fix: missing pures in the Id monad

Cslib/Foundations/Control/Monad/Free/Effects.lean

@@ -100,9 +100,8 @@ The canonical interpreter `toStateM` derived from `liftM` agrees with the hand-w
 recursive interpreter `run` for `FreeState`.
 -/
 @[simp]
-theorem run_toStateM {α : Type u} (comp : FreeState σ α) :
-    (toStateM comp).run = run comp := by
-  ext s₀ : 1
+theorem run_toStateM {α : Type u} (comp : FreeState σ α) (s₀ : σ) :
+    (toStateM comp).run s₀ = pure (run comp s₀) := by
   induction comp generalizing s₀ with
   | pure a => rfl
   | liftBind op cont ih =>
@@ -124,10 +123,9 @@ lemma run_set (s' : σ) (k : PUnit → FreeState σ α) (s₀ : σ) :
 def run' (c : FreeState σ α) (s₀ : σ) : α := (run c s₀).1
 
 @[simp]
-theorem run'_toStateM {α : Type u} (comp : FreeState σ α) :
-    (toStateM comp).run' = run' comp := by
-  ext s₀ : 1
-  rw [run', ← run_toStateM]
+theorem run'_toStateM {α : Type u} (comp : FreeState σ α) (s₀ : σ) :
+    (toStateM comp).run' s₀ = pure (run' comp s₀) := by
+  rw [run', StateT.run'_eq, run_toStateM]
   rfl
 
 @[simp]
@@ -204,18 +202,40 @@ lemma run_pure [Monoid ω] (a : α) :
 lemma run_liftBind_tell [Monoid ω] (w : ω) (k : PUnit → FreeWriter ω α) :
     run (liftBind (.tell w) k) = (let (a, w') := run (k .unit); (a, w * w')) := rfl
 
+
+-- https://github.com/leanprover-community/mathlib4/pull/36497
+section missing_from_mathlib
+
+@[simp]
+theorem _root_.WriterT.run_pure [Monoid ω] [Monad M] (a : α) :
+    WriterT.run (pure a : WriterT ω M α) = pure (a, 1) := rfl
+
+@[simp]
+theorem _root_.WriterT.run_bind [Monoid ω] [Monad M] (x : WriterT ω M α) (f : α → WriterT ω M β) :
+    WriterT.run (x >>= f) = x.run >>= fun (a, w₁) => (fun (b, w₂) => (b, w₁ * w₂)) <$> (f a).run :=
+  rfl
+
+@[simp]
+theorem _root_.WriterT.run_tell [Monad M] (w : ω) :
+    WriterT.run (MonadWriter.tell w : WriterT ω M PUnit) = pure (.unit, w) := rfl
+
+end missing_from_mathlib
+
 /--
 The canonical interpreter `toWriterT` derived from `liftM` agrees with the hand-written
 recursive interpreter `run` for `FreeWriter`.
 -/
 @[simp]
-theorem run_toWriterT {α : Type u} [Monoid ω] :
-    ∀ comp : FreeWriter ω α, (toWriterT comp).run = run comp
-  | .pure _ => by simp only [toWriterT, liftM_pure, run_pure, pure, WriterT.run]
-  | liftBind (.tell w) cont => by
-    simp only [toWriterT, liftM_liftBind, run_liftBind_tell] at *
-    rw [← run_toWriterT]
-    congr
+theorem run_toWriterT {α : Type u} [Monoid ω] (comp : FreeWriter ω α) :
+    (toWriterT comp).run = pure (run comp) := by
+  ext : 1
+  induction comp with
+  | pure _ => simp only [toWriterT, liftM_pure, run_pure, pure, WriterT.run]
+  | liftBind op cont ih =>
+    cases op
+    simp only [toWriterT, liftM_liftBind, run_liftBind_tell, Id.run_pure] at *
+    rw [ ← ih]
+    simp [WriterT.run_bind, writerInterp]
 
 /--
 `listen` captures the log produced by a subcomputation incrementally. It traverses the computation,
@@ -301,9 +321,8 @@ The canonical interpreter `toContT` derived from `liftM` agrees with the hand-wr
 recursive interpreter `run` for `FreeCont`.
 -/
 @[simp]
-theorem run_toContT {α : Type u} (comp : FreeCont r α) :
-    (toContT comp).run = run comp := by
-  ext k
+theorem run_toContT {α : Type u} (comp : FreeCont r α) (k : α → r) :
+    (toContT comp).run k = pure (run comp k) := by
   induction comp with
   | pure a => rfl
   | liftBind op cont ih =>
@@ -387,7 +406,7 @@ The canonical interpreter `toReaderM` derived from `liftM` agrees with the hand-
 recursive interpreter `run` for `FreeReader` -/
 @[simp]
 theorem run_toReaderM {α : Type u} (comp : FreeReader σ α) (s : σ) :
-    (toReaderM comp).run s = run comp s := by
+    (toReaderM comp).run s = pure (run comp s) := by
   induction comp generalizing s with
   | pure a => rfl
   | liftBind op cont ih =>