Define pseudocode for a number of instructions.

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+------------------------------------------------------------------------
+-- Agda Helium
+--
+-- Definition of instructions using the Armv8-M pseudocode.
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --without-K #-}
+
+module Helium.Data.Pseudocode where
+
+open import Data.Bool as Bool using (true; false)
+open import Data.Fin as Fin using (Fin; Fin′; zero; suc; toℕ)
+open import Data.Nat as ℕ using (ℕ; zero; suc)
+import Data.Nat.Properties as ℕₚ
+open import Data.Sum using ([_,_]′)
+open import Data.Vec as Vec using (Vec; []; _∷_)
+open import Data.Vec.Relation.Unary.All using (All; []; _∷_)
+open import Function using (_$_)
+open import Helium.Data.Pseudocode.Core as Core public
+  hiding (module Code)
+import Helium.Instructions as Instr
+import Relation.Binary.PropositionalEquality as P
+open import Relation.Nullary.Decidable.Core using (True)
+
+--- Types
+
+beat : Type
+beat = fin 4
+
+elmtMask : Type
+elmtMask = bits 4
+
+--- State
+
+State : Vec Type _
+State = array (bits 32) 32      -- S
+      ∷ array (bits 32) 16      -- R
+      ∷ bits 16                 -- VPR-P0
+      ∷ bits 8                  -- VPR-mask
+      ∷ bit                     -- FPSCR-QC
+      ∷ bool                    -- _AdvanceVPTState
+      ∷ beat                    -- _BeatId
+      ∷ []
+
+open Core.Code State public
+
+--- References
+
+-- Direct from State
+
+S : ∀ {n Γ} → Expression {n} Γ (array (bits 32) 32)
+S = state 0
+
+R : ∀ {n Γ} → Expression {n} Γ (array (bits 32) 16)
+R = state 1
+
+VPR-P0 : ∀ {n Γ} → Expression {n} Γ (bits 16)
+VPR-P0 = state 2
+
+VPR-mask : ∀ {n Γ} → Expression {n} Γ (bits 8)
+VPR-mask = state 3
+
+FPSCR-QC : ∀ {n Γ} → Expression {n} Γ bit
+FPSCR-QC = state 4
+
+AdvanceVPTState : ∀ {n Γ} → Expression {n} Γ bool
+AdvanceVPTState = state 5
+
+BeatId : ∀ {n Γ} → Expression {n} Γ beat
+BeatId = state 6
+
+-- Indirect
+
+tup⇒array : ∀ {n Γ t m} → Expression {n} Γ (tuple (suc m) (Vec.replicate t)) → Expression Γ (array t (suc m))
+tup⇒array {m = zero}  xs = [ head xs ]
+tup⇒array {m = suc m} xs = [ head xs ] ∶ tup⇒array (tail xs)
+
+group : ∀ {n Γ t k} m → Expression {n} Γ (asType t (k ℕ.* suc m)) → Expression Γ (array (asType t k) (suc m))
+group {k = k} zero    x = [ cast (P.trans (ℕₚ.*-comm k 1) (ℕₚ.+-comm k 0)) x ]
+group {k = k} (suc m) x = [ slice (cast (ℕₚ.+-comm k _) x′) (lit (zero ′f)) ] ∶ group m (slice (cast (P.cong (k ℕ.+_) (ℕₚ.*-comm (suc m) k)) x′) (lit (Fin.fromℕ k ′f)))
+  where
+  x′ = cast (ℕₚ.*-comm k (suc (suc m))) x
+
+join : ∀ {n Γ t k m} → Expression {n} Γ (array (asType t k) (suc m)) → Expression Γ (asType t (k ℕ.* suc m))
+join {k = k} {zero}  x = cast (P.trans (ℕₚ.+-comm 0 k) (ℕₚ.*-comm 1 k)) (unbox x)
+join {k = k} {suc m} x = cast eq (unbox (slice {i = suc m} (cast (ℕₚ.+-comm 1 (suc m)) x) (lit (zero ′f))) ∶ join (slice x (lit (Fin.fromℕ 1 ′f))))
+  where
+  eq = P.trans (P.cong (k ℕ.+_) (ℕₚ.*-comm k (suc m))) (ℕₚ.*-comm (suc (suc m)) k)
+
+index : ∀ {n Γ t m} → Expression {n} Γ (asType t (suc m)) → Expression Γ (fin (suc m)) → Expression Γ (elemType t)
+index {t = bits}    {m} x i = slice (cast (ℕₚ.+-comm 1 m) x) i
+index {t = array _} {m} x i = unbox (slice (cast (ℕₚ.+-comm 1 m) x) i)
+
+Q : ∀ {n Γ} → Expression {n} Γ (array (array (bits 32) 4) 8)
+Q = group 7 S
+
+elem : ∀ {n Γ t k} m → Expression {n} Γ (asType t (k ℕ.* m)) → Expression Γ (fin k) → Expression Γ (asType t m)
