PROOF.ba.lb.body: def of bend-osdev-algo@0.1.0.0 Import: import bend-osdev-algo@0.1.0.0/PROOF.bend as PROOF Lines 1430-1435 of bend-osdev-algo@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&start_line=1430&end_line=1435 import Base import bend-osdev-algo@0.1.0.0/bsearch.bend as S import bend-osdev-array@0.1.0.0/array.bend as AR import bend-osdev-array@0.1.0.0/LAWS.bend as AL import bend-osdev-algo@0.1.0.0/proof_bsearch.bend as Q def ba.lb.body(+d: Nat, -a: Array, +hd: {Nat.is_lt(d, 32n) == True{} : Bool}, +n: U32, +hn: {Nat.is_le(U32.to_nat(n), AR.Arrays.slots(d)) == True{} : Bool}, +key: U32, hs: {S.bsearch.sorted(List.take(&2, U32, AR.Arrays.leaves(U32, a), U32.to_nat(n))) == True{} : Bool}, +l: List<&2, U32>, +e1: {AR.Arrays.of_list(U32, d, l, 0) == a : Array}, +hl: {List.length(&2, U32, l) == AR.Arrays.slots(d) : Nat}) -> {U32.to_nat(Pair.snd(Array, U32, S.bsearch.array_lower_bound(a, n, key))) == S.bsearch.lower_bound(List.take(&2, U32, AR.Arrays.leaves(U32, a), U32.to_nat(n)), key) : Nat}: -ol = AR.Arrays.of_list(U32, d, l, 0) %e1 : {U32.to_nat(Pair.snd(Array, U32, S.bsearch.array_lower_bound(_, n, key))) == S.bsearch.lower_bound(List.take(&2, U32, AR.Arrays.leaves(U32, _), U32.to_nat(n)), key) : Nat} %Equal.sym(List<&2, U32>, AR.Arrays.leaves(U32, ol), l, AL.of_list_leaves(U32, d, l, 0, hl)) : {U32.to_nat(Pair.snd(Array, U32, S.bsearch.array_lower_bound(ol, n, key))) == S.bsearch.lower_bound(List.take(&2, U32, _, U32.to_nat(n)), key) : Nat} %Equal.sym(Array & U32, S.bsearch.array.go(33n, S.BsearchState{ol, 0, n}, key), (ol, Q.bq.run(33n, d, l, 0, n, key)), Q.bq.go(33n, d, l, hl, hd, 0, n, key)) : {U32.to_nat(Pair.snd(Array, U32, _)) == Q.bq.lb(l, n, key) : Nat} ba.answer(d, l, hl, key, n, hn, ba.hs(d, a, l, n, e1, hl, hs)) Uses (28): https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Array https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.length https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.take https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Pair https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Pair.snd https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Sigma https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32 https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=ba.answer https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=ba.hs https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=BsearchState https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.array.go https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.array_lower_bound https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.lower_bound https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.sorted https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/proof_bsearch.bend&name=bq.go https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/proof_bsearch.bend&name=bq.lb https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/proof_bsearch.bend&name=bq.run https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/LAWS.bend&name=of_list_leaves https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.leaves https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.of_list https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.slots