := by
+  simp [ofSlice_le_iff, h₀, h₁]
+
+theorem Pos.slice_le_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Pos.slice q p₀ p₁ h₀ h₁ ≤ p ↔ q ≤ Pos.ofSlice p := by
+  simp [le_ofSlice_iff, h₀, h₁]
+
+theorem Pos.lt_slice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p < Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p < q := by
+  simp [ofSlice_lt_iff, h₀, h₁]
+
+theorem Pos.slice_lt_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    Pos.slice q p₀ p₁ h₀ h₁ < p ↔ q < Pos.ofSlice p := by
+  simp [lt_ofSlice_iff, h₀, h₁]
+
+theorem Pos.le_slice_iff {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos} {q : s.Pos} {h₀ h₁} :
+    p ≤ Pos.slice q p₀ p₁ h₀ h₁ ↔ Pos.ofSlice p ≤ q := by
+  simp [ofSlice_le_iff, h₀, h₁]
+
 theorem Slice.Pos.ofSlice_ne_endPos {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
     (h : p ≠ (s.slice p₀ p₁ h).endPos) : Pos.ofSlice p ≠ s.endPos := by
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₁))
   simpa [← lt_endPos_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Slice.Pos.ofSlice_ne_startPos {s : Slice} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
+    (h : p ≠ (s.slice p₀ p₁ h).startPos) : Pos.ofSlice p ≠ s.startPos := by
+  refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
+  simpa [← startPos_lt_iff, ← ofSlice_lt_ofSlice_iff] using h
+
 theorem Pos.ofSlice_ne_endPos {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
     (h : p ≠ (s.slice p₀ p₁ h).endPos) : Pos.ofSlice p ≠ s.endPos := by
   refine (lt_endPos_iff _).1 (Std.lt_of_lt_of_le ?_ (le_endPos p₁))
   simpa [← Slice.Pos.lt_endPos_iff, ← ofSlice_lt_ofSlice_iff] using h
 
+theorem Pos.ofSlice_ne_startPos {s : String} {p₀ p₁ : s.Pos} {h} {p : (s.slice p₀ p₁ h).Pos}
+    (h : p ≠ (s.slice p₀ p₁ h).startPos) : Pos.ofSlice p ≠ s.startPos := by
+  refine (startPos_lt_iff _).1 (Std.lt_of_le_of_lt (startPos_le p₀) ?_)
+  simpa [← Slice.Pos.startPos_lt_iff, ← ofSlice_lt_ofSlice_iff] using h
+
 @[simp]
 theorem Slice.Pos.offset_le_rawEndPos {s : Slice} {p : s.Pos} :
     p.offset ≤ s.rawEndPos :=

src/Init/Data/String/Lemmas/Pattern/Basic.lean

@@ -19,6 +19,7 @@ import Init.Data.Order.Lemmas
 import Init.ByCases
 import Init.Data.Option.Lemmas
 import Init.Data.Iterators.Lemmas.Consumers.Collect
+import Init.Data.String.Lemmas.FindPos
 
 set_option doc.verso true
 
@@ -31,19 +32,20 @@ This file develops basic theory around searching in strings.
 
 We provide a typeclass for providing semantics to a pattern and then define the relevant notions
 of matching a pattern that let us state compatibility typeclasses for {name}`ForwardPattern` and
-{name}`ToForwardSearcher`. These typeclasses can then be required by correctness results for
-string functions which are implemented using the pattern framework.
+{name}`ToForwardSearcher` as well as their backwards variants. These typeclasses can then be
+required by correctness results for string functions which are implemented using the pattern
+framework.
 -/
 
 /--
 This data-carrying typeclass is used to give semantics to a pattern type that implements
 {name}`ForwardPattern` and/or {name}`ToForwardSearcher` by providing an abstract, not necessarily
-decidable {name}`ForwardPatternModel.Matches` predicate that implementates of {name}`ForwardPattern`
+decidable {name}`PatternModel.Matches` predicate that implementates of {name}`ForwardPattern`
 and {name}`ToForwardSearcher` can be validated against.
 
 Correctness results for generic functions relying on the pattern infrastructure, for example the
 correctness result for {name (scope := "Init.Data.String.Slice")}`String.Slice.split`, are then
-stated in terms of {name}`ForwardPatternModel.Matches`, and can be specialized to specific patterns
+stated in terms of {name}`PatternModel.Matches`, and can be specialized to specific patterns
 from there.
 
 The corresponding compatibility typeclasses are
@@ -59,7 +61,7 @@ searching.
 This means that pattern types that allow searching for the empty string will have to special-case
 the empty string in their correctness statements.
 -/
-class ForwardPatternModel {ρ : Type} (pat : ρ) : Type where
+class PatternModel {ρ : Type} (pat : ρ) : Type where
   /-- The predicate that says which strings match the pattern. -/
   Matches : String → Prop
   not_matches_empty : ¬ Matches ""
@@ -69,49 +71,72 @@ Predicate stating that the region between the start of the slice {name}`s` and t
 {name}`endPos` matches the pattern {name}`pat`. Note that there might be a longer match, see
 {name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.IsLongestMatch`.
 -/
-structure IsMatch (pat : ρ) [ForwardPatternModel pat] {s : Slice} (endPos : s.Pos) : Prop where
-  matches_copy : ForwardPatternModel.Matches pat (s.sliceTo endPos).copy
+structure IsMatch (pat : ρ) [PatternModel pat] {s : Slice} (endPos : s.Pos) : Prop where
+  matches_copy : PatternModel.Matches pat (s.sliceTo endPos).copy
 
-theorem IsMatch.ne_startPos {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos : s.Pos}
+theorem IsMatch.ne_startPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsMatch pat pos) : pos ≠ s.startPos := by
   intro hc
-  apply ForwardPatternModel.not_matches_empty (pat := pat)
+  apply PatternModel.not_matches_empty (pat := pat)
   simpa [hc] using h.matches_copy
 
-theorem isMatch_iff {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos : s.Pos} :
-    IsMatch pat pos ↔ ForwardPatternModel.Matches pat (s.sliceTo pos).copy :=
+theorem isMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsMatch pat pos ↔ PatternModel.Matches pat (s.sliceTo pos).copy :=
   ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩
 
-theorem isMatch_iff_exists_splits {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos : s.Pos} :
-    IsMatch pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ t₂ ∧ ForwardPatternModel.Matches pat t₁ := by
+theorem isMatch_iff_exists_splits {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsMatch pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ t₂ ∧ PatternModel.Matches pat t₁ := by
   rw [isMatch_iff]
   refine ⟨fun h => ⟨_, _, pos.splits, h⟩, fun ⟨t₁, t₂, h₁, h₂⟩ => ?_⟩
   rwa [h₁.eq_left pos.splits] at h₂
 
+/--
+Predicate stating that the region between the position {name}`startPos` and the end of the slice
+{name}`s` matches the pattern {name}`pat`. Note that there might be a longer match.
+-/
+structure IsRevMatch (pat : ρ) [PatternModel pat] {s : Slice} (startPos : s.Pos) : Prop where
+  matches_copy : PatternModel.Matches pat (s.sliceFrom startPos).copy
+
+theorem IsRevMatch.ne_endPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+    (h : IsRevMatch pat pos) : pos ≠ s.endPos := by
+  intro hc
+  apply PatternModel.not_matches_empty (pat := pat)
+  simpa [hc] using h.matches_copy
+
+theorem isRevMatch_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsRevMatch pat pos ↔ PatternModel.Matches pat (s.sliceFrom pos).copy :=
+  ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩
+
+theorem isRevMatch_iff_exists_splits {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsRevMatch pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ t₂ ∧ PatternModel.Matches pat t₂ := by
+  rw [isRevMatch_iff]
+  refine ⟨fun h => ⟨_, _, pos.splits, h⟩, fun ⟨t₁, t₂, h₁, h₂⟩ => ?_⟩
+  rwa [h₁.eq_right pos.splits] at h₂
+
 /--
 Predicate stating that the region between the start of the slice {name}`s` and the position
-{name}`endPos` matches that pattern {name}`pat`, and that there is no longer match starting at the
+{name}`pos` matches the pattern {name}`pat`, and that there is no longer match starting at the
 beginning of the slice. This is what a correct matcher should match.
 
 In some cases, being a match and being a longest match will coincide, see
-{name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.Model.NoPrefixForwardPatternModel`.
+{name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.Model.NoPrefixPatternModel`.
 -/
-structure IsLongestMatch (pat : ρ) [ForwardPatternModel pat] {s : Slice} (pos : s.Pos) where
+structure IsLongestMatch (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.Pos) where
   isMatch : IsMatch pat pos
   not_isMatch : ∀ pos', pos < pos' → ¬ IsMatch pat pos'
 
-theorem IsLongestMatch.ne_startPos {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos : s.Pos}
+theorem IsLongestMatch.ne_startPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
     (h : IsLongestMatch pat pos) : pos ≠ s.startPos :=
   h.isMatch.ne_startPos
 
-theorem IsLongestMatch.eq {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos pos' : s.Pos}
+theorem IsLongestMatch.eq {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
     (h : IsLongestMatch pat pos) (h' : IsLongestMatch pat pos') : pos = pos' := by
   apply Std.le_antisymm
   · exact Std.not_lt.1 (fun hlt => h'.not_isMatch _ hlt h.isMatch)
   · exact Std.not_lt.1 (fun hlt => h.not_isMatch _ hlt h'.isMatch)
 
 open Classical in
-theorem IsMatch.exists_isLongestMatch {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos : s.Pos} :
+theorem IsMatch.exists_isLongestMatch {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
     IsMatch pat pos → ∃ (pos' : s.Pos), IsLongestMatch pat pos' := by
   induction pos using WellFounded.induction Pos.wellFounded_gt with | h pos ih
   intro h₁
@@ -120,61 +145,118 @@ theorem IsMatch.exists_isLongestMatch {pat : ρ} [ForwardPatternModel pat] {s :
     exact ih _ hp₁ hp₂
   · exact ⟨pos, ⟨h₁, fun p' hp₁ hp₂ => h₂ ⟨_, hp₁, hp₂⟩⟩⟩
 
-theorem IsLongestMatch.le_of_isMatch {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos pos' : s.Pos}
+theorem IsLongestMatch.le_of_isMatch {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
     (h : IsLongestMatch pat pos) (h' : IsMatch pat pos') : pos' ≤ pos :=
   Std.not_lt.1 (fun hlt => h.not_isMatch _ hlt h')
 
+/--
+Predicate stating that the region between the start of the slice {name}`s` and the position
+{name}`pos` matches the patten {name}`pat`, and that there is no longer match starting at the
+beginning of the slice. This is what a correct matcher should match.
+
+In some cases, being a match and being a longest match will coincide, see
+{name (scope := "Init.Data.String.Lemmas.Pattern.Basic")}`String.Slice.Pattern.Model.NoPrefixPatternModel`.
+-/
+structure IsLongestRevMatch (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.Pos) where
+  isRevMatch : IsRevMatch pat pos
+  not_isRevMatch : ∀ pos', pos' < pos → ¬ IsRevMatch pat pos'
+
+theorem IsLongestRevMatch.ne_endPos {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos}
+    (h : IsLongestRevMatch pat pos) : pos ≠ s.endPos :=
+  h.isRevMatch.ne_endPos
+
+theorem IsLongestRevMatch.eq {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
+    (h : IsLongestRevMatch pat pos) (h' : IsLongestRevMatch pat pos') : pos = pos' := by
+  apply Std.le_antisymm
+  · exact Std.not_lt.1 (fun hlt => h.not_isRevMatch _ hlt h'.isRevMatch)
+  · exact Std.not_lt.1 (fun hlt => h'.not_isRevMatch _ hlt h.isRevMatch)
+
+open Classical in
+theorem IsRevMatch.exists_isLongestRevMatch {pat : ρ} [PatternModel pat] {s : Slice} {pos : s.Pos} :
+    IsRevMatch pat pos → ∃ (pos' : s.Pos), IsLongestRevMatch pat pos' := by
+  induction pos using WellFounded.induction Pos.wellFounded_lt with | h pos ih
+  intro h₁
+  by_cases h₂ : ∃ pos', pos' < pos ∧ IsRevMatch pat pos'
+  · obtain ⟨pos', hp₁, hp₂⟩ := h₂
+    exact ih _ hp₁ hp₂
+  · exact ⟨pos, ⟨h₁, fun p' hp₁ hp₂ => h₂ ⟨_, hp₁, hp₂⟩⟩⟩
+
+theorem IsLongestRevMatch.le_of_isRevMatch {pat : ρ} [PatternModel pat] {s : Slice} {pos pos' : s.Pos}
+    (h : IsLongestRevMatch pat pos) (h' : IsRevMatch pat pos') : pos ≤ pos' :=
+  Std.not_lt.1 (fun hlt => h.not_isRevMatch _ hlt h')
+
 /--
 Predicate stating that a match for a given pattern is never a proper prefix of another match.
 
 This implies that the notion of match and longest match coincide.
 -/
-class NoPrefixForwardPatternModel {ρ : Type} (pat : ρ) [ForwardPatternModel pat] : Prop where
-  eq_empty (s t) : ForwardPatternModel.Matches pat s → ForwardPatternModel.Matches pat (s ++ t) → t = ""
+class NoPrefixPatternModel {ρ : Type} (pat : ρ) [PatternModel pat] : Prop where
+  eq_empty (s t) : PatternModel.Matches pat s → PatternModel.Matches pat (s ++ t) → t = ""
 
-theorem NoPrefixForwardPatternModel.of_length_eq {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
-    (h : ∀ s t, ForwardPatternModel.Matches pat s → ForwardPatternModel.Matches pat t → s.length = t.length) :
-    NoPrefixForwardPatternModel pat where
+theorem NoPrefixPatternModel.of_length_eq {ρ : Type} {pat : ρ} [PatternModel pat]
+    (h : ∀ s t, PatternModel.Matches pat s → PatternModel.Matches pat t → s.length = t.length) :
+    NoPrefixPatternModel pat where
   eq_empty s t hs ht := by simpa using h s _ hs ht
 
-theorem isLongestMatch_iff_isMatch {ρ : Type} (pat : ρ) [ForwardPatternModel pat] [NoPrefixForwardPatternModel pat]
+theorem isLongestMatch_iff_isMatch {ρ : Type} (pat : ρ) [PatternModel pat] [NoPrefixPatternModel pat]
     {s : Slice} {pos : s.Pos} : IsLongestMatch pat pos ↔ IsMatch pat pos := by
   refine ⟨fun h => h.isMatch, fun h => ⟨h, fun pos' hpos' hm => ?_⟩⟩
   obtain ⟨t₁, t₂, ht₁, ht₂⟩ := isMatch_iff_exists_splits.1 h
   obtain ⟨t₁', t₂', ht₁', ht₂'⟩ := isMatch_iff_exists_splits.1 hm
   obtain ⟨t₅, ht₅, ht₅', ht₅''⟩ := (ht₁.lt_iff_exists_eq_append ht₁').1 hpos'
-  exact ht₅ (NoPrefixForwardPatternModel.eq_empty _ _ ht₂ (ht₅' ▸ ht₂'))
+  exact ht₅ (NoPrefixPatternModel.eq_empty _ _ ht₂ (ht₅' ▸ ht₂'))
+
+/--
+Predicate stating that a match for a given pattern is never a proper suffix of another match.
+
+This implies that the notion of reverse match and longest reverse match coincide.
+-/
+class NoSuffixPatternModel {ρ : Type} (pat : ρ) [PatternModel pat] : Prop where
+  eq_empty (s t) : PatternModel.Matches pat t → PatternModel.Matches pat (s ++ t) → s = ""
+
+theorem NoSuffixPatternModel.of_length_eq {ρ : Type} {pat : ρ} [PatternModel pat]
+    (h : ∀ s t, PatternModel.Matches pat s → PatternModel.Matches pat t → s.length = t.length) :
+    NoSuffixPatternModel pat where
+  eq_empty s t hs ht := by simpa using h t _ hs ht
+
+theorem isLongestRevMatch_iff_isRevMatch {ρ : Type} (pat : ρ) [PatternModel pat] [NoSuffixPatternModel pat]
+    {s : Slice} {pos : s.Pos} : IsLongestRevMatch pat pos ↔ IsRevMatch pat pos := by
+  refine ⟨fun h => h.isRevMatch, fun h => ⟨h, fun pos' hpos' hm => ?_⟩⟩
+  obtain ⟨t₁, t₂, ht₁, ht₂⟩ := isRevMatch_iff_exists_splits.1 h
+  obtain ⟨t₁', t₂', ht₁', ht₂'⟩ := isRevMatch_iff_exists_splits.1 hm
+  obtain ⟨t₅, ht₅, ht₅', ht₅''⟩ := (ht₁'.lt_iff_exists_eq_append ht₁).1 hpos'
+  exact ht₅ (NoSuffixPatternModel.eq_empty _ _ ht₂ (ht₅'' ▸ ht₂'))
 
 /--
 Predicate stating that the slice formed by {name}`startPos` and {name}`endPos` contains is a match
 of {name}`pat` in {name}`s` and it is longest among matches starting at {name}`startPos`.
 -/
-structure IsLongestMatchAt (pat : ρ) [ForwardPatternModel pat] {s : Slice} (startPos endPos : s.Pos) : Prop where
+structure IsLongestMatchAt (pat : ρ) [PatternModel pat] {s : Slice} (startPos endPos : s.Pos) : Prop where
   le : startPos ≤ endPos
   isLongestMatch_sliceFrom : IsLongestMatch pat (Slice.Pos.sliceFrom _ _ le)
 
-theorem isLongestMatchAt_iff {pat : ρ} [ForwardPatternModel pat] {s : Slice} {pos₁ pos₂ : s.Pos} :
+theorem isLongestMatchAt_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt pat pos₁ pos₂ ↔
       ∃ (h : pos₁ ≤ pos₂), IsLongestMatch pat (Slice.Pos.sliceFrom _ _ h) :=
   ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
 
-theorem IsLongestMatchAt.lt {pat : ρ} [ForwardPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+theorem IsLongestMatchAt.lt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
     (h : IsLongestMatchAt pat startPos endPos) : startPos < endPos := by
   have := h.isLongestMatch_sliceFrom.ne_startPos
   rw [← Pos.startPos_lt_iff, ← Slice.Pos.ofSliceFrom_lt_ofSliceFrom_iff] at this
   simpa
 
-theorem IsLongestMatchAt.eq {pat : ρ} [ForwardPatternModel pat] {s : Slice} {startPos endPos endPos' : s.Pos}
+theorem IsLongestMatchAt.eq {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos endPos' : s.Pos}
     (h : IsLongestMatchAt pat startPos endPos) (h' : IsLongestMatchAt pat startPos endPos') :
     endPos = endPos' := by
   simpa using h.isLongestMatch_sliceFrom.eq h'.isLongestMatch_sliceFrom
 
-private theorem isLongestMatch_of_eq {pat : ρ} [ForwardPatternModel pat] {s t : Slice}
+private theorem isLongestMatch_of_eq {pat : ρ} [PatternModel pat] {s t : Slice}
     {pos : s.Pos} {pos' : t.Pos} (h_eq : s = t) (h_pos : pos.offset = pos'.offset)
     (hm : IsLongestMatch pat pos) : IsLongestMatch pat pos' := by
   subst h_eq; exact (Slice.Pos.ext h_pos) ▸ hm
 
-theorem isLongestMatchAt_iff_isLongestMatchAt_ofSliceFrom {pat : ρ} [ForwardPatternModel pat]
+theorem isLongestMatchAt_iff_isLongestMatchAt_ofSliceFrom {pat : ρ} [PatternModel pat]
     {s : Slice} {base : s.Pos} {startPos endPos : (s.sliceFrom base).Pos} :
     IsLongestMatchAt pat startPos endPos ↔ IsLongestMatchAt pat (Pos.ofSliceFrom startPos) (Pos.ofSliceFrom endPos) := by
   constructor
@@ -187,35 +269,88 @@ theorem isLongestMatchAt_iff_isLongestMatchAt_ofSliceFrom {pat : ρ} [ForwardPat
     exact isLongestMatch_of_eq Slice.sliceFrom_sliceFrom.symm
       (by simp [Pos.Raw.ext_iff]; omega) h.isLongestMatch_sliceFrom
 
-theorem IsLongestMatch.isLongestMatchAt_ofSliceFrom {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem IsLongestMatch.isLongestMatchAt_ofSliceFrom {pat : ρ} [PatternModel pat] {s : Slice}
     {p₀ : s.Pos} {pos : (s.sliceFrom p₀).Pos} (h : IsLongestMatch pat pos) :
     IsLongestMatchAt pat p₀ (Slice.Pos.ofSliceFrom pos) where
   le := Slice.Pos.le_ofSliceFrom
   isLongestMatch_sliceFrom := by simpa
 
 @[simp]
-theorem isLongestMatchAt_startPos_iff {pat : ρ} [ForwardPatternModel pat] {s : Slice} {endPos : s.Pos} :
+theorem isLongestMatchAt_startPos_iff {pat : ρ} [PatternModel pat] {s : Slice} {endPos : s.Pos} :
     IsLongestMatchAt pat s.startPos endPos ↔ IsLongestMatch pat endPos := by
   simpa [isLongestMatchAt_iff] using
     ⟨fun h => isLongestMatch_of_eq (by simp) (by simp) h,
      fun h => isLongestMatch_of_eq (by simp) (by simp) h⟩
 