+elem {k = zero}  m       x i = abort i
+elem {k = suc k} zero    x i = cast (ℕₚ.*-comm k 0) x
+elem {k = suc k} (suc m) x i = index (group k (cast (ℕₚ.*-comm (suc k) (suc m)) x)) i
+
+--- Other utiliies
+
+hasBit : ∀ {n Γ m} → Expression {n} Γ (bits (suc m)) → Expression Γ (fin (suc m)) → Expression Γ bool
+hasBit {n} x i = index x i ≟ lit ((true ∷ []) ′x)
+
+sliceⁱ : ∀ {n Γ m} → ℕ → Expression {n} Γ int → Expression Γ (bits m)
+sliceⁱ {m = zero}  n i = lit ([] ′x)
+sliceⁱ {m = suc m} n i = get (m ℕ.+ n) i ∶ sliceⁱ n i
+
+--- Functions
+
+Int : ∀ {n} → Function (bits n ∷ bool ∷ []) int
+Int = skip ∙return (if var 1 then uint (var 0) else sint (var 0))
+
+-- arguments swapped, pred n
+SignedSatQ : ∀ n → Function (int ∷ []) (tuple 2 (bits (suc n) ∷ bool ∷ []))
+SignedSatQ n = declare (lit (true ′b)) (
+  if max <? var 1
+  then
+    var 1 ≔ max
+  else if var 1 <? min
+  then
+    var 1 ≔ min
+  else
+    var 0 ≔ lit (false ′b)
+  ∙return tup (sliceⁱ 0 (var 1) ∷ var 0 ∷ []))
+  where
+  max = lit (2 ′i) ^ lit (n ′i) + - lit (1 ′i)
+  min = - (lit (2 ′i) ^ lit (n ′i))
+
+-- actual shift if 'shift + 1'
+LSL-C : ∀ {n} (shift : ℕ) → Function (bits n ∷ []) (tuple 2 (bits n ∷ bit ∷ []))
+LSL-C {n} shift = declare (var 0 ∶ lit ((Vec.replicate {n = (suc shift)} false) ′x))
+  (skip ∙return tup
+    ( slice (cast (ℕₚ.+-comm n _) (var 0)) (lit (zero ′f))
+    ∷ slice (cast eq₂ (var 0)) (lit (Fin.inject+ shift (Fin.fromℕ n) ′f))
+    ∷ []))
+  where
+  eq₂ = P.trans (P.cong (n ℕ.+_) (ℕₚ.+-comm 1 shift)) (P.sym (ℕₚ.+-assoc n shift 1))
+
+--- Procedures
+
+private
+  div2 : All Fin (4 ∷ []) → Fin 2
+  div2 (zero        ∷ []) = zero
+  div2 (suc zero    ∷ []) = zero
+  div2 (suc (suc i) ∷ []) = suc zero
+
+copyMasked : Procedure (fin 8 ∷ bits 32 ∷ beat ∷ elmtMask ∷ [])
+copyMasked = for 4
+  -- 0:e 1:dest 2:result 3:beat 4:elmtMask
+  ( if hasBit (var 4) (var 0)
+    then
+      elem 8 (index (index Q (var 1)) (var 3)) (var 0) ≔ elem 8 (var 2) (var 0)
+    else skip
+  ) ∙end
+
+VPTAdvance : Procedure (beat ∷ [])
+VPTAdvance = declare (fin div2 (tup (var 0 ∷ []))) (
+  declare (elem 4 VPR-mask (var 0)) (
+    -- 0:vptState 1:maskId 2:beat
+    if var 0 ≟ lit ((true ∷ false ∷ false ∷ false ∷ []) ′x)
+    then
+      var 0 ≔ lit (Vec.replicate false ′x)
+    else if inv (var 0 ≟ lit (Vec.replicate false ′x))
+    then (
+      declare (lit ((false ∷ []) ′x)) (
+        -- 0:inv 1:vptState 2:maskId 3:beat
+        tup (var 1 ∷ var 0 ∷ []) ≔ call (LSL-C 0) (tup (var 1 ∷ [])) ∙
+        if var 0 ≟ lit ((true ∷ []) ′x)
+        then
+          elem 4 VPR-P0 (var 3) ≔ not (elem 4 VPR-P0 (var 3))
+        else skip))
+    else skip ∙
+    if get 0 (asInt (var 2)) ≟ lit ((true ∷ []) ′x)
+    then
+      elem 4 VPR-mask (var 1) ≔ var 0
+    else skip
+    ∙end))
+
+VPTActive : Function (beat ∷ []) bool
+VPTActive = skip ∙return inv (elem 4 VPR-mask (fin div2 (tup (var 0 ∷ []))) ≟ lit (Vec.replicate false ′x))
+
+GetCurInstrBeat : Function [] (tuple 2 (beat ∷ elmtMask ∷ []))
+GetCurInstrBeat = declare (lit (Vec.replicate true ′x)) (
+  -- 0:elmtMask 1:beat
+  if call VPTActive (tup (BeatId ∷ []))
+  then
+    var 0 ≔ var 0 and elem 4 VPR-P0 BeatId
+  else skip
+  ∙return tup (BeatId ∷ var 0 ∷ []))
+
+-- Assumes:
+--   MAX_OVERLAPPING_INSTRS = 1
+--   _InstInfo[0].Valid = 1
+--   BEATS_PER_TICK = 4
+--   procedure argument is action of DecodeExecute
+-- and more!
