1 import Base 2 import ../../lib/logic.bend as L 3 import ../../lib/list.bend as LL 4 import ../../../spec/lib/common.bend as SC 5 import ../../../src/containers/bitlist.bend as BLI 6 import ../../../src/containers/types/bitlist.bend as E 7 import ../dynamic_array/trace.bend as DTR 8 import ../../../spec/containers/bitlist.bend as S 9 import ./state.bend as SS 10 import ./steps.bend as BS 11 12 # Arbitrary finite operation traces: the real runner refines the spec runner 13 # (the same shadow form as the per-step laws), from the constructors new and 14 # with_limit, and from_bools builds exactly the list its bits spell. 15 # (source: proofs/containers/bitlist/trace.src) 16 17 18 def srun_obs(ops: List<&2, E.Op>, m: S.Model) -> List<&2, E.Obs>: 19 Pair.snd(S.Model, List<&2, E.Obs>, S.run(ops, m)) 20 21 def srun_state(ops: List<&2, E.Op>, m: S.Model) -> S.Model: 22 Pair.fst(S.Model, List<&2, E.Obs>, S.run(ops, m)) 23 24 def so_sh(-sh: SS.Sh, -op: E.Op, so: BS.StepOK(sh, op)) -> SS.Sh: 25 match so: 26 case Tuple{sh2, r}: 27 sh2 28 29 def so_obs(-sh: SS.Sh, -op: E.Op, so: BS.StepOK(sh, op)) -> E.Obs: 30 match so: 31 case Tuple{sh2, Tuple{o, r}}: 32 o 33 34 def so_step(-sh: SS.Sh, -op: E.Op, so: BS.StepOK(sh, op)) -> {BLI.step(SS.real(sh), op) == (SS.real(so_sh(sh, op, so)), so_obs(sh, op, so)) : BLI.Bitlist & E.Obs}: 35 match so: 36 case Tuple{sh2, Tuple{o, Tuple{e, r}}}: 37 e 38 39 def so_good(-sh: SS.Sh, -op: E.Op, so: BS.StepOK(sh, op)) -> {SS.good(so_sh(sh, op, so)) == True{} : Bool}: 40 match so: 41 case Tuple{sh2, Tuple{o, Tuple{e, Tuple{g, s}}}}: 42 g 43 44 def so_spec(-sh: SS.Sh, -op: E.Op, so: BS.StepOK(sh, op)) -> {(SS.model(so_sh(sh, op, so)), so_obs(sh, op, so)) == S.step(SS.model(sh), op) : S.Model & E.Obs}: 45 match so: 46 case Tuple{sh2, Tuple{o, Tuple{e, Tuple{g, s}}}}: 47 s 48 49 def spec_run_cons(+op: E.Op, +rest: List<&2, E.Op>, +m: S.Model, +m1: S.Model, +o: E.Obs, +e: {(m1, o) == S.step(m, op) : S.Model & E.Obs}) -> {S.run(Con{op, rest}, m) == (srun_state(rest, m1), Con{o, srun_obs(rest, m1)}) : S.Model & List<&2, E.Obs>}: 50 %e : {S.cons_obs(Pair.snd(S.Model, E.Obs, _), S.run(rest, Pair.fst(S.Model, E.Obs, _))) == (srun_state(rest, m1), Con{o, srun_obs(rest, m1)}) : S.Model & List<&2, E.Obs>} 51 %Equal.sym(S.Model & List<&2, E.Obs>, S.run(rest, m1), (srun_state(rest, m1), srun_obs(rest, m1)), L.pair_eta(S.Model, List<&2, E.Obs>, S.run(rest, m1))) : {S.cons_obs(o, _) == (srun_state(rest, m1), Con{o, srun_obs(rest, m1)}) : S.Model & List<&2, E.Obs>} 52 {==} 53 54 def RunOK(ops: List<&2, E.Op>, sh: SS.Sh, acc: List<&2, E.Obs>) -> Type: 