1	import Base
2	import ../../shape/core.bend as Shape
3	import ../../storage/LAWS.bend as StorageLaws
4	import ./replayed.bend as Replayed
5	import ./pointwise.bend as Pointwise
6	import ./pointwise_proof.bend as PointwiseProof
7	import ./LAWS.bend as Laws
8	
9	# A word appended to two equal word lists, equal words, gives equal lists.
10	def words_snoc(+shape: Shape.Shape, +fill: F32, +ops: List<&2, StorageLaws.Op>, +previous: Nat, +word: F32, +other: F32,
11	  +before: List<&2, F32>, rest: {Laws.words(shape, fill, ops, previous) == before : List<&2, F32>},
12	  last: {word == other : F32}) ->
13	  {List.append(&2, F32, Laws.words(shape, fill, ops, previous), [word]) == List.append(&2, F32, before, [other]) : List<&2, F32>}:
14	  %rest :
15	    {List.append(&2, F32, Laws.words(shape, fill, ops, previous), [word]) == List.append(&2, F32, _, [other]) : List<&2, F32>}
16	  %last :
17	    {List.append(&2, F32, Laws.words(shape, fill, ops, previous), [word]) == List.append(&2, F32, Laws.words(shape, fill, ops, previous), [_]) : List<&2, F32>}
18	  {==}
19	
20	# Reads appended to a history list the same words.
21	def words_reads(~P: Data, ~map: P -> Nat -> Nat, +plan: P, +reads: Nat, +shape: Shape.Shape, +fill: F32, +ops: List<&2, StorageLaws.Op>,
22	  +count: Nat) ->
23	  {Laws.words(shape, fill, List.append(&2, StorageLaws.Op, ops, Pointwise.reads(~P, ~map, plan, reads)), count) ==
24	    Laws.words(shape, fill, ops, count) : List<&2, F32>}:
25	  match count:
26	    case 0n:
27	      {==}
28	    case 1n+previous:
29	      +position = U32.from_nat(previous)
30	      +seen = List.append(&2, StorageLaws.Op, ops, Pointwise.reads(~P, ~map, plan, reads))
31	      words_snoc(shape, fill, seen, previous, Replayed.value(shape, fill, seen, position), Replayed.value(shape, fill, ops, position),
32	        Laws.words(shape, fill, ops, previous), words_reads(~P, ~map, plan, reads, shape, fill, ops, previous),
33	        PointwiseProof.value_reads(~P, ~map, plan, reads, shape, fill, ops, position))
34	
35	# Equal lists with equal words appended are equal.
36	def snoc_equal(+before: List<&2, F32>, +after: List<&2, F32>, +word: F32, +other: F32,
37	  rest: {before == after : List<&2, F32>}, last: {word == other : F32}) ->
38	  {List.append(&2, F32, before, [word]) == List.append(&2, F32, after, [other]) : List<&2, F32>}:
39	  %rest : {List.append(&2, F32, before, [word]) == List.append(&2, F32, _, [other]) : List<&2, F32>}
40	  %last : {List.append(&2, F32, before, [word]) == List.append(&2, F32, before, [_]) : List<&2, F32>}
41	  {==}