+/--
+Predicate stating that the slice formed by {name}`startPos` and {name}`endPos` contains is a match
+of {name}`pat` in {name}`s` and it is longest among matches ending at {name}`endPos`.
+-/
+structure IsLongestRevMatchAt (pat : ρ) [PatternModel pat] {s : Slice} (startPos endPos : s.Pos) : Prop where
+  le : startPos ≤ endPos
+  isLongestRevMatch_sliceTo : IsLongestRevMatch pat (Slice.Pos.sliceTo _ _ le)
+
+theorem isLongestRevMatchAt_iff {pat : ρ} [PatternModel pat] {s : Slice} {pos₁ pos₂ : s.Pos} :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔
+      ∃ (h : pos₁ ≤ pos₂), IsLongestRevMatch pat (Slice.Pos.sliceTo _ _ h) :=
+  ⟨fun ⟨h, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'⟩⟩
+
+theorem IsLongestRevMatchAt.lt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAt pat startPos endPos) : startPos < endPos := by
+  have := h.isLongestRevMatch_sliceTo.ne_endPos
+  rw [← Pos.lt_endPos_iff, ← Slice.Pos.ofSliceTo_lt_ofSliceTo_iff] at this
+  simpa
+
+theorem IsLongestRevMatchAt.eq {pat : ρ} [PatternModel pat] {s : Slice} {startPos startPos' endPos : s.Pos}
+    (h : IsLongestRevMatchAt pat startPos endPos) (h' : IsLongestRevMatchAt pat startPos' endPos) :
+    startPos = startPos' := by
+  simpa using h.isLongestRevMatch_sliceTo.eq h'.isLongestRevMatch_sliceTo
+
+private theorem isLongestRevMatch_of_eq {pat : ρ} [PatternModel pat] {s t : Slice}
+    {pos : s.Pos} {pos' : t.Pos} (h_eq : s = t) (h_pos : pos.offset = pos'.offset)
+    (hm : IsLongestRevMatch pat pos) : IsLongestRevMatch pat pos' := by
+  subst h_eq; exact (Slice.Pos.ext h_pos) ▸ hm
+
+theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_ofSliceTo {pat : ρ} [PatternModel pat]
+    {s : Slice} {base : s.Pos} {startPos endPos : (s.sliceTo base).Pos} :
+    IsLongestRevMatchAt pat startPos endPos ↔ IsLongestRevMatchAt pat (Pos.ofSliceTo startPos) (Pos.ofSliceTo endPos) := by
+  constructor
+  · intro h
+    refine ⟨Slice.Pos.ofSliceTo_le_ofSliceTo_iff.mpr h.le, ?_⟩
+    exact isLongestRevMatch_of_eq Slice.sliceTo_sliceTo (by simp) h.isLongestRevMatch_sliceTo
+  · intro h
+    refine ⟨Slice.Pos.ofSliceTo_le_ofSliceTo_iff.mp h.le, ?_⟩
+    exact isLongestRevMatch_of_eq Slice.sliceTo_sliceTo.symm (by simp) h.isLongestRevMatch_sliceTo
+
+theorem IsLongestRevMatch.isLongestRevMatchAt_ofSliceTo {pat : ρ} [PatternModel pat] {s : Slice}
+    {p₀ : s.Pos} {pos : (s.sliceTo p₀).Pos} (h : IsLongestRevMatch pat pos) :
+    IsLongestRevMatchAt pat (Slice.Pos.ofSliceTo pos) p₀ where
+  le := Slice.Pos.ofSliceTo_le
+  isLongestRevMatch_sliceTo := by simpa
+
+@[simp]
+theorem isLongestRevMatchAt_endPos_iff {pat : ρ} [PatternModel pat] {s : Slice} {startPos : s.Pos} :
+    IsLongestRevMatchAt pat startPos s.endPos ↔ IsLongestRevMatch pat startPos := by
+  simpa [isLongestRevMatchAt_iff] using
+    ⟨fun h => isLongestRevMatch_of_eq (by simp) (by simp) h,
+     fun h => isLongestRevMatch_of_eq (by simp) (by simp) h⟩
+
 /--
 Predicate stating that there is a (longest) match starting at the given position.
 -/
-structure MatchesAt (pat : ρ) [ForwardPatternModel pat] {s : Slice} (pos : s.Pos) : Prop where
+structure MatchesAt (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.Pos) : Prop where
   exists_isLongestMatchAt : ∃ endPos, IsLongestMatchAt pat pos endPos
 
-theorem matchesAt_iff_exists_isLongestMatchAt {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem matchesAt_iff_exists_isLongestMatchAt {pat : ρ} [PatternModel pat] {s : Slice}
     {pos : s.Pos} : MatchesAt pat pos ↔ ∃ endPos, IsLongestMatchAt pat pos endPos :=
   ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩
 
-theorem matchesAt_iff_exists_isLongestMatch {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem matchesAt_iff_exists_isLongestMatch {pat : ρ} [PatternModel pat] {s : Slice}
     {pos : s.Pos} :
     MatchesAt pat pos ↔ ∃ (endPos : s.Pos), ∃ h, IsLongestMatch pat (pos.sliceFrom endPos h) :=
   ⟨fun ⟨p, h⟩ => ⟨p, h.le, h.isLongestMatch_sliceFrom⟩, fun ⟨p, h₁, h₂⟩ => ⟨p, ⟨h₁, h₂⟩⟩⟩
 
-theorem matchesAt_iff_exists_isMatch {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem matchesAt_iff_exists_isMatch {pat : ρ} [PatternModel pat] {s : Slice}
     {pos : s.Pos} :
     MatchesAt pat pos ↔ ∃ (endPos : s.Pos), ∃ h, IsMatch pat (pos.sliceFrom endPos h) := by
   refine ⟨fun ⟨p, h⟩ => ⟨p, h.le, h.isLongestMatch_sliceFrom.isMatch⟩, fun ⟨p, h₁, h₂⟩ => ?_⟩
@@ -225,13 +360,13 @@ theorem matchesAt_iff_exists_isMatch {pat : ρ} [ForwardPatternModel pat] {s : S
      by simpa using hq⟩⟩
 
 @[simp]
-theorem not_matchesAt_endPos {pat : ρ} [ForwardPatternModel pat] {s : Slice} :
+theorem not_matchesAt_endPos {pat : ρ} [PatternModel pat] {s : Slice} :
     ¬ MatchesAt pat s.endPos := by
   simp only [matchesAt_iff_exists_isMatch, Pos.endPos_le, exists_prop_eq]
   intro h
   simpa [← Pos.ofSliceFrom_inj] using h.ne_startPos
 
-theorem matchesAt_iff_matchesAt_ofSliceFrom {pat : ρ} [ForwardPatternModel pat] {s : Slice} {base : s.Pos}
+theorem matchesAt_iff_matchesAt_ofSliceFrom {pat : ρ} [PatternModel pat] {s : Slice} {base : s.Pos}
     {pos : (s.sliceFrom base).Pos} : MatchesAt pat pos ↔ MatchesAt pat (Pos.ofSliceFrom pos) := by
   simp only [matchesAt_iff_exists_isLongestMatchAt]
   constructor
@@ -241,21 +376,66 @@ theorem matchesAt_iff_matchesAt_ofSliceFrom {pat : ρ} [ForwardPatternModel pat]
     exact ⟨base.sliceFrom endPos (Std.le_trans Slice.Pos.le_ofSliceFrom h.le),
            isLongestMatchAt_iff_isLongestMatchAt_ofSliceFrom.mpr (by simpa using h)⟩
 
-theorem IsLongestMatchAt.matchesAt {pat : ρ} [ForwardPatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+theorem IsLongestMatchAt.matchesAt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
     (h : IsLongestMatchAt pat startPos endPos) : MatchesAt pat startPos where
   exists_isLongestMatchAt := ⟨_, h⟩
 
+/--
+Predicate stating that there is a (longest) match ending at the given position.
+-/
+structure RevMatchesAt (pat : ρ) [PatternModel pat] {s : Slice} (pos : s.Pos) : Prop where
+  exists_isLongestRevMatchAt : ∃ startPos, IsLongestRevMatchAt pat startPos pos
+
+theorem revMatchesAt_iff_exists_isLongestRevMatchAt {pat : ρ} [PatternModel pat] {s : Slice}
+    {pos : s.Pos} : RevMatchesAt pat pos ↔ ∃ startPos, IsLongestRevMatchAt pat startPos pos :=
+  ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩
+
+theorem revMatchesAt_iff_exists_isLongestRevMatch {pat : ρ} [PatternModel pat] {s : Slice}
+    {pos : s.Pos} :
+    RevMatchesAt pat pos ↔ ∃ (startPos : s.Pos), ∃ h, IsLongestRevMatch pat (pos.sliceTo startPos h) :=
+  ⟨fun ⟨p, h⟩ => ⟨p, h.le, h.isLongestRevMatch_sliceTo⟩, fun ⟨p, h₁, h₂⟩ => ⟨p, ⟨h₁, h₂⟩⟩⟩
+
+theorem revMatchesAt_iff_exists_isRevMatch {pat : ρ} [PatternModel pat] {s : Slice}
+    {pos : s.Pos} :
+    RevMatchesAt pat pos ↔ ∃ (startPos : s.Pos), ∃ h, IsRevMatch pat (pos.sliceTo startPos h) := by
+  refine ⟨fun ⟨p, h⟩ => ⟨p, h.le, h.isLongestRevMatch_sliceTo.isRevMatch⟩, fun ⟨p, h₁, h₂⟩ => ?_⟩
+  obtain ⟨q, hq⟩ := h₂.exists_isLongestRevMatch
+  exact ⟨Pos.ofSliceTo q,
+    ⟨Std.le_trans (by simpa [← Pos.ofSliceTo_le_ofSliceTo_iff] using hq.le_of_isRevMatch h₂) h₁,
+     by simpa using hq⟩⟩
+
+@[simp]
+theorem not_revMatchesAt_startPos {pat : ρ} [PatternModel pat] {s : Slice} :
+    ¬ RevMatchesAt pat s.startPos := by
+  simp only [revMatchesAt_iff_exists_isRevMatch, Pos.le_startPos, exists_prop_eq]
+  intro h
+  simpa [← Pos.ofSliceTo_inj] using h.ne_endPos
+
+theorem revMatchesAt_iff_revMatchesAt_ofSliceto {pat : ρ} [PatternModel pat] {s : Slice} {base : s.Pos}
+    {pos : (s.sliceTo base).Pos} : RevMatchesAt pat pos ↔ RevMatchesAt pat (Pos.ofSliceTo pos) := by
+  simp only [revMatchesAt_iff_exists_isLongestRevMatchAt]
+  constructor
+  · rintro ⟨startPos, h⟩
+    exact ⟨Pos.ofSliceTo startPos, isLongestRevMatchAt_iff_isLongestRevMatchAt_ofSliceTo.mp h⟩
+  · rintro ⟨startPos, h⟩
+    exact ⟨base.sliceTo startPos (Std.le_trans h.le Slice.Pos.ofSliceTo_le),
+           isLongestRevMatchAt_iff_isLongestRevMatchAt_ofSliceTo.mpr (by simpa using h)⟩
+
+theorem IsLongestRevMatchAt.revMatchesAt {pat : ρ} [PatternModel pat] {s : Slice} {startPos endPos : s.Pos}
+    (h : IsLongestRevMatchAt pat startPos endPos) : RevMatchesAt pat endPos where
+  exists_isLongestRevMatchAt := ⟨_, h⟩
+
 open Classical in
 /--
 Noncomputable model function returning the end point of the longest match starting at the given
 position, or {lean}`none` if there is no match.
 -/
-noncomputable def matchAt? {ρ : Type} (pat : ρ) [ForwardPatternModel pat]
+noncomputable def matchAt? {ρ : Type} (pat : ρ) [PatternModel pat]
     {s : Slice} (startPos : s.Pos) : Option s.Pos :=
   if h : ∃ endPos, IsLongestMatchAt pat startPos endPos then some h.choose else none
 
 @[simp]
-theorem matchAt?_eq_some_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
+theorem matchAt?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat]
     {s : Slice} {startPos endPos : s.Pos} :
     matchAt? pat startPos = some endPos ↔ IsLongestMatchAt pat startPos endPos := by
   fun_cases matchAt? with
@@ -263,40 +443,92 @@ theorem matchAt?_eq_some_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
   | case2 => simp_all
 
 @[simp]
-theorem matchAt?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
+theorem matchAt?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat]
     {s : Slice} {startPos : s.Pos} :
     matchAt? pat startPos = none ↔ ¬ MatchesAt pat startPos := by
   fun_cases matchAt? with
   | case1 h => simpa using ⟨h⟩
   | case2 h => simpa using fun ⟨h'⟩ => h h'
 
+open Classical in
 /--
-Predicate stating compatibility between {name}`ForwardPatternModel` and {name}`ForwardPattern`.
+Noncomputable model function returning the start point of the longest match ending at the given
+position, or {lean}`none` if there is no match.
+-/
+noncomputable def revMatchAt? {ρ : Type} (pat : ρ) [PatternModel pat]
+    {s : Slice} (endPos : s.Pos) : Option s.Pos :=
+  if h : ∃ startPos, IsLongestRevMatchAt pat startPos endPos then some h.choose else none
+
+@[simp]
+theorem revMatchAt?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat]
+    {s : Slice} {startPos endPos : s.Pos} :
+    revMatchAt? pat endPos = some startPos ↔ IsLongestRevMatchAt pat startPos endPos := by
+  fun_cases revMatchAt? with
+  | case1 h => simpa using ⟨by rintro rfl; exact h.choose_spec, fun h' => h.choose_spec.eq h'⟩
+  | case2 => simp_all
+
+@[simp]
+theorem revMatchAt?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat]
+    {s : Slice} {endPos : s.Pos} :
+    revMatchAt? pat endPos = none ↔ ¬ RevMatchesAt pat endPos := by
+  fun_cases revMatchAt? with
+  | case1 h => simpa using ⟨h⟩
+  | case2 h => simpa using fun ⟨h'⟩ => h h'
+
+/--
+Predicate stating compatibility between {name}`PatternModel` and {name}`ForwardPattern`.
 
 This extends {name}`LawfulForwardPattern`, but it is much stronger because it forces the
 {name}`ForwardPattern` to match the longest prefix of the given slice that matches the property
-supplied by the {name}`ForwardPatternModel` instance.
+supplied by the {name}`PatternModel` instance.
 -/
 class LawfulForwardPatternModel {ρ : Type} (pat : ρ) [ForwardPattern pat]
-    [ForwardPatternModel pat] : Prop extends LawfulForwardPattern pat where
+    [PatternModel pat] : Prop extends LawfulForwardPattern pat where
   skipPrefix?_eq_some_iff (pos) : ForwardPattern.skipPrefix? pat s = some pos ↔ IsLongestMatch pat pos
 
-open Classical in
-theorem LawfulForwardPatternModel.skipPrefix?_sliceFrom_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPattern pat] [ForwardPatternModel pat]
+theorem LawfulForwardPatternModel.skipPrefix?_sliceFrom_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPattern pat] [PatternModel pat]
     [LawfulForwardPatternModel pat] {s : Slice} {p₀ : s.Pos} :
     ForwardPattern.skipPrefix? pat (s.sliceFrom p₀) = none ↔ ¬ MatchesAt pat p₀ := by
+  classical
   rw [← Decidable.not_iff_not]
   simp [Option.ne_none_iff_exists', LawfulForwardPatternModel.skipPrefix?_eq_some_iff]
   refine ⟨fun ⟨p, hp⟩ => ?_, fun ⟨p, hp⟩ => ?_⟩
   · exact ⟨Slice.Pos.ofSliceFrom p, hp.isLongestMatchAt_ofSliceFrom⟩
   · exact ⟨p₀.sliceFrom p hp.le, hp.isLongestMatch_sliceFrom⟩
 
-theorem LawfulForwardPatternModel.skipPrefix?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPattern pat] [ForwardPatternModel pat]
+theorem LawfulForwardPatternModel.skipPrefix?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPattern pat] [PatternModel pat]
     [LawfulForwardPatternModel pat] {s : Slice} :
     ForwardPattern.skipPrefix? pat s = none ↔ ¬ MatchesAt pat s.startPos := by
   conv => lhs; rw [← sliceFrom_startPos (s := s)]
   simp [skipPrefix?_sliceFrom_eq_none_iff]
 
+/--
+Predicate stating compatibility between {name}`PatternModel` and {name}`BackwardPattern`.
+
+This extends {name}`LawfulForwardPattern`, but it is much stronger because it forces the
+{name}`ForwardPattern` to match the longest prefix of the given slice that matches the property
+supplied by the {name}`PatternModel` instance.
+-/
+class LawfulBackwardPatternModel {ρ : Type} (pat : ρ) [BackwardPattern pat]
+    [PatternModel pat] : Prop extends LawfulBackwardPattern pat where
+  skipSuffix?_eq_some_iff (pos) : BackwardPattern.skipSuffix? pat s = some pos ↔ IsLongestRevMatch pat pos
+
+theorem LawfulBackwardPatternModel.skipSuffix?_sliceTo_eq_none_iff {ρ : Type} {pat : ρ} [BackwardPattern pat] [PatternModel pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} {p₀ : s.Pos} :
+    BackwardPattern.skipSuffix? pat (s.sliceTo p₀) = none ↔ ¬ RevMatchesAt pat p₀ := by
+  classical
+  rw [← Decidable.not_iff_not]
+  simp [Option.ne_none_iff_exists', LawfulBackwardPatternModel.skipSuffix?_eq_some_iff]
+  refine ⟨fun ⟨p, hp⟩ => ?_, fun ⟨p, hp⟩ => ?_⟩
+  · exact ⟨Slice.Pos.ofSliceTo p, hp.isLongestRevMatchAt_ofSliceTo⟩
+  · exact ⟨p₀.sliceTo p hp.le, hp.isLongestRevMatch_sliceTo⟩
+
+theorem LawfulBackwardPatternModel.skipSuffix?_eq_none_iff {ρ : Type} {pat : ρ} [BackwardPattern pat] [PatternModel pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} :
+    BackwardPattern.skipSuffix? pat s = none ↔ ¬ RevMatchesAt pat s.endPos := by
+  conv => lhs; rw [← sliceTo_endPos (s := s)]
+  simp [skipSuffix?_sliceTo_eq_none_iff]
+
 /--
 Inductive predicate stating that a list of search steps represents a valid search from a given
 position in a slice.
@@ -306,7 +538,7 @@ matches.
 
 Hence, this predicate determines the list of search steps up to grouping of rejections.
 -/
-inductive IsValidSearchFrom (pat : ρ) [ForwardPatternModel pat] {s : Slice} :
+inductive IsValidSearchFrom (pat : ρ) [PatternModel pat] {s : Slice} :
     s.Pos → List (SearchStep s) → Prop where
   | endPos : IsValidSearchFrom pat s.endPos []
   | matched {startPos endPos : s.Pos} :
@@ -316,14 +548,14 @@ inductive IsValidSearchFrom (pat : ρ) [ForwardPatternModel pat] {s : Slice} :
       (∀ pos, startPos ≤ pos → pos < endPos → ¬ MatchesAt pat pos) →
       IsValidSearchFrom pat endPos l → IsValidSearchFrom pat startPos (.rejected startPos endPos :: l)
 
-theorem IsValidSearchFrom.matched_of_eq {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem IsValidSearchFrom.matched_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
     {startPos startPos' endPos : s.Pos} {l : List (SearchStep s)} (h₁ : IsValidSearchFrom pat endPos l)
     (h₂ : IsLongestMatchAt pat startPos' endPos)
     (h₃ : startPos = startPos') : IsValidSearchFrom pat startPos' (.matched startPos endPos :: l) := by
   cases h₃
   exact IsValidSearchFrom.matched h₂ h₁
 
-theorem IsValidSearchFrom.mismatched_of_eq {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem IsValidSearchFrom.mismatched_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
     {startPos startPos' endPos : s.Pos} {l : List (SearchStep s)} (h₁ : IsValidSearchFrom pat endPos l)
       (h₀ : startPos' < endPos)
       (h₂ : ∀ pos, startPos' ≤ pos → pos < endPos → ¬ MatchesAt pat pos) (h₃ : startPos = startPos') :
@@ -331,7 +563,7 @@ theorem IsValidSearchFrom.mismatched_of_eq {pat : ρ} [ForwardPatternModel pat]
   cases h₃
   exact IsValidSearchFrom.mismatched h₀ h₂ h₁
 
-theorem IsValidSearchFrom.endPos_of_eq {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+theorem IsValidSearchFrom.endPos_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
     {p : s.Pos} {l : List (SearchStep s)} (hp : p = s.endPos) (hl : l = []) :
     IsValidSearchFrom pat p l := by
   cases hp
@@ -339,18 +571,18 @@ theorem IsValidSearchFrom.endPos_of_eq {pat : ρ} [ForwardPatternModel pat] {s :
   exact IsValidSearchFrom.endPos
 
 /--
-Predicate stating compatibility between {name}`ForwardPatternModel` and {name}`ToForwardSearcher`.
+Predicate stating compatibility between {name}`PatternModel` and {name}`ToForwardSearcher`.
 
