have ⟨tailRes, tailEnv⟩ := tailR
α₁ σ₁ : Type
ε₁ : σ₁ → Type
δ₁ : Dialect α₁ σ₁ ε₁
α₂ σ₂ : Type
ε₂ : σ₂ → Type
δ₂ : Dialect α₂ σ₂ ε₂
α✝ σ✝ : Type
ε✝ : σ✝ → Type
Δ : Dialect α✝ σ✝ ε✝
S : Semantics Δ
head : TypedSSAVal Δ
tail : List (TypedSSAVal Δ)
HInd :
  ∀ ⦃env : SSAEnv Δ⦄ ⦃r : List (TypedArg Δ)⦄ ⦃env' : SSAEnv Δ⦄,
    mapM
          (fun x => do
            let __do_lift ← TopM.get x.snd x.fst
            pure { fst := x.snd, snd := __do_lift })
          tail env =
        Except.ok (r, env') →
      env' = env
env : SSAEnv Δ
r✝ : List (TypedArg Δ)
env' : SSAEnv Δ
headName : SSAVal
headτ : MLIRType Δ
r : MLIRType.eval headτ × SSAEnv Δ
rRes : MLIRType.eval headτ
left✝ :
  (match SSAEnv.get headName headτ env with
    | some v => v
    | none => default) =
    rRes
x✝ : Except (String × SSAEnv Δ) (List (TypedArg Δ) × SSAEnv Δ)
tailR : List (TypedArg Δ) × SSAEnv Δ
HTailR :
  mapM
      (fun x s =>
        match TopM.get x.snd x.fst s with
        | Except.error err => Except.error err
        | Except.ok v => Except.ok ({ fst := x.snd, snd := v.fst }, v.snd))
      tail (rRes, env).snd =
    Except.ok tailR
H : Except.ok ({ fst := headτ, snd := (rRes, env).fst } :: tailR.fst, tailR.snd) = Except.ok (r✝, env')
⊢ env' = env
α₁ σ₁ : Type
ε₁ : σ₁ → Type
δ₁ : Dialect α₁ σ₁ ε₁
α₂ σ₂ : Type
ε₂ : σ₂ → Type
δ₂ : Dialect α₂ σ₂ ε₂
α✝ σ✝ : Type
ε✝ : σ✝ → Type
Δ : Dialect α✝ σ✝ ε✝
S : Semantics Δ
head : TypedSSAVal Δ
tail : List (TypedSSAVal Δ)
HInd :
  ∀ ⦃env : SSAEnv Δ⦄ ⦃r : List (TypedArg Δ)⦄ ⦃env' : SSAEnv Δ⦄,
    mapM
          (fun x => do
            let __do_lift ← TopM.get x.snd x.fst
            pure { fst := x.snd, snd := __do_lift })
          tail env =
        Except.ok (r, env') →
      env' = env
env : SSAEnv Δ
r✝ : List (TypedArg Δ)
env' : SSAEnv Δ
headName : SSAVal
headτ : MLIRType Δ
r : MLIRType.eval headτ × SSAEnv Δ
rRes : MLIRType.eval headτ
left✝ :
  (match SSAEnv.get headName headτ env with
    | some v => v
    | none => default) =
    rRes
x✝ : Except (String × SSAEnv Δ) (List (TypedArg Δ) × SSAEnv Δ)
tailR : List (TypedArg Δ) × SSAEnv Δ
tailRes : List (TypedArg Δ)
tailEnv : SSAEnv Δ
HTailR :
  mapM
      (fun x s =>
        match TopM.get x.snd x.fst s with
        | Except.error err => Except.error err
        | Except.ok v => Except.ok ({ fst := x.snd, snd := v.fst }, v.snd))
      tail (rRes, env).snd =
    Except.ok (tailRes, tailEnv)
H :
  Except.ok ({ fst := headτ, snd := (rRes, env).fst } :: (tailRes, tailEnv).fst, (tailRes, tailEnv).snd) =
    Except.ok (r✝, env')
⊢ env' = env