# Theory Lift

theory Lift
imports UNITY_Main
```(*  Title:      HOL/UNITY/Simple/Lift.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

The Lift-Control Example.
*)

theory Lift
imports "../UNITY_Main"
begin

record state =
floor :: "int"            ― ‹current position of the lift›
"open" :: "bool"          ― ‹whether the door is opened at floor›
stop  :: "bool"           ― ‹whether the lift is stopped at floor›
req   :: "int set"        ― ‹for each floor, whether the lift is requested›
up    :: "bool"           ― ‹current direction of movement›
move  :: "bool"           ― ‹whether moving takes precedence over opening›

axiomatization
Min :: "int" and   ― ‹least and greatest floors›
Max :: "int"       ― ‹least and greatest floors›
where
Min_le_Max [iff]: "Min ≤ Max"

― ‹Abbreviations: the "always" part›

definition
above :: "state set"
where "above = {s. ∃i. floor s < i & i ≤ Max & i ∈ req s}"

definition
below :: "state set"
where "below = {s. ∃i. Min ≤ i & i < floor s & i ∈ req s}"

definition
queueing :: "state set"
where "queueing = above ∪ below"

definition
goingup :: "state set"
where "goingup = above ∩ ({s. up s}  ∪ -below)"

definition
goingdown :: "state set"
where "goingdown = below ∩ ({s. ~ up s} ∪ -above)"

definition
where "ready = {s. stop s & ~ open s & move s}"

― ‹Further abbreviations›

definition
moving :: "state set"
where "moving = {s. ~ stop s & ~ open s}"

definition
stopped :: "state set"
where "stopped = {s. stop s  & ~ open s & ~ move s}"

definition
opened :: "state set"
where "opened = {s. stop s  &  open s  &  move s}"

definition
closed :: "state set"  ― ‹but this is the same as ready!!›
where "closed = {s. stop s  & ~ open s &  move s}"

definition
atFloor :: "int => state set"
where "atFloor n = {s. floor s = n}"

definition
Req :: "int => state set"
where "Req n = {s. n ∈ req s}"

― ‹The program›

definition
request_act :: "(state*state) set"
where "request_act = {(s,s'). s' = s (|stop:=True, move:=False|)
& ~ stop s & floor s ∈ req s}"

definition
open_act :: "(state*state) set"
where "open_act =
{(s,s'). s' = s (|open :=True,
req  := req s - {floor s},
move := True|)
& stop s & ~ open s & floor s ∈ req s
& ~(move s & s ∈ queueing)}"

definition
close_act :: "(state*state) set"
where "close_act = {(s,s'). s' = s (|open := False|) & open s}"

definition
req_up :: "(state*state) set"
where "req_up =
{(s,s'). s' = s (|stop  :=False,
floor := floor s + 1,
up    := True|)
& s ∈ (ready ∩ goingup)}"

definition
req_down :: "(state*state) set"
where "req_down =
{(s,s'). s' = s (|stop  :=False,
floor := floor s - 1,
up    := False|)
& s ∈ (ready ∩ goingdown)}"

definition
move_up :: "(state*state) set"
where "move_up =
{(s,s'). s' = s (|floor := floor s + 1|)
& ~ stop s & up s & floor s ∉ req s}"

definition
move_down :: "(state*state) set"
where "move_down =
{(s,s'). s' = s (|floor := floor s - 1|)
& ~ stop s & ~ up s & floor s ∉ req s}"

definition
button_press  :: "(state*state) set"
― ‹This action is omitted from prior treatments, which therefore are
action invalidates many of the existing progress arguments: various
"ensures" properties fail. Maybe it should be constrained to only
allow button presses in the current direction of travel, like in a
real lift.›
where "button_press =
{(s,s'). ∃n. s' = s (|req := insert n (req s)|)
& Min ≤ n & n ≤ Max}"

definition
Lift :: "state program"
― ‹for the moment, we OMIT ‹button_press››
where "Lift = mk_total_program
({s. floor s = Min & ~ up s & move s & stop s &
~ open s & req s = {}},
{request_act, open_act, close_act,
req_up, req_down, move_up, move_down},
UNIV)"

― ‹Invariants›

definition
bounded :: "state set"
where "bounded = {s. Min ≤ floor s & floor s ≤ Max}"

definition
open_stop :: "state set"
where "open_stop = {s. open s --> stop s}"

definition