 We require the searcher to always match the longest match at the first position where the pattern
 matches; see {name}`IsValidSearchFrom`.
 -/
-class LawfulToForwardSearcherModel {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {σ : Slice → Type}
+class LawfulToForwardSearcherModel {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] : Prop where
   isValidSearchFrom_toList (s) : IsValidSearchFrom pat s.startPos (ToForwardSearcher.toSearcher pat s).toList
 
 theorem LawfulToForwardSearcherModel.defaultImplementation {pat : ρ} [ForwardPattern pat] [StrictForwardPattern pat]
-    [ForwardPatternModel pat] [LawfulForwardPatternModel pat] :
+    [PatternModel pat] [LawfulForwardPatternModel pat] :
     letI : ToForwardSearcher pat (ToForwardSearcher.DefaultForwardSearcher pat) := .defaultImplementation
     LawfulToForwardSearcherModel pat := by
   let inst : ToForwardSearcher pat (ToForwardSearcher.DefaultForwardSearcher pat) := .defaultImplementation
@@ -390,4 +622,97 @@ theorem LawfulToForwardSearcherModel.defaultImplementation {pat : ρ} [ForwardPa
     · split at heq <;> simp at heq
     · split at heq <;> simp at heq
 
+/--
+Inductive predicate stating that a list of search steps represents a valid backwards search from a
+given position in a slice.
+
+"Searching" here means always taking the longest match at the first position where the pattern
+matches.
+
+Hence, this predicate determines the list of search steps up to grouping of rejections.
+-/
+inductive IsValidRevSearchFrom (pat : ρ) [PatternModel pat] {s : Slice} :
+    s.Pos → List (SearchStep s) → Prop where
+  | startPos : IsValidRevSearchFrom pat s.startPos []
+  | matched {startPos endPos : s.Pos} :
+      IsLongestRevMatchAt pat startPos endPos → IsValidRevSearchFrom pat startPos l →
+      IsValidRevSearchFrom pat endPos (.matched startPos endPos :: l)
+  | mismatched {startPos endPos : s.Pos} : startPos < endPos →
+      (∀ pos, startPos < pos → pos ≤ endPos → ¬ RevMatchesAt pat pos) →
+      IsValidRevSearchFrom pat startPos l → IsValidRevSearchFrom pat endPos (.rejected startPos endPos :: l)
+
+theorem IsValidRevSearchFrom.matched_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos endPos' : s.Pos} {l : List (SearchStep s)} (h₁ : IsValidRevSearchFrom pat startPos l)
+    (h₂ : IsLongestRevMatchAt pat startPos endPos')
+    (h₃ : endPos = endPos') : IsValidRevSearchFrom pat endPos' (.matched startPos endPos :: l) := by
+  cases h₃
+  exact IsValidRevSearchFrom.matched h₂ h₁
+
+theorem IsValidRevSearchFrom.mismatched_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
+    {startPos endPos endPos' : s.Pos} {l : List (SearchStep s)} (h₁ : IsValidRevSearchFrom pat startPos l)
+      (h₀ : startPos < endPos')
+      (h₂ : ∀ pos, startPos < pos → pos ≤ endPos' → ¬ RevMatchesAt pat pos) (h₃ : endPos = endPos') :
+      IsValidRevSearchFrom pat endPos' (.rejected startPos endPos :: l) := by
+  cases h₃
+  exact IsValidRevSearchFrom.mismatched h₀ h₂ h₁
+
+theorem IsValidRevSearchFrom.startPos_of_eq {pat : ρ} [PatternModel pat] {s : Slice}
+    {p : s.Pos} {l : List (SearchStep s)} (hp : p = s.startPos) (hl : l = []) :
+    IsValidRevSearchFrom pat p l := by
+  cases hp
+  cases hl
+  exact IsValidRevSearchFrom.startPos
+
+/--
+Predicate stating compatibility between {name}`PatternModel` and {name}`ToBackwardSearcher`.
+
+We require the searcher to always match the longest match at the first position where the pattern
+matches; see {name}`IsValidRevSearchFrom`.
+-/
+class LawfulToBackwardSearcherModel {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
+    [ToBackwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
+    [∀ s, Std.Iterators.Finite (σ s) Id] : Prop where
+  isValidRevSearchFrom_toList (s) : IsValidRevSearchFrom pat s.endPos (ToBackwardSearcher.toSearcher pat s).toList
+
+theorem LawfulToBackwardSearcherModel.defaultImplementation {pat : ρ} [BackwardPattern pat] [StrictBackwardPattern pat]
+    [PatternModel pat] [LawfulBackwardPatternModel pat] :
+    letI : ToBackwardSearcher pat (ToBackwardSearcher.DefaultBackwardSearcher pat) := .defaultImplementation
+    LawfulToBackwardSearcherModel pat := by
+  let inst : ToBackwardSearcher pat (ToBackwardSearcher.DefaultBackwardSearcher pat) := .defaultImplementation
+  refine ⟨fun s => ?_⟩
+  suffices ∀ (pos : s.Pos),
+      IsValidRevSearchFrom pat pos (Std.Iter.mk (α := ToBackwardSearcher.DefaultBackwardSearcher pat s) ⟨pos⟩).toList from
+    this s.endPos
+  intro pos
+  induction pos using WellFounded.induction Slice.Pos.wellFounded_lt with | h pos ih
+  rw [Std.Iter.toList_eq_match_step, Std.Iter.step_eq]
+  simp only [Std.Iter.toIterM, ne_eq]
+  by_cases h : pos = s.startPos
+  · simpa [h] using IsValidRevSearchFrom.startPos
+  · simp only [h, ↓reduceDIte]
+    split <;> rename_i heq
+    · split at heq <;> rename_i pos' heq'
+      · simp only [Id.run_pure, Std.Shrink.inflate_deflate, Std.IterM.Step.toPure_yield,
+          Std.PlausibleIterStep.yield, Std.IterStep.yield.injEq] at heq
+        rw [← heq.1, ← heq.2]
+        apply IsValidRevSearchFrom.matched
+        · rw [LawfulBackwardPattern.skipSuffixOfNonempty?_eq,
+            LawfulBackwardPatternModel.skipSuffix?_eq_some_iff] at heq'
+          exact heq'.isLongestRevMatchAt_ofSliceTo
+        · simp only [Std.IterM.toIter]
+          apply ih
+          refine Std.lt_of_lt_of_le (Slice.Pos.ofSliceTo_lt_ofSliceTo_iff.2 ?_)
+            (Slice.Pos.ofSliceTo_le (pos := Slice.endPos _))
+          simpa using StrictBackwardPattern.ne_endPos _ _ heq'
+      · simp only [Id.run_pure, Std.Shrink.inflate_deflate, Std.IterM.Step.toPure_yield,
+          Std.PlausibleIterStep.yield, Std.IterStep.yield.injEq] at heq
+        rw [← heq.1, ← heq.2]
+        apply IsValidRevSearchFrom.mismatched (by simp) _ (ih _ (by simp))
+        intro p' hp' hp''
+        obtain rfl : pos = p' := Std.le_antisymm (by simpa using hp') hp''
+        rwa [LawfulBackwardPattern.skipSuffixOfNonempty?_eq,
+          LawfulBackwardPatternModel.skipSuffix?_sliceTo_eq_none_iff] at heq'
+    · split at heq <;> simp at heq
+    · split at heq <;> simp at heq
+
 end String.Slice.Pattern.Model

src/Init/Data/String/Lemmas/Pattern/Char.lean

@@ -20,28 +20,42 @@ import Init.Data.String.Lemmas.Order
 import Init.Data.Order.Lemmas
 import Init.Data.String.OrderInstances
 import Init.Omega
+import Init.Data.String.Lemmas.FindPos
 
 public section
 
 namespace String.Slice.Pattern.Model.Char
 
-instance {c : Char} : ForwardPatternModel c where
+instance {c : Char} : PatternModel c where
   Matches s := s = String.singleton c
   not_matches_empty := by simp
 
-instance {c : Char} : NoPrefixForwardPatternModel c :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+instance {c : Char} : NoPrefixPatternModel c :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
+
+instance {c : Char} : NoSuffixPatternModel c :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
     IsMatch c pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ s.startPos.get h = c := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, sliceTo_copy_eq_iff_exists_splits]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_sliceTo_eq_iff_exists_splits]
   refine ⟨?_, ?_⟩
   · simp only [splits_singleton_iff]
     exact fun ⟨t₂, h, h₁, h₂, h₃⟩ => ⟨h, h₁, h₂⟩
   · rintro ⟨h, rfl, rfl⟩
     exact ⟨_, Slice.splits_next_startPos⟩
 
+theorem isRevMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    IsRevMatch c pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_sliceFrom_eq_iff_exists_splits]
+  refine ⟨?_, ?_⟩
+  · simp only [splits_singleton_right_iff]
+    exact fun ⟨t₂, h, h₁, h₂, h₃⟩ => ⟨h, h₁, h₂⟩
+  · rintro ⟨h, rfl, rfl⟩
+    exact ⟨_, Slice.splits_prev_endPos⟩
+
 theorem isLongestMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
     IsLongestMatch c pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ s.startPos.get h = c := by
@@ -52,21 +66,46 @@ theorem isLongestMatchAt_iff {c : Char} {s : Slice} {pos pos' : s.Pos} :
   simp +contextual [Model.isLongestMatchAt_iff, isLongestMatch_iff, ← Pos.ofSliceFrom_inj,
     Pos.get_eq_get_ofSliceFrom, Pos.ofSliceFrom_next]
 
+theorem isLongestRevMatch_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch c pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
+theorem isLongestRevMatchAt_iff {c : Char} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAt c pos pos' ↔ ∃ h, pos = pos'.prev h ∧ (pos'.prev h).get (by simp) = c := by
+  simp +contextual [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff, ← Pos.ofSliceTo_inj,
+    Pos.get_eq_get_ofSliceTo, Pos.ofSliceTo_prev]
+
 theorem isLongestMatchAt_of_get_eq {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : pos.get h = c) : IsLongestMatchAt c pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get_eq {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : (pos.prev h).get (by simp) = c) : IsLongestRevMatchAt c (pos.prev h) pos :=
+  isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
+
 instance {c : Char} : LawfulForwardPatternModel c where
   skipPrefix?_eq_some_iff {s} pos := by
     simp [isLongestMatch_iff, ForwardPattern.skipPrefix?, and_comm, eq_comm (b := pos)]
 
+instance {c : Char} : LawfulBackwardPatternModel c where
+  skipSuffix?_eq_some_iff {s} pos := by
+    simp [isLongestRevMatch_iff, BackwardPattern.skipSuffix?, and_comm, eq_comm (b := pos)]
+
 theorem toSearcher_eq {c : Char} {s : Slice} :
   ToForwardSearcher.toSearcher c s = ToForwardSearcher.toSearcher (· == c) s := (rfl)
 
+theorem toBackwardSearcher_eq {c : Char} {s : Slice} :
+  ToBackwardSearcher.toSearcher c s = ToBackwardSearcher.toSearcher (· == c) s := (rfl)
+
 theorem matchesAt_iff {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ ∃ (h : pos ≠ s.endPos), pos.get h = c := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ ∃ (h : pos ≠ s.startPos), (pos.prev h).get (by simp) = c := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem matchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ ∃ t₁ t₂, pos.Splits t₁ (singleton c ++ t₂) := by
   rw [matchesAt_iff]
@@ -77,37 +116,81 @@ theorem matchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
     have hne := hs.ne_endPos_of_singleton
     exact ⟨hne, (singleton_append_inj.mp (hs.eq_right (pos.splits_next_right hne))).1.symm⟩
 
+theorem revMatchesAt_iff_splits {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ singleton c) t₂ := by
+  rw [revMatchesAt_iff]
+  refine ⟨?_, ?_⟩
+  · rintro ⟨h, rfl⟩
+    exact ⟨_, _, pos.splits_prev_right h⟩
+  · rintro ⟨t₁, t₂, hs⟩
+    have hne := hs.ne_startPos_of_singleton
+    refine ⟨hne, ?_⟩
+    have := hs.eq_left (pos.splits_prev_right hne)
+    simp only [append_singleton, push_inj] at this
+    exact this.2.symm
+
 theorem not_matchesAt_of_get_ne {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : pos.get h ≠ c) : ¬ MatchesAt c pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get_ne {c : Char} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : (pos.prev h).get (by simp) ≠ c) : ¬ RevMatchesAt c pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {c : Char} :
     matchAt? c pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), pos.get h = c then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {c : Char} :
+    revMatchAt? c pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), (pos.prev h).get (by simp) = c then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 theorem isMatch_iff_isMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
     IsMatch c pos ↔ IsMatch (· == c) pos := by
   simp [isMatch_iff, CharPred.isMatch_iff, beq_iff_eq]
 
+theorem isRevMatch_iff_isRevMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    IsRevMatch c pos ↔ IsRevMatch (· == c) pos := by
+  simp [isRevMatch_iff, CharPred.isRevMatch_iff, beq_iff_eq]
+
 theorem isLongestMatch_iff_isLongestMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
     IsLongestMatch c pos ↔ IsLongestMatch (· == c) pos := by
   simp [isLongestMatch_iff_isMatch, isMatch_iff_isMatch_beq]
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch c pos ↔ IsLongestRevMatch (· == c) pos := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff_isRevMatch_beq]
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_beq {c : Char} {s : Slice}
     {pos pos' : s.Pos} :
     IsLongestMatchAt c pos pos' ↔ IsLongestMatchAt (· == c) pos pos' := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_beq]
 
+theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_beq {c : Char} {s : Slice}
+    {pos pos' : s.Pos} :
+    IsLongestRevMatchAt c pos pos' ↔ IsLongestRevMatchAt (· == c) pos pos' := by
+  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_beq]
+
 theorem matchesAt_iff_matchesAt_beq {c : Char} {s : Slice} {pos : s.Pos} :
     MatchesAt c pos ↔ MatchesAt (· == c) pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_beq]
 
+theorem revMatchesAt_iff_revMatchesAt_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt c pos ↔ RevMatchesAt (· == c) pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+
 theorem matchAt?_eq_matchAt?_beq {c : Char} {s : Slice} {pos : s.Pos} :
     matchAt? c pos = matchAt? (· == c) pos := by
   refine Option.ext (fun pos' => ?_)
   simp [matchAt?_eq_some_iff, isLongestMatchAt_iff_isLongestMatchAt_beq]
 
+theorem revMatchAt?_eq_revMatchAt?_beq {c : Char} {s : Slice} {pos : s.Pos} :
+    revMatchAt? c pos = revMatchAt? (· == c) pos := by
+  refine Option.ext (fun pos' => ?_)
+  simp [revMatchAt?_eq_some_iff, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+
 theorem isValidSearchFrom_iff_isValidSearchFrom_beq {c : Char} {s : Slice} {p : s.Pos}
     {l : List (SearchStep s)} : IsValidSearchFrom c p l ↔ IsValidSearchFrom (· == c) p l := by
   refine ⟨fun h => ?_, fun h => ?_⟩
@@ -120,11 +203,28 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_beq {c : Char} {s : Slice} {p :
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_beq]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_beq]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_beq {c : Char} {s : Slice} {p : s.Pos}
+    {l : List (SearchStep s)} : IsValidRevSearchFrom c p l ↔ IsValidRevSearchFrom (· == c) p l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_beq]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_beq]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_beq]
+
 instance {c : Char} : LawfulToForwardSearcherModel c where
   isValidSearchFrom_toList s := by
     simpa [toSearcher_eq, isValidSearchFrom_iff_isValidSearchFrom_beq] using
       LawfulToForwardSearcherModel.isValidSearchFrom_toList (pat := (· == c)) (s := s)
 
+instance {c : Char} : LawfulToBackwardSearcherModel c where
+  isValidRevSearchFrom_toList s := by
+    simpa [toBackwardSearcher_eq, isValidRevSearchFrom_iff_isValidRevSearchFrom_beq] using
+      LawfulToBackwardSearcherModel.isValidRevSearchFrom_toList (pat := (· == c)) (s := s)
+
 end Pattern.Model.Char
 
 theorem startsWith_char_eq_startsWith_beq {c : Char} {s : Slice} :

src/Init/Data/String/Lemmas/Pattern/Find/Basic.lean

@@ -23,7 +23,7 @@ open Std String.Slice Pattern Pattern.Model
 
 namespace String.Slice
 
-theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos : s.Pos} :
@@ -40,7 +40,7 @@ theorem Pattern.Model.find?_eq_some_iff {ρ : Type} (pat : ρ) [ForwardPatternMo
   | matched h₁ _ _ => have := h₁.matchesAt; grind
   | mismatched => grind
 
-theorem Pattern.Model.find?_eq_none_iff {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.find?_eq_none_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
@@ -65,14 +65,14 @@ theorem find?_eq_none_iff {ρ : Type} (pat : ρ) {σ : Slice → Type}
     [ToForwardSearcher pat σ] {s : Slice} : s.find? pat = none ↔ s.contains pat = false := by
   rw [← Option.isNone_iff_eq_none, ← Option.isSome_eq_false_iff, isSome_find?]
 
-theorem Pattern.Model.contains_eq_false_iff {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.contains_eq_false_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
     s.contains pat = false ↔ ∀ (pos : s.Pos), ¬ MatchesAt pat pos := by
   rw [← find?_eq_none_iff, Slice.find?_eq_none_iff]
 
-theorem Pattern.Model.contains_eq_true_iff {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.contains_eq_true_iff {ρ : Type} (pat : ρ) [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} :
@@ -85,7 +85,7 @@ theorem Pos.find?_eq_find?_sliceFrom {ρ : Type} {pat : ρ} {σ : Slice → Type
     p.find? pat = ((s.sliceFrom p).find? pat).map Pos.ofSliceFrom :=
   (rfl)
 
-theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos pos' : s.Pos} :
@@ -100,7 +100,7 @@ theorem Pattern.Model.posFind?_eq_some_iff {ρ : Type} {pat : ρ} [ForwardPatter
     refine ⟨Pos.sliceFrom _ _ h₁, ⟨by simpa using h₂, fun p hp₁ hp₂ => ?_⟩, by simp⟩
     exact h₃ (Pos.ofSliceFrom p) Slice.Pos.le_ofSliceFrom (Pos.lt_sliceFrom_iff.1 hp₁) hp₂
 
-theorem Pattern.Model.posFind?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] {σ : Slice → Type}
+theorem Pattern.Model.posFind?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat] {σ : Slice → Type}
     [∀ s, Iterator (σ s) Id (SearchStep s)] [∀ s, Iterators.Finite (σ s) Id]
     [∀ s, IteratorLoop (σ s) Id Id] [∀ s, LawfulIteratorLoop (σ s) Id Id]
     [ToForwardSearcher pat σ] [LawfulToForwardSearcherModel pat] {s : Slice} {pos : s.Pos} :

src/Init/Data/String/Lemmas/Pattern/Pred.lean

@@ -19,124 +19,228 @@ import Init.Data.String.Lemmas.Order
 import Init.Data.Order.Lemmas
 import Init.Data.String.OrderInstances
 import Init.Omega
+import Init.Data.String.Lemmas.FindPos
 
 public section
 
 namespace String.Slice.Pattern.Model.CharPred
 
-instance {p : Char → Bool} : ForwardPatternModel p where
+instance {p : Char → Bool} : PatternModel p where
   Matches s := ∃ c, s = singleton c ∧ p c
   not_matches_empty := by
     simp
 
-instance {p : Char → Bool} : NoPrefixForwardPatternModel p :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+instance {p : Char → Bool} : NoPrefixPatternModel p :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
+
+instance {p : Char → Bool} : NoSuffixPatternModel p :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, sliceTo_copy_eq_iff_exists_splits]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_sliceTo_eq_iff_exists_splits]
   refine ⟨?_, ?_⟩
   · simp only [splits_singleton_iff]
     refine fun ⟨c, ⟨t₂, h, h₁, h₂, h₃⟩, hc⟩ => ⟨h, h₁, h₂ ▸ hc⟩
   · rintro ⟨h, rfl, h'⟩
     exact ⟨s.startPos.get h, ⟨_, Slice.splits_next_startPos⟩, h'⟩
 
+theorem isRevMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_sliceFrom_eq_iff_exists_splits]
+  refine ⟨?_, ?_⟩
+  · simp only [splits_singleton_right_iff]
+    refine fun ⟨c, ⟨t₂, h, h₁, h₂, h₃⟩, hc⟩ => ⟨h, h₁, h₂ ▸ hc⟩
+  · rintro ⟨h, rfl, h'⟩
+    exact ⟨(s.endPos.prev h).get (by simp), ⟨_, Slice.splits_prev_endPos⟩, h'⟩
+
 theorem isLongestMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     IsLongestMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
+theorem isLongestRevMatch_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
 theorem isLongestMatchAt_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ ∃ h, pos' = pos.next h ∧ p (pos.get h) := by
   simp +contextual [Model.isLongestMatchAt_iff, isLongestMatch_iff, ← Pos.ofSliceFrom_inj,
     Pos.get_eq_get_ofSliceFrom, Pos.ofSliceFrom_next]
 
+theorem isLongestRevMatchAt_iff {p : Char → Bool} {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAt p pos pos' ↔ ∃ h, pos = pos'.prev h ∧ p ((pos'.prev h).get (by simp)) := by
+  simp +contextual [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff, ← Pos.ofSliceTo_inj,
+    Pos.get_eq_get_ofSliceTo, Pos.ofSliceTo_prev]
+
 theorem isLongestMatchAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : p (pos.get h)) : IsLongestMatchAt p pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : p ((pos.prev h).get (by simp))) : IsLongestRevMatchAt p (pos.prev h) pos :=
+  isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
+
 instance {p : Char → Bool} : LawfulForwardPatternModel p where
   skipPrefix?_eq_some_iff {s} pos := by
     simp [isLongestMatch_iff, ForwardPattern.skipPrefix?, and_comm, eq_comm (b := pos)]
 
