proofs/traversal_witness.bend

stelliferous@0.0.2.0 · 287 lines · 30 definitions

1# Proof-only preservation of affine traversal owners. Reader values are handed
2# back unchanged; output witnesses keep their original depth. Numerical loops
3# call none of these functions: certificate equations use them under rewrites.
4import Base
5import stelliferous@0.0.2.0/traversal_loop.bend as Loop
6import stelliferous@0.0.2.0/traversal_array.bend as Traversal
7import stelliferous@0.0.2.0/proofs/storage_witness.bend as Witness
8
9type Sample<-Reader: Type,-Element: Data,-reader: Reader,-result: Reader & Element> is Data:
10  Sample{-value: Element,equation: {result == (reader,value) : Reader & Element}}
11
12type StateEvidence<-Reader: Type,-Element: Data,-Context: Data,-reader: Reader,-depth: Nat,
13  -state: Traversal.State<&1,&1,Reader,Array<Element>,Context>> is Data:
14  StateEvidence{-output: Array<Element>,position: U32,context: Context,
15    equation: {state == Traversal.State{reader,output,position,context} : Traversal.State<&1,&1,Reader,Array<Element>,Context>},
16    output_evidence: Witness.Witness(Element,depth,output)}
17
18def store(~Reader: Type,~Element: Data,~Context: Data,~destination: Context -> U32 -> U32,
19  +depth: Nat,-reader: Reader,+position: U32,+context: Context,-output: Array<Element>,-result: Reader & Element,
20  output_evidence: Witness.Witness(Element,depth,output),sample: Sample<Reader,Element,reader,result>) ->
21  StateEvidence<Reader,Element,Context,reader,depth,Traversal.store(~Reader,~Element,~Context,~destination,position,context,output,result)>:
22  match sample:
23    case Sample{value,equation}:
24      %Equal.sym(Reader & Element,result,(reader,value),equation) :
25        StateEvidence<Reader,Element,Context,reader,depth,Traversal.store(~Reader,~Element,~Context,~destination,position,context,output,_)>
26      StateEvidence{Array.set(Element,output,destination(context,position),value),(position + 1 : U32),context,{==},
27        Witness.write(Element,depth,output,destination(context,position),value,output_evidence)}
28
29def step(~Reader: Type,~Element: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
30  ~read: Reader -> Context -> U32 -> Reader & Element,~destination: Context -> U32 -> U32,
31  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
32    Sample<Reader,Element,reader,read(reader,context,index)>,
33  +depth: Nat,-reader: Reader,-state: Traversal.State<&1,&1,Reader,Array<Element>,Context>,
34  reader_evidence: ReaderEvidence(reader),evidence: StateEvidence<Reader,Element,Context,reader,depth,state>) ->
35  StateEvidence<Reader,Element,Context,reader,depth,Traversal.step(~Reader,~Element,~Context,~read,~destination,state)>:
36  match evidence:
37    case StateEvidence{output,+position,+context,equation,output_evidence}:
38      %Equal.sym(Traversal.State<&1,&1,Reader,Array<Element>,Context>,state,Traversal.State{reader,output,position,context},equation) :
39        StateEvidence<Reader,Element,Context,reader,depth,Traversal.step(~Reader,~Element,~Context,~read,~destination,_)>
40      store(~Reader,~Element,~Context,~destination,depth,reader,position,context,output,read(reader,context,position),output_evidence,
41        read_preserves(reader,context,position,reader_evidence))
42
43def run(~Reader: Type,~Element: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
44  ~read: Reader -> Context -> U32 -> Reader & Element,~destination: Context -> U32 -> U32,
45  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
46    Sample<Reader,Element,reader,read(reader,context,index)>,
47  +depth: Nat,count: Nat,-reader: Reader,-state: Traversal.State<&1,&1,Reader,Array<Element>,Context>,
