~/bend-docscommunity

proofs/containers/doubly_linked_list/ins.bend checks

raw source on the hub · import bend-collections-laws-containers@1.0.0.0/proofs/containers/doubly_linked_list/ins.bend as Ins

22 imports
import Base
import ../../lib/logic.bend as L
import ../../lib/nat.bend as N
import ../../lib/u32.bend as U
import ../../lib/u32alg.bend as A
import ../../lib/array.bend as AR
import ../../lib/list.bend as LL
import ../../../spec/lib/common.bend as SC
import ../../../spec/containers/doubly_linked_list.bend as S
import ../../lib/u32div.bend as UD
import ../../../src/containers/doubly_linked_list.bend as D
import ../../../src/containers/internal/dlist_storage.bend as R
import ../../../src/containers/types/internal_dlist.bend as I
import ../../../src/containers/types/doubly_linked_list.bend as E
import ./state.bend as ST
import ./rel.bend as RL
import ./link.bend as LN
import ./ok.bend as OK
import ./insp.bend as IP
import ./insf.bend as IF
import ./grow.bend as GR
import ../../lib/links.bend as LK

Definitions

def lt_nat source · line 29 · raw

@+a:U32 -> @+b:U32 -> @+c:Bool -> @+h:{U32.is_lt(a, b) == c : Bool} -> {Nat.is_lt(0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(a), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(b)) == c : Bool}

Templates

template ins_room source · line 34 · raw

@-T:Data -> @+one:Nat -> @+h1:{one == 1n : Nat} -> @+tag:U32 -> @+cap:U32 -> @+fresh:U32 -> @+head:U32 -> @+tail:U32 -> @+depth:Nat -> @+vT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<Maybe<&2, T>> -> @+pT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+nT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+gT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+a:List<&2, Nat> -> @+b:List<&2, Nat> -> @+hg:{0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/state.goodF(T, tag, cap, fresh, 0, head, tail, depth, vT, pT, nT, gT, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), []) == True{} : Bool} -> @+hr:{U32.is_lt(fresh, cap) == True{} : Bool} -> @+x:T -> 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/ok.POK(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/types/doubly_linked_list.Handle, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/insp.insAB(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.alloc(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.DS{tag, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(Maybe<&2, T>, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(U32, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), []}), x, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/doubly_linked_list.inserted(T, tag, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/internal/dlist_storage.insert_between_free(T, tag, fresh, 0, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.length(Nat, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b)), head, tail, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, pT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, nT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.last_or(a, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(b, 0), x)))

room in the blocks: the fresh id is linked in place

template ins_grow source · line 48 · raw

@-T:Data -> @+one:Nat -> @+h1:{one == 1n : Nat} -> @+tag:U32 -> @+cap:U32 -> @+fresh:U32 -> @+head:U32 -> @+tail:U32 -> @+depth:Nat -> @+vT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<Maybe<&2, T>> -> @+pT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+nT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+gT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+a:List<&2, Nat> -> @+b:List<&2, Nat> -> @+cz:Nat -> @+hcz:{cz == 29n : Nat} -> @+hg:{0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/state.goodF(T, tag, cap, fresh, 0, head, tail, depth, vT, pT, nT, gT, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), []) == True{} : Bool} -> @+hr:{U32.is_lt(fresh, cap) == False{} : Bool} -> @+hroom:{Nat.is_lt(0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.pow2(cz)) == True{} : Bool} -> @+x:T -> 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/ok.POK(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/types/doubly_linked_list.Handle, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/insp.insAB(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.alloc(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.DS{tag, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(Maybe<&2, T>, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(U32, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), []}), x, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/doubly_linked_list.inserted(T, tag, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/internal/dlist_storage.insert_between_free(T, tag, fresh, 0, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.length(Nat, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b)), head, tail, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, pT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, nT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.last_or(a, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(b, 0), x)))