+instance {p : Char → Bool} : LawfulBackwardPatternModel p where
+  skipSuffix?_eq_some_iff {s} pos := by
+    simp [isLongestRevMatch_iff, BackwardPattern.skipSuffix?, and_comm, eq_comm (b := pos)]
+
 instance {p : Char → Bool} : LawfulToForwardSearcherModel p :=
   .defaultImplementation
 
+instance {p : Char → Bool} : LawfulToBackwardSearcherModel p :=
+  .defaultImplementation
+
 theorem matchesAt_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
     MatchesAt p pos ↔ ∃ (h : pos ≠ s.endPos), p (pos.get h) := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt p pos ↔ ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem not_matchesAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.endPos}
     (hc : p (pos.get h) = false) : ¬ MatchesAt p pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get {p : Char → Bool} {s : Slice} {pos : s.Pos} {h : pos ≠ s.startPos}
+    (hc : p ((pos.prev h).get (by simp)) = false) : ¬ RevMatchesAt p pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Bool} :
     matchAt? p pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), p (pos.get h) then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Bool} :
+    revMatchAt? p pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 namespace Decidable
 
-instance {p : Char → Prop} [DecidablePred p] : ForwardPatternModel p where
-  Matches := ForwardPatternModel.Matches (decide <| p ·)
-  not_matches_empty := ForwardPatternModel.not_matches_empty (pat := (decide <| p ·))
+instance {p : Char → Prop} [DecidablePred p] : PatternModel p where
+  Matches := PatternModel.Matches (decide <| p ·)
+  not_matches_empty := PatternModel.not_matches_empty (pat := (decide <| p ·))
 
-instance {p : Char → Prop} [DecidablePred p] : NoPrefixForwardPatternModel p where
-  eq_empty := NoPrefixForwardPatternModel.eq_empty (pat := (decide <| p ·))
+instance {p : Char → Prop} [DecidablePred p] : NoPrefixPatternModel p where
+  eq_empty := NoPrefixPatternModel.eq_empty (pat := (decide <| p ·))
+
+instance {p : Char → Prop} [DecidablePred p] : NoSuffixPatternModel p where
+  eq_empty := NoSuffixPatternModel.eq_empty (pat := (decide <| p ·))
 
 theorem isMatch_iff_isMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔ IsMatch (decide <| p ·) pos :=
   ⟨fun ⟨h⟩ => ⟨h⟩, fun ⟨h⟩ => ⟨h⟩⟩
 
+theorem isRevMatch_iff_isRevMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔ IsRevMatch (decide <| p ·) pos :=
+  ⟨fun ⟨h⟩ => ⟨h⟩, fun ⟨h⟩ => ⟨h⟩⟩
+
 theorem isMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   simp [isMatch_iff_isMatch_decide, CharPred.isMatch_iff]
 
+theorem isRevMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp [isRevMatch_iff_isRevMatch_decide, CharPred.isRevMatch_iff]
+
 theorem isLongestMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     IsLongestMatch p pos ↔
       ∃ (h : s.startPos ≠ s.endPos), pos = s.startPos.next h ∧ p (s.startPos.get h) := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff]
 
+theorem isLongestRevMatch_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch p pos ↔
+      ∃ (h : s.endPos ≠ s.startPos), pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff]
+
 theorem isLongestMatch_iff_isLongestMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : IsLongestMatch p pos ↔ IsLongestMatch (decide <| p ·) pos := by
   simp [isLongestMatch_iff_isMatch, isMatch_iff_isMatch_decide]
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : IsLongestRevMatch p pos ↔ IsLongestRevMatch (decide <| p ·) pos := by
+  simp [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff_isRevMatch_decide]
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ IsLongestMatchAt (decide <| p ·) pos pos' := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_decide]
 
+theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos pos' : s.Pos} :
+    IsLongestRevMatchAt p pos pos' ↔ IsLongestRevMatchAt (decide <| p ·) pos pos' := by
+  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_decide]
+
 theorem isLongestMatchAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos pos' : s.Pos} :
     IsLongestMatchAt p pos pos' ↔ ∃ h, pos' = pos.next h ∧ p (pos.get h) := by
   simp [isLongestMatchAt_iff_isLongestMatchAt_decide, CharPred.isLongestMatchAt_iff]
 
+theorem isLongestRevMatchAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos pos' : s.Pos} :
+    IsLongestRevMatchAt p pos pos' ↔ ∃ h, pos = pos'.prev h ∧ p ((pos'.prev h).get (by simp)) := by
+  simp [isLongestRevMatchAt_iff_isLongestRevMatchAt_decide, CharPred.isLongestRevMatchAt_iff]
+
 theorem isLongestMatchAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
     {h : pos ≠ s.endPos} (hc : p (pos.get h)) : IsLongestMatchAt p pos (pos.next h) :=
   isLongestMatchAt_iff.2 ⟨h, by simp [hc]⟩
 
+theorem isLongestRevMatchAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
+    {h : pos ≠ s.startPos} (hc : p ((pos.prev h).get (by simp))) :
+    IsLongestRevMatchAt p (pos.prev h) pos :=
+  isLongestRevMatchAt_iff.2 ⟨h, by simp [hc]⟩
+
 theorem matchesAt_iff_matchesAt_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : MatchesAt p pos ↔ MatchesAt (decide <| p ·) pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_decide]
 
+theorem revMatchesAt_iff_revMatchesAt_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : RevMatchesAt p pos ↔ RevMatchesAt (decide <| p ·) pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+
 theorem matchAt?_eq_matchAt?_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
     {pos : s.Pos} : matchAt? p pos = matchAt? (decide <| p ·) pos := by
   ext endPos
   simp [isLongestMatchAt_iff_isLongestMatchAt_decide]
 
+theorem revMatchAt?_eq_revMatchAt?_decide {p : Char → Prop} [DecidablePred p] {s : Slice}
+    {pos : s.Pos} : revMatchAt? p pos = revMatchAt? (decide <| p ·) pos := by
+  ext startPos
+  simp [isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+
 theorem skipPrefix?_eq_skipPrefix?_decide {p : Char → Prop} [DecidablePred p] :
     ForwardPattern.skipPrefix? p = ForwardPattern.skipPrefix? (decide <| p ·) := rfl
 
+theorem skipSuffix?_eq_skipSuffix?_decide {p : Char → Prop} [DecidablePred p] :
+    BackwardPattern.skipSuffix? p = BackwardPattern.skipSuffix? (decide <| p ·) := rfl
+
 instance {p : Char → Prop} [DecidablePred p] : LawfulForwardPatternModel p where
   skipPrefix?_eq_some_iff {s} pos := by
     rw [skipPrefix?_eq_skipPrefix?_decide, isLongestMatch_iff_isLongestMatch_decide]
     exact LawfulForwardPatternModel.skipPrefix?_eq_some_iff ..
 
+instance {p : Char → Prop} [DecidablePred p] : LawfulBackwardPatternModel p where
+  skipSuffix?_eq_some_iff {s} pos := by
+    rw [skipSuffix?_eq_skipSuffix?_decide, isLongestRevMatch_iff_isLongestRevMatch_decide]
+    exact LawfulBackwardPatternModel.skipSuffix?_eq_some_iff ..
+
 theorem toSearcher_eq {p : Char → Prop} [DecidablePred p] {s : Slice} :
     ToForwardSearcher.toSearcher p s = ToForwardSearcher.toSearcher (decide <| p ·) s := (rfl)
 
+theorem toBackwardSearcher_eq {p : Char → Prop} [DecidablePred p] {s : Slice} :
+    ToBackwardSearcher.toSearcher p s = ToBackwardSearcher.toSearcher (decide <| p ·) s := (rfl)
+
 theorem isValidSearchFrom_iff_isValidSearchFrom_decide {p : Char → Prop} [DecidablePred p]
     {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
     IsValidSearchFrom p pos l ↔ IsValidSearchFrom (decide <| p ·) pos l := by
@@ -150,24 +254,55 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_decide {p : Char → Prop} [Deci
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_decide]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_decide]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_decide {p : Char → Prop} [DecidablePred p]
+    {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
+    IsValidRevSearchFrom p pos l ↔ IsValidRevSearchFrom (decide <| p ·) pos l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_decide]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched, isLongestRevMatchAt_iff_isLongestRevMatchAt_decide]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_decide]
+
 instance {p : Char → Prop} [DecidablePred p] : LawfulToForwardSearcherModel p where
   isValidSearchFrom_toList s := by
     simpa [toSearcher_eq, isValidSearchFrom_iff_isValidSearchFrom_decide] using
       LawfulToForwardSearcherModel.isValidSearchFrom_toList (pat := (decide <| p ·)) (s := s)
 
+instance {p : Char → Prop} [DecidablePred p] : LawfulToBackwardSearcherModel p where
+  isValidRevSearchFrom_toList s := by
+    simpa [toBackwardSearcher_eq, isValidRevSearchFrom_iff_isValidRevSearchFrom_decide] using
+      LawfulToBackwardSearcherModel.isValidRevSearchFrom_toList (pat := (decide <| p ·)) (s := s)
+
 theorem matchesAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
     MatchesAt p pos ↔ ∃ (h : pos ≠ s.endPos), p (pos.get h) := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff, exists_comm]
 
+theorem revMatchesAt_iff {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos} :
+    RevMatchesAt p pos ↔ ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff, exists_comm]
+
 theorem not_matchesAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
     {h : pos ≠ s.endPos} (hc : ¬ p (pos.get h)) : ¬ MatchesAt p pos := by
   simp [matchesAt_iff, hc]
 
+theorem not_revMatchesAt_of_get {p : Char → Prop} [DecidablePred p] {s : Slice} {pos : s.Pos}
+    {h : pos ≠ s.startPos} (hc : ¬ p ((pos.prev h).get (by simp))) : ¬ RevMatchesAt p pos := by
+  simp [revMatchesAt_iff, hc]
+
 theorem matchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Prop} [DecidablePred p] :
     matchAt? p pos =
       if h₀ : ∃ (h : pos ≠ s.endPos), p (pos.get h) then some (pos.next h₀.1) else none := by
   split <;> simp_all [isLongestMatchAt_iff, matchesAt_iff]
 
+theorem revMatchAt?_eq {s : Slice} {pos : s.Pos} {p : Char → Prop} [DecidablePred p] :
+    revMatchAt? p pos =
+      if h₀ : ∃ (h : pos ≠ s.startPos), p ((pos.prev h).get (by simp)) then some (pos.prev h₀.1) else none := by
+  split <;> simp_all [isLongestRevMatchAt_iff, revMatchesAt_iff]
+
 end Decidable
 
 end Pattern.Model.CharPred

src/Init/Data/String/Lemmas/Pattern/Split/Basic.lean

@@ -28,7 +28,7 @@ set_option doc.verso true
 # Verification of {name}`String.Slice.splitToSubslice`
 
 This PR verifies the {name}`String.Slice.splitToSubslice` function by relating it to a model
-implementation based on the {name}`String.Slice.Pattern.Model.ForwardPatternModel` class.
+implementation based on the {name}`String.Slice.Pattern.Model.PatternModel` class.
 
 This gives a low-level correctness proof from which higher-level API lemmas can be derived.
 -/
@@ -36,7 +36,7 @@ This gives a low-level correctness proof from which higher-level API lemmas can
 namespace String.Slice.Pattern.Model
 
 @[cbv_opaque]
-public protected noncomputable def split {ρ : Type} (pat : ρ) [ForwardPatternModel pat] {s : Slice}
+public protected noncomputable def split {ρ : Type} (pat : ρ) [PatternModel pat] {s : Slice}
     (firstRejected curr : s.Pos) (hle : firstRejected ≤ curr) : List s.Subslice :=
   if h : curr = s.endPos then
     [s.subslice _ _ hle]
@@ -49,12 +49,12 @@ public protected noncomputable def split {ρ : Type} (pat : ρ) [ForwardPatternM
 termination_by curr
 
 @[simp]
-public theorem split_endPos {ρ : Type} {pat : ρ} [ForwardPatternModel pat] {s : Slice}
+public theorem split_endPos {ρ : Type} {pat : ρ} [PatternModel pat] {s : Slice}
     {firstRejected : s.Pos} :
     Model.split (s := s) pat firstRejected s.endPos (by simp) = [s.subslice firstRejected s.endPos (by simp)] := by
   simp [Model.split]
 
-public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
+public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [PatternModel pat]
     {s : Slice} {firstRejected start stop : s.Pos} {hle} (h : IsLongestMatchAt pat start stop) :
     Model.split pat firstRejected start hle =
       s.subslice _ _ hle :: Model.split pat stop stop (by exact Std.le_refl _) := by
@@ -63,7 +63,7 @@ public theorem split_eq_of_isLongestMatchAt {ρ : Type} {pat : ρ} [ForwardPatte
   · congr <;> exact (matchAt?_eq_some_iff.1 ‹_›).eq h
   · simp [matchAt?_eq_some_iff.2 ‹_›] at *
 
-public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
+public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat]
     {s : Slice} {firstRejected start} (stop : s.Pos) (h₀ : start ≤ stop) {hle}
     (h : ∀ p, start ≤ p → p < stop → ¬ MatchesAt pat p) :
     Model.split pat firstRejected start hle =
@@ -80,7 +80,7 @@ public theorem split_eq_of_not_matchesAt {ρ : Type} {pat : ρ} [ForwardPatternM
   · obtain rfl : start = stop := Std.le_antisymm h₀ (Std.not_lt.1 h')
     simp
 
-public theorem split_eq_next_of_not_matchesAt {ρ : Type} {pat : ρ} [ForwardPatternModel pat]
+public theorem split_eq_next_of_not_matchesAt {ρ : Type} {pat : ρ} [PatternModel pat]
     {s : Slice} {firstRejected start} {hle} (hs : start ≠ s.endPos) (h : ¬ MatchesAt pat start) :
     Model.split pat firstRejected start hle =
       Model.split pat firstRejected (start.next hs) (by exact Std.le_trans hle (by simp)) := by
@@ -103,7 +103,7 @@ def splitFromSteps {s : Slice} (currPos : s.Pos) (l : List (SearchStep s)) : Lis
   | .matched p q :: l => s.subslice! currPos p :: splitFromSteps q l
 
 theorem IsValidSearchFrom.splitFromSteps_eq_extend_split {ρ : Type} (pat : ρ)
-    [ForwardPatternModel pat] (l : List (SearchStep s)) (pos pos' : s.Pos) (h₀ : pos ≤ pos')
+    [PatternModel pat] (l : List (SearchStep s)) (pos pos' : s.Pos) (h₀ : pos ≤ pos')
     (h' : ∀ p, pos ≤ p → p < pos' → ¬ MatchesAt pat p)
     (h : IsValidSearchFrom pat pos' l) :
     splitFromSteps pos l = Model.split pat pos pos' h₀ := by
@@ -155,7 +155,7 @@ end Model
 open Model
 
 @[cbv_eval]
-public theorem toList_splitToSubslice_eq_modelSplit {ρ : Type} (pat : ρ) [ForwardPatternModel pat]
+public theorem toList_splitToSubslice_eq_modelSplit {ρ : Type} (pat : ρ) [PatternModel pat]
     {σ : Slice → Type} [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [LawfulToForwardSearcherModel pat] (s : Slice) :
     (s.splitToSubslice pat).toList = Model.split pat s.startPos s.startPos (by exact Std.le_refl _) := by
@@ -168,7 +168,7 @@ end Pattern
 open Pattern
 
 public theorem toList_splitToSubslice_of_isEmpty {ρ : Type} (pat : ρ)
-    [Model.ForwardPatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] {s : Slice}
     (h : s.isEmpty = true) :
@@ -182,7 +182,7 @@ public theorem toList_split_eq_splitToSubslice {ρ : Type} (pat : ρ) {σ : Slic
   simp [split, Std.Iter.toList_map]
 
 public theorem toList_split_of_isEmpty {ρ : Type} (pat : ρ)
-    [Model.ForwardPatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] {s : Slice}
     (h : s.isEmpty = true) :
@@ -200,7 +200,7 @@ public theorem split_eq_split_toSlice {ρ : Type} {pat : ρ} {σ : Slice → Typ
 
 @[simp]
 public theorem toList_split_empty {ρ : Type} (pat : ρ)
-    [Model.ForwardPatternModel pat] {σ : Slice → Type}
+    [Model.PatternModel pat] {σ : Slice → Type}
     [ToForwardSearcher pat σ] [∀ s, Std.Iterator (σ s) Id (SearchStep s)]
     [∀ s, Std.Iterators.Finite (σ s) Id] [Model.LawfulToForwardSearcherModel pat] :
     ("".split pat).toList.map Slice.copy = [""] := by

src/Init/Data/String/Lemmas/Pattern/String/Basic.lean

@@ -19,12 +19,12 @@ namespace String.Slice.Pattern.Model
 
 namespace ForwardSliceSearcher
 
-instance {pat : Slice} : ForwardPatternModel pat where
+instance {pat : Slice} : PatternModel pat where
   /-
-  See the docstring of `ForwardPatternModel` for an explanation about why we disallow matching the
+  See the docstring of `PatternModel` for an explanation about why we disallow matching the
   empty string.
 
-  Requiring `s ≠ ""` is a trick that allows us to give a `ForwardPatternModel` instance
+  Requiring `s ≠ ""` is a trick that allows us to give a `PatternModel` instance
   unconditionally, without forcing `pat.copy` to be non-empty (which would make it very awkward
   to state theorems about the instance). It does not change anything about the fact that all lemmas
   about this instance require `pat.isEmpty = false`.
@@ -32,34 +32,60 @@ instance {pat : Slice} : ForwardPatternModel pat where
   Matches s := s ≠ "" ∧ s = pat.copy
   not_matches_empty := by simp
 
-instance {pat : Slice} : NoPrefixForwardPatternModel pat :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+instance {pat : Slice} : NoPrefixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
+
+instance {pat : Slice} : NoSuffixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     IsMatch pat pos ↔ (s.sliceTo pos).copy = pat.copy := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, ne_eq, copy_eq_empty_iff,
+  simp only [Model.isMatch_iff, PatternModel.Matches, ne_eq, copy_eq_empty_iff,
     Bool.not_eq_true, and_iff_right_iff_imp]
   intro h'
   rw [← isEmpty_copy (s := s.sliceTo pos), h', isEmpty_copy, h]
 
+theorem isRevMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+    IsRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat.copy := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, ne_eq, copy_eq_empty_iff,
+    Bool.not_eq_true, and_iff_right_iff_imp]
+  intro h'
+  rw [← isEmpty_copy (s := s.sliceFrom pos), h', isEmpty_copy, h]
+
 theorem isLongestMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatch pat pos ↔ (s.sliceTo pos).copy = pat.copy := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff h]
 
+theorem isLongestRevMatch_iff {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+    IsLongestRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat.copy := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff h]
+
 theorem isLongestMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat.copy := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff h]
 
+theorem isLongestRevMatchAt_iff {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat.copy := by
+  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff h]
+
 theorem isLongestMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ t₁ t₂, pos₁.Splits t₁ (pat.copy ++ t₂) ∧
       pos₂.Splits (t₁ ++ pat.copy) t₂ := by
   simp only [isLongestMatchAt_iff h, copy_slice_eq_iff_splits]
 
+theorem isLongestRevMatchAt_iff_splits {pat s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat.isEmpty = false) :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ t₁ t₂, pos₁.Splits t₁ (pat.copy ++ t₂) ∧
+      pos₂.Splits (t₁ ++ pat.copy) t₂ := by
+  simp only [isLongestRevMatchAt_iff h, copy_slice_eq_iff_splits]
+
 theorem isLongestMatch_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatch pat pos ↔ ∃ t, pos.Splits pat.copy t := by
-  simp only [← isLongestMatchAt_startPos_iff, isLongestMatchAt_iff_splits h, splits_startPos_iff,
-    and_assoc, exists_and_left, exists_eq_left, empty_append]
-  exact ⟨fun ⟨h, _, h'⟩ => ⟨h, h'⟩, fun ⟨h, h'⟩ => ⟨h, h'.eq_append.symm, h'⟩⟩
+  rw [isLongestMatch_iff h, copy_sliceTo_eq_iff_exists_splits]
+
+theorem isLongestRevMatch_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+    IsLongestRevMatch pat pos ↔ ∃ t, pos.Splits t pat.copy := by
+  rw [isLongestRevMatch_iff h, copy_sliceFrom_eq_iff_exists_splits]
 
 theorem isLongestMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false) :
     IsLongestMatchAt pat pos₁ pos₂ ↔
@@ -71,6 +97,18 @@ theorem isLongestMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos} (h
     exact ⟨by simp [Pos.le_iff, Pos.Raw.le_iff]; omega,
       by simp [← h', ← toByteArray_inj, toByteArray_copy_slice]⟩
 
+theorem isLongestRevMatchAt_iff_extract {pat s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat.isEmpty = false) :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔
+      s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx =
+        pat.copy.toByteArray := by
+  rw [isLongestRevMatchAt_iff h]
+  refine ⟨fun ⟨h, h'⟩ => ?_, fun h' => ?_⟩
+  · simp [← h', toByteArray_copy_slice]
+  · rw [← Slice.toByteArray_copy_ne_empty_iff, ← h', ne_eq, ByteArray.extract_eq_empty_iff] at h
+    exact ⟨by simp [Pos.le_iff, Pos.Raw.le_iff]; omega,
+      by simp [← h', ← toByteArray_inj, toByteArray_copy_slice]⟩
+
 theorem offset_of_isLongestMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos} (h : pat.isEmpty = false)
     (h' : IsLongestMatchAt pat pos₁ pos₂) : pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
   simp only [Pos.Raw.ext_iff, Pos.Raw.byteIdx_increaseBy]
@@ -81,12 +119,29 @@ theorem offset_of_isLongestMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos} (h :
   suffices pos₂.offset.byteIdx ≤ s.utf8ByteSize by omega
   simpa [Pos.le_iff, Pos.Raw.le_iff] using pos₂.le_endPos
 
