Theorem: SSAEnv.set_commutes
Proof state before:
α σ : Type
ε : σ → Type
α✝ σ✝ : Type
ε✝ : σ✝ → Type
δ : Dialect α✝ σ✝ ε✝
v v' : SSAVal
Hne : v' ≠ v
env : SSAEnv δ
τ τ' : MLIRType δ
val : MLIRType.eval τ
val' : MLIRType.eval τ'
v₂ : SSAVal
τ₂ : MLIRType δ
⊢ (if v = v₂ then
      if H : τ = τ₂ then
        some
          (cast
            (_ :
              (match τ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val)
      else none
    else
      if v' = v₂ then
        if H : τ' = τ₂ then
          some
            (cast
              (_ :
                (match τ' with
                  | MLIRType.float a => Float
                  | MLIRType.int signedness sz => FinInt sz
                  | MLIRType.tensor1d => Tensor1D
                  | MLIRType.tensor2d => Tensor2D
                  | MLIRType.tensor4d => Tensor4D
                  | MLIRType.index => ℤ
                  | MLIRType.undefined a => Unit
                  | MLIRType.extended σ => ε✝ σ
                  | MLIRType.erased => Unit) =
                  match τ₂ with
                  | MLIRType.float a => Float
                  | MLIRType.int signedness sz => FinInt sz
                  | MLIRType.tensor1d => Tensor1D
                  | MLIRType.tensor2d => Tensor2D
                  | MLIRType.tensor4d => Tensor4D
                  | MLIRType.index => ℤ
                  | MLIRType.undefined a => Unit
                  | MLIRType.extended σ => ε✝ σ
                  | MLIRType.erased => Unit)
              val')
        else none
      else get v₂ τ₂ env) =
    if v' = v₂ then
      if H : τ' = τ₂ then
        some
          (cast
            (_ :
              (match τ' with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val')
      else none
    else
      if v = v₂ then
        if H : τ = τ₂ then
          some
            (cast
              (_ :
                (match τ with
                  | MLIRType.float a => Float
                  | MLIRType.int signedness sz => FinInt sz
                  | MLIRType.tensor1d => Tensor1D
                  | MLIRType.tensor2d => Tensor2D
                  | MLIRType.tensor4d => Tensor4D
                  | MLIRType.index => ℤ
                  | MLIRType.undefined a => Unit
                  | MLIRType.extended σ => ε✝ σ
                  | MLIRType.erased => Unit) =
                  match τ₂ with
                  | MLIRType.float a => Float
                  | MLIRType.int signedness sz => FinInt sz
                  | MLIRType.tensor1d => Tensor1D
                  | MLIRType.tensor2d => Tensor2D
                  | MLIRType.tensor4d => Tensor4D
                  | MLIRType.index => ℤ
                  | MLIRType.undefined a => Unit
                  | MLIRType.extended σ => ε✝ σ
                  | MLIRType.erased => Unit)
              val)
        else none
      else get v₂ τ₂ env
Tactic:
split
Proof state after:
case inl
α σ : Type
ε : σ → Type
α✝ σ✝ : Type
ε✝ : σ✝ → Type
δ : Dialect α✝ σ✝ ε✝
v v' : SSAVal
Hne : v' ≠ v
env : SSAEnv δ
τ τ' : MLIRType δ
val : MLIRType.eval τ
val' : MLIRType.eval τ'
v₂ : SSAVal
τ₂ : MLIRType δ
h✝ : v = v₂
⊢ (if H : τ = τ₂ then
      some
        (cast
          (_ :
            (match τ with
              | MLIRType.float a => Float
              | MLIRType.int signedness sz => FinInt sz
              | MLIRType.tensor1d => Tensor1D
              | MLIRType.tensor2d => Tensor2D
              | MLIRType.tensor4d => Tensor4D
              | MLIRType.index => ℤ
              | MLIRType.undefined a => Unit
              | MLIRType.extended σ => ε✝ σ
              | MLIRType.erased => Unit) =
              match τ₂ with
              | MLIRType.float a => Float
              | MLIRType.int signedness sz => FinInt sz
              | MLIRType.tensor1d => Tensor1D
              | MLIRType.tensor2d => Tensor2D
              | MLIRType.tensor4d => Tensor4D
              | MLIRType.index => ℤ
              | MLIRType.undefined a => Unit
              | MLIRType.extended σ => ε✝ σ
              | MLIRType.erased => Unit)
          val)
    else none) =
    if v' = v₂ then
      if H : τ' = τ₂ then
        some
          (cast
            (_ :
              (match τ' with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val')
      else none
    else
      if H : τ = τ₂ then
        some
          (cast
            (_ :
              (match τ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val)
      else none

case inr
α σ : Type
ε : σ → Type
α✝ σ✝ : Type
ε✝ : σ✝ → Type
δ : Dialect α✝ σ✝ ε✝
v v' : SSAVal
Hne : v' ≠ v
env : SSAEnv δ
τ τ' : MLIRType δ
val : MLIRType.eval τ
val' : MLIRType.eval τ'
v₂ : SSAVal
τ₂ : MLIRType δ
h✝ : ¬v = v₂
⊢ (if v' = v₂ then
      if H : τ' = τ₂ then
        some
          (cast
            (_ :
              (match τ' with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val')
      else none
    else get v₂ τ₂ env) =
    if v' = v₂ then
      if H : τ' = τ₂ then
        some
          (cast
            (_ :
              (match τ' with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit) =
                match τ₂ with
                | MLIRType.float a => Float
                | MLIRType.int signedness sz => FinInt sz
                | MLIRType.tensor1d => Tensor1D
                | MLIRType.tensor2d => Tensor2D
                | MLIRType.tensor4d => Tensor4D
                | MLIRType.index => ℤ
                | MLIRType.undefined a => Unit
                | MLIRType.extended σ => ε✝ σ
                | MLIRType.erased => Unit)
            val')
      else none
    else get v₂ τ₂ env