no room: the blocks double and the fresh id is the first of the new half

template ins_nil source · line 88 · raw

@-T:Data -> @+one:Nat -> @+h1:{one == 1n : Nat} -> @+tag:U32 -> @+cap:U32 -> @+fresh:U32 -> @+head:U32 -> @+tail:U32 -> @+depth:Nat -> @+vT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<Maybe<&2, T>> -> @+pT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+nT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+gT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+a:List<&2, Nat> -> @+b:List<&2, Nat> -> @+cz:Nat -> @+hcz:{cz == 29n : Nat} -> @+hg:{0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/state.goodF(T, tag, cap, fresh, 0, head, tail, depth, vT, pT, nT, gT, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), []) == True{} : Bool} -> @+hroom:{Nat.is_lt(0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.pow2(cz)) == True{} : Bool} -> @+x:T -> @+c:Bool -> @+hr:{U32.is_lt(fresh, cap) == c : Bool} -> 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/ok.POK(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/types/doubly_linked_list.Handle, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/insp.insAB(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.alloc(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.DS{tag, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(Maybe<&2, T>, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(U32, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), []}), x, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/doubly_linked_list.inserted(T, tag, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/internal/dlist_storage.insert_between_free(T, tag, fresh, 0, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.length(Nat, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b)), head, tail, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, pT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, nT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.last_or(a, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(b, 0), x)))

template ins_fl source · line 96 · raw

@-T:Data -> @+one:Nat -> @+h1:{one == 1n : Nat} -> @+tag:U32 -> @+cap:U32 -> @+fresh:U32 -> @+head:U32 -> @+tail:U32 -> @+depth:Nat -> @+vT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<Maybe<&2, T>> -> @+pT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+nT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+gT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+a:List<&2, Nat> -> @+b:List<&2, Nat> -> @+cz:Nat -> @+hcz:{cz == 29n : Nat} -> @+fl:List<&2, Nat> -> @+hg:{0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/state.goodF(T, tag, cap, fresh, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(fl, 0), head, tail, depth, vT, pT, nT, gT, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), fl) == True{} : Bool} -> @+hroom:{Nat.is_lt(0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.pow2(cz)) == True{} : Bool} -> @+x:T -> 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/ok.POK(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/types/doubly_linked_list.Handle, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/insp.insAB(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.alloc(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.DS{tag, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(Maybe<&2, T>, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(U32, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), fl}), x, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/doubly_linked_list.inserted(T, tag, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/internal/dlist_storage.insert_between_free(T, tag, fresh, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(fl, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.length(Nat, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b)), head, tail, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, pT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, nT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.last_or(a, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(b, 0), x)))

template ins_ok source · line 104 · raw

@-T:Data -> @+one:Nat -> @+h1:{one == 1n : Nat} -> @+tag:U32 -> @+cap:U32 -> @+fresh:U32 -> @+head:U32 -> @+tail:U32 -> @+depth:Nat -> @+vT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<Maybe<&2, T>> -> @+pT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+nT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+gT:0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.Tree<U32> -> @+a:List<&2, Nat> -> @+b:List<&2, Nat> -> @+cz:Nat -> @+hcz:{cz == 29n : Nat} -> @+free:U32 -> @+fl:List<&2, Nat> -> @+hg:{0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/state.goodF(T, tag, cap, fresh, free, head, tail, depth, vT, pT, nT, gT, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), fl) == True{} : Bool} -> @+hroom:{Nat.is_lt(0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.pow2(cz)) == True{} : Bool} -> @+x:T -> 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/ok.POK(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/types/doubly_linked_list.Handle, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/containers/doubly_linked_list/insp.insAB(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.alloc(T, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/containers/doubly_linked_list.DS{tag, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(Maybe<&2, T>, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.take(U32, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.slots(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/u32div.v(fresh)), fl}), x, a, b), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/doubly_linked_list.inserted(T, tag, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, gT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/src/containers/internal/dlist_storage.insert_between_free(T, tag, fresh, free, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.length(Nat, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/spec/lib/common.append(Nat, a, b)), head, tail, depth, cap, 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(Maybe<&2, T>, vT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, pT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/array.thaw(U32, nT), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.last_or(a, 0), 0x5c489f5d9646d7cc9aa3dd8137e9dc07/proofs/lib/links.fst_or(b, 0), x)))

THEOREM (insert between a and b)