+ExecBeats : Procedure [] → Procedure []
+ExecBeats DecodeExec =
+  for 4 (
+    -- 0:beatId
+    BeatId ≔ var 0 ∙
+    AdvanceVPTState ≔ lit (true ′b) ∙
+    invoke DecodeExec (tup []) ∙
+    if AdvanceVPTState
+    then
+      invoke VPTAdvance (tup (var 0 ∷ []))
+    else skip)
+  ∙end
+
+from32 : ∀ size {n Γ} → Expression {n} Γ (bits 32) → Expression Γ (array (bits (toℕ (Instr.Size.esize size))) (toℕ (Instr.Size.elements size)))
+from32 Instr.8bit  = group 3
+from32 Instr.16bit = group 1
+from32 Instr.32bit = group 0
+
+to32 : ∀ size {n Γ} → Expression {n} Γ (array (bits (toℕ (Instr.Size.esize size))) (toℕ (Instr.Size.elements size))) → Expression Γ (bits 32)
+to32 Instr.8bit  = join
+to32 Instr.16bit = join
+to32 Instr.32bit = join
+
+module _ (d : Instr.VecOp₂) where
+  open Instr.VecOp₂ d
+
+  vec-op₂ : Function (bits (toℕ esize) ∷ bits (toℕ esize) ∷ []) (bits (toℕ esize)) → Procedure []
+  vec-op₂ op =
+    declare (lit (zero ′f)) (
+    declare (lit (Vec.replicate false ′x)) (
+    -- 0:elmtMask 1:curBeat
+    tup (var 1 ∷ var 0 ∷ []) ≔ call GetCurInstrBeat (tup []) ∙
+    declare (lit (Vec.replicate (Vec.replicate false ′x) ′a)) (
+    declare (from32 size (index (index Q (lit (src₁ ′f))) (var 2))) (
+    -- 0:op₁ 1:result 2:elmtMask 3:curBeat
+    for (toℕ elements) (
+       -- 0:e 1:op₁ 2:result 3:elmtMask 4:curBeat
+      declare op₂ (
+      -- 0:op₂ 1:e 2:op₁ 3:result 4:elmtMask 5:curBeat
+      index (var 3) (var 1) ≔ call op (tup (index (var 2) (var 1) ∷ var 0 ∷ [])))) ∙
+    -- 0:op₁ 1:result 2:elmtMask 3:curBeat
+    invoke copyMasked (tup (lit (dest ′f) ∷ to32 size (var 1) ∷ var 3 ∷ var 2 ∷ []))))
+    ∙end))
+    where
+    -- 0:e 1:op₁ 2:result 3:elmtMask 4:curBeat
+    op₂ =
+      [ (λ src₂ → index (from32 size (index R (lit (src₂ ′f)))) (lit (zero ′f)))
+      , (λ src₂ → index (from32 size (index (index Q (lit (src₂ ′f))) (var 4))) (var 0))
+      ]′ src₂
+
+vadd : Instr.VAdd → Procedure []
+vadd d = vec-op₂ d (skip ∙return sliceⁱ 0 (uint (var 0) + uint (var 1)))
+
+vsub : Instr.VSub → Procedure []
+vsub d = vec-op₂ d (skip ∙return sliceⁱ 0 (uint (var 0) - uint (var 1)))
+
+vhsub : Instr.VHSub → Procedure []
+vhsub d = vec-op₂ op₂ (skip ∙return sliceⁱ 1 (toInt (var 0) - toInt (var 1)))
+  where open Instr.VHSub d; toInt = λ i → call Int (tup (i ∷ lit (unsigned ′b) ∷ []))
+
+vmul : Instr.VMul → Procedure []
+vmul d = vec-op₂ d (skip ∙return sliceⁱ 0 (sint (var 0) * sint (var 1)))
+
+vmulh : Instr.VMulH → Procedure []
+vmulh d = vec-op₂ op₂ (skip ∙return sliceⁱ (toℕ esize) (toInt (var 0) * toInt (var 1)))
+  where
