hcju/isabelleps huggingface dataset:
matchless_def: shows "matchless = inv_image (measure size_mset) fst"
defect_def: fixes N :: "'c quasinorm" shows "defect N = fst (Rep_quasinorm N)"
aux: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "(lookup f = (\<lambda>_. 0::'b)) = (f = 0)"
inner_eq_zero_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(inner x x = 0) = (x = zero)"
comp_surj: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'e" assumes "top_pres (\<P> f)"   and "top_pres (\<P> g)" shows "top_pres (\<P> (g \<circ> f))"
prems: shows "subtract (Node l x b r) t\<^sub>2 = Some t"
ifte_thms_1: fixes e1 :: "'a"   and e2 :: "'a" shows "If \<bottom> then e1 else e2 = \<bottom>"
in_set_conv_decomp: fixes x :: "'g"   and xs :: "'g list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
last_append: fixes xs :: "'d list"   and ys :: "'d list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
finite_in_span: fixes S :: "'a vec set"   and v :: "'a vec" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "v \<in> local.span S" shows "\<exists>a. True \<and> lincomb a S = v"
decide_univ_lem_helper_R: fixes fml_struct :: "nat fml"   and polys :: "real poly list"   and fml :: "real poly fml" assumes "(fml_struct, polys) = convert fml" shows "(\<forall>x. lookup_sem fml_struct (map (\<lambda>p. poly p x) polys)) = decide_universal_R fml"
SUP_le_iff: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and f :: "'i \<Rightarrow> 'h"   and A :: "'i set"   and u :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
pCons_0_as_mult: fixes p :: "'c poly" shows "pCons (0::'c) p = [:0::'c, 1::'c:] * p"
g2: shows " s2 \<longrightarrow>\<^sub>1\<^sup>* t4"
clear_locals_sip_not_ip: fixes \<xi> :: "state" shows "sip (clear_locals \<xi>) \<noteq> ip \<xi>"
axis_median_def: fixes k :: "'a"   and ps :: "(real, 'a) vec list" shows "axis_median k ps = (let n = (length ps - 1) div 2 in fast_select n (map (\<lambda>p. p $ k) ps))"
distr_completion: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "X \<in> M \<rightarrow>\<^sub>M N" shows "distr (completion M) N X = distr M N X"
filterlim_filtercomap_iff: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and G :: "'d filter"   and F :: "'b filter" shows "filterlim f (filtercomap g G) F = filterlim (g \<circ> f) G F"
sup_left_commute: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
nonzero_divide_eq_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and c :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(divide b c = a) = (b = times a c)"
prems_2: shows "apply_operations (xs @ [x]) = Some ys"
Ana_t: shows "Ana t = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (!) S, map ((!) S) N)"
existence_ivl_reverse: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "0 - t \<in> existence_ivl0 (flow0 x0 t)"
swapAlts_in_set_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c set"   and u :: "'c" assumes "{a, b} \<subseteq> A" shows "(swapAlts a b u \<in> A) = (u \<in> A)"
finite: fixes m :: "'s" assumes "invar m" shows "finite (dom (\<alpha> m))"
simps: fixes cs :: "atom list"   and x :: "'a" shows "supp_lst ([cs]lst. x) = supp x - set cs"
abs_eq: fixes xd :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'c"   and xc :: "'d \<Rightarrow> 'e"   and xb :: "'d \<Rightarrow> 'f"   and xa :: "'d \<Rightarrow> 'g"   and x :: "'d \<Rightarrow> 'h" shows "qtop xd (mk\<^sub>e xc) (mk\<^sub>e xb) (mk\<^sub>e xa) (mk\<^sub>e x) = mk\<^sub>e (\<lambda>b. xd (xc b) (xb b) (xa b) (x b))"
tfr\<^sub>s\<^sub>t\<^sub>p_is_comp_tfr\<^sub>s\<^sub>t\<^sub>p: fixes a :: "('fun, ('fun, 'atom) Term.term \<times> nat) strand_step" shows "tfr\<^sub>s\<^sub>t\<^sub>p a = comp_tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma> a"