open_move :: "state set"
where "open_move = {s. open s --> move s}"

definition
stop_floor :: "state set"
where "stop_floor = {s. stop s & ~ move s --> floor s ∈ req s}"

definition
moving_up :: "state set"
where "moving_up = {s. ~ stop s & up s -->
(∃f. floor s ≤ f & f ≤ Max & f ∈ req s)}"

definition
moving_down :: "state set"
where "moving_down = {s. ~ stop s & ~ up s -->
(∃f. Min ≤ f & f ≤ floor s & f ∈ req s)}"

definition
metric :: "[int,state] => int"
where "metric =
(%n s. if floor s < n then (if up s then n - floor s
else (floor s - Min) + (n-Min))
else
if n < floor s then (if up s then (Max - floor s) + (Max-n)
else floor s - n)
else 0)"

locale Floor =
fixes n
assumes Min_le_n [iff]: "Min ≤ n"
and n_le_Max [iff]: "n ≤ Max"

lemma not_mem_distinct: "[| x ∉ A;  y ∈ A |] ==> x ≠ y"
by blast

declare Lift_def [THEN def_prg_Init, simp]

declare request_act_def [THEN def_act_simp, simp]
declare open_act_def [THEN def_act_simp, simp]
declare close_act_def [THEN def_act_simp, simp]
declare req_up_def [THEN def_act_simp, simp]
declare req_down_def [THEN def_act_simp, simp]
declare move_up_def [THEN def_act_simp, simp]
declare move_down_def [THEN def_act_simp, simp]
declare button_press_def [THEN def_act_simp, simp]

(*The ALWAYS properties*)
declare above_def [THEN def_set_simp, simp]
declare below_def [THEN def_set_simp, simp]
declare queueing_def [THEN def_set_simp, simp]
declare goingup_def [THEN def_set_simp, simp]
declare goingdown_def [THEN def_set_simp, simp]

(*Basic definitions*)
declare bounded_def [simp]
open_stop_def [simp]
open_move_def [simp]
stop_floor_def [simp]
moving_up_def [simp]
moving_down_def [simp]

lemma open_stop: "Lift ∈ Always open_stop"
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety)
done

lemma stop_floor: "Lift ∈ Always stop_floor"
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety)
done

(*This one needs open_stop, which was proved above*)
lemma open_move: "Lift ∈ Always open_move"
apply (cut_tac open_stop)
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety)
done

lemma moving_up: "Lift ∈ Always moving_up"
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety)
done

lemma moving_down: "Lift ∈ Always moving_down"
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety)
apply (blast dest: order_le_imp_less_or_eq)
done

lemma bounded: "Lift ∈ Always bounded"
apply (cut_tac moving_up moving_down)
apply (rule AlwaysI, force)
apply (unfold Lift_def, safety, auto)
apply (drule not_mem_distinct, assumption, arith)+
done

subsection‹Progress›

declare moving_def [THEN def_set_simp, simp]
declare stopped_def [THEN def_set_simp, simp]
declare opened_def [THEN def_set_simp, simp]
declare closed_def [THEN def_set_simp, simp]
declare atFloor_def [THEN def_set_simp, simp]
declare Req_def [THEN def_set_simp, simp]

text‹The HUG'93 paper mistakenly omits the Req n from these!›

(** Lift_1 **)
lemma E_thm01: "Lift ∈ (stopped ∩ atFloor n) LeadsTo (opened ∩ atFloor n)"
apply (cut_tac stop_floor)
apply (unfold Lift_def, ensures_tac "open_act")
done  (*lem_lift_1_5*)

lemma E_thm02: "Lift ∈ (Req n ∩ stopped - atFloor n) LeadsTo
(Req n ∩ opened - atFloor n)"
apply (cut_tac stop_floor)
apply (unfold Lift_def, ensures_tac "open_act")
done  (*lem_lift_1_1*)

lemma E_thm03: "Lift ∈ (Req n ∩ opened - atFloor n) LeadsTo
(Req n ∩ closed - (atFloor n - queueing))"
apply (unfold Lift_def, ensures_tac "close_act")
done  (*lem_lift_1_2*)

lemma E_thm04: "Lift ∈ (Req n ∩ closed ∩ (atFloor n - queueing))
apply (unfold Lift_def, ensures_tac "open_act")
done  (*lem_lift_1_7*)

(** Lift 2.  Statements of thm05a and thm05b were wrong! **)

lemmas linorder_leI = linorder_not_less [THEN iffD1]

context Floor
begin

lemmas le_MinD = Min_le_n [THEN order_antisym]
and Max_leD = n_le_Max [THEN [2] order_antisym]

declare le_MinD [dest!]
