specs/olist/proofs/1623070846000del1-01 ⊦ ordered≤(x) → ordered≤(x -1l a)le-rle-oole-recapp-01cons-02le-e617del1-01-proofdel1-01-proof-infoslsordered-prefix ⊦ x ⊑ x0 ∧ ordered≤(x0) → (ordered≤(x) ↔ true)k-le-rapp-01cons-02le-e417ordered-prefix-proofordered-prefix-proof-infoaslasdel1 ⊦ ordered<(x) → ordered<(x -1l a)orderedls-lsls-ools-recconsls-e515del1-proofdel1-proof-infoslsole-last-01 ⊦ ordered≤(x) ∧ x ≠ [] → (ordered≤(x + a ') ↔ ¬ a < x.last)ole-last13ole-last-01-proofole-last-01-proof-infoslseq ⊦ ins≤(a, x) = ins≤(a, y) ↔ x = yin-04ins-recins-yemptyins-nins-e421eq-proofeq-proof-infoslssw ⊦ ins≤(a, ins≤(b, x)) = ins≤(b, ins≤(a, x))ins-y-01ins-yins-nins-one-02ins-one-01ins-oneins-e318sw-proofsw-proof-infomerge-empty ⊦ merge([], x) = x00merge-empty-proofmerge-empty-proof-infoslsinsperm-02 ⊦ perm(ins≤(a, x + y), ins≤(a, x) + y)ins-yins-ninspermins-e28insperm-02-proofinsperm-02-proof-infoslsole-in ⊦ ordered≤(a ' + x) ∧ b ∈ x → ¬ b < ale-rapp-01cons-0215ole-in-proofole-in-proof-infoforwardlocalforwardins-one-01 ⊦ ¬ b < a → ins≤(a, b ') = a ' + b 'ins-y13ins-one-01-proofins-one-01-proof-infoslsle-r ⊦ ordered≤(a ' + b ' + x) ↔ ¬ b < a ∧ ordered≤(b ' + x)00le-r-proofle-r-proof-infowslwsnokodkodaxole-last ⊦ ordered≤(x + a ') → (ordered≤(x + a ' + b ') ↔ ¬ b < a)le-ole-role-inole-recapp-01cons-02ole-two523ole-last-proofole-last-proof-infoslseqdefperm(x, y), ordered≤(x), ordered≤(y) ⊦ x = ydel1-01-01del1-01app-01cons-0227eqdef-proofeqdef-proof-infocons ⊦ ordered<(a ' + x) → ordered<(x)ls-rls-els-o03cons-proofcons-proof-infoslsordered-append ⊦ ordered≤(x) ∧ ordered≤(x0) ∧ (x ≠ [] ∧ x0 ≠ [] → ¬ x0.head < x.last) → ordered≤(x + x0)k-le-rle-rle-oapp-01cons-02le-e313ordered-append-proofordered-append-proof-infoapp-01 ⊦ ordered≤(x + y) → ordered≤(y)cons-0205app-01-proofapp-01-proof-infoslsins-less-first ⊦ ordered≤(x) ∧ x ≠ [] ∧ ¬ a < x.head → ins≤(a, x) = x.head + ins≤(a, x.tail)cons-03ins-yins-napp-01cons-0205ins-less-first-proofins-less-first-proof-infole-o ⊦ ordered≤(a ')00le-o-proofle-o-proof-infoslsnokodkodaxsort-possible ⊦ ∃ y. perm(x, y) ∧ ordered≤(y) ∧ # x = # ydel1-02in-04lenolele-e35sort-possible-proofsort-possible-proof-infoszle1-orderedls ⊦ # x ≤ 1 → ordered<(x)ls-ols-e12szle1-orderedls-proofszle1-orderedls-proof-infoslsdel1-01-02 ⊦ a ≠ b ∧ ordered≤(x) → ins≤(a, x -1l b) = ins≤(a, x) -1l bole-recins-yeqins-napp-01cons-02ins-e417del1-01-02-proofdel1-01-02-proof-infoorderedls-ls ⊦ b < a ∧ b ∈ x → ¬ ordered<(a ' + x)ls-r15orderedls-ls-prooforderedls-ls-proof-infoslsole-01 ⊦ ordered≤(ins≤(a, x)) ↔ ordered≤(x)app-01cons-02le-rins-yemptyole-recins-nle-oins-ele-e416ole-01-proofole-01-proof-infoslsordered-postfix ⊦ x ⊒ x0 ∧ ordered≤(x0) → (ordered≤(x) ↔ true)le-rk-le-rle-ele-oole-recapp-01cons-02817ordered-postfix-proofordered-postfix-proof-infoaslasorderedle-ls ⊦ b < a ∧ b ∈ x → ¬ ordered≤(a ' + x)cons-02ole-in02orderedle-ls-prooforderedle-ls-proof-infoslsordered-ex-divide-lsleordered≤(x) ⊦ ∃ y, z. x = y + z ∧ (∀ b. b ∈ y → b < a) ∧ (∀ b. b ∈ z → b = a ∨ a < b)orderedle-lsapp-01cons-02le-e715ordered-ex-divide-lsle-proofordered-ex-divide-lsle-proof-infols-o ⊦ ordered<(a ')00ls-o-proofls-o-proof-infoslsnokodkodaxempty ⊦ ins≤(a, x) ≠ []ins-yins-nins-e05empty-proofempty-proof-infoslsinsperm-01 ⊦ perm(ins≤(a, x), ins≤(a, y)) ↔ perm(x, y)insperm59insperm-01-proofinsperm-01-proof-infoslslen ⊦ # ins≤(a, x) = # x + 1ins-yins-nins-e06len-prooflen-proof-infoslsls-r ⊦ ordered<(a ' + b ' + x) ↔ a < b ∧ ordered<(b ' + x)00ls-r-proofls-r-proof-infowslwsnokodkodaxols-rec ⊦ x ≠ [] ∧ x.tail ≠ [] → (ordered<(x) ↔ x.head < x.tail.head ∧ ordered<(x.tail))ls-rcons05ols-rec-proofols-rec-proof-infoins-n ⊦ b < a → ins≤(a, b ' + x) = b + ins≤(a, x)00ins-n-proofins-n-proof-infolocalcutslsins-e ⊦ ins≤(a, []) = a '00ins-e-proofins-e-proof-infoslsordered-ex-divide-lelsordered≤(x) ⊦ ∃ y, z. x = y + z ∧ (∀ b. b ∈ y → b < a ∨ b = a) ∧ (∀ b. b ∈ z → a < b)orderedle-lsapp-01cons-02le-e819ordered-ex-divide-lels-proofordered-ex-divide-lels-proof-infoin-04 ⊦ a ∈ ins≤(a, x)ins-yins-nins-e06in-04-proofin-04-proof-infoslscons-03 ⊦ ordered≤(a ' + x) → ins≤(a, x) = a ' + xins-yle-rins-eapp-01cons-0203cons-03-proofcons-03-proof-infoslsone-02 ⊦ ins≤(a, a ') = a ' + a 'ins-one-02ins-one-0112one-02-proofone-02-proof-infoslsins2 ⊦ ins≤(a, ins≤(b, x)) = ins≤(b, ins≤(a, x)) ↔ truesw13ins2-proofins2-proof-infoslsole-two ⊦ ordered≤(a ' + b ') ↔ ¬ b < ale-ole-r12ole-two-proofole-two-proof-infoslsorderedls-all ⊦ ordered<(x) ↔ (∀ m, n. m < n ∧ n < # x → x[m] < x[n])orderedls-lsls-ools-recls-e1131orderedls-all-prooforderedls-all-proof-infoin-01-02 ⊦ a ≠ b → (a ∈ ins≤(b, x) ↔ a ∈ x)ins-yins-nins-e27in-01-02-proofin-01-02-proof-infoslsins-y-01 ⊦ a < b → ins≤(a, b ' + x) = a + b + xins-nins-y12ins-y-01-proofins-y-01-proof-infoslsapp ⊦ ordered≤(x + y) → ordered≤(x)le-rle-oole-recle-e29app-proofapp-proof-infoslsdel1-01-01 ⊦ ordered≤(a ' + (x -1l a)) ∧ ordered≤(x) ∧ a ∈ x → a ' + (x -1l a) = xorderedle-lsle-rapp-01del1-01cons-0213del1-01-01-proofdel1-01-01-proof-infoslsle-e ⊦ ordered≤([])00le-e-proofle-e-proof-infoslsnokodkodaxins-rec ⊦ x ≠ [] ∧ x.head < a → ins≤(a, x) = x.head + ins≤(a, x.tail)ins-n03ins-rec-proofins-rec-proof-infoins-one ⊦ b < a → ins≤(a, b ') = b ' + a 'ins-eins-n13ins-one-proofins-one-proof-infoslsinsperm-03 ⊦ perm(ins≤(a, x), y) ↔ perm(a + x, y)insperm25insperm-03-proofinsperm-03-proof-infosls-e ⊦ ordered<([])00ls-e-proofls-e-proof-infoslsnokodkodaxins-y ⊦ ¬ b < a → ins≤(a, b ' + x) = a + b + x00ins-y-proofins-y-proof-infoslsordered-shorter-sublist ⊦ ordered≤(sublist(m0, n1 + 1, x)) ∧ n2 < n1 ∧ n1 + m0 < # x → (ordered≤(sublist(m0, n2 + 1, x)) ↔ true)ordered-prefix36ordered-shorter-sublist-proofordered-shorter-sublist-proof-infosslssmerge-rec ⊦ merge(a ' + x0, x) = ins≤(a, merge(x0, x))00merge-rec-proofmerge-rec-proof-infoslsrest-01 ⊦ ordered≤(x) ∧ x ≠ [] → ordered≤(x.tail)app-0103rest-01-proofrest-01-proof-infoslsins-less-last ⊦ ¬ a0 < a → ins≤(a, x + a0 ') = ins≤(a, x) + a0 'ins-yins-nins-eins-one-0117ins-less-last-proofins-less-last-proof-infoslsole-rec ⊦ x ≠ [] ∧ x.tail ≠ [] → (ordered≤(x) ↔ ¬ x.tail.head < x.head ∧ ordered≤(x.tail))le-rapp-0105ole-rec-proofole-rec-proof-infok-le-r ⊦ ordered≤(y) ↔ # y ≤ 1 ∨ ¬ y.tail.head < y.head ∧ ordered≤(y.tail)le-rapp-01le-ole-e05k-le-r-proofk-le-r-proof-infokodkodaxins-yes ⊦ x ≠ [] ∧ ¬ x.head < a → ins≤(a, x) = a + xins-y03ins-yes-proofins-yes-proof-infodel1-02 ⊦ ins≤(a, x) -1l a = xins-yins-nins-e28del1-02-proofdel1-02-proof-infoslsex ⊦ ∃ y, z. ins≤(a, x) = y + a ' + z ∧ x = y + zins-nins-yins-e411ex-proofex-proof-infoperm-ins ⊦ perm((x + ins≤(a, y)) -1l a, z) ↔ perm(x + y, z)del1-02in-04insperm-0147perm-ins-proofperm-ins-proof-infoslsapp-02 ⊦ ordered≤(x + y + z) → ordered≤(y)app-01cons-02app25app-02-proofapp-02-proof-infoslssame ⊦ ins≤(a, a ' + x) = a ' + a ' + xins-yins-y-0102same-proofsame-proof-infoslsrest ⊦ ordered<(x) ∧ x ≠ [] → ordered<(x.tail)cons03rest-proofrest-proof-infoslsole ⊦ ordered≤(x) → ordered≤(ins≤(a, x))ole-0102ole-proofole-proof-infoslsk-ls-r ⊦ ordered<(y) ↔ # y ≤ 1 ∨ y.head < y.tail.head ∧ ordered<(y.tail)ls-rconsls-ols-e04k-ls-r-proofk-ls-r-proof-infokodkodaxins-one-02 ⊦ a < b → ins≤(a, b ') = a ' + b 'ins-oneins-y13ins-one-02-proofins-one-02-proof-infoslsorderedle-all ⊦ ordered≤(x) ↔ (∀ m, n. m ≤ n ∧ n < # x → ¬ x[n] < x[m])orderedle-lsapp-01cons-02le-oole-recle-e1029orderedle-all-prooforderedle-all-proof-infoinsperm ⊦ perm(a ' + x, ins≤(a, x))del1-02in-0412insperm-proofinsperm-proof-infosszle1-orderedle ⊦ # x ≤ 1 → ordered≤(x)le-ole-e12szle1-orderedle-proofszle1-orderedle-proof-infoslscons-02 ⊦ ordered≤(a ' + x) → ordered≤(x)le-rle-ele-o03cons-02-proofcons-02-proof-infoslocalforward0