hcju/isabelleps huggingface dataset:
conjug_trans: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<sim> v"   and "v \<sim> w" shows "u \<sim> w"
f1: shows "\<forall>m n na. (\<not> MPoly_Type.degree m n < na \<or> 0 < na \<and> (\<forall>p. p \<notin> monomials m \<or> lookup p n < na)) \<and> (MPoly_Type.degree m n < na \<or> \<not> 0 < na \<or> pp na n m \<in> monomials m \<and> \<not> lookup (pp na n m) n < na)"
compactI: fixes s :: "'a set" assumes "\<And>C. \<lbrakk>Ball C open; s \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'" shows "compact s"
strip_guards_simps_2: fixes F :: "'f set"   and f :: "'d \<Rightarrow> 'd" shows "strip_guards F (Basic f) = Basic f"
eln_tendsto: fixes u :: "'a \<Rightarrow> ennreal"   and l :: "ennreal"   and F :: "'a filter" assumes "(u \<longlongrightarrow> l) F" shows "((\<lambda>n. eln (u n)) \<longlongrightarrow> eln l) F"
lm021: fixes X :: "'c set"   and mx :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite X"   and "mx \<in> X"   and "\<forall>x\<in>X - {mx}. f x < f mx" shows "argmax f X \<subseteq> {mx}"
Goal: fixes Pa :: "pi"   and Qa :: "pi"   and xa :: "name"   and \<sigma>' :: "(name \<times> name) list"   and aa :: "name" assumes "Pa \<approx>\<^sup>s Qa"   and "xa \<sharp> \<sigma>'" shows "aa<xa>.Pa[<\<sigma>'>] \<approx> aa<xa>.Qa[<\<sigma>'>]"
fvs_simp_3: fixes A :: "'b trm"   and B :: "'b trm" shows "fvs (App A B) = fvs A \<union> fvs B"
struct_spec_def: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "struct_spec sel ap ab compl = (sel_spec sel \<and> ap_spec ap \<and> ab_spec ab \<and> compl_struct compl)"
dg_def: shows "op_dg \<BB> = [op_dg \<BB>\<lparr>Obj\<rparr>, op_dg \<BB>\<lparr>Arr\<rparr>, op_dg \<BB>\<lparr>Dom\<rparr>, op_dg \<BB>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
peek_st_split: fixes P :: "st \<Rightarrow> 'b assn" shows "peek_st P = (\<lambda>Y s. P (snd s) Y s)"
ext: fixes g :: "'a2"   and f :: "'a2" assumes "g \<cdot>\<^sub>2 f \<noteq> C2.null" shows "C2.seq g f"
R1: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and x :: "'c"   and xs :: "'c list"   and ys :: "'c list"   and X :: "('c, 'd) ipurge_rec" assumes "\<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X"   and "D x \<in> sources I D u (x # xs)" shows "\<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr> \<in> ipurge_tr_rev_t_set X"
PREC: shows "h \<Turnstile> P x * F1 \<and>\<^sub>A P y * F2 \<longrightarrow> x = y"
fds_mangoldt': shows "fds mangoldt = fds_zeta * fds_deriv (fds moebius_mu)"
\<mu>_bound_row_def: fixes n :: "nat"   and fs :: "int vec list"   and bnd :: "rat"   and i :: "nat" shows "LLL.\<mu>_bound_row n fs bnd i = (\<forall>j\<le>i. (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)\<^sup>2 \<le> bnd)"
preserves_ide: fixes a :: "'a1" assumes "A1.ide a" shows "A2_B.ide (F a)"
non_span_det_zero: fixes G :: "'a vec list"   and n :: "nat" assumes "length G = n"   and "\<not> carrier_vec n \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set G)"   and "set G \<subseteq> carrier_vec n" shows "det (mat_of_rows n G) = (0::'a)"
distinct_35: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x31@[x32] \<noteq> Stop"
distinct_2: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
Wbis_coind2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "theta \<subseteq> Wretr (theta \<union> Wbis)"   and "theta\<inverse> \<subseteq> Wretr (theta\<inverse> \<union> Wbis)" shows "theta \<subseteq> Wbis"
boundedI: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> a" shows "sup b c \<le> a"
imirror_mem_conv: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<le> iMin I + Max I" shows "(mirror_elem n I \<in> imirror I) = (n \<in> I)"
matchers_def: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) set" shows "matchers P = {\<sigma>. \<forall>e\<in>P. fst e \<cdot> \<sigma> = snd e}"