Gp_smult_nsg: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<natural>H \<diamondop> N \<triangleright> N"
vector_space_pair_axioms: shows "vector_space_pair (*\<^sub>C) (*\<^sub>C)"
cases: fixes a :: "fm" assumes "se_fm a" obtains t :: "tm" and u :: "tm" where "a = t__ IN u"   | A :: "fm" and B :: "fm" where "a = A__ OR B__" and "se_fm A__" and "se_fm B__"   | A :: "fm" and B :: "fm" where "a = Aa__ AND B" and "se_fm Aa__" and "se_fm B"   | A :: "fm" and i :: "name" where "a = SyntaxN.Ex i__ Ab__" and "se_fm Ab__"   | A :: "fm" and i :: "name" and t :: "tm" where "a = All2 i t A" and "se_fm A" and "atom i \<sharp> t"
simps: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi" shows "triple.pairs_sup (x, y) (z, w) = (x \<squnion> z, y \<sqinter> w)"
assms_1: shows "e1 = (y + (q - b)) mod q"
assms_5: fixes h :: "'a"   and h' :: "'a" shows "\<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')"
Span_mem_iff: fixes Us :: "'a list"   and a :: "'a" assumes "set Us \<subseteq> carrier R"   and "a \<in> carrier R" shows "(a \<in> Span K Us) = (\<exists>k\<in>K - {\<zero>}. \<exists>Ks. set Ks \<subseteq> K \<and> combine (k # Ks) (a # Us) = \<zero>)"
assoc_ide_simp: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "C'.src f = C'.trg g"   and "C'.src g = C'.trg h" shows "C'.\<a> f g h = \<a>\<^sub>C\<^sup>-\<^sup>1[h, g, f]"
prv_all_imp_trans: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (all x (imp \<phi> \<chi>))"   and "bprv (all x (imp \<chi> \<psi>))" shows "bprv (all x (imp \<phi> \<psi>))"
isTable_def: fixes f :: "'c \<Rightarrow> 'd"   and vs :: "'c list"   and t :: "('c \<times> 'd) list" shows "isTable f vs t \<equiv> \<forall>p. p \<in> set t \<longrightarrow> snd p = f (fst p) \<and> fst p \<in> set vs"
is_path_simps_1: fixes v :: "'V" shows "is_path v [] v = (v \<in> V)"
QmpTr5: fixes H :: "'a set"   and N :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N"   and "X \<in> carrier (G / H)"   and "Y \<in> carrier (G / H)" shows "(Qm\<^bsub>G H,N\<^esub>) (X \<cdot>\<^bsub>G / H\<^esub> Y) = (Qm\<^bsub>G H,N\<^esub>) X \<cdot>\<^bsub>G / N\<^esub> (Qm\<^bsub>G H,N\<^esub>) Y"
mem_ant: fixes a :: "ant" shows "a = - \<infinity> \<or> (\<exists>z. a = ant z) \<or> a = \<infinity>"
f_inv_f: fixes y :: "'index \<times> 'trap"   and x :: "'range" shows "y \<in> set_spmf I \<longrightarrow> x \<in> range (fst y) \<longrightarrow> F\<^sub>i\<^sub>n\<^sub>v (fst y) (snd y) (F (fst y) x) = x"
in_idem: fixes A :: "'a multiset set"   and x :: "'a multiset" assumes "finite A"   and "x \<in> A" shows "x \<union># subset_mset.Sup_fin A = subset_mset.Sup_fin A"
decomp_size_mset_less: fixes ss :: "('d, 'e) Term.term list"   and ts :: "('d, 'f) Term.term list"   and M :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and f :: "'d" assumes "length ss = length ts" shows "size_mset (M + mset (zip ss ts)) < size_mset (M + {#(Fun f ss, Fun f ts)#})"
disc2_noProt: shows "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"
assms_1: shows "fun_eq_on g f (\<Union> A)"
strong_low_bisim_mm_def: fixes \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "strong_low_bisim_mm \<R> \<equiv> sym \<R> \<and> closed_glob_consistent \<R> \<and> (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<and> (\<forall>c\<^sub>1' mds' mem\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> (\<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>, \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>) \<in> \<R>)))"
ordIso_reflexive: fixes r :: "('d \<times> 'd) set" assumes "Well_order r" shows "r =o r"
nempty_stateSpace_nempty_ss_proj: fixes ss :: "('d, 'e) fmap set"   and vs :: "'d set" assumes "ss \<noteq> {}" shows "ss_proj ss vs \<noteq> {}"
init_z1: shows "local.adverse z i j \<triangleright> xmin = local.adverse 1 i j \<triangleright> xmin"