+theorem offset_of_isLongestRevMatchAt {pat s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat.isEmpty = false) (h' : IsLongestRevMatchAt pat pos₁ pos₂) :
+    pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
+  simp only [Pos.Raw.ext_iff, Pos.Raw.byteIdx_increaseBy]
+  rw [isLongestRevMatchAt_iff_extract h] at h'
+  rw [← Slice.toByteArray_copy_ne_empty_iff, ← h', ne_eq, ByteArray.extract_eq_empty_iff] at h
+  replace h' := congrArg ByteArray.size h'
+  simp only [ByteArray.size_extract, size_toByteArray, utf8ByteSize_copy] at h'
+  suffices pos₂.offset.byteIdx ≤ s.utf8ByteSize by omega
+  simpa [Pos.le_iff, Pos.Raw.le_iff] using pos₂.le_endPos
+
 theorem matchesAt_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     MatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ (pat.copy ++ t₂) := by
   simp only [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_splits h]
   exact ⟨fun ⟨e, t₁, t₂, ht₁, ht₂⟩ => ⟨t₁, t₂, ht₁⟩,
     fun ⟨t₁, t₂, ht⟩ => ⟨ht.rotateRight, t₁, t₂, ht, ht.splits_rotateRight⟩⟩
 
+theorem revMatchesAt_iff_splits {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
+    RevMatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ pat.copy) t₂ := by
+  simp only [revMatchesAt_iff_exists_isLongestRevMatchAt, isLongestRevMatchAt_iff_splits h]
+  exact ⟨fun ⟨e, t₁, t₂, ht₁, ht₂⟩ => ⟨t₁, t₂, ht₂⟩,
+    fun ⟨t₁, t₂, ht⟩ => ⟨ht.rotateLeft, t₁, t₂, ht.splits_rotateLeft, ht⟩⟩
+
 theorem exists_matchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false) :
     (∃ (pos : s.Pos), MatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat.copy ++ t₂ := by
   simp only [matchesAt_iff_splits h]
@@ -99,6 +154,18 @@ theorem exists_matchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false)
         ⟨t₁, pat.copy ++ t₂, by rw [← append_assoc]; exact heq, rfl⟩
     exact ⟨s.pos _ hvalid, t₁, t₂, ⟨by rw [← append_assoc]; exact heq, by simp⟩⟩
 
+theorem exists_revMatchesAt_iff_eq_append {pat s : Slice} (h : pat.isEmpty = false) :
+    (∃ (pos : s.Pos), RevMatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat.copy ++ t₂ := by
+  simp only [revMatchesAt_iff_splits h]
+  constructor
+  · rintro ⟨pos, t₁, t₂, hsplit⟩
+    exact ⟨t₁, t₂, by rw [hsplit.eq_append, append_assoc]⟩
+  · rintro ⟨t₁, t₂, heq⟩
+    have hvalid : (t₁ ++ pat.copy).rawEndPos.IsValidForSlice s :=
+      Pos.Raw.isValidForSlice_iff_exists_append.mpr
+        ⟨t₁ ++ pat.copy, t₂, heq, rfl⟩
+    exact ⟨s.pos _ hvalid, t₁, t₂, ⟨heq, by simp⟩⟩
+
 theorem matchesAt_iff_isLongestMatchAt {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     MatchesAt pat pos ↔ ∃ (h : (pos.offset.increaseBy pat.utf8ByteSize).IsValidForSlice s),
       IsLongestMatchAt pat pos (s.pos _ h) := by
@@ -108,6 +175,25 @@ theorem matchesAt_iff_isLongestMatchAt {pat s : Slice} {pos : s.Pos} (h : pat.is
   obtain rfl : p = s.pos _ this := by simpa [Pos.ext_iff] using offset_of_isLongestMatchAt h h'
   exact h'
 
+theorem revMatchesAt_iff_isLongestRevMatchAt {pat s : Slice} {pos : s.Pos}
+    (h : pat.isEmpty = false) :
+    RevMatchesAt pat pos ↔
+      ∃ (h : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s),
+        IsLongestRevMatchAt pat (s.pos _ h) pos := by
+  refine ⟨fun ⟨⟨p, h'⟩⟩ => ?_, fun ⟨_, h⟩ => ⟨⟨_, h⟩⟩⟩
+  have hoff := offset_of_isLongestRevMatchAt h h'
+  have hvalid : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s := by
+    rw [show pos.offset.decreaseBy pat.utf8ByteSize = p.offset from by
+      simp [Pos.Raw.ext_iff, Pos.Raw.byteIdx_decreaseBy, Pos.Raw.byteIdx_increaseBy] at hoff ⊢
+      omega]
+    exact p.isValidForSlice
+  refine ⟨hvalid, ?_⟩
+  obtain rfl : p = s.pos _ hvalid := by
+    simp only [Pos.ext_iff, offset_pos]
+    simp [Pos.Raw.ext_iff, Pos.Raw.byteIdx_decreaseBy, Pos.Raw.byteIdx_increaseBy] at hoff ⊢
+    omega
+  exact h'
+
 theorem matchesAt_iff_getElem {pat s : Slice} {pos : s.Pos} (h : pat.isEmpty = false) :
     MatchesAt pat pos ↔
       ∃ (h : pos.offset.byteIdx + pat.copy.toByteArray.size ≤ s.copy.toByteArray.size),
@@ -146,31 +232,56 @@ end ForwardSliceSearcher
 
 namespace ForwardStringSearcher
 
-instance {pat : String} : ForwardPatternModel pat where
+instance {pat : String} : PatternModel pat where
   Matches s := s ≠ "" ∧ s = pat
   not_matches_empty := by simp
 
-instance {pat : String} : NoPrefixForwardPatternModel pat :=
-  .of_length_eq (by simp +contextual [ForwardPatternModel.Matches])
+instance {pat : String} : NoPrefixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
+
+instance {pat : String} : NoSuffixPatternModel pat :=
+  .of_length_eq (by simp +contextual [PatternModel.Matches])
 
 theorem isMatch_iff_slice {pat : String} {s : Slice} {pos : s.Pos} :
     IsMatch (ρ := String) pat pos ↔ IsMatch (ρ := Slice) pat.toSlice pos := by
-  simp only [Model.isMatch_iff, ForwardPatternModel.Matches, copy_toSlice]
+  simp only [Model.isMatch_iff, PatternModel.Matches, copy_toSlice]
+
+theorem isRevMatch_iff_slice {pat : String} {s : Slice} {pos : s.Pos} :
+    IsRevMatch (ρ := String) pat pos ↔ IsRevMatch (ρ := Slice) pat.toSlice pos := by
+  simp only [Model.isRevMatch_iff, PatternModel.Matches, copy_toSlice]
 
 theorem isLongestMatch_iff_isLongestMatch_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
     IsLongestMatch (ρ := String) pat pos ↔ IsLongestMatch (ρ := Slice) pat.toSlice pos where
   mp h := ⟨isMatch_iff_slice.1 h.isMatch, fun p hp hm => h.not_isMatch p hp (isMatch_iff_slice.2 hm)⟩
   mpr h := ⟨isMatch_iff_slice.2 h.isMatch, fun p hp hm => h.not_isMatch p hp (isMatch_iff_slice.1 hm)⟩
 
+theorem isLongestRevMatch_iff_isLongestRevMatch_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    IsLongestRevMatch (ρ := String) pat pos ↔ IsLongestRevMatch (ρ := Slice) pat.toSlice pos where
+  mp h := ⟨isRevMatch_iff_slice.1 h.isRevMatch,
+    fun p hp hm => h.not_isRevMatch p hp (isRevMatch_iff_slice.2 hm)⟩
+  mpr h := ⟨isRevMatch_iff_slice.2 h.isRevMatch,
+    fun p hp hm => h.not_isRevMatch p hp (isRevMatch_iff_slice.1 hm)⟩
+
 theorem isLongestMatchAt_iff_isLongestMatchAt_toSlice {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} :
     IsLongestMatchAt (ρ := String) pat pos₁ pos₂ ↔
       IsLongestMatchAt (ρ := Slice) pat.toSlice pos₁ pos₂ := by
   simp [Model.isLongestMatchAt_iff, isLongestMatch_iff_isLongestMatch_toSlice]
 
+theorem isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice {pat : String} {s : Slice}
+    {pos₁ pos₂ : s.Pos} :
+    IsLongestRevMatchAt (ρ := String) pat pos₁ pos₂ ↔
+      IsLongestRevMatchAt (ρ := Slice) pat.toSlice pos₁ pos₂ := by
+  simp [Model.isLongestRevMatchAt_iff, isLongestRevMatch_iff_isLongestRevMatch_toSlice]
+
 theorem matchesAt_iff_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
     MatchesAt (ρ := String) pat pos ↔ MatchesAt (ρ := Slice) pat.toSlice pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
 
+theorem revMatchesAt_iff_toSlice {pat : String} {s : Slice} {pos : s.Pos} :
+    RevMatchesAt (ρ := String) pat pos ↔ RevMatchesAt (ρ := Slice) pat.toSlice pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt,
+    isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+
 private theorem toSlice_isEmpty (h : pat ≠ "") : pat.toSlice.isEmpty = false := by
   rwa [isEmpty_toSlice, isEmpty_eq_false_iff]
 
@@ -179,16 +290,31 @@ theorem isMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
   rw [isMatch_iff_slice, ForwardSliceSearcher.isMatch_iff (toSlice_isEmpty h)]
   simp
 
+theorem isRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+    IsRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat := by
+  rw [isRevMatch_iff_slice, ForwardSliceSearcher.isRevMatch_iff (toSlice_isEmpty h)]
+  simp
+
 theorem isLongestMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
     IsLongestMatch pat pos ↔ (s.sliceTo pos).copy = pat := by
   rw [isLongestMatch_iff_isMatch, isMatch_iff h]
 
+theorem isLongestRevMatch_iff {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+    IsLongestRevMatch pat pos ↔ (s.sliceFrom pos).copy = pat := by
+  rw [isLongestRevMatch_iff_isRevMatch, isRevMatch_iff h]
+
 theorem isLongestMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} (h : pat ≠ "") :
     IsLongestMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat := by
   rw [isLongestMatchAt_iff_isLongestMatchAt_toSlice,
     ForwardSliceSearcher.isLongestMatchAt_iff (toSlice_isEmpty h)]
   simp
 
+theorem isLongestRevMatchAt_iff {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos} (h : pat ≠ "") :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔ ∃ h, (s.slice pos₁ pos₂ h).copy = pat := by
+  rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
+    ForwardSliceSearcher.isLongestRevMatchAt_iff (toSlice_isEmpty h)]
+  simp
+
 theorem isLongestMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
     (h : pat ≠ "") :
     IsLongestMatchAt pat pos₁ pos₂ ↔
@@ -197,6 +323,14 @@ theorem isLongestMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ :
     ForwardSliceSearcher.isLongestMatchAt_iff_splits (toSlice_isEmpty h)]
   simp
 
+theorem isLongestRevMatchAt_iff_splits {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat ≠ "") :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔
+      ∃ t₁ t₂, pos₁.Splits t₁ (pat ++ t₂) ∧ pos₂.Splits (t₁ ++ pat) t₂ := by
+  rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
+    ForwardSliceSearcher.isLongestRevMatchAt_iff_splits (toSlice_isEmpty h)]
+  simp
+
 theorem isLongestMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
     (h : pat ≠ "") :
     IsLongestMatchAt pat pos₁ pos₂ ↔
@@ -205,6 +339,14 @@ theorem isLongestMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ :
     ForwardSliceSearcher.isLongestMatchAt_iff_extract (toSlice_isEmpty h)]
   simp
 
+theorem isLongestRevMatchAt_iff_extract {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat ≠ "") :
+    IsLongestRevMatchAt pat pos₁ pos₂ ↔
+      s.copy.toByteArray.extract pos₁.offset.byteIdx pos₂.offset.byteIdx = pat.toByteArray := by
+  rw [isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice,
+    ForwardSliceSearcher.isLongestRevMatchAt_iff_extract (toSlice_isEmpty h)]
+  simp
+
 theorem offset_of_isLongestMatchAt {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
     (h : pat ≠ "") (h' : IsLongestMatchAt pat pos₁ pos₂) :
     pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
@@ -212,12 +354,25 @@ theorem offset_of_isLongestMatchAt {pat : String} {s : Slice} {pos₁ pos₂ : s
   exact ForwardSliceSearcher.offset_of_isLongestMatchAt (toSlice_isEmpty h)
     (isLongestMatchAt_iff_isLongestMatchAt_toSlice.1 h')
 
+theorem offset_of_isLongestRevMatchAt {pat : String} {s : Slice} {pos₁ pos₂ : s.Pos}
+    (h : pat ≠ "") (h' : IsLongestRevMatchAt pat pos₁ pos₂) :
+    pos₂.offset = pos₁.offset.increaseBy pat.utf8ByteSize := by
+  rw [show pat.utf8ByteSize = pat.toSlice.utf8ByteSize from utf8ByteSize_toSlice.symm]
+  exact ForwardSliceSearcher.offset_of_isLongestRevMatchAt (toSlice_isEmpty h)
+    (isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice.1 h')
+
 theorem matchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
     MatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits t₁ (pat ++ t₂) := by
   rw [matchesAt_iff_toSlice,
     ForwardSliceSearcher.matchesAt_iff_splits (toSlice_isEmpty h)]
   simp
 
+theorem revMatchesAt_iff_splits {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
+    RevMatchesAt pat pos ↔ ∃ t₁ t₂, pos.Splits (t₁ ++ pat) t₂ := by
+  rw [revMatchesAt_iff_toSlice,
+    ForwardSliceSearcher.revMatchesAt_iff_splits (toSlice_isEmpty h)]
+  simp
+
 theorem exists_matchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "") :
     (∃ (pos : s.Pos), MatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat ++ t₂ := by
   simp only [matchesAt_iff_splits h]
@@ -230,6 +385,14 @@ theorem exists_matchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "
         ⟨t₁, pat ++ t₂, by rw [← append_assoc]; exact heq, rfl⟩
     exact ⟨s.pos _ hvalid, t₁, t₂, ⟨by rw [← append_assoc]; exact heq, by simp⟩⟩
 
+theorem exists_revMatchesAt_iff_eq_append {pat : String} {s : Slice} (h : pat ≠ "") :
+    (∃ (pos : s.Pos), RevMatchesAt pat pos) ↔ ∃ t₁ t₂, s.copy = t₁ ++ pat ++ t₂ := by
+  rw [show (∃ (pos : s.Pos), RevMatchesAt (ρ := String) pat pos) ↔
+      (∃ (pos : s.Pos), RevMatchesAt (ρ := Slice) pat.toSlice pos) from by
+    simp [revMatchesAt_iff_toSlice],
+    ForwardSliceSearcher.exists_revMatchesAt_iff_eq_append (toSlice_isEmpty h)]
+  simp
+
 theorem matchesAt_iff_isLongestMatchAt {pat : String} {s : Slice} {pos : s.Pos}
     (h : pat ≠ "") :
     MatchesAt pat pos ↔ ∃ (h : (pos.offset.increaseBy pat.utf8ByteSize).IsValidForSlice s),
@@ -239,6 +402,16 @@ theorem matchesAt_iff_isLongestMatchAt {pat : String} {s : Slice} {pos : s.Pos}
   simp only [utf8ByteSize_toSlice, ← isLongestMatchAt_iff_isLongestMatchAt_toSlice] at key
   rwa [matchesAt_iff_toSlice]
 
+theorem revMatchesAt_iff_isLongestRevMatchAt {pat : String} {s : Slice} {pos : s.Pos}
+    (h : pat ≠ "") :
+    RevMatchesAt pat pos ↔
+      ∃ (h : (pos.offset.decreaseBy pat.utf8ByteSize).IsValidForSlice s),
+        IsLongestRevMatchAt pat (s.pos _ h) pos := by
+  have key := ForwardSliceSearcher.revMatchesAt_iff_isLongestRevMatchAt (pat := pat.toSlice)
+    (toSlice_isEmpty h) (pos := pos)
+  simp only [utf8ByteSize_toSlice, ← isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice] at key
+  rwa [revMatchesAt_iff_toSlice]
+
 theorem matchesAt_iff_getElem {pat : String} {s : Slice} {pos : s.Pos} (h : pat ≠ "") :
     MatchesAt pat pos ↔
       ∃ (h : pos.offset.byteIdx + pat.toByteArray.size ≤ s.copy.toByteArray.size),
@@ -259,6 +432,11 @@ theorem matchesAt_iff_matchesAt_toSlice {pat : String} {s : Slice}
     {pos : s.Pos} : MatchesAt pat pos ↔ MatchesAt pat.toSlice pos := by
   simp [matchesAt_iff_exists_isLongestMatchAt, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
 
+theorem revMatchesAt_iff_revMatchesAt_toSlice {pat : String} {s : Slice}
+    {pos : s.Pos} : RevMatchesAt pat pos ↔ RevMatchesAt pat.toSlice pos := by
+  simp [revMatchesAt_iff_exists_isLongestRevMatchAt,
+    isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+
 theorem toSearcher_eq {pat : String} {s : Slice} :
     ToForwardSearcher.toSearcher pat s = ToForwardSearcher.toSearcher pat.toSlice s := (rfl)
 
@@ -275,6 +453,21 @@ theorem isValidSearchFrom_iff_isValidSearchFrom_toSlice {pat : String}
     | matched => simp_all [IsValidSearchFrom.matched, isLongestMatchAt_iff_isLongestMatchAt_toSlice]
     | mismatched => simp_all [IsValidSearchFrom.mismatched, matchesAt_iff_matchesAt_toSlice]
 
+theorem isValidRevSearchFrom_iff_isValidRevSearchFrom_toSlice {pat : String}
+    {s : Slice} {pos : s.Pos} {l : List (SearchStep s)} :
+    IsValidRevSearchFrom pat pos l ↔ IsValidRevSearchFrom pat.toSlice pos l := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched,
+        isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_toSlice]
+  · induction h with
+    | startPos => simpa using IsValidRevSearchFrom.startPos
+    | matched => simp_all [IsValidRevSearchFrom.matched,
+        isLongestRevMatchAt_iff_isLongestRevMatchAt_toSlice]
+    | mismatched => simp_all [IsValidRevSearchFrom.mismatched, revMatchesAt_iff_revMatchesAt_toSlice]
+
 end ForwardStringSearcher
 
 end String.Slice.Pattern.Model

src/Init/Data/String/Lemmas/Pattern/String/ForwardPattern.lean

@@ -76,10 +76,12 @@ namespace Model.ForwardSliceSearcher
 
 open Pattern.ForwardSliceSearcher
 
+public instance {pat : Slice} : LawfulForwardPattern pat where
+  skipPrefixOfNonempty?_eq _ := rfl
+  startsWith_eq _ := isSome_skipPrefix?.symm
+
 public theorem lawfulForwardPatternModel {pat : Slice} (hpat : pat.isEmpty = false) :
     LawfulForwardPatternModel pat where
-  skipPrefixOfNonempty?_eq h := rfl
-  startsWith_eq s := isSome_skipPrefix?.symm
   skipPrefix?_eq_some_iff pos := by
     simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff hpat]
 