48  +reader_evidence: ReaderEvidence(reader),evidence: StateEvidence<Reader,Element,Context,reader,depth,state>) ->
49  StateEvidence<Reader,Element,Context,reader,depth,Traversal.run(~Reader,~Element,~Context,~read,~destination,count,state)>:
50  match count:
51    case 0n: evidence
52    case 1n+rest:
53      run(~Reader,~Element,~Context,~ReaderEvidence,~read,~destination,~read_preserves,depth,rest,reader,
54        Traversal.step(~Reader,~Element,~Context,~read,~destination,state),reader_evidence,
55        step(~Reader,~Element,~Context,~ReaderEvidence,~read,~destination,~read_preserves,depth,reader,state,reader_evidence,evidence))
56
57type ResultEvidence<-Reader: Type,-Element: Data,-reader: Reader,-depth: Nat,-result: Reader & Array<Element>> is Data:
58  ResultEvidence{-output: Array<Element>,equation: {result == (reader,output) : Reader & Array<Element>},
59    output_evidence: Witness.Witness(Element,depth,output)}
60
61def finish(~Reader: Type,~Element: Data,~Context: Data,-reader: Reader,-depth: Nat,
62  -state: Traversal.State<&1,&1,Reader,Array<Element>,Context>,evidence: StateEvidence<Reader,Element,Context,reader,depth,state>) ->
63  ResultEvidence<Reader,Element,reader,depth,Traversal.finish(~Reader,~Element,~Context,state)>:
64  match evidence:
65    case StateEvidence{output,position,context,equation,output_evidence}:
66      %Equal.sym(Traversal.State<&1,&1,Reader,Array<Element>,Context>,state,Traversal.State{reader,output,position,context},equation) :
67        ResultEvidence<Reader,Element,reader,depth,Traversal.finish(~Reader,~Element,~Context,_)>
68      ResultEvidence{output,{==},output_evidence}
69
70def array_evidence(~Element: Data,-storage: Array<Element>) -> Data:
71  Witness.ArrayWitness<Element,storage>
72
73def array_sample(~Element: Data,-storage: Array<Element>,-result: Array<Element> & Element,
74  read: Witness.Read<Element,storage,result>) -> Sample<Array<Element>,Element,storage,result>:
75  match read:
76    case Witness.Read{value,equation}: Sample{value,equation}
77
78def offset_read(~Element: Data,-storage: Array<Element>,context: Traversal.Offsets,index: U32,
79  evidence: Witness.ArrayWitness<Element,storage>) ->
80  Sample<Array<Element>,Element,storage,Traversal.source_offset(~Element,storage,context,index)>:
81  Traversal.Offsets{source,destination} = context
82  array_sample(~Element,storage,Array.get(Element,storage,(source + index : U32)),Witness.observed(Element,storage,(source + index : U32),evidence))
83
84def first(~Left: Type,~Right: Type,pair: Left & Right) -> Left:
85  (left,right) = pair
86  left
87
88def second(~Left: Type,~Right: Type,pair: Left & Right) -> Right:
89  (left,right) = pair
90  right
91
92def source_unchanged(~Reader: Type,~Element: Data,-reader: Reader,-depth: Nat,-result: Reader & Array<Element>,
93  evidence: ResultEvidence<Reader,Element,reader,depth,result>) -> {first(~Reader,~Array<Element>,result) == reader : Reader}:
94  match evidence:
95    case ResultEvidence{output,equation,output_evidence}:
96      Equal.cong(Reader & Array<Element>,Reader,pair => first(~Reader,~Array<Element>,pair),result,(reader,output),equation)
97
98def output_witness(~Reader: Type,~Element: Data,-reader: Reader,-depth: Nat,-result: Reader & Array<Element>,
99  evidence: ResultEvidence<Reader,Element,reader,depth,result>) -> Witness.Witness(Element,depth,second(~Reader,~Array<Element>,result)):
100  match evidence:
101    case ResultEvidence{output,equation,output_evidence}:
102      %Equal.sym(Reader & Array<Element>,result,(reader,output),equation) : Witness.Witness(Element,depth,second(~Reader,~Array<Element>,_))
103      output_evidence