simps_1: fixes uu :: "'a list" shows "chop 0 uu = []"
direct_sum_is_module: fixes R :: "('g, 'h) ring_scheme"   and M1 :: "('g, 'i, 'j) module_scheme"   and M2 :: "('g, 'k, 'l) module_scheme" assumes "module R M1"   and "module R M2" shows "module R (direct_sum M1 M2)"
is_interval_def: fixes s :: "'c set" shows "is_interval s = (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i) \<longrightarrow> x \<in> s)"
is_dghm_axioms: shows "\<FF> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_prod I \<AA>"
is_regular_def: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" shows "is_regular s U \<equiv> \<forall>\<pp>. \<pp> \<in> U \<longrightarrow> (\<exists>V. is_zariski_open V \<and> V \<subseteq> U \<and> \<pp> \<in> V \<and> is_locally_frac s V)"
f: shows "\<down>\<^sub>s (c1, s) = t"
cnf_of_encode_all_operator_effects_structure: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t) = (\<Union>(k, op)\<in>{0..<t} \<times> set (\<Pi>\<^sub>\<O>). \<Union>v\<in>set (add_effects_of op). {{(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}}) \<union> (\<Union>(k, op)\<in>{0..<t} \<times> set (\<Pi>\<^sub>\<O>). \<Union>v\<in>set (delete_effects_of op). {{(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>}})"
simps_3: fixes a :: "'b"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G p(a) w = {}"
of_nat_eq_1_iff: fixes x :: "nat" shows "(of_nat x = (1::'a)) = (x = 1)"
simps_1: fixes uu :: "'g \<Rightarrow> 'f dres" shows "dFAIL \<bind> uu = dFAIL"
eq_fin_concat_start: fixes w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" shows "(w @ w\<^sub>1 =\<^sub>F w @ w\<^sub>2) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
set_subset_Cons: fixes xs :: "'a list"   and x :: "'a" shows "set xs \<subseteq> set (x # xs)"
inv_in_hom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.iso f"   and "VoV.in_hom f a b" shows "VoV.in_hom (VoV.inv f) b a"
simps_4: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
dropWhile_exp1: fixes n :: "nat"   and xs :: "cell list" shows "dropWhile (\<lambda>a. a = Oc) (Oc \<up> n @ xs) = dropWhile (\<lambda>a. a = Oc) xs"
M_3: shows "\<Union> (ik\<^sub>e\<^sub>s\<^sub>t ` M) = \<Union> (ik\<^sub>e\<^sub>s\<^sub>t ` decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>)"
sparse_row_vector_add: fixes a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list" shows "sparse_row_vector (add_spvec a b) = sparse_row_vector a + sparse_row_vector b"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<approx> P \<oplus> Q \<oplus> R"
bij_betw: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> bij_betw (F (fst y)) (domain (fst y)) (range (fst y))"
countable_Un: fixes A :: "'b set"   and B :: "'b set" assumes "countable A"   and "countable B" shows "countable (A \<union> B)"
antimonoE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" obtains "f y \<le> f x"
iFROM_inext: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "inext t [n\<dots>] = Suc t"
t_star_elim: fixes x :: "'a \<times> ('b \<times> 'a) list"   and X :: "('a \<times> ('b \<times> 'a) list) set" shows "(x \<in> t_star X) = (\<exists>n. x \<in> trace_dioid.power X n)"
MakeFtorPreFtor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "PreFunctor F" shows "PreFunctor (MakeFtor F)"
pr_pri_agrk_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and x :: "rat" assumes "(evs, S, A, U) \<in> protocol" shows "(Pri_AgrK x \<in> analz (A \<union> spies evs)) = (Pri_AgrK x \<in> A)"
rcosets_mult_eq: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and M :: "'c set" assumes "H \<lhd> G"   and "M \<in> rcosets\<^bsub>G\<^esub> H" shows "H <#>\<^bsub>G\<^esub> M = M"
prems_3: shows "D \<in> f \<turnstile> folding_g.\<C>"
