1	import Base
2	import ../../../spec/containers/queue.bend as S
3	import ../../../src/containers/queue.bend as Q
4	import ../../../src/containers/types/queue.bend as E
5	import ./state.bend as ST
6	import ./steps.bend as PS
7	import ./trace.bend as FT
8	import ../../lib/logic.bend as L
9	import ../../../spec/lib/common.bend as SC
10	import ../../../spec/lib/sequence.bend as V
11	import ../../lib/sequence.bend as VL
12	
13	# FIFO queue (two lists, src/containers/queue.bend): public proof entry point.
14	#   shadow       ST.Sh{front, back}: the queue QU{front, back, |front| +
15	#                |back|}
16	#   abstraction  ST.model(sh) = front ++ reverse(back), oldest first
17	#   ready        PS.ready: reversing the back into an empty front keeps the
18	#                model and leaves the front nonempty unless the queue is empty
19	#   operations   PS.step_ok: every public operation, errors included
20	#                (dequeue or peek on an empty queue is Fail{EmptyQueue}, queue
21	#                unchanged)
22	#   traces       trace_from / trace_new: any finite operation list, no
23	#                premise on its length or on the queue's size
24	
25	def new_abs(~T: Data) -> {ST.model(T, ST.initial(T)) == Nil{} : List<&2, T>}:
26	  {==}
27	
28	def new_real(~T: Data) -> {Q.new(~T) == ST.real(T, ST.initial(T)) : Q.Queue<T>}:
29	  {==}
30	
31	def step_ok(~T: Data, +sh: ST.Shadow<T>, +op: E.Op<T>) -> PS.StepOK(~T, sh, op):
32	  PS.step_ok(~T, sh, op)
33	
34	# a queue determines its shadow
35	def shadow_unique(~T: Data, +a: ST.Shadow<T>, +b: ST.Shadow<T>, +e: {ST.real(T, a) == ST.real(T, b) : Q.Queue<T>}) -> {a == b : ST.Shadow<T>}:
36	  ST.real_inj(T, a, b, e)
37	
38	def trace_from(~T: Data, +ops: List<&2, E.Op<T>>, +sh0: ST.Shadow<T>) -> FT.TraceOK(~T, ops, sh0):
39	  FT.trace_from(~T, ops, sh0)
40	
41	# arbitrary finite operation traces from the real constructor
42	def trace_new(~T: Data, +ops: List<&2, E.Op<T>>) -> FT.TraceOK(~T, ops, ST.initial(T)):
43	  FT.trace_from(~T, ops, ST.initial(T))
44	
45	# ==== the contract of queue (stated in spec/containers/queue.bend) ====================
46	
47	# ---- the implementation ----
48	def Impl(~T: Data, sh: ST.Shadow<T>, op: E.Op<T>, Post: (List<&2, T> & E.Obs<T>) -> Type) -> Type:
49	  Sigma<&1, &1, ST.Shadow<T>, sh2 => Sigma<&1, &1, E.Obs<T>, o => {Q.step(~T, ST.real(T, sh), op) == (ST.real(T, sh2), o) : Q.Queue<T> & E.Obs<T>} & Post((ST.model(T, sh2), o))>>
50	
51	def impl_of(~T: Data, -sh: ST.Shadow<T>, -op: E.Op<T>, -Post: (List<&2, T> & E.Obs<T>) -> Type, k: PS.StepOK(~T, sh, op), pf: Post(S.step(T, ST.model(T, sh), op))) -> Impl(~T, sh, op, Post):
52	  match k:
53	    case Tuple{sh2, Tuple{o, Tuple{e1, e2}}}:
54	      (sh2, (o, (e1, L.subst(List<&2, T> & E.Obs<T>, Post, S.step(T, ST.model(T, sh), op), (ST.model(T, sh2), o), Equal.sym(List<&2, T> & E.Obs<T>, (ST.model(T, sh2), o), S.step(T, ST.model(T, sh), op), e2), pf))))
55	
56	def impl(~T: Data, +sh: ST.Shadow<T>, +op: E.Op<T>, -Post: (List<&2, T> & E.Obs<T>) -> Type, pf: Post(S.step(T, ST.model(T, sh), op))) -> Impl(~T, sh, op, Post):
57	  impl_of(~T, sh, op, Post, step_ok(~T, sh, op), pf)
58	
59	# ---- Length, iteration ----
60	def length_result(-T: Data, +xs: List<&2, T>) -> S.Length.length_result(T, xs):
61	  {==}
62	
63	def length_frame(-T: Data, +xs: List<&2, T>) -> S.Length.length_frame(T, xs):
64	  {==}
65	
66	def to_list_model(-T: Data, +xs: List<&2, T>) -> S.Iteration.to_list_model(T, xs):
67	  {==}
68	
69	def to_list_frame(-T: Data, +xs: List<&2, T>) -> S.Iteration.to_list_frame(T, xs):
70	  {==}
71	
72	# ---- Empty_Vector ----
73	def new_empty(~T: Data) -> {SC.length(T, ST.model(T, ST.initial(T))) == 0n : Nat}:
74	  %Equal.sym(List<&2, T>, ST.model(T, ST.initial(T)), Nil{}, new_abs(~T)) : {SC.length(T, _) == 0n : Nat}
75	  {==}
76	
77	def new_impl(~T: Data) -> {Q.new(~T) == ST.real(T, ST.initial(T)) : Q.Queue<T>}:
78	  new_real(~T)
79	
80	# ---- Append: Length + 1, Equal_Prefix (Model'Old, Model), Element (Last'Old + 1) = New_Item ----
81	def enqueue_length(-T: Data, +xs: List<&2, T>, +v: T) -> S.Append.enqueue_length(T, xs, v):
82	  VL.snoc_length(T, xs, v)
83	
84	def enqueue_prefix(-T: Data, +xs: List<&2, T>, +v: T) -> S.Append.enqueue_prefix(T, xs, v):
85	  VL.snoc_prefix(T, xs, v)
86	
87	def enqueue_element(-T: Data, +xs: List<&2, T>, +v: T) -> S.Append.enqueue_element(T, xs, v):
88	  VL.snoc_last(T, xs, v)
89	
90	# ---- Delete_First: Length - 1, Range_Shifted (New, Old, First, Last, 1); dequeue returns First_Element'Old ----
91	def dequeue_length(-T: Data, +h: T, +t: List<&2, T>) -> S.Delete_First.dequeue_length(T, h, t):
92	  {==}
93	
94	def dequeue_shifted(-T: Data, +h: T, +t: List<&2, T>) -> S.Delete_First.dequeue_shifted(T, h, t):
95	  VL.tail_shifted(T, h, t)
96	
97	def dequeue_result(-T: Data, +h: T, +t: List<&2, T>) -> S.Delete_First.dequeue_result(T, h, t):
98	  {==}
99	
100	def dequeue_empty(-T: Data) -> S.Delete_First.dequeue_empty(T):
101	  {==}
102	
103	# ---- First_Element: Element (Model, First); nothing changes ----
104	def peek_first(-T: Data, +h: T, +t: List<&2, T>) -> S.First_Element.peek_first(T, h, t):
105	  ({==}, {==})
106	
107	def peek_frame(-T: Data, +xs: List<&2, T>) -> S.First_Element.peek_frame(T, xs):
108	  {==}
109	
110	def peek_empty(-T: Data) -> S.First_Element.peek_empty(T):
111	  {==}