104
105def update_sample(~Reader: Type,~Element: Data,~Value: Data,~Context: Data,
106  ~combine: Element -> Value -> Element,~destination: Context -> U32 -> U32,
107  +depth: Nat,-reader: Reader,+position: U32,+context: Context,-output: Array<Element>,-current: Element,
108  -result: Reader & Value,output_evidence: Witness.Witness(Element,depth,output),
109  sample: Sample<Reader,Value,reader,result>) ->
110  StateEvidence<Reader,Element,Context,reader,depth,
111    Traversal.update_sample(~Reader,~Element,~Value,~Context,~combine,~destination,position,context,output,current,result)>:
112  match sample:
113    case Sample{value,equation}:
114      %Equal.sym(Reader & Value,result,(reader,value),equation) :
115        StateEvidence<Reader,Element,Context,reader,depth,
116          Traversal.update_sample(~Reader,~Element,~Value,~Context,~combine,~destination,position,context,output,current,_)>
117      StateEvidence{Array.set(Element,output,destination(context,position),combine(current,value)),(position + 1 : U32),context,{==},
118        Witness.write(Element,depth,output,destination(context,position),combine(current,value),output_evidence)}
119
120def update_current(~Reader: Type,~Element: Data,~Value: Data,~Context: Data,
121  ~read: Reader -> Context -> U32 -> Reader & Value,~combine: Element -> Value -> Element,
122  ~destination: Context -> U32 -> U32,+depth: Nat,-reader: Reader,+position: U32,+context: Context,
123  -output: Array<Element>,-result: Array<Element> & Element,
124  output_evidence: Witness.Witness(Element,depth,output),observed: Witness.Read<Element,output,result>,
125  sample: Sample<Reader,Value,reader,read(reader,context,position)>) ->
126  StateEvidence<Reader,Element,Context,reader,depth,
127    Traversal.update_current(~Reader,~Element,~Value,~Context,~read,~combine,~destination,reader,position,context,result)>:
128  match observed:
129    case Witness.Read{value,equation}:
130      %Equal.sym(Array<Element> & Element,result,(output,value),equation) :
131        StateEvidence<Reader,Element,Context,reader,depth,
132          Traversal.update_current(~Reader,~Element,~Value,~Context,~read,~combine,~destination,reader,position,context,_)>
133      update_sample(~Reader,~Element,~Value,~Context,~combine,~destination,depth,reader,position,context,output,value,
134        read(reader,context,position),output_evidence,sample)
135
136def update_step(~Reader: Type,~Element: Data,~Value: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
137  ~read: Reader -> Context -> U32 -> Reader & Value,~combine: Element -> Value -> Element,~destination: Context -> U32 -> U32,
138  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
139    Sample<Reader,Value,reader,read(reader,context,index)>,
140  +depth: Nat,-reader: Reader,-state: Traversal.State<&1,&1,Reader,Array<Element>,Context>,
141  reader_evidence: ReaderEvidence(reader),evidence: StateEvidence<Reader,Element,Context,reader,depth,state>) ->
142  StateEvidence<Reader,Element,Context,reader,depth,
143    Traversal.update_step(~Reader,~Element,~Value,~Context,~read,~combine,~destination,state)>:
144  match evidence:
145    case StateEvidence{output,+position,+context,equation,+output_evidence}:
146      %Equal.sym(Traversal.State<&1,&1,Reader,Array<Element>,Context>,state,Traversal.State{reader,output,position,context},equation) :
147        StateEvidence<Reader,Element,Context,reader,depth,
148          Traversal.update_step(~Reader,~Element,~Value,~Context,~read,~combine,~destination,_)>
149      update_current(~Reader,~Element,~Value,~Context,~read,~combine,~destination,depth,reader,position,context,output,