invalid_assn_distrib: shows "invalid_assn monom_assn \<times>\<^sub>a invalid_assn int_assn = invalid_assn monomial_assn"
csol_eq: fixes s :: "real" assumes "s \<le> t + e"   and "t0 \<le> s" shows "csol t0 x0 (t + e) s = flow t0 x0 s"
mono: fixes "consider" :: "'d \<Rightarrow> bool"   and x :: "('b, 'd, 'e) gpv" shows "lfp.mono_body (\<lambda>interaction_bound. \<Squnion>generat\<in>set_spmf (the_gpv x). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound (c input)) else \<Squnion>input. interaction_bound (c input))"
rtf_is_nbr: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right" shows "neighbor_invar c s t f l n"
Un_insert_right: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "A \<union> insert a B = insert a (A \<union> B)"
possible_assignments_for_set_is: fixes v :: "'c"   and \<Psi> :: "('c, 'd) sas_plus_problem" assumes "v \<in> dom (range_of \<Psi>)" shows "set (possible_assignments_for \<Psi> v) = {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v}"
ChamberSubcomplexD_complex: fixes Y :: "'a set set" assumes "ChamberSubcomplex Y" shows "ChamberComplex Y"
finprod_closed_subgroup: fixes H :: "'a set"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "subgroup H G"   and "f \<in> A \<rightarrow> H" shows "finprod G f A \<in> H"
cut_greater_Un: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<union> B) \<down>> t = A \<down>> t \<union> B \<down>> t"
graph_def: fixes G :: "('c, 'd) pre_digraph" shows "graph G \<equiv> digraph G \<and> pseudo_graph G"
assms_2: shows "\<forall>w. w \<in> Range \<mu>"
equalOnI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>A. f x = g x" shows "equalOn A f g"
fps_inv_fps_exp_compose_1: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "fps_inv (fps_exp a - 1) oo fps_exp a - 1 = fps_X"
inj_on_restrict_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "inj_on (restrict f A) A = inj_on f A"
hyp2_incident_in_middle: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and l :: "proj2_line" assumes "p \<noteq> q"   and "p \<in> S"   and "q \<in> S"   and "a \<in> hyp2 \<union> S"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident a l" shows "B\<^sub>\<real> cart2_pt p cart2_pt a cart2_pt q"
nhds_neq_bot: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.topological_space open" shows "topological_space.nhds open a \<noteq> bot"
adjunct_ord_selfD: fixes a :: "'a" assumes "a \<notin> carrier D" shows "D = Iod (adjunct_ord D a) (carrier D)"
E: shows "(A, B, C, D) = split_block E 1 1"
T_nonTickFree_imp_decomp: fixes t :: "'b event list"   and P :: "'b process" assumes "t \<in> T P"   and "\<not> tickFree t" shows "\<exists>s. t = s @ [tick]"
eval_monom_vars_list: fixes xs :: "'c monom"   and \<alpha> :: "'c \<Rightarrow> 'd"   and \<beta> :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> set (monom_vars_list xs) \<Longrightarrow> \<alpha> x = \<beta> x" shows "eval_monom \<alpha> xs = eval_monom \<beta> xs"
p: shows "\<forall>i\<in>I. p i \<in> S i"
show_vec_real_def: fixes v :: "(real, 'a) vec" shows "show_vec_real v = (\<chi>i. show_real (v $ i))"
1: shows "\<guillemotleft>h : F y \<rightarrow>\<^sub>C x\<guillemotright>"
collapse_2: fixes atom :: "'a atom" assumes "\<not> is_predAtm atom" shows "Eq (lhs atom) (rhs atom) = atom"
simps_2: fixes a :: "nat"   and b :: "nat"   and p :: "real mpoly" shows "swap_atom a b (Less p) = Less (swap_poly a b p)"
simps_1: fixes I :: "interp"   and p :: "char"   and \<theta> :: "trm" shows "fml_sem I (Pred p \<theta>) = {\<omega>. Preds I p (term_sem I \<theta> \<omega>)}"
simps_1: fixes x :: "'a" shows "0 \<otimes> x = x"
fbox_export2: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows " |x] p \<le>  |x \<cdot> ad q] (d p \<cdot> ad q)"
adjoined_in_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G" shows "a \<in> adjoin G H a"