55 Sigma<&1, &1, SS.Sh, sh_2 => {BLI.run_acc(ops, (SS.real(sh), acc)) == (SS.real(sh_2), List.reverse.go(&2, E.Obs, srun_obs(ops, SS.model(sh)), acc)) : BLI.Bitlist & List<&2, E.Obs>} & ({SS.good(sh_2) == True{} : Bool} & {SS.model(sh_2) == srun_state(ops, SS.model(sh)) : S.Model})> 56 57 def ro_sh(-ops: List<&2, E.Op>, -sh: SS.Sh, -acc: List<&2, E.Obs>, r: RunOK(ops, sh, acc)) -> SS.Sh: 58 match r: 59 case Tuple{sh2, x}: 60 sh2 61 62 def ro_run(-ops: List<&2, E.Op>, -sh: SS.Sh, -acc: List<&2, E.Obs>, r: RunOK(ops, sh, acc)) -> {BLI.run_acc(ops, (SS.real(sh), acc)) == (SS.real(ro_sh(ops, sh, acc, r)), List.reverse.go(&2, E.Obs, srun_obs(ops, SS.model(sh)), acc)) : BLI.Bitlist & List<&2, E.Obs>}: 63 match r: 64 case Tuple{sh2, Tuple{e, x}}: 65 e 66 67 def ro_good(-ops: List<&2, E.Op>, -sh: SS.Sh, -acc: List<&2, E.Obs>, r: RunOK(ops, sh, acc)) -> {SS.good(ro_sh(ops, sh, acc, r)) == True{} : Bool}: 68 match r: 69 case Tuple{sh2, Tuple{e, Tuple{g, m}}}: 70 g 71 72 def ro_model(-ops: List<&2, E.Op>, -sh: SS.Sh, -acc: List<&2, E.Obs>, r: RunOK(ops, sh, acc)) -> {SS.model(ro_sh(ops, sh, acc, r)) == srun_state(ops, SS.model(sh)) : S.Model}: 73 match r: 74 case Tuple{sh2, Tuple{e, Tuple{g, m}}}: 75 m 76 77 def run_ok(+ops: List<&2, E.Op>, +sh: SS.Sh, +acc: List<&2, E.Obs>, +g: {SS.good(sh) == True{} : Bool}) -> RunOK(ops, sh, acc): 78 match ops: 79 case Nil{}: 80 (sh, ({==}, (g, {==}))) 81 case Con{+op, +rest}: 82 +m = SS.model(sh) 83 +sh1 = so_sh(sh, op, BS.step_ok(sh, op, g)) 84 +o = so_obs(sh, op, BS.step_ok(sh, op, g)) 85 +g1 = so_good(sh, op, BS.step_ok(sh, op, g)) 86 +m1 = SS.model(sh1) 87 +acc1 = {Con{o, acc} : List<&2, E.Obs>} 88 +sh2 = ro_sh(rest, sh1, acc1, run_ok(rest, sh1, acc1, g1)) 89 +es = spec_run_cons(op, rest, m, m1, o, so_spec(sh, op, BS.step_ok(sh, op, g))) 90 -R = S.run(rest, m1) 91 (sh2, ( 92 %Equal.sym(BLI.Bitlist & E.Obs, BLI.step(SS.real(sh), op), (SS.real(sh1), o), so_step(sh, op, BS.step_ok(sh, op, g))) : {BLI.run_acc(rest, BLI.record(acc, _)) == (SS.real(sh2), List.reverse.go(&2, E.Obs, srun_obs(Con{op, rest}, m), acc)) : BLI.Bitlist & List<&2, E.Obs>} 93 %Equal.sym(BLI.Bitlist & List<&2, E.Obs>, BLI.run_acc(rest, (SS.real(sh1), acc1)), (SS.real(sh2), List.reverse.go(&2, E.Obs, srun_obs(rest, m1), acc1)), ro_run(rest, sh1, acc1, run_ok(rest, sh1, acc1, g1))) : {_ == (SS.real(sh2), List.reverse.go(&2, E.Obs, srun_obs(Con{op, rest}, m), acc)) : BLI.Bitlist & List<&2, E.Obs>} 94 %Equal.sym(S.Model & List<&2, E.Obs>, S.run(Con{op, rest}, m), (Pair.fst(S.Model, List<&2, E.Obs>, R), Con{o, Pair.snd(S.Model, List<&2, E.Obs>, R)}), es) : {(SS.real(sh2), List.reverse.go(&2, E.Obs, srun_obs(rest, m1), acc1)) == (SS.real(sh2), List.reverse.go(&2, E.Obs, Pair.snd(S.Model, List<&2, E.Obs>, _), acc)) : BLI.Bitlist & List<&2, E.Obs>} 95 {==}, 96 (ro_good(rest, sh1, acc1, run_ok(rest, sh1, acc1, g1)), 97 %Equal.sym(S.Model & List<&2, E.Obs>, S.run(Con{op, rest}, m), (Pair.fst(S.Model, List<&2, E.Obs>, R), Con{o, Pair.snd(S.Model, List<&2, E.Obs>, R)}), es) : {SS.model(sh2) == Pair.fst(S.Model, List<&2, E.Obs>, _) : S.Model} 98 ro_model(rest, sh1, acc1, run_ok(rest, sh1, acc1, g1))))) 99 100 def run_from(+ops: List<&2, E.Op>, +sh0: SS.Sh, +g0: {SS.good(sh0) == True{} : Bool}) -> {BLI.run(ops, SS.real(sh0)) == (SS.real(ro_sh(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0))), srun_obs(ops, SS.model(sh0))) : BLI.Bitlist & List<&2, E.Obs>}: 101 +sh2 = ro_sh(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0)) 102 +os = srun_obs(ops, SS.model(sh0)) 103 %Equal.sym(BLI.Bitlist & List<&2, E.Obs>, BLI.run_acc(ops, (SS.real(sh0), Nil{})), (SS.real(sh2), List.reverse.go(&2, E.Obs, os, Nil{})), ro_run(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0))) : {BLI.finish(_) == (SS.real(sh2), os) : BLI.Bitlist & List<&2, E.Obs>} 104 %Equal.sym(List<&2, E.Obs>, List.reverse(&2, E.Obs, List.reverse.go(&2, E.Obs, os, Nil{})), os, LL.rev_rev(E.Obs, os)) : {(SS.real(sh2), _) == (SS.real(sh2), os) : BLI.Bitlist & List<&2, E.Obs>} 105 {==} 106 107 def TraceOK(ops: List<&2, E.Op>, sh0: SS.Sh) -> Type: 108 Sigma<&1, &1, SS.Sh, sh_2 => {BLI.run(ops, SS.real(sh0)) == (SS.real(sh_2), srun_obs(ops, SS.model(sh0))) : BLI.Bitlist & List<&2, E.Obs>} & ({SS.good(sh_2) == True{} : Bool} & {SS.model(sh_2) == srun_state(ops, SS.model(sh0)) : S.Model})> 109 110 def trace_from(+ops: List<&2, E.Op>, +sh0: SS.Sh, +g0: {SS.good(sh0) == True{} : Bool}) -> TraceOK(ops, sh0): 111 (ro_sh(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0)), 112 (run_from(ops, sh0, g0), 113 (ro_good(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0)), 114 ro_model(ops, sh0, Nil{}, run_ok(ops, sh0, Nil{}, g0))))) 115 116 # ---- the constructors ---- 117 118 def initial(+l: Maybe<&2, Nat>) -> SS.Sh: 119 {SS.BSh{l, 0n, DTR.initial(U32)} : SS.Sh} 120 121 def new_real() -> {BLI.new() == SS.real(initial(None{})) : BLI.Bitlist}: 122 {==} 123 124 def with_limit_real(+n: Nat) -> {BLI.with_limit(n) == SS.real(initial(Some{n})) : BLI.Bitlist}: 125 {==} 126 127 def initial_good(+l: Maybe<&2, Nat>) -> {SS.good(initial(l)) == True{} : Bool}: 128 {==} 129 130 def new_model() -> {SS.model(initial(None{})) == S.new() : S.Model}: 