1	import Base
2	import ../tensor_indexed.bend as Indexed
3	import ../traversal_array.bend as Traversal
4	import ../traversal_loop.bend as Loop
5	import ./storage_tree_model.bend as Tree
6	import ./storage_certificate.bend as Certificate
7	import ./storage_witness.bend as Witness
8	import ./traversal_witness.bend as TraversalWitness
9	import ./traversal_certificate.bend as TraversalCertificate
10	
11	def sample(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element,
12	  -input: Array<Element>,+context: Context,+position: U32,-result: Array<Element> & Element,
13	  evidence: TraversalWitness.Sample<Array<Element>,Element,input,result>) ->
14	  TraversalWitness.Sample<Array<Element>,Element,input,Indexed.value(~Element,~Context,~index,~operation,context,position,result)>:
15	  match evidence:
16	    case TraversalWitness.Sample{value,equation}:
17	      %Equal.sym(Array<Element> & Element,result,(input,value),equation) :
18	        TraversalWitness.Sample<Array<Element>,Element,input,Indexed.value(~Element,~Context,~index,~operation,context,position,_)>
19	      TraversalWitness.Sample{operation(index(context,position),value),{==}}
20	
21	def read(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element,
22	  -input: Array<Element>,context: Context,position: U32,evidence: Witness.ArrayWitness<Element,input>) ->
23	  TraversalWitness.Sample<Array<Element>,Element,input,Indexed.read(~Element,~Context,~index,~operation,input,context,position)>:
24	  +position = position
25	  sample(~Element,~Context,~index,~operation,input,context,position,Array.get(Element,input,position),
26	    TraversalWitness.array_sample(~Element,input,Array.get(Element,input,position),Witness.observed(Element,input,position,evidence)))
27	
28	def preserves(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element,
29	  +depth: Nat,call: Indexed.Call<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>,
30	  equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) ->
31	  TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,Indexed.run(~Element,~Context,~index,~operation,call,input,output)>:
32	  Indexed.Call{count,context} = call
33	  TraversalCertificate.completed(~Array<Element>,~Element,~Context,~TraversalWitness.array_evidence(~Element),
34	    ~Traversal.run(~Array<Element>,~Element,~Context,~Indexed.read(~Element,~Context,~index,~operation),~Indexed.destination(~Context)),
35	    ~TraversalWitness.run(~Array<Element>,~Element,~Context,~TraversalWitness.array_evidence(~Element),~Indexed.read(~Element,~Context,~index,~operation),
36	      ~Indexed.destination(~Context),~read(~Element,~Context,~index,~operation)),
37	    count,0,context,input,output,Certificate.witness(Element,input,input_certificate),depth,equation)
38	
39	def certificates(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element,
40	  +call: Indexed.Call<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>,
41	  output_certificate: Certificate.Certificate<Element,output>) ->
42	  TraversalCertificate.Certificates<Array<Element>,Element,Certificate.family(~Element),Indexed.run(~Element,~Context,~index,~operation,call,input,output)>:
43	  Certificate.Certificate{+depth,+equation} = input_certificate
44	  TraversalCertificate.operation(~Array<Element>,~Element,~Indexed.Call<Context>,~Certificate.family(~Element),
45	    ~Indexed.run(~Element,~Context,~index,~operation),~preserves(~Element,~Context,~index,~operation),
46	    call,input,output,Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},output_certificate)
47	
48	def next_block(~Element: Data,~Context: Data,+depth: Nat,block: U32,size: U32,context: Context,-input: Array<Element>,-state: Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Block>,
49	  evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Block,input,depth,state>) ->
50	  TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.next_block(~Element,~Context,block,size,context,state)>:
51	  match evidence:
52	    case TraversalWitness.StateEvidence{output,position,inner,equation,output_evidence}:
53	      %Equal.sym(Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Block>,state,Traversal.State{input,output,position,inner},equation) :
54	        TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.next_block(~Element,~Context,block,size,context,_)>
55	      TraversalWitness.StateEvidence{output,position,Indexed.Blocks{(block + 1 : U32),size,context},{==},output_evidence}
56	
57	def block_step(~Element: Data,~Context: Data,~base: Context -> U32 -> U32,~operation: U32 -> Element -> Element,+depth: Nat,-input: Array<Element>,-state: Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,
58	  +input_evidence: Witness.ArrayWitness<Element,input>,evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,state>) ->
59	  TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.block_step(~Element,~Context,~base,~operation,state)>:
60	  match evidence:
61	    case TraversalWitness.StateEvidence{output,+position,Indexed.Blocks{+block,+size,+context},equation,output_evidence}:
62	      %Equal.sym(Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,state,Traversal.State{input,output,position,Indexed.Blocks{block,size,context}},equation) :
63	        TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.block_step(~Element,~Context,~base,~operation,_)>