intros_1: fixes c :: "classes1" assumes "c \<noteq> Object" shows "subclass1 c Object"
that: fixes y :: "real" assumes "y \<in> closed_segment a b"   and "f b - f a = (b - a) * f' y" shows "thesis"
assms_1: fixes p :: "'a"   and q :: "'a" assumes "prime p"   and "prime q" shows "P p q"
le_imp_less_or_eq: fixes x :: "'c list"   and y :: "'c list" assumes "suffix x y" shows "strict_suffix x y \<or> x = y"
mtx_vec_mult_add_rdistr: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx"   and x :: "(real, 'a) vec" shows "(A + B) *\<^sub>V x = A *\<^sub>V x + B *\<^sub>V x"
D_NEW_H: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and C :: "Class" assumes "CONTEXT x = high" shows "(G, New x C, HighSec) \<in> Deriv"
plus_of_int: fixes x :: "int"   and n :: "int" shows "gauss_sum_int (x + of_int n * int n) = gauss_sum_int x"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "near_ordered_set A less_eq" shows "antisymmetric A less_eq"
disc_nodes_document_ptr_h: shows "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h3"
complex_roots_of_int_poly3_def: fixes p :: "int poly" shows "complex_roots_of_int_poly3 p \<equiv> let n = degree p; rrts = real_roots_of_int_poly p; nr = length rrts; crts = map (\<lambda>r. Complex r 0) rrts in if n = nr then crts else let nr_crts = n - nr in if nr_crts = 2 then let pp = real_of_int_poly p div (\<Prod>x\<leftarrow>rrts. [:- x, 1:]); cpp = map_poly (\<lambda>r. Complex r 0) pp in crts @ croots2 cpp else let nr_pos_crts = nr_crts div 2; rxs = real_parts_3 p; ixs = pos_imaginary_parts_3 p; rts = concat (map (\<lambda>rx. map (Pair rx) ixs) rxs); crts' = map pair_to_complex (filter_list_length (map_prod tighten_bounds_3 tighten_bounds_3) (\<lambda>(r, i). 0 \<in>\<^sub>c ipoly_complex_interval p (Complex_Interval (get_itvl_3 r) (get_itvl_3 i))) nr_pos_crts rts) in crts @ concat (map (\<lambda>x. [x, cnj x]) crts')"
mr_cosmOpTr: fixes H :: "'a set"   and m :: "'a" assumes "submodule R M H"   and "m \<in> carrier M" shows "mr_cosmOp M H (m \<uplus>\<^bsub>M\<^esub> H) = (-\<^sub>a m) \<uplus>\<^bsub>M\<^esub> H"
add_increasing2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
arrE: fixes f :: "'d" assumes "D'.arr f" shows "D.arr f \<and> Obj (src\<^sub>D f) \<and> Obj (trg\<^sub>D f)"
fundfacets_2: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "C0 \<inter> s `\<rightarrow> C0 \<lhd> s `\<rightarrow> C0"
"3_1": fixes i :: "nat" assumes "\<And>i. i < length zs \<Longrightarrow> zs ! i < length xs + Suc ia__ \<and> (Suc ia__ \<le> zs ! i \<longrightarrow> xs ! (zs ! i - Suc ia__) = Some i)"   and "\<And>j1 j2. \<lbrakk>j1 < j2; j2 < length zs\<rbrakk> \<Longrightarrow> zs ! j1 < zs ! j2"   and "\<And>z. z \<in> set zs \<Longrightarrow> z < Suc ia__"   and "i < length (snd (mk_eqcl xs zs (Suc ia__) T))"   and "length xs + Suc ia__ \<le> length T + 1" shows "snd (mk_eqcl xs zs (Suc ia__) T) ! i < length (fst (mk_eqcl xs zs (Suc ia__) T)) + Suc ia__ \<and> (Suc ia__ \<le> snd (mk_eqcl xs zs (Suc ia__) T) ! i \<longrightarrow> fst (mk_eqcl xs zs (Suc ia__) T) ! (snd (mk_eqcl xs zs (Suc ia__) T) ! i - Suc ia__) = i)"
indep_system_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier" shows "indep_system \<E> (indep_system.indep_in indep \<E>)"
inv_inv: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.inv (VV.inv f) = f"
leftmost_cons_terminal: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol"   and i :: "nat"   and xs :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.is_terminal \<TT> x" shows "CFG.leftmost \<NN> \<TT> i (x # xs) = (0 < i \<and> CFG.leftmost \<NN> \<TT> (i - 1) xs)"
match_pres_def: fixes pres :: "(nat \<times> nat) list"   and s :: "nat \<Rightarrow> nat option" shows "match_pres pres s \<equiv> \<forall>pre\<in>set pres. match_pre pre s"
