hcju/isabelleps huggingface dataset:
of_nat_power_le_of_nat_cancel_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (semiring_1.of_nat one plus zero x) (power.power one times (semiring_1.of_nat one plus zero b) w) = (x \<le> b ^ w)"
eval_fps_inverse: fixes r :: "ereal"   and f :: "complex fps"   and z :: "complex" assumes "\<And>z. ereal (cmod z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"   and "ereal (cmod z) < fps_conv_radius f"   and "ereal (cmod z) < r" shows "eval_fps (inverse f) z = inverse (eval_fps f z)"
decomps\<^sub>e\<^sub>s\<^sub>t_exist_aux: fixes D :: "('fun, 'var) extstrand_step list"   and M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>"   and "M \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile> t"   and "\<not> M \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile>\<^sub>c t" obtains D' :: "('fun, 'var) extstrand_step list" where "D @ D' \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>" and "M \<union> ik\<^sub>e\<^sub>s\<^sub>t (D @ D') \<turnstile>\<^sub>c t" and "M \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<subset> M \<union> ik\<^sub>e\<^sub>s\<^sub>t (D @ D')"
trss_xchange_context: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and csp :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and cn :: "'a list multiset" assumes "((s, c), w, s', csp + c) \<in> trcl (trss fg)"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "((s, cn), w, s', csp + cn) \<in> trcl (trss fg)"
assms_4: shows "set (labels \<tau>) \<subseteq> r \<down>s {\<beta>}"
nat_of_num_inverse: fixes x :: "num" shows "num_of_nat (nat_of_num x) = x"
ext_cf_lo_zero: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat"   and x :: "nat" assumes "pol_coeff S c"   and "0 < n"   and "x \<le> n - Suc 0" shows "snd (ext_cf S n c) x = \<zero>\<^bsub>S\<^esub>"
nodes_root_height_lower_bound: fixes k :: "nat"   and t :: "'b btree" assumes "root_order k t"   and "bal t" shows "2 * bound k (height t - 1) + of_bool (t \<noteq> Leaf) * k \<le> nodes t * k"
alphaAll_qSkelAll: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(\<forall>X'. X #= X' \<longrightarrow> qSkel X = qSkel X') \<and> (\<forall>A'. A $= A' \<longrightarrow> qSkelAbs A = qSkelAbs A')"
preserves_arr: fixes f :: "'a" assumes "B.arr f" shows "B.arr (src\<^sub>B f)"
permutes_implies_multiset_of_partition_cards_eq: fixes p\<^sub>A :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and p\<^sub>B :: "'d \<Rightarrow> 'd"   and B :: "'d set"   and f' :: "'c \<Rightarrow> 'd" assumes "p\<^sub>A permutes A"   and "p\<^sub>B permutes B" shows "image_mset card (mset_set ((\<lambda>b. {x \<in> A. p\<^sub>B (f' (p\<^sub>A x)) = b}) ` B - {{}})) = image_mset card (mset_set ((\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}))"
path_image_subpath_subset: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" assumes "u \<in> {0..1}"   and "v \<in> {0..1}" shows "path_image (subpath u v g) \<subseteq> path_image g"
Brouwer_degree2_homotopic: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f g" shows "Brouwer_degree2 p f = Brouwer_degree2 p g"
finite_fst_ts_tuple_rel: fixes xs :: "(nat \<times> 'a option list set) list"   and P :: "nat \<times> 'a option list \<Rightarrow> bool" shows "finite (fst ` {tas \<in> ts_tuple_rel (set xs). P tas})"
map_pmf_rep_eq: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" shows "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
scf_d_pol: fixes p :: "'a"   and d :: "nat" assumes "p \<in> carrier R"   and "deg R S X p \<le> an d" shows "pol_coeff S (scf_d R S X p d) \<and> fst (scf_d R S X p d) = d \<and> p = polyn_expr R X d (scf_d R S X p d)"
l3_defs_8: shows "l3_trans \<equiv> \<Union>M N X Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B N X \<union> l3_step3 Ra A B N X \<union> l3_step4 Rb A B N X \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
assms_6: fixes x :: "'a"   and Z :: "'a set" assumes "B (f x)"   and "A (sum f Z)" shows "A (f x + sum f Z)"
ide_char': fixes a :: "'b" shows "C'.ide a = (C'.arr a \<and> (C'.dom a = a \<or> C'.cod a = a))"
eq_for_lvar: fixes x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "x \<in> lvars t" shows "eq_for_lvar t x \<in> set t \<and> lhs (eq_for_lvar t x) = x"
diff_eq_diff_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.group_add minus plus zero uminus"   and "minus a b = minus c d" shows "(a = b) = (c = d)"
B: shows "(a, c) \<in> (unlabel ars)\<^sup>*"
matrix_eq: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "(A = B) = (\<forall>x. A *v x = B *v x)"
hitting_paths_def: shows "G.hitting_paths \<equiv> \<lambda>x. x \<noteq> v0 \<and> ((\<exists>xs\<in>paths. x \<in> set xs) \<or> x = v1)"