64	      next_block(~Element,~Context,depth,block,size,context,input,
65	        Traversal.run(~Array<Element>,~Element,~Indexed.Block,~Indexed.read(~Element,~Indexed.Block,~Indexed.block_index,~operation),~Indexed.destination(~Indexed.Block),U32.to_nat(size),
66	          Traversal.State{input,output,position,Indexed.Block{base(context,block),position}}),
67	        TraversalWitness.run(~Array<Element>,~Element,~Indexed.Block,~TraversalWitness.array_evidence(~Element),~Indexed.read(~Element,~Indexed.Block,~Indexed.block_index,~operation),
68	          ~Indexed.destination(~Indexed.Block),~read(~Element,~Indexed.Block,~Indexed.block_index,~operation),depth,U32.to_nat(size),input,
69	          Traversal.State{input,output,position,Indexed.Block{base(context,block),position}},input_evidence,
70	          TraversalWitness.StateEvidence{output,position,Indexed.Block{base(context,block),position},{==},output_evidence}))
71	
72	def block_loop(~Element: Data,~Context: Data,~base: Context -> U32 -> U32,~operation: U32 -> Element -> Element,+depth: Nat,count: Nat,-input: Array<Element>,-state: Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,
73	  +input_evidence: Witness.ArrayWitness<Element,input>,evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,state>) ->
74	  TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,
75	    Loop.run(~Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,~Indexed.block_step(~Element,~Context,~base,~operation),count,state)>:
76	  match count:
77	    case 0n: evidence
78	    case 1n+rest:
79	      block_loop(~Element,~Context,~base,~operation,depth,rest,input,Indexed.block_step(~Element,~Context,~base,~operation,state),input_evidence,
80	        block_step(~Element,~Context,~base,~operation,depth,input,state,input_evidence,evidence))
81	
82	type Tiling<-Context: Data> is Data:
83	  Tiling{total: U32,context: Context}
84	
85	def tiled_operation(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~base: Context -> U32 -> U32,~blocks: Context -> U32,~size: Context -> U32,~operation: U32 -> Element -> Element,
86	  tiling: Tiling<Context>,input: Array<Element>,output: Array<Element>) -> Array<Element> & Array<Element>:
87	  Tiling{total,context} = tiling
88	  Indexed.run_tiled(~Element,~Context,~index,~base,~blocks,~size,~operation,total,context,input,output)
89	
90	def whole_preserves(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~base: Context -> U32 -> U32,~operation: U32 -> Element -> Element,
91	  +depth: Nat,whole: Bool,blocks: U32,size: U32,total: U32,context: Context,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>,
92	  equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) ->
93	  TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,Indexed.run_whole(~Element,~Context,~index,~base,~operation,whole,blocks,size,total,context,input,output)>:
94	  match whole:
95	    case True{}:
96	      TraversalCertificate.completed(~Array<Element>,~Element,~Indexed.Blocks<Context>,~TraversalWitness.array_evidence(~Element),
97	        ~Loop.run(~Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,~Indexed.block_step(~Element,~Context,~base,~operation)),~block_loop(~Element,~Context,~base,~operation),
98	        U32.to_nat(blocks),0,Indexed.Blocks{0,size,context},input,output,Certificate.witness(Element,input,input_certificate),depth,equation)
99	    case False{}:
100	      preserves(~Element,~Context,~index,~operation,depth,Indexed.Call{U32.to_nat(total),context},input,output,input_certificate,equation)
101	
102	def tiled_preserves(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~base: Context -> U32 -> U32,~blocks: Context -> U32,~size: Context -> U32,~operation: U32 -> Element -> Element,
103	  +depth: Nat,tiling: Tiling<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>,
104	  equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) ->
105	  TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,tiled_operation(~Element,~Context,~index,~base,~blocks,~size,~operation,tiling,input,output)>:
106	  Tiling{+total,+context} = tiling
107	  whole_preserves(~Element,~Context,~index,~base,~operation,depth,U32.is_le(total,Indexed.block_limit()) && Nat.is_eq(Nat.mul(U32.to_nat(blocks(context)),U32.to_nat(size(context))),U32.to_nat(total)),
108	    blocks(context),size(context),total,context,input,output,input_certificate,equation)
109	
110	def tiled_certificates(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~base: Context -> U32 -> U32,~blocks: Context -> U32,~size: Context -> U32,~operation: U32 -> Element -> Element,
111	  +total: U32,+context: Context,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>,
112	  output_certificate: Certificate.Certificate<Element,output>) ->
113	  TraversalCertificate.Certificates<Array<Element>,Element,Certificate.family(~Element),Indexed.run_tiled(~Element,~Context,~index,~base,~blocks,~size,~operation,total,context,input,output)>:
114	  Certificate.Certificate{+depth,+equation} = input_certificate
115	  TraversalCertificate.operation(~Array<Element>,~Element,~Tiling<Context>,~Certificate.family(~Element),
116	    ~tiled_operation(~Element,~Context,~index,~base,~blocks,~size,~operation),~tiled_preserves(~Element,~Context,~index,~base,~blocks,~size,~operation),
117	    Tiling{total,context},input,output,Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},output_certificate)