Theorem: denoteOp_equiv
Proof state before:
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
op_name : String
res args : List (TypedSSAVal Δ)
regions : List (Region Δ)
attrs : AttrDict Δ
env : SSAEnv Δ
r : TypedArgs Δ
env' : SSAEnv Δ
H :
  (match denoteOpArgs Δ args env with
    | Except.error err => Except.error err
    | Except.ok v =>
      match
        OpM.toTopM (TopM.mapDenoteRegion Δ regions)
          (Semantics.semantics_op (IOp.mk op_name (List.map Prod.snd res) v.fst (OpM.denoteRegions regions 0) attrs))
          v.snd with
      | Except.error err => Except.error err
      | Except.ok v =>
        (match res with
          | [] => fun s => Except.ok (v.fst, s)
          | [res] =>
            match v.fst with
            | [{ fst := τ, snd := v_1 }] => fun s =>
              match TopM.set τ res.fst v_1 s with
              | Except.error err => Except.error err
              | Except.ok v_2 => Except.ok (v.fst, v_2.snd)
            | x => fun s =>
              match
                TopM.raiseUB (toString "denoteOp: expected 1 return value, got '" ++ toString v.fst ++ toString "'")
                  s with
              | Except.error err => Except.error err
              | Except.ok v_1 => Except.ok (v.fst, v_1.snd)
          | x => fun s =>
            match
              TopM.raiseUB (toString "denoteOp: expected 0 or 1 results, got '" ++ toString res ++ toString "'") s with
            | Except.error err => Except.error err
            | Except.ok v_1 => Except.ok (v.fst, v_1.snd))
          v.snd) =
    Except.ok (r, env')
env₂ : SSAEnv Δ
Henv₂ : SSAEnv.equiv env env₂
⊢ ∃ env₂',
    SSAEnv.equiv env' env₂' ∧
      (match denoteOpArgs Δ args env₂ with
        | Except.error err => Except.error err
        | Except.ok v =>
          match
            OpM.toTopM (TopM.mapDenoteRegion Δ regions)
              (Semantics.semantics_op
                (IOp.mk op_name (List.map Prod.snd res) v.fst (OpM.denoteRegions regions 0) attrs))
              v.snd with
          | Except.error err => Except.error err
          | Except.ok v =>
            (match res with
              | [] => fun s => Except.ok (v.fst, s)
              | [res] =>
                match v.fst with
                | [{ fst := τ, snd := v_1 }] => fun s =>
                  match TopM.set τ res.fst v_1 s with
                  | Except.error err => Except.error err
                  | Except.ok v_2 => Except.ok (v.fst, v_2.snd)
                | x => fun s =>
                  match
                    TopM.raiseUB (toString "denoteOp: expected 1 return value, got '" ++ toString v.fst ++ toString "'")
                      s with
                  | Except.error err => Except.error err
                  | Except.ok v_1 => Except.ok (v.fst, v_1.snd)
              | x => fun s =>
                match
                  TopM.raiseUB (toString "denoteOp: expected 0 or 1 results, got '" ++ toString res ++ toString "'")
                    s with
                | Except.error err => Except.error err
                | Except.ok v_1 => Except.ok (v.fst, v_1.snd))
              v.snd) =
        Except.ok (r, env₂')
Tactic:
split at H <;> try contradiction
Proof state after:
case h_2
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
op_name : String
res args : List (TypedSSAVal Δ)
regions : List (Region Δ)
attrs : AttrDict Δ
env : SSAEnv Δ
r : TypedArgs Δ
env' env₂ : SSAEnv Δ
Henv₂ : SSAEnv.equiv env env₂
x✝ : Except (String × SSAEnv Δ) (List (TypedArg Δ) × SSAEnv Δ)
v✝ : List (TypedArg Δ) × SSAEnv Δ
heq✝ : denoteOpArgs Δ args env = Except.ok v✝
H :
  (match
      OpM.toTopM (TopM.mapDenoteRegion Δ regions)
        (Semantics.semantics_op (IOp.mk op_name (List.map Prod.snd res) v✝.fst (OpM.denoteRegions regions 0) attrs))
        v✝.snd with
    | Except.error err => Except.error err
    | Except.ok v =>
      (match res with
        | [] => fun s => Except.ok (v.fst, s)
        | [res] =>
          match v.fst with
          | [{ fst := τ, snd := v_1 }] => fun s =>
            match TopM.set τ res.fst v_1 s with
            | Except.error err => Except.error err
            | Except.ok v_2 => Except.ok (v.fst, v_2.snd)
          | x => fun s =>
            match
              TopM.raiseUB (toString "denoteOp: expected 1 return value, got '" ++ toString v.fst ++ toString "'")
                s with
            | Except.error err => Except.error err
            | Except.ok v_1 => Except.ok (v.fst, v_1.snd)
        | x => fun s =>
          match
            TopM.raiseUB (toString "denoteOp: expected 0 or 1 results, got '" ++ toString res ++ toString "'") s with
          | Except.error err => Except.error err
          | Except.ok v_1 => Except.ok (v.fst, v_1.snd))
        v.snd) =
    Except.ok (r, env')
⊢ ∃ env₂',
    SSAEnv.equiv env' env₂' ∧
      (match denoteOpArgs Δ args env₂ with
        | Except.error err => Except.error err
        | Except.ok v =>
          match
            OpM.toTopM (TopM.mapDenoteRegion Δ regions)
              (Semantics.semantics_op
                (IOp.mk op_name (List.map Prod.snd res) v.fst (OpM.denoteRegions regions 0) attrs))
              v.snd with
          | Except.error err => Except.error err
          | Except.ok v =>
            (match res with
              | [] => fun s => Except.ok (v.fst, s)
              | [res] =>
                match v.fst with
                | [{ fst := τ, snd := v_1 }] => fun s =>
                  match TopM.set τ res.fst v_1 s with
                  | Except.error err => Except.error err
                  | Except.ok v_2 => Except.ok (v.fst, v_2.snd)
                | x => fun s =>
                  match
                    TopM.raiseUB (toString "denoteOp: expected 1 return value, got '" ++ toString v.fst ++ toString "'")
                      s with
                  | Except.error err => Except.error err
                  | Except.ok v_1 => Except.ok (v.fst, v_1.snd)
              | x => fun s =>
                match
                  TopM.raiseUB (toString "denoteOp: expected 0 or 1 results, got '" ++ toString res ++ toString "'")
                    s with
                | Except.error err => Except.error err
                | Except.ok v_1 => Except.ok (v.fst, v_1.snd))
              v.snd) =
        Except.ok (r, env₂')