module Hoare
        (PrimComm : Set)
        (Cond : Set)
        (Axiom : Cond @$\rightarrow$@ PrimComm @$\rightarrow$@ Cond @$\rightarrow$@ Set)
        (Tautology : Cond @$\rightarrow$@ Cond @$\rightarrow$@ Set)
        (_and_ :  Cond @$\rightarrow$@ Cond @$\rightarrow$@ Cond)
        (neg :  Cond @$\rightarrow$@ Cond )
  where

data Comm : Set where
  Skip  : Comm
  Abort : Comm
  PComm : PrimComm @$\rightarrow$@ Comm
  Seq   : Comm @$\rightarrow$@ Comm @$\rightarrow$@ Comm
  If    : Cond @$\rightarrow$@ Comm @$\rightarrow$@ Comm @$\rightarrow$@ Comm
  While : Cond @$\rightarrow$@ Comm @$\rightarrow$@ Comm

-- Hoare Triple
data HT : Set where
  ht : Cond @$\rightarrow$@ Comm @$\rightarrow$@ Cond @$\rightarrow$@ HT

{-
                prPre              pr              prPost
             -------------  ------------------  ----------------
             bPre => bPre'  {bPre'} c {bPost'}  bPost' => bPost
Weakening : ----------------------------------------------------
                       {bPre} c {bPost}

Assign: ----------------------------
         {bPost[v<-e]} v:=e {bPost}

             pr1                pr2
      -----------------  ------------------
      {bPre} cm1 {bMid}  {bMid} cm2 {bPost}
Seq: ---------------------------------------
      {bPre} cm1 ; cm2 {bPost}

               pr1                         pr2
     -----------------------  ---------------------------
     {bPre @$\wedge$@ c} cm1 {bPost}  {bPre @$\wedge$@ neg c} cm2 {bPost}
If: ------------------------------------------------------
     {bPre} If c then cm1 else cm2 fi {bPost}

                          pr
                 -------------------
                 {inv @$\wedge$@ c} cm {inv}
While: ---------------------------------------
        {inv} while c do cm od {inv @$\wedge$@ neg c}
-}


data HTProof : Cond @$\rightarrow$@ Comm @$\rightarrow$@ Cond @$\rightarrow$@ Set where
  PrimRule : {bPre : Cond} @$\rightarrow$@ {pcm : PrimComm} @$\rightarrow$@ {bPost : Cond} @$\rightarrow$@
             (pr : Axiom bPre pcm bPost) @$\rightarrow$@
             HTProof bPre (PComm pcm) bPost
  SkipRule : (b : Cond) @$\rightarrow$@ HTProof b Skip b
  AbortRule : (bPre : Cond) @$\rightarrow$@ (bPost : Cond) @$\rightarrow$@
              HTProof bPre Abort bPost
  WeakeningRule : {bPre : Cond} @$\rightarrow$@ {bPre' : Cond} @$\rightarrow$@ {cm : Comm} @$\rightarrow$@
                {bPost' : Cond} @$\rightarrow$@ {bPost : Cond} @$\rightarrow$@
                Tautology bPre bPre' @$\rightarrow$@
                HTProof bPre' cm bPost' @$\rightarrow$@
                Tautology bPost' bPost @$\rightarrow$@
                HTProof bPre cm bPost
  SeqRule : {bPre : Cond} @$\rightarrow$@ {cm1 : Comm} @$\rightarrow$@ {bMid : Cond} @$\rightarrow$@
            {cm2 : Comm} @$\rightarrow$@ {bPost : Cond} @$\rightarrow$@
            HTProof bPre cm1 bMid @$\rightarrow$@
            HTProof bMid cm2 bPost @$\rightarrow$@
            HTProof bPre (Seq cm1 cm2) bPost
  IfRule : {cmThen : Comm} @$\rightarrow$@ {cmElse : Comm} @$\rightarrow$@
           {bPre : Cond} @$\rightarrow$@ {bPost : Cond} @$\rightarrow$@
           {b : Cond} @$\rightarrow$@
           HTProof (bPre and b) cmThen bPost @$\rightarrow$@
           HTProof (bPre and neg b) cmElse bPost @$\rightarrow$@
           HTProof bPre (If b cmThen cmElse) bPost
  WhileRule : {cm : Comm} @$\rightarrow$@ {bInv : Cond} @$\rightarrow$@ {b : Cond} @$\rightarrow$@
              HTProof (bInv and b) cm bInv @$\rightarrow$@
              HTProof bInv (While b cm) (bInv and neg b)