150        Array.get(Element,output,destination(context,position)),output_evidence,
151        Witness.read(Element,depth,output,destination(context,position),output_evidence),
152        read_preserves(reader,context,position,reader_evidence))
153
154def update(~Reader: Type,~Element: Data,~Value: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
155  ~read: Reader -> Context -> U32 -> Reader & Value,~combine: Element -> Value -> Element,~destination: Context -> U32 -> U32,
156  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
157    Sample<Reader,Value,reader,read(reader,context,index)>,
158  +depth: Nat,count: Nat,-reader: Reader,-state: Traversal.State<&1,&1,Reader,Array<Element>,Context>,
159  +reader_evidence: ReaderEvidence(reader),evidence: StateEvidence<Reader,Element,Context,reader,depth,state>) ->
160  StateEvidence<Reader,Element,Context,reader,depth,
161    Traversal.update(~Reader,~Element,~Value,~Context,~read,~combine,~destination,count,state)>:
162  match count:
163    case 0n: evidence
164    case 1n+rest:
165      update(~Reader,~Element,~Value,~Context,~ReaderEvidence,~read,~combine,~destination,~read_preserves,depth,rest,reader,
166        Traversal.update_step(~Reader,~Element,~Value,~Context,~read,~combine,~destination,state),reader_evidence,
167        update_step(~Reader,~Element,~Value,~Context,~ReaderEvidence,~read,~combine,~destination,~read_preserves,
168          depth,reader,state,reader_evidence,evidence))
169
170type MapEvidence<-Element: Data,-Context: Data,-depth: Nat,
171  -state: Traversal.State<&2,&1,Unit,Array<Element>,Context>> is Data:
172  MapEvidence{-output: Array<Element>,position: U32,context: Context,
173    equation: {state == Traversal.State{Unit{},output,position,context} : Traversal.State<&2,&1,Unit,Array<Element>,Context>},
174    output_evidence: Witness.Witness(Element,depth,output)}
175
176def mapped_value(~Element: Data,~Context: Data,~map: Context -> Element -> Element,
177  +depth: Nat,+position: U32,+context: Context,-output: Array<Element>,-result: Array<Element> & Element,
178  output_evidence: Witness.Witness(Element,depth,output),observed: Witness.Read<Element,output,result>) ->
179  MapEvidence<Element,Context,depth,Traversal.mapped_value(~Element,~Context,~map,position,context,result)>:
180  match observed:
181    case Witness.Read{value,equation}:
182      %Equal.sym(Array<Element> & Element,result,(output,value),equation) :
183        MapEvidence<Element,Context,depth,Traversal.mapped_value(~Element,~Context,~map,position,context,_)>
184      MapEvidence{Array.set(Element,output,position,map(context,value)),(position + 1 : U32),context,{==},
185        Witness.write(Element,depth,output,position,map(context,value),output_evidence)}
186
187def map_step(~Element: Data,~Context: Data,~map: Context -> Element -> Element,+depth: Nat,
188  -state: Traversal.State<&2,&1,Unit,Array<Element>,Context>,evidence: MapEvidence<Element,Context,depth,state>) ->
189  MapEvidence<Element,Context,depth,Traversal.map_step(~Element,~Context,~map,state)>:
190  match evidence:
191    case MapEvidence{output,+position,+context,equation,+output_evidence}:
192      %Equal.sym(Traversal.State<&2,&1,Unit,Array<Element>,Context>,state,Traversal.State{Unit{},output,position,context},equation) :
193        MapEvidence<Element,Context,depth,Traversal.map_step(~Element,~Context,~map,_)>
194      mapped_value(~Element,~Context,~map,depth,position,context,output,Array.get(Element,output,position),output_evidence,
195        Witness.read(Element,depth,output,position,output_evidence))
196
197def map_run(~Element: Data,~Context: Data,~map: Context -> Element -> Element,+depth: Nat,count: Nat,