sat_ineq_subterm_inj_subst: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a set"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "subterm_inj_on \<theta> (subst_domain \<theta>)"   and "ground (subst_range \<theta>)"   and "subst_domain \<theta> \<inter> X = {}"   and "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<theta>) \<inter> (subterms s \<union> subterms t) = {}"   and "fv s \<union> fv t - subst_domain \<theta> \<subseteq> X"   and "(\<forall>x\<in>fv s \<union> fv t - X. \<exists>c. \<theta> x = Fun c []) \<or> (\<forall>f U. Fun f U \<in> subterms s \<union> subterms t \<longrightarrow> U = [] \<or> (\<exists>u\<in>set U. u \<notin> Var ` X))"   and "\<forall>\<delta>. subst_domain \<delta> = X \<and> ground (subst_range \<delta>) \<longrightarrow> s \<cdot> \<delta> \<cdot> \<I> \<noteq> t \<cdot> \<delta> \<cdot> \<I>"   and "fv s \<union> fv t - X \<subseteq> subst_domain \<I>"   and "subst_domain \<I> \<inter> X = {}"   and "ground (subst_range \<I>)"   and "subst_domain \<I> = subst_domain \<theta>"   and "subst_domain \<delta> = X"   and "ground (subst_range \<delta>)" shows "s \<cdot> \<delta> \<cdot> \<theta> \<noteq> t \<cdot> \<delta> \<cdot> \<theta>"
rel_distinct_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y21 :: "'a"   and y22 :: "'a list" shows "\<not> list_all2 R (y21 # y22) []"
simps_6: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and q :: "char list"   and es' :: "expr list"   and rets' :: "char list list"   and ps :: "char list list"   and p :: "char list" shows "containsCall procs (Call q es' rets') ps p = (p = q \<and> ps = [] \<or> (\<exists>ins outs c ps'. ps = q # ps' \<and> (q, ins, outs, c) \<in> set procs \<and> containsCall procs c ps' p))"
eqvtI: fixes f :: "'e" assumes "\<And>p. p \<bullet> f \<equiv> f" shows "eqvt f"
word_roti_eq_word_rotr_word_rotl: fixes i :: "int"   and w :: "'b word" shows "word_roti i w = (if 0 \<le> i then word_rotr (nat i) w else word_rotl (nat (- i)) w)"
prems_4: shows "\<I> \<turnstile>g gpv \<surd>"
SD_strict_singleton_left: fixes x :: "'alt"   and q :: "'alt pmf" assumes "x \<in> alts"   and "q \<in> lotteries_on alts" shows "return_pmf x \<prec>[SD (Pareto R)] q = ((\<forall>y\<in>set_pmf q. Pareto R x y) \<and> (\<exists>y\<in>set_pmf q. x \<prec>[Pareto R] y))"
assms_2: shows "homotopic_with q X2 Y2 g g'"
lookup_reorder: fixes ko :: "'o"   and xs :: "'x"   and k :: "'a" shows "lookup (reorder ko xs) k = lookup xs k"
cut_bound_max: fixes n :: "nat"   and m :: "nat"   and F :: "'b formula" shows "max n (cut_bound' (max n m) F) = cut_bound' (max n m) F"
X1: fixes i :: "nat" shows "alstep (cl i) (arun cl cs i) (arun cl cs (Suc i)) \<and> A.reachable (arun cl cs i) \<and> R (arun cl cs (Suc i)) (cs (Suc i))"
elims_2: fixes x :: "sat_plan_variable" assumes "valid_state_var x" obtains t :: "nat" and k :: "nat" where "x = Operator t__ k__" and "t__ < h \<and> k__ < n_ops"   | t :: "nat" and k :: "nat" where "x = State t k" and "t < h"
in_components_connected: fixes c :: "'c set"   and s :: "'c set" assumes "c \<in> components s" shows "connected c"
rel_converter_pos_distr: fixes A :: "'a \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'l \<Rightarrow> bool"   and A' :: "'i \<Rightarrow> 'e \<Rightarrow> bool"   and B' :: "'j \<Rightarrow> 'f \<Rightarrow> bool"   and C' :: "'k \<Rightarrow> 'g \<Rightarrow> bool"   and R' :: "'l \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_converter A B C R OO rel_converter A' B' C' R' \<le> rel_converter (A OO A') (B OO B') (C OO C') (R OO R')"
ide_def: fixes a :: "'a" shows "H.ide a \<equiv> a \<star> a \<noteq> H.null \<and> (\<forall>f. (f \<star> a \<noteq> H.null \<longrightarrow> f \<star> a = f) \<and> (a \<star> f \<noteq> H.null \<longrightarrow> a \<star> f = f))"
norm_PX2: fixes P :: "pol"   and i :: "nat"   and Q :: "pol" assumes "isnorm (PX P i Q)" shows "isnorm Q"
strict_order_var: fixes x :: "'a" shows "(x \<le> - top \<and> x \<sqinter> x \<le> x) = (x \<sqinter> id x = bot \<and> x \<sqinter> x \<le> x)"
unit_dvdE: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains c :: "'a" where "a \<noteq> zero" and "b = times a c"
leg1_simps_3: shows "src (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) = src (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0)"
Exec'_2: shows "prefixList exec exec' \<and> prefixList trace trace' \<or> exec = exec' \<and> trace = trace'"
trd_irred: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "\<not> punit.is_red (set fs) (punit.trd fs p)"
divide_le_0_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide one a) zero = less_eq a zero"
dghm_comp_components_3: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>HomDom\<rparr> = \<FF>\<lparr>HomDom\<rparr>"
cases: fixes x :: "('d, 'c, 'c) matrix" obtains A :: "'c atMost \<Rightarrow> 'c atMost \<Rightarrow> 'd" where "x = Matrix A"
runit_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.arr V (bicategory.runit V H \<a> \<i> src f)"
Prim1I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat option" assumes "prim_recfn 1 r"   and "\<And>x. f x = eval r [x]" shows "f \<in> Prim1"
Abs_preal_induct: fixes P :: "preal \<Rightarrow> bool"   and x :: "preal" assumes "\<And>x. Dedekind_Real.cut x \<Longrightarrow> P (Abs_preal x)" shows "P x"
vI_fin: shows "- \<infinity> < I.subcocycle_avg_ereal vI"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_Nil: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s [] = {}"
converse_reachable_induct: fixes u :: "'a"   and v :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "u \<rightarrow>\<^sup>* v"   and "v \<in> verts G \<Longrightarrow> P v"   and "\<And>x y. \<lbrakk>x \<rightarrow> y; y \<rightarrow>\<^sup>* v; P y\<rbrakk> \<Longrightarrow> P x" shows "P u"
prime_harmonic_aux1: fixes A :: "'a set" assumes "finite A" shows "(\<Prod>x\<in>A. (1::'a) + (1::'a) / x) = (\<Sum>x\<in>Pow A. (1::'a) / \<Prod>x)"
WT_gpv_\<I>_mono: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('a, 'b) \<I>" assumes "\<I> \<turnstile>g gpv \<surd>"   and "\<I> \<le> \<I>'" shows "\<I>' \<turnstile>g gpv \<surd>"
Gets_ticket_parts: fixes A :: "agent"   and K :: "nat"   and SesKey :: "msg"   and Peer :: "msg"   and Ta :: "msg"   and Ticket :: "msg"   and evs :: "event list" assumes "Gets A (Crypt K \<lbrace>SesKey, Peer, Ta, Ticket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "Ticket \<in> parts (knows Spy evs)"
IH: fixes i1 :: "'index"   and s1 :: "'state1" assumes "match i1 s1 s2''" shows "\<exists>i2 s1'. L1.eval s1 s1' \<and> match i2 s1' s2'"
simps_7: fixes p :: "('a, 'b) form" shows "FOL_Fitting.size_form (Neg p) = FOL_Fitting.size_form p + 1"
redex_Seq_False: fixes c :: "('a, 'b, 'c) com"   and c'' :: "('a, 'b, 'c) com"   and c' :: "('a, 'b, 'c) com" shows "(redex c = Seq c'' c') = False"
drop_takefill: fixes k :: "nat"   and fill :: "'b"   and m :: "nat"   and w :: "'b list" shows "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
intro: fixes G :: "('a, 'b) Graph_scheme" assumes "\<And>v w. \<lbrakk>v \<in> V\<^bsub>G\<^esub>; w \<in> V\<^bsub>G\<^esub>\<rbrakk> \<Longrightarrow> Graph.connected G v w"   and "\<And>xs. \<not> Graph.cycle G xs" shows "Tree_axioms G"
Abs_bit1_def: fixes x :: "int" shows "Abs x = Abs_bit1' x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "real \<Rightarrow> real list \<Rightarrow> 'a"   and f2 :: "real \<Rightarrow> real list \<Rightarrow> 'a"   and atom :: "atom" shows "P (case atom of Less x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12. atom = Less x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. atom = Eq x21 x22 \<longrightarrow> P (f2 x21 x22)))"
finlstsp_finlsts_eq: fixes A :: "'b set" shows "finlstsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> A\<^sup>\<star>)"