@@ -89,15 +91,116 @@ namespace Model.ForwardStringSearcher
 
 open Pattern.ForwardSliceSearcher
 
+public instance {pat : String} : LawfulForwardPattern pat where
+  skipPrefixOfNonempty?_eq _ := rfl
+  startsWith_eq _ := isSome_skipPrefix?.symm
+
 public theorem lawfulForwardPatternModel {pat : String} (hpat : pat ≠ "") :
     LawfulForwardPatternModel pat where
-  skipPrefixOfNonempty?_eq h := rfl
-  startsWith_eq s := isSome_skipPrefix?.symm
   skipPrefix?_eq_some_iff pos := by
     simp [ForwardPattern.skipPrefix?, skipPrefix?_eq_some_iff, isLongestMatch_iff hpat]
 
 end Model.ForwardStringSearcher
 
+namespace BackwardSliceSearcher
+
+theorem endsWith_iff {pat s : Slice} : endsWith pat s ↔ ∃ t, s.copy = t ++ pat.copy := by
+  rw [endsWith]
+  simp [Internal.memcmpSlice_eq_true_iff, utf8ByteSize_eq_size_toByteArray_copy, -size_toByteArray]
+  generalize pat.copy = pat
+  generalize s.copy = s
+  refine ⟨fun ⟨h₁, h₂⟩ => ?_, ?_⟩
+  · rw [Nat.sub_add_cancel h₁] at h₂
+    suffices (s.rawEndPos.unoffsetBy pat.rawEndPos).IsValid s by
+      have h₃ : (s.sliceFrom (s.pos _ this)).copy = pat := by
+        rw [← toByteArray_inj, (s.pos _ this).splits.toByteArray_right_eq]
+        simpa [offset_pos, Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos]
+      have := (s.pos _ this).splits
+      rw [h₃] at this
+      exact ⟨_, this.eq_append⟩
+    rw [Pos.Raw.isValid_iff_isValidUTF8_extract_utf8ByteSize]
+    refine ⟨by simp [Pos.Raw.le_iff, Pos.Raw.byteIdx_unoffsetBy], ?_⟩
+    simp only [size_toByteArray] at h₂
+    simpa [Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos, h₂] using pat.isValidUTF8
+  · rintro ⟨t, rfl⟩
+    exact ⟨by simp, by rw [Nat.sub_add_cancel (by simp)]; exact
+      ByteArray.extract_append_eq_right (by simp) (by simp)⟩
+
+theorem skipSuffix?_eq_some_iff {pat s : Slice} {pos : s.Pos} :
+    skipSuffix? pat s = some pos ↔ (s.sliceFrom pos).copy = pat.copy := by
+  fun_cases skipSuffix? with
+  | case1 h =>
+    simp only [Option.some.injEq]
+    obtain ⟨t, ht⟩ := endsWith_iff.1 h
+    have hpc : pat.copy.utf8ByteSize = pat.utf8ByteSize := Slice.utf8ByteSize_copy
+    have hsz : s.utf8ByteSize = t.utf8ByteSize + pat.utf8ByteSize := by
+      have := congrArg String.utf8ByteSize ht
+      simp only [utf8ByteSize_append, Slice.utf8ByteSize_copy] at this
+      exact this
+    have hoff : (s.endPos.offset.unoffsetBy pat.rawEndPos) = t.rawEndPos := by
+      ext
+      simp only [offset_endPos, Pos.Raw.byteIdx_unoffsetBy, byteIdx_rawEndPos,
+        String.byteIdx_rawEndPos]
+      omega
+    have hval : (s.endPos.offset.unoffsetBy pat.rawEndPos).IsValidForSlice s :=
+      Pos.Raw.isValidForSlice_iff_exists_append.mpr ⟨t, pat.copy, ht, hoff⟩
+    have hsp : (s.pos _ hval).Splits t pat.copy := ⟨ht, hoff⟩
+    rw [Slice.pos!_eq_pos hval]
+    exact ⟨(· ▸ hsp.copy_sliceFrom_eq),
+      fun h => hsp.pos_eq_of_eq_right (h ▸ pos.splits)⟩
+  | case2 h =>
+    simp only [endsWith_iff, not_exists] at h
+    simp only [reduceCtorEq, false_iff]
+    intro heq
+    have := h (s.sliceTo pos).copy
+    simp [← heq, pos.splits.eq_append] at this
+
+theorem isSome_skipSuffix? {pat s : Slice} : (skipSuffix? pat s).isSome = endsWith pat s := by
+  fun_cases skipSuffix? <;> simp_all
+
+public theorem endsWith_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    BackwardPattern.endsWith pat s = true := by
+  suffices pat.copy = "" by simp [BackwardPattern.endsWith, endsWith_iff, this]
+  simpa
+
+public theorem skipSuffix?_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    BackwardPattern.skipSuffix? pat s = some s.endPos := by
+  simpa [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff]
+
+end BackwardSliceSearcher
+
+namespace Model.BackwardSliceSearcher
+
+open Pattern.BackwardSliceSearcher
+
+public instance {pat : Slice} : LawfulBackwardPattern pat where
+  skipSuffixOfNonempty?_eq _ := rfl
+  endsWith_eq _ := isSome_skipSuffix?.symm
+
+public theorem lawfulBackwardPatternModel {pat : Slice} (hpat : pat.isEmpty = false) :
+    LawfulBackwardPatternModel pat where
+  skipSuffix?_eq_some_iff pos := by
+    simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
+      ForwardSliceSearcher.isLongestRevMatch_iff hpat]
+
+end Model.BackwardSliceSearcher
+
+namespace Model.BackwardStringSearcher
+
+open Pattern.BackwardSliceSearcher
+
+public instance {pat : String} : LawfulBackwardPattern pat where
+  skipSuffixOfNonempty?_eq _ := rfl
+  endsWith_eq _ := isSome_skipSuffix?.symm
+
+public theorem lawfulBackwardPatternModel {pat : String} (hpat : pat ≠ "") :
+    LawfulBackwardPatternModel pat where
+  skipSuffix?_eq_some_iff pos := by
+    simp [BackwardPattern.skipSuffix?, skipSuffix?_eq_some_iff,
+      ForwardStringSearcher.isLongestRevMatch_iff hpat]
+
+end Model.BackwardStringSearcher
+
 end Pattern
 
 public theorem startsWith_string_eq_startsWith_toSlice {pat : String} {s : Slice} :

src/Init/Data/String/Lemmas/Pattern/TakeDrop/Basic.lean

@@ -29,12 +29,12 @@ theorem startsWith_eq_forwardPatternStartsWith {ρ : Type} {pat : ρ} [ForwardPa
 theorem dropPrefix?_eq_map_skipPrefix? {ρ : Type} {pat : ρ} [ForwardPattern pat] {s : Slice} :
     s.dropPrefix? pat = (s.skipPrefix? pat).map s.sliceFrom := (rfl)
 
-theorem Pattern.Model.skipPrefix?_eq_some_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] [ForwardPattern pat]
+theorem Pattern.Model.skipPrefix?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat] [ForwardPattern pat]
     [LawfulForwardPatternModel pat] {s : Slice} {pos : s.Pos} :
     s.skipPrefix? pat = some pos ↔ IsLongestMatch pat pos := by
   rw [skipPrefix?_eq_forwardPatternSkipPrefix?, LawfulForwardPatternModel.skipPrefix?_eq_some_iff]
 
-theorem Pattern.Model.skipPrefix?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] [ForwardPattern pat]
+theorem Pattern.Model.skipPrefix?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat] [ForwardPattern pat]
     [LawfulForwardPatternModel pat] {s : Slice} :
     s.skipPrefix? pat = none ↔ ¬ MatchesAt pat s.startPos := by
   rw [skipPrefix?_eq_forwardPatternSkipPrefix?, LawfulForwardPatternModel.skipPrefix?_eq_none_iff]
@@ -44,13 +44,13 @@ theorem isSome_skipPrefix? {ρ : Type} {pat : ρ} [ForwardPattern pat] [LawfulFo
     (s.skipPrefix? pat).isSome = s.startsWith pat := by
   rw [startsWith_eq_forwardPatternStartsWith, skipPrefix?, LawfulForwardPattern.startsWith_eq]
 
-theorem Pattern.Model.startsWith_eq_false_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] [ForwardPattern pat]
+theorem Pattern.Model.startsWith_eq_false_iff {ρ : Type} {pat : ρ} [PatternModel pat] [ForwardPattern pat]
     [LawfulForwardPatternModel pat] {s : Slice} :
     s.startsWith pat = false ↔ ¬ MatchesAt pat s.startPos := by
   rw [← Pattern.Model.skipPrefix?_eq_none_iff, ← Option.isNone_iff_eq_none,
     ← isSome_skipPrefix?, Option.isSome_eq_false_iff]
 
-theorem Pattern.Model.startsWith_iff {ρ : Type} {pat : ρ} [ForwardPatternModel pat] [ForwardPattern pat]
+theorem Pattern.Model.startsWith_iff {ρ : Type} {pat : ρ} [PatternModel pat] [ForwardPattern pat]
     [LawfulForwardPatternModel pat] {s : Slice} :
     s.startsWith pat = true ↔ MatchesAt pat s.startPos := by
   rw [← Bool.not_eq_false, startsWith_eq_false_iff, Classical.not_not]
@@ -65,13 +65,65 @@ theorem dropPrefix?_eq_none_iff {ρ : Type} {pat : ρ} [ForwardPattern pat] [Law
     {s : Slice} : s.dropPrefix? pat = none ↔ s.startsWith pat = false := by
   simp [dropPrefix?_eq_map_skipPrefix?]
 
-theorem Pattern.Model.eq_append_of_dropPrefix?_eq_some {ρ : Type} {pat : ρ} [ForwardPatternModel pat] [ForwardPattern pat]
+theorem Pattern.Model.eq_append_of_dropPrefix?_eq_some {ρ : Type} {pat : ρ} [PatternModel pat] [ForwardPattern pat]
     [LawfulForwardPatternModel pat] {s res : Slice} (h : s.dropPrefix? pat = some res) :
-    ∃ t, ForwardPatternModel.Matches pat t ∧ s.copy = t ++ res.copy := by
+    ∃ t, PatternModel.Matches pat t ∧ s.copy = t ++ res.copy := by
   simp only [dropPrefix?_eq_map_skipPrefix?, Option.map_eq_some_iff, skipPrefix?_eq_some_iff] at h
   obtain ⟨pos, h₁, h₂⟩ := h
   exact ⟨(s.sliceTo pos).copy, h₁.isMatch.matches_copy, by simp [← h₂, ← copy_eq_copy_sliceTo]⟩
 
+theorem skipSuffix?_eq_backwardPatternSkipSuffix? {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : Slice} :
+    s.skipSuffix? pat = BackwardPattern.skipSuffix? pat s := (rfl)
+
+theorem endsWith_eq_backwardPatternEndsWith {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : Slice} :
+    s.endsWith pat = BackwardPattern.endsWith pat s := (rfl)
+
+theorem dropSuffix?_eq_map_skipSuffix? {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : Slice} :
+    s.dropSuffix? pat = (s.skipSuffix? pat).map s.sliceTo := (rfl)
+
+theorem Pattern.Model.skipSuffix?_eq_some_iff {ρ : Type} {pat : ρ} [PatternModel pat] [BackwardPattern pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? pat = some pos ↔ IsLongestRevMatch pat pos := by
+  rw [skipSuffix?_eq_backwardPatternSkipSuffix?, LawfulBackwardPatternModel.skipSuffix?_eq_some_iff]
+
+theorem Pattern.Model.skipSuffix?_eq_none_iff {ρ : Type} {pat : ρ} [PatternModel pat] [BackwardPattern pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} :
+    s.skipSuffix? pat = none ↔ ¬ RevMatchesAt pat s.endPos := by
+  rw [skipSuffix?_eq_backwardPatternSkipSuffix?, LawfulBackwardPatternModel.skipSuffix?_eq_none_iff]
+
+@[simp]
+theorem isSome_skipSuffix? {ρ : Type} {pat : ρ} [BackwardPattern pat] [LawfulBackwardPattern pat] {s : Slice} :
+    (s.skipSuffix? pat).isSome = s.endsWith pat := by
+  rw [endsWith_eq_backwardPatternEndsWith, skipSuffix?, LawfulBackwardPattern.endsWith_eq]
+
+theorem Pattern.Model.endsWith_eq_false_iff {ρ : Type} {pat : ρ} [PatternModel pat] [BackwardPattern pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} :
+    s.endsWith pat = false ↔ ¬ RevMatchesAt pat s.endPos := by
+  rw [← Pattern.Model.skipSuffix?_eq_none_iff, ← Option.isNone_iff_eq_none,
+    ← isSome_skipSuffix?, Option.isSome_eq_false_iff]
+
+theorem Pattern.Model.endsWith_iff {ρ : Type} {pat : ρ} [PatternModel pat] [BackwardPattern pat]
+    [LawfulBackwardPatternModel pat] {s : Slice} :
+    s.endsWith pat = true ↔ RevMatchesAt pat s.endPos := by
+  rw [← Bool.not_eq_false, endsWith_eq_false_iff, Classical.not_not]
+
+@[simp]
+theorem skipSuffix?_eq_none_iff {ρ : Type} {pat : ρ} [BackwardPattern pat] [LawfulBackwardPattern pat]
+    {s : Slice} : s.skipSuffix? pat = none ↔ s.endsWith pat = false := by
+  rw [← Option.isNone_iff_eq_none, ← Option.isSome_eq_false_iff, isSome_skipSuffix?]
+
+@[simp]
+theorem dropSuffix?_eq_none_iff {ρ : Type} {pat : ρ} [BackwardPattern pat] [LawfulBackwardPattern pat]
+    {s : Slice} : s.dropSuffix? pat = none ↔ s.endsWith pat = false := by
+  simp [dropSuffix?_eq_map_skipSuffix?]
+
+theorem Pattern.Model.eq_append_of_dropSuffix?_eq_some {ρ : Type} {pat : ρ} [PatternModel pat] [BackwardPattern pat]
+    [LawfulBackwardPatternModel pat] {s res : Slice} (h : s.dropSuffix? pat = some res) :
+    ∃ t, PatternModel.Matches pat t ∧ s.copy = res.copy ++ t := by
+  simp only [dropSuffix?_eq_map_skipSuffix?, Option.map_eq_some_iff, skipSuffix?_eq_some_iff] at h
+  obtain ⟨pos, h₁, h₂⟩ := h
+  exact ⟨(s.sliceFrom pos).copy, h₁.isRevMatch.matches_copy, by simp [← h₂, ← copy_eq_copy_sliceTo]⟩
+
 end Slice
 
 theorem skipPrefix?_eq_skipPrefix?_toSlice {ρ : Type} {pat : ρ} [ForwardPattern pat] {s : String} :
@@ -83,4 +135,13 @@ theorem startsWith_eq_startsWith_toSlice {ρ : Type} {pat : ρ} [ForwardPattern
 theorem dropPrefix?_eq_dropPrefix?_toSlice {ρ : Type} {pat : ρ} [ForwardPattern pat] {s : String} :
     s.dropPrefix? pat = s.toSlice.dropPrefix? pat := (rfl)
 
+theorem skipSuffix?_eq_skipSuffix?_toSlice {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : String} :
+    s.skipSuffix? pat = (s.toSlice.skipSuffix? pat).map Pos.ofToSlice := (rfl)
+
+theorem endsWith_eq_endsWith_toSlice {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : String} :
+    s.endsWith pat = s.toSlice.endsWith pat := (rfl)
+
+theorem dropSuffix?_eq_dropSuffix?_toSlice {ρ : Type} {pat : ρ} [BackwardPattern pat] {s : String} :
+    s.dropSuffix? pat = s.toSlice.dropSuffix? pat := (rfl)
+
 end String

src/Init/Data/String/Lemmas/Pattern/TakeDrop/Char.lean

@@ -11,6 +11,8 @@ public import Init.Data.String.TakeDrop
 import Init.Data.String.Lemmas.Pattern.TakeDrop.Basic
 import Init.Data.String.Lemmas.Pattern.Char
 import Init.Data.Option.Lemmas
+import Init.Data.String.Lemmas.FindPos
+import Init.Data.List.Sublist
 
 public section
 
@@ -52,7 +54,42 @@ theorem startsWith_char_eq_false_iff_forall_append {c : Char} {s : Slice} :
 
 theorem eq_append_of_dropPrefix?_char_eq_some {c : Char} {s res : Slice} (h : s.dropPrefix? c = some res) :
     s.copy = singleton c ++ res.copy := by
-  simpa [ForwardPatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+  simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+
+theorem skipSuffix?_char_eq_some_iff {c : Char} {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? c = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  rw [Pattern.Model.skipSuffix?_eq_some_iff, Char.isLongestRevMatch_iff]
+
+theorem endsWith_char_iff_get {c : Char} {s : Slice} :
+    s.endsWith c ↔ ∃ h, (s.endPos.prev h).get (by simp) = c := by
+  simp [Pattern.Model.endsWith_iff, Char.revMatchesAt_iff]
+
+theorem endsWith_char_eq_false_iff_get {c : Char} {s : Slice} :
+    s.endsWith c = false ↔ ∀ h, (s.endPos.prev h).get (by simp) ≠ c := by
+  simp [Pattern.Model.endsWith_eq_false_iff, Char.revMatchesAt_iff]
+
+theorem endsWith_char_iff_exists_append {c : Char} {s : Slice} :
+    s.endsWith c ↔ ∃ t, s.copy = t ++ singleton c := by
+  rw [Pattern.Model.endsWith_iff, Char.revMatchesAt_iff_splits]
+  simp only [splits_endPos_iff, exists_eq_right, eq_comm (a := s.copy)]
+
+theorem endsWith_char_eq_getLast? {c : Char} {s : Slice} :
+    s.endsWith c = (s.copy.toList.getLast? == some c) := by
+  rw [Bool.eq_iff_iff, endsWith_char_iff_exists_append, beq_iff_eq,
+    ← List.singleton_suffix_iff_getLast?_eq_some, List.suffix_iff_exists_eq_append]
+  constructor
+  · rintro ⟨t, ht⟩
+    exact ⟨t.toList, by rw [ht, toList_append, toList_singleton]⟩
+  · rintro ⟨l, hl⟩
+    exact ⟨ofList l, by rw [← toList_inj, toList_append, toList_singleton, toList_ofList]; exact hl⟩
+
+theorem endsWith_char_eq_false_iff_forall_append {c : Char} {s : Slice} :
+    s.endsWith c = false ↔ ∀ t, s.copy ≠ t ++ singleton c := by
+  simp [← Bool.not_eq_true, endsWith_char_iff_exists_append]
+
+theorem eq_append_of_dropSuffix?_char_eq_some {c : Char} {s res : Slice} (h : s.dropSuffix? c = some res) :
+    s.copy = res.copy ++ singleton c := by
+  simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropSuffix?_eq_some h
 
 end Slice
 
@@ -86,4 +123,34 @@ theorem eq_append_of_dropPrefix?_char_eq_some {c : Char} {s : String} {res : Sli
   rw [dropPrefix?_eq_dropPrefix?_toSlice] at h
   simpa using Slice.eq_append_of_dropPrefix?_char_eq_some h
 
+theorem skipSuffix?_char_eq_some_iff {c : Char} {s : String} {pos : s.Pos} :
+    s.skipSuffix? c = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c := by
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_char_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+theorem endsWith_char_iff_get {c : Char} {s : String} :
+    s.endsWith c ↔ ∃ h, (s.endPos.prev h).get (by simp) = c := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_char_eq_false_iff_get {c : Char} {s : String} :
+    s.endsWith c = false ↔ ∀ h, (s.endPos.prev h).get (by simp) ≠ c := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_eq_false_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_char_eq_getLast? {c : Char} {s : String} :
+    s.endsWith c = (s.toList.getLast? == some c) := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_eq_getLast?]
+
+theorem endsWith_char_iff_exists_append {c : Char} {s : String} :
+    s.endsWith c ↔ ∃ t, s = t ++ singleton c := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_char_iff_exists_append]
+
+theorem endsWith_char_eq_false_iff_forall_append {c : Char} {s : String} :
+    s.endsWith c = false ↔ ∀ t, s ≠ t ++ singleton c := by
+  simp [← Bool.not_eq_true, endsWith_char_iff_exists_append]
+
+theorem eq_append_of_dropSuffix?_char_eq_some {c : Char} {s : String} {res : Slice} (h : s.dropSuffix? c = some res) :
+    s = res.copy ++ singleton c := by
+  rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
+  simpa using Slice.eq_append_of_dropSuffix?_char_eq_some h
+
 end String

src/Init/Data/String/Lemmas/Pattern/TakeDrop/Pred.lean

@@ -11,6 +11,7 @@ public import Init.Data.String.TakeDrop
 import Init.Data.String.Lemmas.Pattern.TakeDrop.Basic
 import Init.Data.String.Lemmas.Pattern.Pred
 import Init.Data.Option.Lemmas
+import Init.Data.String.Lemmas.FindPos
 import Init.ByCases
 
 public section
@@ -45,7 +46,7 @@ theorem startsWith_bool_eq_head? {p : Char → Bool} {s : Slice} :
 
 theorem eq_append_of_dropPrefix?_bool_eq_some {p : Char → Bool} {s res : Slice} (h : s.dropPrefix? p = some res) :
     ∃ c, s.copy = singleton c ++ res.copy ∧ p c = true := by
-  obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [ForwardPatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
+  obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropPrefix?_eq_some h
   exact ⟨_, h₂, h₁⟩
 
 theorem skipPrefix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
@@ -69,6 +70,54 @@ theorem eq_append_of_dropPrefix_prop_eq_some {P : Char → Prop} [DecidablePred
   rw [dropPrefix?_prop_eq_dropPrefix?_decide] at h
   simpa using eq_append_of_dropPrefix?_bool_eq_some h
 