and linorder_leI [THEN le_MinD, dest!]
and Max_leD [dest!]
and linorder_leI [THEN Max_leD, dest!]

(*lem_lift_2_0
NOT an ensures_tac property, but a mere inclusion
don't know why script lift_2.uni says ENSURES*)
lemma E_thm05c:
"Lift ∈ (Req n ∩ closed - (atFloor n - queueing))
LeadsTo ((closed ∩ goingup ∩ Req n)  ∪
(closed ∩ goingdown ∩ Req n))"

(*lift_2*)
lemma lift_2: "Lift ∈ (Req n ∩ closed - (atFloor n - queueing))
apply (unfold Lift_def)
apply (ensures_tac [2] "req_down")
apply (ensures_tac "req_up", auto)
done

(** Towards lift_4 ***)

declare if_split_asm [split]

(*lem_lift_4_1 *)
lemma E_thm12a:
"0 < N ==>
Lift ∈ (moving ∩ Req n ∩ {s. metric n s = N} ∩
{s. floor s ∉ req s} ∩ {s. up s})
(moving ∩ Req n ∩ {s. metric n s < N})"
apply (cut_tac moving_up)
apply (unfold Lift_def, ensures_tac "move_up", safe)
(*this step consolidates two formulae to the goal  metric n s' ≤ metric n s*)
apply (erule linorder_leI [THEN order_antisym, symmetric])
done

(*lem_lift_4_3 *)
lemma E_thm12b: "0 < N ==>
Lift ∈ (moving ∩ Req n ∩ {s. metric n s = N} ∩
{s. floor s ∉ req s} - {s. up s})
LeadsTo (moving ∩ Req n ∩ {s. metric n s < N})"
apply (cut_tac moving_down)
apply (unfold Lift_def, ensures_tac "move_down", safe)
(*this step consolidates two formulae to the goal  metric n s' ≤ metric n s*)
apply (erule linorder_leI [THEN order_antisym, symmetric])
done

(*lift_4*)
lemma lift_4:
"0<N ==> Lift ∈ (moving ∩ Req n ∩ {s. metric n s = N} ∩
{s. floor s ∉ req s}) LeadsTo
(moving ∩ Req n ∩ {s. metric n s < N})"
done

(** towards lift_5 **)

(*lem_lift_5_3*)
lemma E_thm16a: "0<N
==> Lift ∈ (closed ∩ Req n ∩ {s. metric n s = N} ∩ goingup) LeadsTo
(moving ∩ Req n ∩ {s. metric n s < N})"
apply (cut_tac bounded)
apply (unfold Lift_def, ensures_tac "req_up")
done

(*lem_lift_5_1 has ~goingup instead of goingdown*)
lemma E_thm16b: "0<N ==>
Lift ∈ (closed ∩ Req n ∩ {s. metric n s = N} ∩ goingdown) LeadsTo
(moving ∩ Req n ∩ {s. metric n s < N})"
apply (cut_tac bounded)
apply (unfold Lift_def, ensures_tac "req_down")
done

(*lem_lift_5_0 proves an intersection involving ~goingup and goingup,
i.e. the trivial disjunction, leading to an asymmetrical proof.*)
lemma E_thm16c:
"0<N ==> Req n ∩ {s. metric n s = N} ⊆ goingup ∪ goingdown"