assms_1: shows "proj_stoch_proc f n x = proj_stoch_proc f n y"
CNV_def: fixes x :: "'e"   and y :: "'e" shows "CNV x y \<equiv> x = y"
completenessForSAT: fixes F0 :: "Literal list list"   and state0 :: "State"   and state :: "State"   and decisionVars :: "nat set" assumes "satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars"   and "isFinalState state decisionVars" shows "\<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state))"
iso_assoc: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.iso \<a>\<^sub>D[f, g, h]"
intro: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')" shows "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs"
assms_1: shows "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = c"
alphaAll_refl: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> X #= X) \<and> (qGoodAbs A \<longrightarrow> A $= A)"
preserves_arr: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "D.arr (FoH\<^sub>C.map f)"
monofun_cfun_arg: fixes x :: "'e"   and y :: "'e"   and f :: "'e \<rightarrow> 'f" assumes "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> f\<cdot>y"
length_init: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "e_length (f \<triangleright> n) = Suc n"
IH_1: assumes "e \<in> set (cycle_edges (enum_cycle j))" shows "h_minus_aux j e \<le> h' e"
iwlsFSwSTR_iterSTR_ipresCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "ipresCons (iterSTR MOD) (iterAbsSTR MOD) MOD"
inf_bres: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "times_prod' x y = Inf_prod {z. less_eq_prod y (prod.bres x z)}"
suffix_eval: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "(\<sigma> |\<^sub>s i) j = \<sigma> (j + i)"
lincomb_Nil: fixes rs :: "'r list"   and ms :: "'m list" assumes "rs = [] \<or> ms = []" shows "rs \<bullet>\<cdot> ms = (0::'m)"
solve_factored_ratfps_def: fixes p :: "'a poly"   and cs :: "('a \<times> nat) list" shows "solve_factored_ratfps p cs = (let n = length cs in case poly_pfd_simple p cs of (a, zs) \<Rightarrow> (a, zip_with (\<lambda>zs (c, n). (sum_list (map2 (\<lambda>z j. inverse_irred_power_poly z (n + 1 - j)) zs [0..<Suc n]), c)) zs cs))"
distinct_hd_tl: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x = hd xs" shows "x \<notin> set (tl xs)"
get_root_node_is_l_get_root_node: shows "l_get_root_node get_root_node get_parent"
Suc_1: fixes \<sigma>' :: "'state"   and coll' :: "'coll"   and \<sigma> :: "'state"   and coll'' :: "'coll"   and n' :: "nat" assumes "(\<sigma>', coll') \<in> cbig P \<sigma>"   and "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> n1"   and "(\<sigma>', coll'') \<in> csmall_nstep P \<sigma> n'" shows "n1 = n'"
dmu_i_im1: shows "rat_of_int (d fs' (Suc (k - 1))) * fs'.gs.\<mu> k (k - 1) = rat_of_int (d fs (Suc (k - 1))) * fs.gs.\<mu> k (k - 1)"
pmf_embed_pmf: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "pmf (embed_pmf f) x = f x"
closest_pair_rec_dist: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length xs"   and "sorted_fst xs"   and "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs" shows "sparse (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set xs)"
MVNROKBA_def: fixes v :: "Vals" shows "MVNROKBA v = (v \<in> MemVal \<or> v = NotAResult \<or> v = OK \<or> v = BadArg)"
before_dist_eq_fst: fixes vs :: "'a list"   and r2 :: "'a"   and r1 :: "'a" assumes "distinct vs"   and "r2 \<in> set vs" shows "(r1 \<in> set (fst (splitAt r2 vs))) = before vs r1 r2"
append_Nil: fixes ys :: "'c list" shows "[] \<cdot> ys = ys"
monomial_decomp_exact: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "monomial_decomp qs" shows "monomial_decomp (exact k qs)"
prefix_code_2: fixes x :: "'a"   and xs :: "'a list" shows "prefix (x # xs) [] = False"