131 {==} 132 133 def with_limit_model(+n: Nat) -> {SS.model(initial(Some{n})) == S.with_limit(n) : S.Model}: 134 {==} 135 136 # ---- from_bools ---- 137 138 def fpush(p: BLI.Bitlist & Result<&2, &2, E.Error, Unit>) -> {BLI.from_push(p) == Pair.fst(BLI.Bitlist, E.Obs, BLI.obs_unit(p)) : BLI.Bitlist}: 139 match p: 140 case Tuple{s, r}: 141 {==} 142 143 def FromOK(bs: List<&2, Bool>, sh: SS.Sh) -> Type: 144 Sigma<&1, &1, SS.Sh, sh_2 => {BLI.from_go(bs, SS.real(sh)) == SS.real(sh_2) : BLI.Bitlist} & ({SS.good(sh_2) == True{} : Bool} & {SS.model(sh_2) == S.push_all(bs, SS.model(sh)) : S.Model})> 145 146 def from_lift(+b: Bool, +t: List<&2, Bool>, sh: SS.Sh, +sh1: SS.Sh, +e1: {BLI.from_push(BLI.push(SS.real(sh), b)) == SS.real(sh1) : BLI.Bitlist}, +m1: {SS.model(sh1) == Pair.fst(S.Model, E.Obs, S.step(SS.model(sh), E.Push{b})) : S.Model}, ih: FromOK(t, sh1)) -> FromOK(Con{b, t}, sh): 147 match ih: 148 case Tuple{sh2, Tuple{e2, Tuple{g2, mm}}}: 149 (sh2, (%Equal.sym(BLI.Bitlist, BLI.from_push(BLI.push(SS.real(sh), b)), SS.real(sh1), e1) : {BLI.from_go(t, _) == SS.real(sh2) : BLI.Bitlist} 150 e2, 151 (g2, %Equal.sym(S.Model, Pair.fst(S.Model, E.Obs, S.step(SS.model(sh), E.Push{b})), SS.model(sh1), Equal.sym(S.Model, SS.model(sh1), Pair.fst(S.Model, E.Obs, S.step(SS.model(sh), E.Push{b})), m1)) : {SS.model(sh2) == S.push_all(t, _) : S.Model} 152 mm))) 153 154 def from_ok(+bs: List<&2, Bool>, +sh: SS.Sh, +g: {SS.good(sh) == True{} : Bool}) -> FromOK(bs, sh): 155 match bs: 156 case Nil{}: 157 (sh, ({==}, (g, {==}))) 158 case Con{+b, +t}: 159 +sh1 = so_sh(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g)) 160 +e1 = Equal.trans(BLI.Bitlist, BLI.from_push(BLI.push(SS.real(sh), b)), Pair.fst(BLI.Bitlist, E.Obs, BLI.step(SS.real(sh), E.Push{b})), SS.real(sh1), fpush(BLI.push(SS.real(sh), b)), 161 Equal.cong(BLI.Bitlist & E.Obs, BLI.Bitlist, p => Pair.fst(BLI.Bitlist, E.Obs, p), BLI.step(SS.real(sh), E.Push{b}), (SS.real(sh1), so_obs(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g))), so_step(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g)))) 162 +m1 = Equal.cong(S.Model & E.Obs, S.Model, p => Pair.fst(S.Model, E.Obs, p), (SS.model(sh1), so_obs(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g))), S.step(SS.model(sh), E.Push{b}), so_spec(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g))) 163 from_lift(b, t, sh, sh1, e1, m1, from_ok(t, sh1, so_good(sh, E.Push{b}, BS.step_ok(sh, E.Push{b}, g)))) 164 165 def from_bools_ok(+bs: List<&2, Bool>) -> FromOK(bs, initial(None{})): 166 from_ok(bs, initial(None{}), initial_good(None{}))