(*lift_5*)
lemma lift_5:
"0<N ==> Lift ∈ (closed ∩ Req n ∩ {s. metric n s = N}) LeadsTo
(moving ∩ Req n ∩ {s. metric n s < N})"
apply (drule E_thm16c, auto)
done

(** towards lift_3 **)

(*lemma used to prove lem_lift_3_1*)
lemma metric_eq_0D [dest]:
"[| metric n s = 0;  Min ≤ floor s;  floor s ≤ Max |] ==> floor s = n"

(*lem_lift_3_1*)
lemma E_thm11: "Lift ∈ (moving ∩ Req n ∩ {s. metric n s = 0}) LeadsTo
(stopped ∩ atFloor n)"
apply (cut_tac bounded)
apply (unfold Lift_def, ensures_tac "request_act", auto)
done

(*lem_lift_3_5*)
lemma E_thm13:
"Lift ∈ (moving ∩ Req n ∩ {s. metric n s = N} ∩ {s. floor s ∈ req s})
LeadsTo (stopped ∩ Req n ∩ {s. metric n s = N} ∩ {s. floor s ∈ req s})"
apply (unfold Lift_def, ensures_tac "request_act")
done

(*lem_lift_3_6*)
lemma E_thm14: "0 < N ==>
Lift ∈
(stopped ∩ Req n ∩ {s. metric n s = N} ∩ {s. floor s ∈ req s})
LeadsTo (opened ∩ Req n ∩ {s. metric n s = N})"
apply (unfold Lift_def, ensures_tac "open_act")
done

(*lem_lift_3_7*)
lemma E_thm15: "Lift ∈ (opened ∩ Req n ∩ {s. metric n s = N})
LeadsTo (closed ∩ Req n ∩ {s. metric n s = N})"
apply (unfold Lift_def, ensures_tac "close_act")
done

(** the final steps **)

lemma lift_3_Req: "0 < N ==>
Lift ∈
(moving ∩ Req n ∩ {s. metric n s = N} ∩ {s. floor s ∈ req s})
LeadsTo (moving ∩ Req n ∩ {s. metric n s < N})"
apply (blast intro!: E_thm13 E_thm14 E_thm15 lift_5 intro: LeadsTo_Trans)
done

(*Now we observe that our integer metric is really a natural number*)
lemma Always_nonneg: "Lift ∈ Always {s. 0 ≤ metric n s}"
apply (rule bounded [THEN Always_weaken])
done

lemmas R_thm11 = Always_LeadsTo_weaken [OF Always_nonneg E_thm11]

lemma lift_3: "Lift ∈ (moving ∩ Req n) LeadsTo (stopped ∩ atFloor n)"
apply (rule Always_nonneg [THEN integ_0_le_induct])
apply (case_tac "0 < z")
(*If z ≤ 0 then actually z = 0*)
prefer 2 apply (force intro: R_thm11 order_antisym simp add: linorder_not_less)
apply (rule LeadsTo_weaken_R [OF asm_rl Un_upper1])
done

lemma lift_1: "Lift ∈ (Req n) LeadsTo (opened ∩ atFloor n)"
prefer 2
apply (rule E_thm01 [THEN [2] LeadsTo_Trans_Un])
apply (rule lift_3 [THEN [2] LeadsTo_Trans_Un])
apply (rule lift_2 [THEN [2] LeadsTo_Trans_Un])
apply (rule LeadsTo_Trans_Un [OF E_thm02 E_thm03])