proofs/tensor_indexed_certificate.bend source
proofs/tensor_indexed_certificate.bend on the hub · documented module
import Baseimport ../tensor_indexed.bend as Indexedimport ../traversal_array.bend as Traversalimport ../traversal_loop.bend as Loopimport ./storage_tree_model.bend as Treeimport ./storage_certificate.bend as Certificateimport ./storage_witness.bend as Witnessimport ./traversal_witness.bend as TraversalWitnessimport ./traversal_certificate.bend as TraversalCertificatedef sample(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element, -input: Array<Element>,+context: Context,+position: U32,-result: Array<Element> & Element, evidence: TraversalWitness.Sample<Array<Element>,Element,input,result>) -> TraversalWitness.Sample<Array<Element>,Element,input,Indexed.value(~Element,~Context,~index,~operation,context,position,result)>: match evidence: case TraversalWitness.Sample{value,equation}: %Equal.sym(Array<Element> & Element,result,(input,value),equation) : TraversalWitness.Sample<Array<Element>,Element,input,Indexed.value(~Element,~Context,~index,~operation,context,position,_)> TraversalWitness.Sample{operation(index(context,position),value),{==}}def read(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element, -input: Array<Element>,context: Context,position: U32,evidence: Witness.ArrayWitness<Element,input>) -> TraversalWitness.Sample<Array<Element>,Element,input,Indexed.read(~Element,~Context,~index,~operation,input,context,position)>: +position = position sample(~Element,~Context,~index,~operation,input,context,position,Array.get(Element,input,position), TraversalWitness.array_sample(~Element,input,Array.get(Element,input,position),Witness.observed(Element,input,position,evidence)))def preserves(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element, +depth: Nat,call: Indexed.Call<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>, equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) -> TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,Indexed.run(~Element,~Context,~index,~operation,call,input,output)>: Indexed.Call{count,context} = call TraversalCertificate.completed(~Array<Element>,~Element,~Context,~TraversalWitness.array_evidence(~Element), ~Traversal.run(~Array<Element>,~Element,~Context,~Indexed.read(~Element,~Context,~index,~operation),~Indexed.destination(~Context)), ~TraversalWitness.run(~Array<Element>,~Element,~Context,~TraversalWitness.array_evidence(~Element),~Indexed.read(~Element,~Context,~index,~operation), ~Indexed.destination(~Context),~read(~Element,~Context,~index,~operation)), count,0,context,input,output,Certificate.witness(Element,input,input_certificate),depth,equation)def certificates(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~operation: U32 -> Element -> Element, +call: Indexed.Call<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>, output_certificate: Certificate.Certificate<Element,output>) -> TraversalCertificate.Certificates<Array<Element>,Element,Certificate.family(~Element),Indexed.run(~Element,~Context,~index,~operation,call,input,output)>: Certificate.Certificate{+depth,+equation} = input_certificate TraversalCertificate.operation(~Array<Element>,~Element,~Indexed.Call<Context>,~Certificate.family(~Element), ~Indexed.run(~Element,~Context,~index,~operation),~preserves(~Element,~Context,~index,~operation), call,input,output,Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},output_certificate)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>, evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Block,input,depth,state>) -> TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.next_block(~Element,~Context,block,size,context,state)>: match evidence: case TraversalWitness.StateEvidence{output,position,inner,equation,output_evidence}: %Equal.sym(Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Block>,state,Traversal.State{input,output,position,inner},equation) : TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.next_block(~Element,~Context,block,size,context,_)> TraversalWitness.StateEvidence{output,position,Indexed.Blocks{(block + 1 : U32),size,context},{==},output_evidence}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>>, +input_evidence: Witness.ArrayWitness<Element,input>,evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,state>) -> TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.block_step(~Element,~Context,~base,~operation,state)>: match evidence: case TraversalWitness.StateEvidence{output,+position,Indexed.Blocks{+block,+size,+context},equation,output_evidence}: %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) : TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,Indexed.block_step(~Element,~Context,~base,~operation,_)> next_block(~Element,~Context,depth,block,size,context,input, Traversal.run(~Array<Element>,~Element,~Indexed.Block,~Indexed.read(~Element,~Indexed.Block,~Indexed.block_index,~operation),~Indexed.destination(~Indexed.Block),U32.to_nat(size), Traversal.State{input,output,position,Indexed.Block{base(context,block),position}}), TraversalWitness.run(~Array<Element>,~Element,~Indexed.Block,~TraversalWitness.array_evidence(~Element),~Indexed.read(~Element,~Indexed.Block,~Indexed.block_index,~operation), ~Indexed.destination(~Indexed.Block),~read(~Element,~Indexed.Block,~Indexed.block_index,~operation),depth,U32.to_nat(size),input, Traversal.State{input,output,position,Indexed.Block{base(context,block),position}},input_evidence, TraversalWitness.StateEvidence{output,position,Indexed.Block{base(context,block),position},{==},output_evidence}))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>>, +input_evidence: Witness.ArrayWitness<Element,input>,evidence: TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth,state>) -> TraversalWitness.StateEvidence<Array<Element>,Element,Indexed.Blocks<Context>,input,depth, Loop.run(~Traversal.State<&1,&1,Array<Element>,Array<Element>,Indexed.Blocks<Context>>,~Indexed.block_step(~Element,~Context,~base,~operation),count,state)>: match count: case 0n: evidence case 1n+rest: block_loop(~Element,~Context,~base,~operation,depth,rest,input,Indexed.block_step(~Element,~Context,~base,~operation,state),input_evidence, block_step(~Element,~Context,~base,~operation,depth,input,state,input_evidence,evidence))type Tiling<-Context: Data> is Data: Tiling{total: U32,context: Context}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, tiling: Tiling<Context>,input: Array<Element>,output: Array<Element>) -> Array<Element> & Array<Element>: Tiling{total,context} = tiling Indexed.run_tiled(~Element,~Context,~index,~base,~blocks,~size,~operation,total,context,input,output)def whole_preserves(~Element: Data,~Context: Data,~index: Context -> U32 -> U32,~base: Context -> U32 -> U32,~operation: U32 -> Element -> Element, +depth: Nat,whole: Bool,blocks: U32,size: U32,total: U32,context: Context,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>, equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) -> TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,Indexed.run_whole(~Element,~Context,~index,~base,~operation,whole,blocks,size,total,context,input,output)>: match whole: case True{}: TraversalCertificate.completed(~Array<Element>,~Element,~Indexed.Blocks<Context>,~TraversalWitness.array_evidence(~Element), ~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), U32.to_nat(blocks),0,Indexed.Blocks{0,size,context},input,output,Certificate.witness(Element,input,input_certificate),depth,equation) case False{}: preserves(~Element,~Context,~index,~operation,depth,Indexed.Call{U32.to_nat(total),context},input,output,input_certificate,equation)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, +depth: Nat,tiling: Tiling<Context>,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>, equation: {output == Tree.pack(Element,Certificate.normalized(Element,depth,Tree.reflect(Element,output))) : Array<Element>}) -> TraversalWitness.ResultEvidence<Array<Element>,Element,input,depth,tiled_operation(~Element,~Context,~index,~base,~blocks,~size,~operation,tiling,input,output)>: Tiling{+total,+context} = tiling 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)), blocks(context),size(context),total,context,input,output,input_certificate,equation)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, +total: U32,+context: Context,-input: Array<Element>,-output: Array<Element>,input_certificate: Certificate.Certificate<Element,input>, output_certificate: Certificate.Certificate<Element,output>) -> TraversalCertificate.Certificates<Array<Element>,Element,Certificate.family(~Element),Indexed.run_tiled(~Element,~Context,~index,~base,~blocks,~size,~operation,total,context,input,output)>: Certificate.Certificate{+depth,+equation} = input_certificate TraversalCertificate.operation(~Array<Element>,~Element,~Tiling<Context>,~Certificate.family(~Element), ~tiled_operation(~Element,~Context,~index,~base,~blocks,~size,~operation),~tiled_preserves(~Element,~Context,~index,~base,~blocks,~size,~operation), Tiling{total,context},input,output,Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},Certificate.Certificate{depth,equation},output_certificate)