+theorem skipSuffix?_bool_eq_some_iff {p : Char → Bool} {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? p = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) = true := by
+  rw [Pattern.Model.skipSuffix?_eq_some_iff, CharPred.isLongestRevMatch_iff]
+
+theorem endsWith_bool_iff_get {p : Char → Bool} {s : Slice} :
+    s.endsWith p ↔ ∃ h, p ((s.endPos.prev h).get (by simp)) = true := by
+  simp [Pattern.Model.endsWith_iff, CharPred.revMatchesAt_iff]
+
+theorem endsWith_bool_eq_false_iff_get {p : Char → Bool} {s : Slice} :
+    s.endsWith p = false ↔ ∀ h, p ((s.endPos.prev h).get (by simp)) = false := by
+  simp [Pattern.Model.endsWith_eq_false_iff, CharPred.revMatchesAt_iff]
+
+theorem endsWith_bool_eq_getLast? {p : Char → Bool} {s : Slice} :
+    s.endsWith p = s.copy.toList.getLast?.any p := by
+  rw [Bool.eq_iff_iff, Pattern.Model.endsWith_iff, CharPred.revMatchesAt_iff]
+  by_cases h : s.endPos = s.startPos
+  · refine ⟨fun ⟨h', _⟩ => by simp_all, ?_⟩
+    have : s.copy = "" := by simp_all [Slice.startPos_eq_endPos_iff.mp h.symm]
+    simp [this]
+  · obtain ⟨t, ht⟩ := s.splits_endPos.exists_eq_append_singleton_of_ne_startPos h
+    simp [h, ht]
+
+theorem eq_append_of_dropSuffix?_bool_eq_some {p : Char → Bool} {s res : Slice} (h : s.dropSuffix? p = some res) :
+    ∃ c, s.copy = res.copy ++ singleton c ∧ p c = true := by
+  obtain ⟨_, ⟨c, ⟨rfl, h₁⟩⟩, h₂⟩ := by simpa [PatternModel.Matches] using Pattern.Model.eq_append_of_dropSuffix?_eq_some h
+  exact ⟨_, h₂, h₁⟩
+
+theorem skipSuffix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? P = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ P ((s.endPos.prev h).get (by simp)) := by
+  simp [skipSuffix?_prop_eq_skipSuffix?_decide, skipSuffix?_bool_eq_some_iff]
+
+theorem endsWith_prop_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P ↔ ∃ h, P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_iff_get]
+
+theorem endsWith_prop_eq_false_iff_get {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P = false ↔ ∀ h, ¬ P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_eq_false_iff_get]
+
+theorem endsWith_prop_eq_getLast? {P : Char → Prop} [DecidablePred P] {s : Slice} :
+    s.endsWith P = s.copy.toList.getLast?.any (decide <| P ·) := by
+  simp [endsWith_prop_eq_endsWith_decide, endsWith_bool_eq_getLast?]
+
+theorem eq_append_of_dropSuffix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s res : Slice} (h : s.dropSuffix? P = some res) :
+    ∃ c, s.copy = res.copy ++ singleton c ∧ P c := by
+  rw [dropSuffix?_prop_eq_dropSuffix?_decide] at h
+  simpa using eq_append_of_dropSuffix?_bool_eq_some h
+
 end Slice
 
 theorem skipPrefix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
@@ -115,4 +164,48 @@ theorem eq_append_of_dropPrefix?_prop_eq_some {P : Char → Prop} [DecidablePred
   rw [dropPrefix?_eq_dropPrefix?_toSlice] at h
   simpa using Slice.eq_append_of_dropPrefix_prop_eq_some h
 
+theorem skipSuffix?_bool_eq_some_iff {p : Char → Bool} {s : String} {pos : s.Pos} :
+    s.skipSuffix? p = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ p ((s.endPos.prev h).get (by simp)) = true := by
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_bool_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+theorem endsWith_bool_iff_get {p : Char → Bool} {s : String} :
+    s.endsWith p ↔ ∃ h, p ((s.endPos.prev h).get (by simp)) = true := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_bool_eq_false_iff_get {p : Char → Bool} {s : String} :
+    s.endsWith p = false ↔ ∀ h, p ((s.endPos.prev h).get (by simp)) = false := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_eq_false_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_bool_eq_getLast? {p : Char → Bool} {s : String} :
+    s.endsWith p = s.toList.getLast?.any p := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_bool_eq_getLast?]
+
+theorem eq_append_of_dropSuffix?_bool_eq_some {p : Char → Bool} {s : String} {res : Slice} (h : s.dropSuffix? p = some res) :
+    ∃ c, s = res.copy ++ singleton c ∧ p c = true := by
+  rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
+  simpa using Slice.eq_append_of_dropSuffix?_bool_eq_some h
+
+theorem skipSuffix?_prop_eq_some_iff {P : Char → Prop} [DecidablePred P] {s : String} {pos : s.Pos} :
+    s.skipSuffix? P = some pos ↔ ∃ h, pos = s.endPos.prev h ∧ P ((s.endPos.prev h).get (by simp)) := by
+  simp [skipSuffix?_eq_skipSuffix?_toSlice, Slice.skipSuffix?_prop_eq_some_iff, ← Pos.toSlice_inj,
+    Pos.prev_toSlice]
+
+theorem endsWith_prop_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
+    s.endsWith P ↔ ∃ h, P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_prop_eq_false_iff_get {P : Char → Prop} [DecidablePred P] {s : String} :
+    s.endsWith P = false ↔ ∀ h, ¬ P ((s.endPos.prev h).get (by simp)) := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_eq_false_iff_get, Pos.prev_toSlice]
+
+theorem endsWith_prop_eq_getLast? {P : Char → Prop} [DecidablePred P] {s : String} :
+    s.endsWith P = s.toList.getLast?.any (decide <| P ·) := by
+  simp [endsWith_eq_endsWith_toSlice, Slice.endsWith_prop_eq_getLast?]
+
+theorem eq_append_of_dropSuffix?_prop_eq_some {P : Char → Prop} [DecidablePred P] {s : String} {res : Slice}
+    (h : s.dropSuffix? P = some res) : ∃ c, s = res.copy ++ singleton c ∧ P c := by
+  rw [dropSuffix?_eq_dropSuffix?_toSlice] at h
+  simpa using Slice.eq_append_of_dropSuffix?_prop_eq_some h
+
 end String

src/Init/Data/String/Lemmas/Pattern/TakeDrop/String.lean

@@ -67,7 +67,7 @@ theorem eq_append_of_dropPrefix?_slice_eq_some {pat s res : Slice} (h : s.dropPr
   | false =>
     have := ForwardSliceSearcher.lawfulForwardPatternModel hpat
     have := Pattern.Model.eq_append_of_dropPrefix?_eq_some h
-    simp only [ForwardPatternModel.Matches] at this
+    simp only [PatternModel.Matches] at this
     obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
     exact h
   | true => simp [Option.some.inj (h ▸ dropPrefix?_slice_of_isEmpty hpat), (show pat.copy = "" by simpa)]
@@ -104,6 +104,87 @@ theorem eq_append_of_dropPrefix?_string_eq_some {pat : String} {s res : Slice} (
   rw [dropPrefix?_string_eq_dropPrefix?_toSlice] at h
   simpa using eq_append_of_dropPrefix?_slice_eq_some h
 
+theorem skipSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    s.skipSuffix? pat = some s.endPos := by
+  rw [skipSuffix?_eq_backwardPatternSkipSuffix?, BackwardSliceSearcher.skipSuffix?_of_isEmpty hpat]
+
+@[simp]
+theorem skipSuffix?_slice_eq_some_iff {pat s : Slice} {pos : s.Pos} :
+    s.skipSuffix? pat = some pos ↔ ∃ t, pos.Splits t pat.copy := by
+  match h : pat.isEmpty with
+  | false =>
+    have := BackwardSliceSearcher.lawfulBackwardPatternModel h
+    rw [Pattern.Model.skipSuffix?_eq_some_iff, ForwardSliceSearcher.isLongestRevMatch_iff_splits h]
+  | true => simp [skipSuffix?_slice_of_isEmpty h, (show pat.copy = "" by simpa), eq_comm]
+
+theorem endsWith_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    s.endsWith pat = true := by
+  rw [endsWith_eq_backwardPatternEndsWith, BackwardSliceSearcher.endsWith_of_isEmpty hpat]
+
+@[simp]
+theorem endsWith_slice_iff {pat s : Slice} :
+    s.endsWith pat ↔ pat.copy.toList <:+ s.copy.toList := by
+  match h : pat.isEmpty with
+  | false =>
+    have := BackwardSliceSearcher.lawfulBackwardPatternModel h
+    simp only [Model.endsWith_iff, ForwardSliceSearcher.revMatchesAt_iff_splits h,
+      splits_endPos_iff, exists_eq_right]
+    simp only [← toList_inj, toList_append, List.suffix_iff_exists_append_eq]
+    exact ⟨fun ⟨t, ht⟩ => ⟨t.toList, by simp [ht]⟩, fun ⟨t, ht⟩ => ⟨String.ofList t, by simp [← ht]⟩⟩
+  | true => simp [endsWith_slice_of_isEmpty h, (show pat.copy = "" by simpa)]
+
+@[simp]
+theorem endsWith_slice_eq_false_iff {pat s : Slice} :
+    s.endsWith pat = false ↔ ¬ (pat.copy.toList <:+ s.copy.toList) := by
+  simp [← Bool.not_eq_true, endsWith_slice_iff]
+
+theorem dropSuffix?_slice_of_isEmpty {pat s : Slice} (hpat : pat.isEmpty = true) :
+    s.dropSuffix? pat = some s := by
+  simp [dropSuffix?_eq_map_skipSuffix?, skipSuffix?_slice_of_isEmpty hpat]
+
+theorem eq_append_of_dropSuffix?_slice_eq_some {pat s res : Slice} (h : s.dropSuffix? pat = some res) :
+    s.copy = res.copy ++ pat.copy := by
+  match hpat : pat.isEmpty with
+  | false =>
+    have := BackwardSliceSearcher.lawfulBackwardPatternModel hpat
+    have := Pattern.Model.eq_append_of_dropSuffix?_eq_some h
+    simp only [PatternModel.Matches] at this
+    obtain ⟨_, ⟨-, rfl⟩, h⟩ := this
+    exact h
+  | true => simp [Option.some.inj (h ▸ dropSuffix?_slice_of_isEmpty hpat), (show pat.copy = "" by simpa)]
+
+@[simp]
+theorem skipSuffix?_string_eq_some_iff' {pat : String} {s : Slice} {pos : s.Pos} :
+    s.skipSuffix? pat = some pos ↔ ∃ t, pos.Splits t pat := by
+  simp [skipSuffix?_string_eq_skipSuffix?_toSlice]
+
+@[simp]
+theorem skipSuffix?_string_empty {s : Slice} : s.skipSuffix? "" = some s.endPos := by
+  simp
+
+@[simp]
+theorem endsWith_string_iff {pat : String} {s : Slice} :
+    s.endsWith pat ↔ pat.toList <:+ s.copy.toList := by
+  simp [endsWith_string_eq_endsWith_toSlice]
+
+@[simp]
+theorem endsWith_string_empty {s : Slice} : s.endsWith "" = true := by
+  simp
+
+@[simp]
+theorem endsWith_string_eq_false_iff {pat : String} {s : Slice} :
+    s.endsWith pat = false ↔ ¬ (pat.toList <:+ s.copy.toList) := by
+  simp [endsWith_string_eq_endsWith_toSlice]
+
+@[simp]
+theorem dropSuffix?_string_empty {s : Slice} : s.dropSuffix? "" = some s := by
+  simpa [dropSuffix?_string_eq_dropSuffix?_toSlice] using dropSuffix?_slice_of_isEmpty (by simp)
+
+theorem eq_append_of_dropSuffix?_string_eq_some {pat : String} {s res : Slice} (h : s.dropSuffix? pat = some res) :
+    s.copy = res.copy ++ pat := by
+  rw [dropSuffix?_string_eq_dropSuffix?_toSlice] at h
+  simpa using eq_append_of_dropSuffix?_slice_eq_some h
+
 end Slice
 
 theorem skipPrefix?_slice_of_isEmpty {pat : Slice} {s : String} (hpat : pat.isEmpty = true) :

src/Init/Data/String/Lemmas/Splits.lean

@@ -367,7 +367,7 @@ theorem Slice.Pos.Splits.of_prev {s : Slice} {p : s.Pos} {hp}
   obtain ⟨rfl, rfl, rfl⟩ := by simpa using h.eq (splits_prev p hp)
   exact splits_prev_right p hp
 
-theorem Slice.sliceTo_copy_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ : String} :
+theorem Slice.copy_sliceTo_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ : String} :
     (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
   refine ⟨?_, ?_⟩
   · rintro rfl
@@ -375,13 +375,21 @@ theorem Slice.sliceTo_copy_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₁ :
   · rintro ⟨t₂, h⟩
     exact p.splits.eq_left h
 
-theorem sliceTo_copy_eq_iff_exists_splits {s : String} {p : s.Pos} {t₁ : String} :
-    (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
+theorem Slice.copy_sliceFrom_eq_iff_exists_splits {s : Slice} {p : s.Pos} {t₂ : String} :
+    (s.sliceFrom p).copy = t₂ ↔ ∃ t₁, p.Splits t₁ t₂ := by
   refine ⟨?_, ?_⟩
   · rintro rfl
     exact ⟨_, p.splits⟩
   · rintro ⟨t₂, h⟩
-    exact p.splits.eq_left h
+    exact p.splits.eq_right h
+
+theorem copy_sliceTo_eq_iff_exists_splits {s : String} {p : s.Pos} {t₁ : String} :
+    (s.sliceTo p).copy = t₁ ↔ ∃ t₂, p.Splits t₁ t₂ := by
+  simp [← Pos.splits_toSlice_iff, ← Slice.copy_sliceTo_eq_iff_exists_splits]
+
+theorem copy_sliceFrom_eq_iff_exists_splits {s : String} {p : s.Pos} {t₂ : String} :
+    (s.sliceFrom p).copy = t₂ ↔ ∃ t₁, p.Splits t₁ t₂ := by
+  simp [← Pos.splits_toSlice_iff, ← Slice.copy_sliceFrom_eq_iff_exists_splits]
 
 theorem Pos.Splits.offset_eq_decreaseBy {s : String} {p : s.Pos} (h : p.Splits t₁ t₂) :
     p.offset = s.rawEndPos.decreaseBy t₂.utf8ByteSize := by
@@ -427,8 +435,7 @@ theorem Slice.splits_singleton_iff {s : Slice} {p : s.Pos} {c : Char} {t : Strin
       simp [startPos_ne_endPos_iff, ← copy_ne_empty_iff, h.eq_append]
     have spl : (s.startPos.next this).Splits (singleton c) t := by
       rw [← empty_append (s := singleton c)]
-      apply Pos.Splits.next
-      simp [h.eq_append]
+      exact Pos.Splits.next (by simp [h.eq_append])
     refine ⟨this, ⟨h.pos_eq spl, ?_, h.eq_append⟩⟩
     rw [← empty_append (s := singleton c)] at spl
     exact spl.get_eq_of_singleton
@@ -442,6 +449,27 @@ theorem splits_singleton_iff {s : String} {p : s.Pos} {c : Char} {t : String} :
   rw [← Pos.splits_toSlice_iff, Slice.splits_singleton_iff]
   simp [← Pos.ofToSlice_inj]
 
+theorem Slice.splits_singleton_right_iff {s : Slice} {p : s.Pos} {c : Char} {t : String} :
+    p.Splits t (singleton c) ↔
+      ∃ h, p = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c ∧ s.copy = t ++ singleton c := by
+  refine ⟨fun h => ?_, ?_⟩
+  · have : s.endPos ≠ s.startPos := by
+      simp [ne_comm (a := s.endPos), startPos_ne_endPos_iff, ← copy_ne_empty_iff, h.eq_append]
+    have spl : (s.endPos.prev this).Splits t (singleton c) := by
+      rw [← append_empty (s := singleton c)]
+      exact Pos.Splits.prev (by simp [h.eq_append])
+    refine ⟨this, ⟨h.pos_eq spl, ?_, h.eq_append⟩⟩
+    exact (h.eq_append ▸ Pos.next_prev (h := this) ▸ s.splits_endPos).get_eq_of_singleton
+  · rintro ⟨h, ⟨rfl, rfl, h'⟩⟩
+    rw [← String.append_empty (s := singleton _)]
+    exact Pos.Splits.prev (by simp [h'])
+
+theorem splits_singleton_right_iff {s : String} {p : s.Pos} {c : Char} {t : String} :
+    p.Splits t (singleton c) ↔
+      ∃ h, p = s.endPos.prev h ∧ (s.endPos.prev h).get (by simp) = c ∧ s = t ++ singleton c := by
+  rw [← Pos.splits_toSlice_iff, Slice.splits_singleton_right_iff]
+  simp [← Pos.ofToSlice_inj, Pos.prev_toSlice]
+
 theorem Slice.splits_next_startPos {s : Slice} {h : s.startPos ≠ s.endPos} :
     (s.startPos.next h).Splits
       (singleton (s.startPos.get h)) (s.sliceFrom (s.startPos.next h)).copy := by
@@ -456,6 +484,20 @@ theorem splits_next_startPos {s : String} {h : s.startPos ≠ s.endPos} :
   rw [← Pos.splits_toSlice_iff]
   apply (Slice.splits_next_startPos).of_eq <;> simp [String.Pos.next_toSlice]
 
+theorem Slice.splits_prev_endPos {s : Slice} {h : s.endPos ≠ s.startPos} :
+    (s.endPos.prev h).Splits
+      (s.sliceTo (s.endPos.prev h)).copy (singleton ((s.endPos.prev h).get (by simp))) := by
+  rw [← String.append_empty (s := singleton _)]
+  apply Slice.Pos.Splits.prev
+  have := Slice.Pos.splits_prev_right s.endPos h
+  rwa [copy_sliceFrom_endPos] at this
+
+theorem splits_prev_endPos {s : String} {h : s.endPos ≠ s.startPos} :
+    (s.endPos.prev h).Splits
+      (s.sliceTo (s.endPos.prev h)).copy (singleton ((s.endPos.prev h).get (by simp))) := by
+  rw [← Pos.splits_toSlice_iff]
+  apply (Slice.splits_prev_endPos).of_eq <;> simp [String.Pos.prev_toSlice, h]
+
 theorem Slice.Pos.Splits.toByteArray_eq_left {s : Slice} {p : s.Pos} {t₁ t₂ : String} (h : p.Splits t₁ t₂) :
     t₁.toByteArray = s.copy.toByteArray.extract 0 p.offset.byteIdx := by
   rw [h.eq_left p.splits]

src/Init/Data/String/Pattern/Basic.lean

@@ -117,7 +117,7 @@ class ForwardPattern {ρ : Type} (pat : ρ) where
   -/
   startsWith : (s : Slice) → Bool := fun s => (skipPrefix? s).isSome
 
-@[deprecated ForwardPattern.dropPrefix? (since := "2026-03-19")]
+@[deprecated ForwardPattern.skipPrefix? (since := "2026-03-19")]
 def ForwardPattern.dropPrefix? {ρ : Type} (pat : ρ) [ForwardPattern pat] (s : Slice) : Option s.Pos :=
   ForwardPattern.skipPrefix? pat s
 

src/Init/Data/String/Pattern/Char.lean

@@ -47,8 +47,8 @@ instance {c : Char} : LawfulBackwardPattern c where
   skipSuffixOfNonempty?_eq h := LawfulBackwardPattern.skipSuffixOfNonempty?_eq (pat := (· == c)) h
   endsWith_eq s := LawfulBackwardPattern.endsWith_eq (pat := (· == c)) s
 
-instance {c : Char} : ToBackwardSearcher c (ToBackwardSearcher.DefaultBackwardSearcher c) :=
-  .defaultImplementation
+instance {c : Char} : ToBackwardSearcher c (ToBackwardSearcher.DefaultBackwardSearcher (· == c)) where
+  toSearcher s := ToBackwardSearcher.toSearcher (· == c) s
 
 end Char
 

src/Init/Data/String/Pattern/Pred.lean

@@ -139,8 +139,9 @@ instance {p : Char → Prop} [DecidablePred p] : LawfulBackwardPattern p where
   skipSuffixOfNonempty?_eq h := LawfulBackwardPattern.skipSuffixOfNonempty?_eq (pat := (decide <| p ·)) h
   endsWith_eq s := LawfulBackwardPattern.endsWith_eq (pat := (decide <| p ·)) s
 
-instance {p : Char → Prop} [DecidablePred p] : ToBackwardSearcher p (ToBackwardSearcher.DefaultBackwardSearcher p) :=
-  .defaultImplementation
+instance {p : Char → Prop} [DecidablePred p] :
+    ToBackwardSearcher p (ToBackwardSearcher.DefaultBackwardSearcher (decide <| p ·)) where
+  toSearcher s := ToBackwardSearcher.toSearcher (decide <| p ·) s
 
 end Decidable
 

tests/bench/mvcgen/sym/lib/VCGen.lean

@@ -451,10 +451,18 @@ meta def mkBackwardRuleForSplit (splitInfo : SplitInfo) (m σs ps instWP : Expr)
 VC generation
 -/
 
-/-- Configuration specific to grind-mode VCGen. -/
-public structure GrindContext where
-  /-- Simp methods used to pre-simplify hypotheses before grind internalization. -/
-  hypSimpMethods : Sym.Simp.Methods
+/-- Pre-tactic to try on each emitted VC before returning it to the user. -/
+public inductive VCGen.PreTac where
+  /-- No pre-tactic; VCs are returned as-is. -/
+  | none
+  /-- Use grind with the given hypothesis simplification methods. -/
+  | grind
+  /-- Use a user-provided tactic syntax. -/
+  | tactic (tac : Syntax)
+
+meta def VCGen.PreTac.isGrind : VCGen.PreTac → Bool
+  | .grind => true
+  | _ => false
 
 public structure VCGen.Context where
   specThms : SpecTheoremsNew
@@ -466,8 +474,10 @@ public structure VCGen.Context where
   exceptCondsEntailsRflRule : BackwardRule
   /-- The backward rule for `Triple.of_entails_wp`. -/
   tripleOfEntailsWPRule : BackwardRule
-  /-- If `some`, VCGen runs in grind mode with the given configuration. -/
-  grindCtx? : Option GrindContext := none
+  /-- User-customizable simp methods used to pre-simplify hypotheses. -/
+  hypSimpMethods : Option Sym.Simp.Methods := none
+  /-- Pre-tactic to try on each emitted VC. -/
+  preTac : PreTac := .none
 
 public structure VCGen.State where
   /--
@@ -618,16 +628,6 @@ meta partial def reduceProjBeta? (e : Expr) : SymM (Option Expr) :=
           pure (some (.letE x ty val body' nondep))
       | _ => pure lastReduction
 