+  open Instr.VMulH d; toInt = λ i → call Int (tup (i ∷ lit (unsigned ′b) ∷ []))
+
+vrmulh : Instr.VRMulH → Procedure []
+vrmulh d = vec-op₂ op₂ (skip ∙return sliceⁱ (toℕ esize) (toInt (var 0) * toInt (var 1) + lit (1 ′i) << lit (toℕ esize-1 ′i)))
+  where
+  open Instr.VRMulH d; toInt = λ i → call Int (tup (i ∷ lit (unsigned ′b) ∷ []))
+
+private
+  vqr?dmulh : Instr.VQDMulH → Function (int ∷ int ∷ []) int → Procedure []
+  vqr?dmulh d f =
+    declare (lit (zero ′f)) (
+    declare (lit (Vec.replicate false ′x)) (
+    -- 0:elmtMask 1:curBeat
+    tup (var 1 ∷ var 0 ∷ []) ≔ call GetCurInstrBeat (tup []) ∙
+    declare (lit (Vec.replicate (Vec.replicate false ′x) ′a)) (
+    declare (from32 size (index (index Q (lit (src₁ ′f))) (var 2))) (
+    -- 0:op₁ 1:result 2:elmtMask 3:curBeat
+    for (toℕ elements) (
+       -- 0:e 1:op₁ 2:result 3:elmtMask 4:curBeat
+      declare op₂ (
+      -- 0:op₂ 1:e 2:op₁ 3:result 4:elmtMask 5:curBeat
+      declare (call f (tup (sint (index (var 2) (var 1)) ∷ sint (var 0) ∷ []))) (
+      -- 0:value 1:op₂ 2:e 3:op₁ 4:result 5:elmtMask 6:curBeat
+      declare (lit (false ′b)) (
+      -- 0:sat 1:value 2:op₂ 3:e 4:op₁ 5:result 6:elmtMask 7:curBeat
+      tup (index (var 5) (var 3) ∷ var 0 ∷ []) ≔ call (SignedSatQ (toℕ esize-1)) (tup (var 1 ∷ [])) ∙
+      if var 0 && hasBit (var 6) (fin e*esize>>3 (tup (var 3 ∷ [])))
+      then
+        FPSCR-QC ≔ lit ((true ∷ []) ′x)
+      else skip)))) ∙
+    -- 0:op₁ 1:result 2:elmtMask 3:curBeat
+    invoke copyMasked (tup (lit (dest ′f) ∷ to32 size (var 1) ∷ var 3 ∷ var 2 ∷ []))))
+    ∙end))
+      where
+      open Instr.VecOp₂ d
+      -- 0:e 1:op₁ 2:result 3:elmtMask 4:curBeat
+      op₂ =
+        [ (λ src₂ → index (from32 size (index R (lit (src₂ ′f)))) (lit (zero ′f)))
+        , (λ src₂ → index (from32 size (index (index Q (lit (src₂ ′f))) (var 4))) (var 0))
+        ]′ src₂
+
+      e*esize>>3 : All Fin (toℕ elements ∷ []) → Fin 4
+      e*esize>>3 (x ∷ []) = helper size x
+        where
+        helper : ∀ size → Fin′ (Instr.Size.elements size) → Fin 4
+        helper Instr.8bit  i = Fin.combine i (zero {0})
+        helper Instr.16bit i = Fin.combine i (zero {1})
+        helper Instr.32bit i = Fin.combine i zero
+
+vqdmulh : Instr.VQDMulH → Procedure []
+vqdmulh d = vqr?dmulh d (skip ∙return lit (2 ′i) * var 0 * var 1 >> lit (toℕ esize ′i))
+  where open Instr.VecOp₂ d using (esize)
+
+vqrdmulh : Instr.VQRDMulH → Procedure []
+vqrdmulh d = vqr?dmulh d (skip ∙return (lit (2 ′i) * var 0 * var 1 + lit (1 ′i) << lit (toℕ esize-1 ′i)) >> lit (toℕ esize ′i))
+  where open Instr.VecOp₂ d using (esize; esize-1)