198  -state: Traversal.State<&2,&1,Unit,Array<Element>,Context>,evidence: MapEvidence<Element,Context,depth,state>) ->
199  MapEvidence<Element,Context,depth,Loop.run(~Traversal.State<&2,&1,Unit,Array<Element>,Context>,~Traversal.map_step(~Element,~Context,~map),count,state)>:
200  match count:
201    case 0n: evidence
202    case 1n+rest:
203      map_run(~Element,~Context,~map,depth,rest,Traversal.map_step(~Element,~Context,~map,state),
204        map_step(~Element,~Context,~map,depth,state,evidence))
205
206def mapped_output(~Element: Data,~Context: Data,-depth: Nat,-state: Traversal.State<&2,&1,Unit,Array<Element>,Context>,
207  evidence: MapEvidence<Element,Context,depth,state>) -> Witness.Witness(Element,depth,Traversal.mapped_output(~Element,~Context,state)):
208  match evidence:
209    case MapEvidence{output,position,context,equation,output_evidence}:
210      %Equal.sym(Traversal.State<&2,&1,Unit,Array<Element>,Context>,state,Traversal.State{Unit{},output,position,context},equation) :
211        Witness.Witness(Element,depth,Traversal.mapped_output(~Element,~Context,_))
212      output_evidence
213
214def map_in_place(~Element: Data,~Context: Data,~map: Context -> Element -> Element,+depth: Nat,count: Nat,
215  position: U32,context: Context,-output: Array<Element>,evidence: Witness.Witness(Element,depth,output)) ->
216  Witness.Witness(Element,depth,Traversal.map_in_place(~Element,~Context,~map,count,position,context,output)):
217  mapped_output(~Element,~Context,depth,
218    Loop.run(~Traversal.State<&2,&1,Unit,Array<Element>,Context>,~Traversal.map_step(~Element,~Context,~map),count,
219      Traversal.State{Unit{},output,position,context}),
220    map_run(~Element,~Context,~map,depth,count,Traversal.State{Unit{},output,position,context},MapEvidence{output,position,context,{==},evidence}))
221
222type FoldEvidence<-Reader: Type,-Accumulator: Data,-Context: Data,-reader: Reader,
223  -state: Traversal.State<&1,&2,Reader,Accumulator,Context>> is Data:
224  FoldEvidence{-accumulator: Accumulator,position: U32,context: Context,
225    equation: {state == Traversal.State{reader,accumulator,position,context} : Traversal.State<&1,&2,Reader,Accumulator,Context>}}
226
227def fold_value(~Reader: Type,~Element: Data,~Accumulator: Data,~Context: Data,
228  ~combine: Accumulator -> Element -> Accumulator,-reader: Reader,-accumulator: Accumulator,+position: U32,context: Context,
229  -result: Reader & Element,sample: Sample<Reader,Element,reader,result>) ->
230  FoldEvidence<Reader,Accumulator,Context,reader,
231    Traversal.fold_value(~Reader,~Element,~Accumulator,~Context,~combine,accumulator,position,context,result)>:
232  match sample:
233    case Sample{value,equation}:
234      %Equal.sym(Reader & Element,result,(reader,value),equation) :
235        FoldEvidence<Reader,Accumulator,Context,reader,
236          Traversal.fold_value(~Reader,~Element,~Accumulator,~Context,~combine,accumulator,position,context,_)>
237      FoldEvidence{combine(accumulator,value),(position + 1 : U32),context,{==}}
238
239def fold_step(~Reader: Type,~Element: Data,~Accumulator: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
240  ~read: Reader -> Context -> U32 -> Reader & Element,~combine: Accumulator -> Element -> Accumulator,
241  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
242    Sample<Reader,Element,reader,read(reader,context,index)>,
243  -reader: Reader,-state: Traversal.State<&1,&2,Reader,Accumulator,Context>,reader_evidence: ReaderEvidence(reader),
244  evidence: FoldEvidence<Reader,Accumulator,Context,reader,state>) ->
245  FoldEvidence<Reader,Accumulator,Context,reader,Traversal.fold_step(~Reader,~Element,~Accumulator,~Context,~read,~combine,state)>:
246  match evidence:
247    case FoldEvidence{accumulator,+position,+context,equation}:
248      %Equal.sym(Traversal.State<&1,&2,Reader,Accumulator,Context>,state,Traversal.State{reader,accumulator,position,context},equation) :
249        FoldEvidence<Reader,Accumulator,Context,reader,Traversal.fold_step(~Reader,~Element,~Accumulator,~Context,~read,~combine,_)>
250      fold_value(~Reader,~Element,~Accumulator,~Context,~combine,reader,accumulator,position,context,read(reader,context,position),
251        read_preserves(reader,context,position,reader_evidence))
252
253def fold_run(~Reader: Type,~Element: Data,~Accumulator: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
254  ~read: Reader -> Context -> U32 -> Reader & Element,~combine: Accumulator -> Element -> Accumulator,
255  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
256    Sample<Reader,Element,reader,read(reader,context,index)>,
257  count: Nat,-reader: Reader,-state: Traversal.State<&1,&2,Reader,Accumulator,Context>,+reader_evidence: ReaderEvidence(reader),
258  evidence: FoldEvidence<Reader,Accumulator,Context,reader,state>) ->
259  FoldEvidence<Reader,Accumulator,Context,reader,
260    Loop.run(~Traversal.State<&1,&2,Reader,Accumulator,Context>,~Traversal.fold_step(~Reader,~Element,~Accumulator,~Context,~read,~combine),count,state)>:
261  match count:
262    case 0n: evidence
263    case 1n+rest:
264      fold_run(~Reader,~Element,~Accumulator,~Context,~ReaderEvidence,~read,~combine,~read_preserves,rest,reader,
265        Traversal.fold_step(~Reader,~Element,~Accumulator,~Context,~read,~combine,state),reader_evidence,
266        fold_step(~Reader,~Element,~Accumulator,~Context,~ReaderEvidence,~read,~combine,~read_preserves,reader,state,reader_evidence,evidence))
267
268def fold_finished(~Reader: Type,~Accumulator: Data,~Context: Data,-reader: Reader,
269  -state: Traversal.State<&1,&2,Reader,Accumulator,Context>,evidence: FoldEvidence<Reader,Accumulator,Context,reader,state>) ->
270  {first(~Reader,~Accumulator,Traversal.finish_fold(~Reader,~Accumulator,~Context,state)) == reader : Reader}:
271  match evidence:
272    case FoldEvidence{accumulator,position,context,equation}:
273      %Equal.sym(Traversal.State<&1,&2,Reader,Accumulator,Context>,state,Traversal.State{reader,accumulator,position,context},equation) :
274        {first(~Reader,~Accumulator,Traversal.finish_fold(~Reader,~Accumulator,~Context,_)) == reader : Reader}
275      {==}
276
277def fold(~Reader: Type,~Element: Data,~Accumulator: Data,~Context: Data,~ReaderEvidence: @-reader: Reader -> Data,
278  ~read: Reader -> Context -> U32 -> Reader & Element,~combine: Accumulator -> Element -> Accumulator,
279  ~read_preserves: @-reader: Reader -> @context: Context -> @index: U32 -> ReaderEvidence(reader) ->
280    Sample<Reader,Element,reader,read(reader,context,index)>,
281  count: Nat,-reader: Reader,position: U32,context: Context,-initial: Accumulator,reader_evidence: ReaderEvidence(reader)) ->
282  {first(~Reader,~Accumulator,Traversal.fold(~Reader,~Element,~Accumulator,~Context,~read,~combine,count,position,reader,initial,context)) == reader : Reader}:
283  fold_finished(~Reader,~Accumulator,~Context,reader,
284    Loop.run(~Traversal.State<&1,&2,Reader,Accumulator,Context>,~Traversal.fold_step(~Reader,~Element,~Accumulator,~Context,~read,~combine),count,
285      Traversal.State{reader,initial,position,context}),
286    fold_run(~Reader,~Element,~Accumulator,~Context,~ReaderEvidence,~read,~combine,~read_preserves,count,reader,
287      Traversal.State{reader,initial,position,context},reader_evidence,FoldEvidence{initial,position,context,{==}}))