-structure WorkItem where
-  mvarId : MVarId
-  grindGoal? : Option Grind.Goal := none
-
-@[inline] meta def WorkItem.withMVarId (item : WorkItem) (newGoal : MVarId) : WorkItem :=
-  { item with mvarId := newGoal, grindGoal? := item.grindGoal?.map fun g => { g with mvarId := newGoal } }
-
-@[inline] meta def WorkItem.forkTo (item : WorkItem) (subgoals : List MVarId) : List WorkItem :=
-  subgoals.map item.withMVarId
-
 inductive SolveResult where
   /-- `target` was not of the form `H ⊢ₛ T`. -/
   | noEntailment (target : Expr)
@@ -640,8 +640,8 @@ inductive SolveResult where
   Candidates were `thms`, but none of them matched the monad.
   -/
   | noSpecFoundForProgram (e : Expr) (monad : Expr) (thms : Array SpecTheoremNew)
-  /-- Successfully discharged the goal. These are the subgoals. -/
-  | goals (subgoals : List WorkItem)
+  /-- Successfully decomposed the goal. These are the subgoals. -/
+  | goals (subgoals : List MVarId)
 
 open Sym Sym.Internal
 -- The following function is vendored until it is made public:
@@ -674,29 +674,11 @@ open Sym Sym.Internal
 meta def mkAppNS [Monad m] [Internal.MonadShareCommon m] (f : Expr) (args : Array Expr) : m Expr :=
   mkAppRangeS f 0 args.size args
 
-private meta def getNatLit? (e : Expr) : Option Nat := do
-  let_expr OfNat.ofNat _ n _ := e | failure
-  let .lit (.natVal n) := n | failure
-  return n
-
-/--
-A `Sym.Simp` post-simproc that reassociates Nat addition to fold nested literal additions.
-Rewrites `(a + m) + n` → `a + (m + n)` when `m` and `n` are Nat literals, using `Nat.add_assoc`.
-Since `m + n` reduces to a literal by kernel computation, this collapses chains like
-`s + 1 + 1 + 1` into `s + 3` in a single step.
--/
-meta def reassocNatAdd : Sym.Simp.Simproc := fun e => do
-  let_expr HAdd.hAdd α _ _ inst ab n := e | return .rfl
-  let_expr Nat := α | return .rfl
-  let some nVal := getNatLit? n | return .rfl
-  let_expr HAdd.hAdd _ _ _ _ a m := ab | return .rfl
-  let some mVal := getNatLit? m | return .rfl
-  -- (a + m) + n → a + (m + n), with m + n folded to a literal
-  let sumExpr ← share <| toExpr (mVal + nVal)
-  let result ← share <| mkApp6 (mkConst ``HAdd.hAdd [0, 0, 0]) α α α inst a sumExpr
-  -- Proof: Nat.add_assoc a m n : (a + m) + n = a + (m + n)
-  let proof := mkApp3 (mkConst ``Nat.add_assoc) a m n
-  return .step result proof
+meta def simp (e : Expr) (methods : Sym.Simp.Methods) : VCGenM Sym.Simp.Result := do
+  let simpState := (← get).simpState
+  let (result, simpState') ← Sym.Simp.SimpM.run (Sym.simp e) methods {} simpState
+  modify fun s => { s with simpState := simpState' }
+  return result
 
 /--
 Simplify types of not-yet-internalized hypotheses in the grind goal using `Sym.simp`.
@@ -709,45 +691,54 @@ meta def simpNewHyps (mvarId : MVarId) (nextDeclIdx : Nat) (methods : Sym.Simp.M
   let lctx ← getLCtx
   let mut mvarId := mvarId
   for decl in lctx do
+    -- TODO: Start this loop at index nextDeclIdx. Yields much less convenient code.
     if decl.index < nextDeclIdx then continue
     if decl.isImplementationDetail then continue
-    let simpState := (← get).simpState
-    let (result, simpState') ← Sym.Simp.SimpM.run (Sym.Simp.simp decl.type) methods {} simpState
-    modify fun s => { s with simpState := simpState' }
-    match result with
+    match ← simp decl.type methods with
     | .rfl .. => pure ()
     | .step newType _proof .. =>
       mvarId ← mvarId.replaceLocalDeclDefEq decl.fvarId newType
   return mvarId
 
-/-- Internalize pending hypotheses in the grind state before forking to multiple subgoals.
+/-- Internalize pending hypotheses into the E-graph for sharing before forking to multiple subgoals.
+Skips `simpNewHyps` so it is safe to call even when the mvar is assigned (e.g., after a
+backward rule has been applied). `simpNewHyps` will run later per-VC in `emitVC`.
 If `processHypotheses` discovers a contradiction (`inconsistent = true`), the E-graph state
 contains stale proof data (the contradiction proof targets the parent's mvar, not the children's).
 In that case, restore the pre-internalization state so each child can discover the contradiction
 independently and construct its own proof via `closeGoal`.
-
 -/
-meta def internalizePending (item : WorkItem) : VCGenM WorkItem := do
-  if let some grindGoal := item.grindGoal? then
-    let some grindCtx := (← read).grindCtx? | unreachable!
-    let mvarId ← simpNewHyps item.mvarId grindGoal.nextDeclIdx grindCtx.hypSimpMethods
-    let grindGoal := { grindGoal with mvarId }
-    let saved := grindGoal
-    let grindGoal ← Grind.processHypotheses grindGoal
-    if grindGoal.inconsistent then
-      return { item with grindGoal? := some saved }
-    return { item with grindGoal? := some grindGoal }
-  return item
+meta def PreTac.processHypotheses (preTac : PreTac) (goal : Grind.Goal) : VCGenM Grind.Goal := do
+  let mut goal := goal
+  if let some methods := (← read).hypSimpMethods then
+    let mvarId ← simpNewHyps goal.mvarId goal.nextDeclIdx methods
+    goal := { goal with mvarId }
+  if preTac.isGrind then
+    goal ← Grind.processHypotheses goal
+  return goal
 
 /--
-The main VC generation function.
-Looks at a goal of the form `P ⊢ₛ T`. Then
-* If `T` is a lambda, introduce another state variable.
-* If `T` is of the form `wp⟦e⟧ Q s₁ ... sₙ`, look up a spec theorem for `e`. Produce the backward
-  rule to apply this spec theorem and then apply it ot the goal.
+The main VC generation step. Operates on a plain `MVarId` with no knowledge of grind.
+Returns `.goals subgoals` when the goal was decomposed, or a classification result
+(`.noEntailment`, `.noProgramFoundInTarget`, etc.) when no further decomposition is possible.
+
+The function performs the following steps in order:
+
+1. **Forall introduction**: If the target is a `∀`, introduce binders via `Sym.intros`.
+2. **Triple unfolding**: If the target is `⦃P⦄ x ⦃Q⦄`, unfold into `P ⊢ₛ wp⟦x⟧ Q`.
+3. **PostCond.entails decomposition**: Split `PostCond.entails` into its components.
+4. **Lambda introduction**: If the RHS `T` in `H ⊢ₛ T` is a lambda, eta-expand via
+   `SPred.entails_cons_intro` (introduces an extra state variable).
+5. **Proj/beta reduction**: Reduce `Prod.fst`/`Prod.snd` projections and beta redexes in
+   both `H` and `T` (e.g., `(fun _ => T, Q.snd).fst s` → `T`).
+6. **Syntactic rfl**: If `T` is not a `PredTrans.apply`, try closing by `SPred.entails.refl`.
+7. **Let-zeta**: Zeta-reduce let-expressions in the program head.
+8. **Iota reduction**: Reduce matchers/recursors with concrete discriminants.
+9. **ite/dite/match splitting**: Apply the appropriate split backward rule.
+10. **Spec application**: Look up a registered `@[spec]` theorem (triple or simp) and apply
+    its cached backward rule.
 -/
-meta def solve (item : WorkItem) : VCGenM SolveResult := item.mvarId.withContext do
-  let goal := item.mvarId
+meta def solve (goal : MVarId) : VCGenM SolveResult := goal.withContext do
   let target ← goal.getType
   trace[Elab.Tactic.Do.vcgen] "target: {target}"
   -- There are two layers of preprocessing before we get to taking apart program syntax.
@@ -757,16 +748,16 @@ meta def solve (item : WorkItem) : VCGenM SolveResult := item.mvarId.withContext
 
   if target.isForall then
     let IntrosResult.goal _ goal ← Sym.intros goal | throwError "Failed to introduce binders for {target}"
-    return .goals [item.withMVarId goal]
+    return .goals [goal]
 
   let f := target.getAppFn
   if f.isConstOf ``Triple then
     let goal ← tripleOfWP goal
-    return .goals [item.withMVarId goal]
+    return .goals [goal]
 
   if f.isConstOf ``PostCond.entails then
     let goal ← decomposePostCondEntails goal
-    return .goals [item.withMVarId goal]
+    return .goals [goal]
 
   let_expr ent@SPred.entails σs H T := target | return .noEntailment target
   -- The goal is of the form `H ⊢ₛ T`. Try some reductions to expose `wp⟦e⟧ Q s₁ ... sₙ` in `T`.
@@ -777,7 +768,7 @@ meta def solve (item : WorkItem) : VCGenM SolveResult := item.mvarId.withContext
     -- extra state arg `s` to reduce away the lambda.
     let .goals [goal] ← (← read).entailsConsIntroRule.apply goal
       | throwError "Applying {.ofConstName ``SPred.entails_cons_intro} to {target} failed. It should not."
-    return .goals [item.withMVarId goal]
+    return .goals [goal]
 
   /-
   Do a very targeted simplification to turn `H ⊢ₛ (fun _ => T, Q.snd).fst s` into `H ⊢ₛ T`, and
@@ -798,7 +789,7 @@ meta def solve (item : WorkItem) : VCGenM SolveResult := item.mvarId.withContext
   let T? ← reduceProjBeta? T
   if H?.isSome || T?.isSome then
     let goal ← goal.replaceTargetDefEq (← Sym.Internal.mkAppS₃ ent σs (H?.getD H) (T?.getD T))
-    return .goals [item.withMVarId goal]
+    return .goals [goal]
 
   -- Look for program syntax in `T`.
   T.withApp fun head args => do
@@ -832,74 +823,80 @@ meta def solve (item : WorkItem) : VCGenM SolveResult := item.mvarId.withContext
   if let .letE _x _ty val body _nonDep := f then
     let body' ← Sym.instantiateRevBetaS body #[val]
     let e' ← mkAppRevS body' e.getAppRevArgs
-    return .goals [item.withMVarId (← replaceProgDefEq e')]
+    return .goals [← replaceProgDefEq e']
 
   -- Split ite/dite/match
   if let some info ← liftMetaM <| Lean.Elab.Tactic.Do.getSplitInfo? e then
     -- Try iota reduction first (reduces matcher/recursor with concrete discriminant)
     if let some e' ← liftMetaM <| reduceRecMatcher? e then
-      return .goals [item.withMVarId (← replaceProgDefEq (← shareCommonInc e'))]
-    -- Internalize pending hypotheses before forking
-    let item ← internalizePending item
+      return .goals [← replaceProgDefEq (← shareCommonInc e')]
     let rule ← mkBackwardRuleFromSplitInfoCached info m σs ps instWP excessArgs
-    let ApplyResult.goals goals ← rule.apply item.mvarId
+    let ApplyResult.goals goals ← rule.apply goal
       | throwError "Failed to apply split rule for {indentExpr e}"
-    return .goals (item.forkTo goals)
+    return .goals goals
 
   -- Apply registered specifications (both triple and simp specs use cached backward rules).
   if f.isConst || f.isFVar then
-    -- Internalize pending hypotheses before potential multi-goal fork
-    let item ← internalizePending item
     trace[Elab.Tactic.Do.vcgen] "Applying a spec for {e}. Excess args: {excessArgs}"
     match ← (← read).specThms.findSpecs e with
     | .error thms => return .noSpecFoundForProgram e m thms
     | .ok thm =>
     trace[Elab.Tactic.Do.vcgen] "Spec for {e}: {thm.proof}"
     let rule ← mkBackwardRuleFromSpecCached thm m σs ps instWP excessArgs
-    let ApplyResult.goals goals ← rule.apply item.mvarId
+    let ApplyResult.goals goals ← rule.apply goal
       | throwError "Failed to apply rule {thm.proof} for {indentExpr e}"
-    return .goals (item.forkTo goals)
+    return .goals goals
 
   return .noStrategyForProgram e
 
 /--
-Called when decomposing the goal further did not succeed; in this case we emit a VC for the goal.
-In grind mode, tries to solve the VC using the accumulated `Grind.Goal` state (E-graph) via
-`Grind.withProtectedMCtx` + `Grind.processHypotheses` + `Grind.solve`.
+Runs the `preTac` on the VC:
+- `.grind`: tries to solve the VC using the accumulated `Grind.Goal` state via `Grind.Goal.grind`.
+- `.tactic`: runs the user-provided tactic on the VC, potentially emitting multiple subgoals.
+- `.none`: returns the VC as-is.
 -/
-meta def emitVC (item : WorkItem) : VCGenM Unit := do
-  let ty ← item.mvarId.getType
-  if ty.isAppOf ``Std.Do.Invariant then
-    item.mvarId.setKind .syntheticOpaque
-    modify fun s => { s with invariants := s.invariants.push item.mvarId }
-  else if let some grindGoal := item.grindGoal? then
-    let some grindCtx := (← read).grindCtx? | unreachable!
-    let mvarId ← simpNewHyps item.mvarId grindGoal.nextDeclIdx grindCtx.hypSimpMethods
-    let grindGoal := { grindGoal with mvarId }
-    let config ← Grind.getConfig
-    Grind.withProtectedMCtx config mvarId fun mvarId' => do
-      let grindGoal' := { grindGoal with mvarId := mvarId' }
-      let grindGoal' ← Grind.processHypotheses grindGoal'
-      unless ← mvarId'.isAssigned do
-        discard <| Grind.solve grindGoal'
-    unless ← mvarId.isAssigned do
-      mvarId.setKind .syntheticOpaque
-      modify fun s => { s with vcs := s.vcs.push mvarId }
-  else
-    item.mvarId.setKind .syntheticOpaque
-    modify fun s => { s with vcs := s.vcs.push item.mvarId }
+meta def PreTac.run : PreTac →  Grind.Goal → VCGenM (List MVarId)
+  | .none, goal => return [goal.mvarId]
+  | .grind, goal => do
+    let savedMCtx ← getMCtx
+    let goal ← goal.internalizeAll
+    match ← goal.grind with
+    | .closed => return []
+    | .failed .. =>
+      setMCtx savedMCtx
+      return [goal.mvarId]
+  | .tactic tac, goal =>
+    try
+      let (gs, _) ← Lean.Elab.runTactic goal.mvarId tac {} {}
+      pure gs
+    catch _ =>
+      pure [goal.mvarId]
 
-meta def work (item : WorkItem) : VCGenM Unit := do
-  let goal ← preprocessMVar item.mvarId
-  let item := item.withMVarId goal
-  let mut worklist := Std.Queue.empty.enqueue item
+/--
+Called when decomposing the goal further did not succeed; in this case we emit a VC for the goal.
+-/
+meta def emitVC (goal : Grind.Goal) : VCGenM Unit := do
+  let ty ← goal.mvarId.getType
+  if ty.isAppOf ``Std.Do.Invariant then
+    goal.mvarId.setKind .syntheticOpaque
+    modify fun s => { s with invariants := s.invariants.push goal.mvarId }
+    return
+  let goals ← (← read).preTac.run goal
+  for g in goals do g.setKind .syntheticOpaque
+  modify fun s => { s with vcs := s.vcs ++ goals.toArray }
+
+meta def work (goal : Grind.Goal) : VCGenM Unit := do
+  let mvarId ← preprocessMVar goal.mvarId
+  let goal := { goal with mvarId }
+  let mut worklist := Std.Queue.empty.enqueue goal
   repeat do
-    let some (item, worklist') := worklist.dequeue? | break
+    let some (goal, worklist') := worklist.dequeue? | break
+    let mut goal := goal
     worklist := worklist'
-    let res ← solve item
+    let res ← solve goal.mvarId
     match res with
     | .noEntailment .. | .noProgramFoundInTarget .. =>
-      emitVC item
+      emitVC goal
     | .noSpecFoundForProgram prog _ #[] =>
       throwError "No spec found for program {prog}."
     | .noSpecFoundForProgram prog monad thms =>
@@ -907,7 +904,11 @@ meta def work (item : WorkItem) : VCGenM Unit := do
     | .noStrategyForProgram prog =>
       throwError "Did not know how to decompose weakest precondition for {prog}"
     | .goals subgoals =>
-      worklist := worklist.enqueueAll subgoals
+      -- In grind mode with multiple subgoals, preprocess pending hypotheses
+      -- to share E-graph context before forking.
+      if (← read).preTac.isGrind && subgoals.length > 1 then
+        goal ← Grind.processHypotheses goal
+      worklist := worklist.enqueueAll (subgoals.map ({ goal with mvarId := · }))
 
 public structure Result where
   invariants : Array MVarId
@@ -921,13 +922,11 @@ When `grindMode` is true, integrates grind into the VCGen loop for incremental c
 internalization, avoiding O(n) re-internalization per VC.
 -/
 public meta partial def main (goal : MVarId) (ctx : Context) : Grind.GrindM Result := do
-  let grindGoal? ←
-    if ctx.grindCtx?.isSome then
-      let g ← Grind.mkGoalCore goal
-      some <$> Grind.processHypotheses g
-    else pure none
-  let item : WorkItem := { mvarId := goal, grindGoal? }
-  let ((), state) ← StateRefT'.run (ReaderT.run (work item) ctx) {}
+  let grindGoal ← Grind.mkGoalCore goal
+  let grindGoal ← if ctx.preTac.isGrind then
+    Grind.processHypotheses grindGoal
+  else pure grindGoal
+  let ((), state) ← StateRefT'.run (ReaderT.run (work grindGoal) ctx) {}
   _ ← state.invariants.mapIdxM fun idx mv => do
     mv.setTag (Name.mkSimple ("inv" ++ toString (idx + 1)))
   _ ← state.vcs.mapIdxM fun idx mv => do
@@ -1008,6 +1007,30 @@ syntax (name := mvcgen') "mvcgen'"
   (" [" withoutPosition((simpStar <|> simpErase <|> simpLemma),*,?) "] ")?
   (&" with " tactic)? : tactic
 
+private meta def getNatLit? (e : Expr) : Option Nat := do
+  let_expr OfNat.ofNat _ n _ := e | failure
+  let .lit (.natVal n) := n | failure
+  return n
+
+/--
+A `Sym.Simp` post-simproc that reassociates Nat addition to fold nested literal additions.
+Rewrites `(a + m) + n` → `a + (m + n)` when `m` and `n` are Nat literals, using `Nat.add_assoc`.
+Since `m + n` reduces to a literal by kernel computation, this collapses chains like
+`s + 1 + 1 + 1` into `s + 3` in a single step.
+-/
+private meta def reassocNatAdd : Sym.Simp.Simproc := fun e => do
+  let_expr HAdd.hAdd α _ _ inst ab n := e | return .rfl
+  let_expr Nat := α | return .rfl
+  let some nVal := getNatLit? n | return .rfl
+  let_expr HAdd.hAdd _ _ _ _ a m := ab | return .rfl
+  let some mVal := getNatLit? m | return .rfl
+  -- (a + m) + n → a + (m + n), with m + n folded to a literal
+  let sumExpr ← share <| toExpr (mVal + nVal)
+  let result ← share <| mkApp6 (mkConst ``HAdd.hAdd [0, 0, 0]) α α α inst a sumExpr
+  -- Proof: Nat.add_assoc a m n : (a + m) + n = a + (m + n)
+  let proof := mkApp3 (mkConst ``Nat.add_assoc) a m n
+  return .step result proof
+
 /-- Parse grind configuration from the `with grind ...` clause and build `Grind.Params`.
 Overrides the internal simp step limit to accommodate large unrolled goals. -/
 private meta def mkGrindParamsFromSyntax (grindStx : Syntax) (goal : MVarId) : TacticM Grind.Params := do
@@ -1029,22 +1052,19 @@ public meta def elabMVCGen' : Tactic := fun stx => withMainContext do
   let withTac? := if stx[2].getNumArgs != 0 then some stx[2][1] else none
   let isGrind := withTac?.any (·.getKind == ``Lean.Parser.Tactic.grind)
   let mut params ← Grind.mkDefaultParams {}
-  let mut grindCtx? : Option GrindContext := none
+  let mut preTac : VCGen.PreTac := .none
+  let mut hypSimpMethods : Option Sym.Simp.Methods := none
   if isGrind then
     params ← mkGrindParamsFromSyntax withTac?.get! goal
-    grindCtx? := some { hypSimpMethods := { post := VCGen.reassocNatAdd } }
-  let ctx := { ctx with grindCtx? }
+    preTac := .grind
+    hypSimpMethods := some { post := reassocNatAdd } -- TODO: Make this user-extensible
+  else if let some tac := withTac? then
+    preTac := .tactic tac
+  let ctx := { ctx with preTac, hypSimpMethods }
 
   let result ← Grind.GrindM.run (VCGen.main goal ctx) params
 
-  let mut vcs := result.vcs
-  if let some tac := withTac? then
-    if !isGrind then
-      let mut remaining : Array MVarId := #[]
-      for vc in result.vcs do
-        remaining := remaining ++ (← try evalTacticAt tac vc catch _ => pure [vc]).toArray
-      vcs := remaining
-  replaceMainGoal (result.invariants ++ vcs).toList
+  replaceMainGoal (result.invariants ++ result.vcs).toList
 
 /-!
 Local tests for faster iteration: