proofs/traversal_batch_refinement.bend source
proofs/traversal_batch_refinement.bend on the hub · documented module
# A batch writes the mapped values of consecutive positions after reading all# of them from the original tree. Inside the tree it is the ordered in-place# map of the same positions: no step of that map reads a position an earlier# step wrote. For every element type and every batch length.import Baseimport ../traversal_array.bend as Traversalimport ../traversal_loop.bend as Loopimport ./traversal_array_refinement.bend as ArrayLawsimport ./traversal_map_readback.bend as MapReadbackimport ./storage_tree_model.bend as Treeimport ./storage_shape.bend as Shapeimport ./nat_to_u32_bounds.bend as Boundsimport ./nat_order.bend as Orderimport ./natural_addition.bend as Additionimport ./nat_interval.bend as Intervaldef batch(~Element: Data,~Context: Data,~map: Context -> U32 -> Element -> Element,count: Nat,+position: U32,+context: Context,+source: Tree.Tree<Element>, target: Tree.Tree<Element>) -> Tree.Tree<Element>: match count: case 0n: target case 1n+rest: batch(~Element,~Context,~map,rest,(position + 1 : U32),context,source,Tree.write(Element,target,position,map(context,position,Tree.read(Element,source,position))))def ModelState(~Element: Data,~Context: Data) -> Data: Traversal.State<&2,&2,Unit,Tree.Tree<Element>,Context># A loop's last step, for every state and count.def last_step(~State: Data,~step: State -> State,+count: Nat,+state: State) -> {Loop.run(~State,~step,1n+count,state) == step(Loop.run(~State,~step,count,state)) : State}: match count: case 0n: {==} case 1n+ +rest: last_step(~State,~step,rest,step(state))# The whole state after count steps of the model map.def state_after(~Element: Data,~Context: Data,~map: Context -> U32 -> Element -> Element,+count: Nat,+start: Nat,+context: Context,+input: Tree.Tree<Element>) -> {Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),count,Traversal.State{Unit{},input,U32.from_nat(start),context}) == Traversal.State{Unit{},MapReadback.run_indexed(~Element,~Context,~map,count,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,count)),context} : ModelState(~Element,~Context)}: match count: case 0n: %Equal.sym(Nat,Nat.add(start,0n),start,Addition.zero_right(start)) : {Traversal.State{Unit{},input,U32.from_nat(start),context} == Traversal.State{Unit{},input,U32.from_nat(_),context} : ModelState(~Element,~Context)} {==} case 1n+ +rest: +written = Tree.write(Element,input,U32.from_nat(start),map(context,U32.from_nat(start),Tree.read(Element,input,U32.from_nat(start)))) %Equal.sym(Nat,Nat.add(start,1n+rest),1n+Nat.add(start,rest),Addition.successor_right(start,rest)) : {Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),rest,Traversal.State{Unit{},written,(U32.from_nat(start) + 1 : U32),context}) == Traversal.State{Unit{},MapReadback.run_indexed(~Element,~Context,~map,rest,(U32.from_nat(start) + 1 : U32),context,written),U32.from_nat(_),context} : ModelState(~Element,~Context)} %Equal.sym(U32,(U32.from_nat(start) + 1 : U32),U32.from_nat(1n+start),Bounds.add_one(U32.from_nat(start))) : {Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),rest,Traversal.State{Unit{},written,_,context}) == Traversal.State{Unit{},MapReadback.run_indexed(~Element,~Context,~map,rest,_,context,written),U32.from_nat(1n+Nat.add(start,rest)),context} : ModelState(~Element,~Context)} state_after(~Element,~Context,~map,rest,1n+start,context,written)# One more step of the model map writes the position after the ones it mapped.def extended(~Element: Data,~Context: Data,~map: Context -> U32 -> Element -> Element,+done: Nat,+start: Nat,+context: Context,+input: Tree.Tree<Element>) -> {MapReadback.run_indexed(~Element,~Context,~map,1n+done,U32.from_nat(start),context,input) == Tree.write(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done)), map(context,U32.from_nat(Nat.add(start,done)),Tree.read(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done))))) : Tree.Tree<Element>}: +initial = {Traversal.State{Unit{},input,U32.from_nat(start),context} : ModelState(~Element,~Context)} %Equal.sym(ModelState(~Element,~Context),Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),1n+done,initial), ArrayLaws.model_map_step(~Element,~Context,~map,Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),done,initial)), last_step(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),done,initial)) : {MapReadback.output(~Element,~Context,_) == Tree.write(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done)), map(context,U32.from_nat(Nat.add(start,done)),Tree.read(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done))))) : Tree.Tree<Element>} %Equal.sym(ModelState(~Element,~Context),Loop.run(~ModelState(~Element,~Context),~ArrayLaws.model_map_step(~Element,~Context,~map),done,initial), Traversal.State{Unit{},MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done)),context}, state_after(~Element,~Context,~map,done,start,context,input)) : {MapReadback.output(~Element,~Context,ArrayLaws.model_map_step(~Element,~Context,~map,_)) == Tree.write(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done)), map(context,U32.from_nat(Nat.add(start,done)),Tree.read(Element,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input),U32.from_nat(Nat.add(start,done))))) : Tree.Tree<Element>} {==}# A batch that continues done steps of the map completes the map.def continued(~Element: Data,~Context: Data,~map: Context -> U32 -> Element -> Element,+remaining: Nat,+done: Nat,+start: Nat,+context: Context,+input: Tree.Tree<Element>, +depth: Nat,+capacity: U32,+capacity_equal: {Tree.capacity(Element,input) == capacity : U32},+balanced: Shape.Balanced(Element,depth,input), +limit: {Nat.is_le(depth,31n) == True{} : Bool},+capacity_ok: {U32.is_le(capacity,2147483648) == True{} : Bool}, +room: {Nat.is_le(Nat.add(Nat.add(start,done),remaining),U32.to_nat(capacity)) == True{} : Bool}) -> {batch(~Element,~Context,~map,remaining,U32.from_nat(Nat.add(start,done)),context,input,MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input)) == MapReadback.run_indexed(~Element,~Context,~map,Nat.add(done,remaining),U32.from_nat(start),context,input) : Tree.Tree<Element>}: match remaining: case 0n: %Equal.sym(Nat,Nat.add(done,0n),done,Addition.zero_right(done)) : {MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input) == MapReadback.run_indexed(~Element,~Context,~map,_,U32.from_nat(start),context,input) : Tree.Tree<Element>} {==} case 1n+ +rest: +reached = Nat.add(start,done) +query = U32.from_nat(reached) +mapped = MapReadback.run_indexed(~Element,~Context,~map,done,U32.from_nat(start),context,input) +inside = Order.room_index(reached,rest,U32.to_nat(capacity),room) +within = Bounds.strict_le(reached,U32.to_nat(capacity),inside) +unread = MapReadback.outside_indexed(~Element,~Context,~map,done,start,query,context,input,depth,capacity,capacity_equal,balanced,limit,capacity_ok,within, MapReadback.address_inside(reached,capacity,capacity_ok,inside), Interval.After{ %Equal.sym(Nat,U32.to_nat(query),reached,Bounds.roundtrip(reached,capacity,capacity_ok,within)) : {Nat.is_le(reached,_) == True{} : Bool} Order.reflexive(reached)}) %Equal.sym(Nat,Nat.add(done,1n+rest),1n+Nat.add(done,rest),Addition.successor_right(done,rest)) : {batch(~Element,~Context,~map,rest,(query + 1 : U32),context,input,Tree.write(Element,mapped,query,map(context,query,Tree.read(Element,input,query)))) == MapReadback.run_indexed(~Element,~Context,~map,_,U32.from_nat(start),context,input) : Tree.Tree<Element>} %unread : {batch(~Element,~Context,~map,rest,(query + 1 : U32),context,input,Tree.write(Element,mapped,query,map(context,query,_))) == MapReadback.run_indexed(~Element,~Context,~map,1n+Nat.add(done,rest),U32.from_nat(start),context,input) : Tree.Tree<Element>} %extended(~Element,~Context,~map,done,start,context,input) : {batch(~Element,~Context,~map,rest,(query + 1 : U32),context,input,_) == MapReadback.run_indexed(~Element,~Context,~map,1n+Nat.add(done,rest),U32.from_nat(start),context,input) : Tree.Tree<Element>} %Equal.sym(U32,(query + 1 : U32),U32.from_nat(1n+reached),Bounds.add_one(query)) : {batch(~Element,~Context,~map,rest,_,context,input,MapReadback.run_indexed(~Element,~Context,~map,1n+done,U32.from_nat(start),context,input)) == MapReadback.run_indexed(~Element,~Context,~map,1n+Nat.add(done,rest),U32.from_nat(start),context,input) : Tree.Tree<Element>} %Addition.successor_right(start,done) : {batch(~Element,~Context,~map,rest,U32.from_nat(_),context,input,MapReadback.run_indexed(~Element,~Context,~map,1n+done,U32.from_nat(start),context,input)) == MapReadback.run_indexed(~Element,~Context,~map,1n+Nat.add(done,rest),U32.from_nat(start),context,input) : Tree.Tree<Element>} continued(~Element,~Context,~map,rest,1n+done,start,context,input,depth,capacity,capacity_equal,balanced,limit,capacity_ok, %Equal.sym(Nat,Nat.add(start,1n+done),1n+reached,Addition.successor_right(start,done)) : {Nat.is_le(Nat.add(_,rest),U32.to_nat(capacity)) == True{} : Bool} Order.advance_room(reached,rest,U32.to_nat(capacity),room))# A batch from the original tree is the map of its positions.def mapped(~Element: Data,~Context: Data,~map: Context -> U32 -> Element -> Element,+count: Nat,+start: Nat,+context: Context,+input: Tree.Tree<Element>, +depth: Nat,+capacity: U32,+capacity_equal: {Tree.capacity(Element,input) == capacity : U32},+balanced: Shape.Balanced(Element,depth,input), +limit: {Nat.is_le(depth,31n) == True{} : Bool},+capacity_ok: {U32.is_le(capacity,2147483648) == True{} : Bool}, +room: {Nat.is_le(Nat.add(start,count),U32.to_nat(capacity)) == True{} : Bool}) -> {batch(~Element,~Context,~map,count,U32.from_nat(start),context,input,input) == MapReadback.run_indexed(~Element,~Context,~map,count,U32.from_nat(start),context,input) : Tree.Tree<Element>}: %Addition.zero_right(start) : {batch(~Element,~Context,~map,count,U32.from_nat(_),context,input,input) == MapReadback.run_indexed(~Element,~Context,~map,count,U32.from_nat(start),context,input) : Tree.Tree<Element>} continued(~Element,~Context,~map,count,0n,start,context,input,depth,capacity,capacity_equal,balanced,limit,capacity_ok, %Equal.sym(Nat,Nat.add(start,0n),start,Addition.zero_right(start)) : {Nat.is_le(Nat.add(_,count),U32.to_nat(capacity)) == True{} : Bool} room)