inj_map_strong: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'e"   and fa :: "'c \<Rightarrow> 'e" assumes "\<And>z za. \<lbrakk>z \<in> range x; za \<in> range xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "f \<circ> x = fa \<circ> xa" shows "x = xa"
y_axis_line_integral_is_only_vertical: shows "one_chain_line_integral F {j} (two_chain_boundary two_chain) = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
exists_unique_pirreducible_gen: fixes K :: "'a set"   and h :: "'a list \<Rightarrow> 'a" assumes "subfield K R"   and "ring_hom_ring (K [X]) R h"   and "a_kernel (K [X]) R h \<noteq> {[]}"   and "a_kernel (K [X]) R h \<noteq> carrier (K [X])" shows "\<exists>!p. p \<in> carrier (K [X]) \<and> pirreducible K p \<and> lead_coeff p = \<one> \<and> a_kernel (K [X]) R h = PIdl\<^bsub>K [X]\<^esub> p"
presheaf_of_rings_axioms: shows "presheaf_of_rings S is_open id.im_sheaf id.im_sheaf_morphisms b id.add_im_sheaf id.mult_im_sheaf id.zero_im_sheaf id.one_im_sheaf"
table_def: fixes n :: "nat"   and V :: "nat set"   and X :: "'c option list set" shows "table n V X = (\<forall>x\<in>X. wf_tuple n V x)"
idl: shows "(\<Otimes>a\<in>insert a' A. finprod G (g a) B) = finprod G (g a') B \<otimes> (\<Otimes>(a, b)\<in>A \<times> B. g a b)"
distinct_6: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Assign x31 x32 \<noteq> Assume x2"
bind_commute_pmf: fixes A :: "'f pmf"   and B :: "'g pmf"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> 'e pmf" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
Id_onI: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a"   and ko :: "'a key_order" assumes "a \<noteq> b"   and "le_of_key_order ko b a" shows "lt_of_key_order ko b a"
mult_left_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
bij_comp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "bij f"   and "bij g" shows "bij (g \<circ> f)"
srtc_conversion: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow>\<^sup>* b) = ((a, b) \<in> {(x, y). x \<rightarrow> y}\<^sup>\<leftrightarrow>\<^sup>*)"
closedin_singleton: fixes U :: "'c set"   and a :: "'c" shows "closedin (top_of_set U) {a} = (a \<in> U)"
qdelta_constraint_to_atom_monom: fixes h :: "QDelta ns_constraint"   and v :: "nat \<Rightarrow> QDelta"   and var :: "nat" assumes "is_monom (poly h)" shows "v \<Turnstile>\<^sub>a qdelta_constraint_to_atom h var = v \<Turnstile>\<^sub>n\<^sub>s h"
b_in: shows "b \<in> segment G.face_cycle_succ a a"
C7_a: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (-l y \<le> -l x)"
cf_pos_poly_dvd: fixes f :: "int poly" shows "cf_pos_poly f dvd f"
changes_sign_var: fixes a :: "real"   and x :: "real"   and b :: "real"   and c :: "real"   and y :: "real" assumes "sign_num (a * x\<^sup>2 + b * x + c) \<noteq> sign_num (a * y\<^sup>2 + b * y + c) \<and> x < y" shows "\<exists>q. a * q\<^sup>2 + b * q + c = 0 \<and> x \<le> q \<and> q \<le> y"
Suc_in_fimage_Suc: fixes x :: "nat"   and X :: "nat fset" shows "(Suc x |\<in>| Suc |`| X) = (x |\<in>| X)"
abs_summable_on_uminus_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "((\<lambda>x. - f x) abs_summable_on A) = (f abs_summable_on A)"
assms_5: fixes x :: "'a"   and s :: "'b" assumes "I x"   and "S s" shows "S (f x s)"
UN_singleton: fixes A :: "'a set" shows "\<mu> (\<P> \<eta> A) = A"
coeff_poly_of_pm: fixes x :: "'d"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and k :: "nat" shows "poly.coeff (poly_of_pm x p) k = lookup p (monomial k x)"
lemma7: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and ve_a :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and f :: "val"   and \<beta> :: "nat \<Rightarrow> contour option" assumes "|ve| \<lessapprox> ve_a" shows "|\<A> f \<beta> ve| \<lessapprox> \<aA> f |\<beta>| ve_a"
assms_1: shows "(\<forall>\<^sub>cx. \<gamma> x) t n"