enum_less: fixes P :: "'a \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "infinitely_many1 P"   and "i < j" shows "infinitely_many1.enum P i < infinitely_many1.enum P j"
epiE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "VV.epi f"   and "VV.seq g f"   and "VV.seq g' f"   and "VV.comp g f = VV.comp g' f" shows "g = g'"
extender_deeper: fixes F :: "'b formula"   and k :: "nat"   and S :: "'b formula set"   and l :: "nat" assumes "F \<in> extender k S"   and "k \<le> l" shows "F \<in> extender l S"
minimally_liberal_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set" shows "minimally_liberal scf A Is \<equiv> \<exists>i\<in>Is. \<exists>j\<in>Is. i \<noteq> j \<and> (\<exists>x\<in>A. \<exists>y\<in>A. x \<noteq> y \<and> decisive scf A Is {i} x y \<and> decisive scf A Is {i} y x) \<and> (\<exists>x\<in>A. \<exists>y\<in>A. x \<noteq> y \<and> decisive scf A Is {j} x y \<and> decisive scf A Is {j} y x)"
simps_6: fixes a :: "Cooper.num"   and b :: "Cooper.num" shows "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel"
not_in_kD_nsqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<notin> kD rt" shows "nsqn rt dip = 0"
PData_def: fixes L :: "'b option list"   and D :: "'b dataspace" shows "PData L D \<equiv> length L = PartNum D \<and> (\<forall>i\<in>{n. n < PartNum D}. L ! i \<noteq> None \<longrightarrow> the (L ! i) \<in> D !D! i)"
stutter_equiv_refl: fixes \<sigma> :: "nat \<Rightarrow> 'a" shows "\<sigma> \<approx> \<sigma>"
Aenc_Exp: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Aenc X Y \<noteq> Exp X' Y'"
C_D_i: shows "C \<in> Sup_upto_llist Ns (enat i)"
Program: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')" shows "direct_computation.concurrent_step (ts, m, \<S>) (ts[i := (p', is @ is', \<theta>, sb, \<D>, \<O>, \<R>)], m, \<S>)"
OAlist_tc_map2_val_rneutr_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'e"   and xs :: "('d, 'e) oalist_tc"   and ys :: "('d, 'f) oalist_tc" shows "OAlist_tc_map2_val_rneutr f xs ys = OAlist_tc (map2_val_pair_tc f id (map_val_pair (\<lambda>k. f k (0::'e))) (list_of_oalist_tc xs) (list_of_oalist_tc ys))"
in_enum: fixes enum :: "'c list"   and enum_all :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'c" assumes "class.enum enum enum_all enum_ex" shows "x \<in> set enum"
span_finite: fixes S :: "('a, 'b) vec set" assumes "finite S" shows "vec.span S = range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
pp_dist_inf: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (uminus (inf x y)) = inf (uminus (uminus x)) (uminus (uminus y))"
eq: fixes bl :: "'a set" shows "(bl \<in># multiple_blocks n) = (bl \<in># \<B>)"
FrameStatEqClosed: fixes F :: "'b frame"   and G :: "'b frame"   and p :: "(name \<times> name) list" assumes "F \<simeq>\<^sub>F G" shows "p \<bullet> F \<simeq>\<^sub>F p \<bullet> G"
simps: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" shows "aeseq_eq f g = finite {n. f n \<noteq> g n}"
mergesort_is_sort: shows "mergesort = Sorting_Algorithms.sort"
x_leq_c_complement_general: fixes c :: "'a"   and h :: "'a"   and x :: "'a" assumes "vector c"   and "c * c\<^sup>T \<le> forest_components h"   and "x \<le> c\<^sup>T"   and "x \<le> - forest_components h" shows "x \<le> - c"
noFault_def': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and f :: "'f" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Fault f} = (\<not> \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f)"
sup_def: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "\<Squnion>\<^bsub>L\<^esub>A = (SOME x. is_lub L x A)"
select_unit_norm: fixes u :: "'b" shows "\<exists>v. norm v = 1 \<and> v \<bullet> u = - norm u"
induct: fixes x1 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {x1} x2 { x3}"   and "P ($ 1) SKIP ($ 0)"   and "\<And>x a Q. P ((\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1) (x ::= a) Q"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s} c\<^sub>1 { Q}; P (\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s) c\<^sub>1 Q; \<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s} c\<^sub>2 { Q}; P (\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P ((\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1) (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P C Q F. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {P} C { Q}; P P C Q\<rbrakk> \<Longrightarrow> P (P \<and>* F) C (Q \<and>* F)"   and "\<And>P C\<^sub>1 Q C\<^sub>2 R. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {P} C\<^sub>1 { Q}; P P C\<^sub>1 Q; \<turnstile>\<^sub>3\<^sub>a {Q} C\<^sub>2 { R}; P Q C\<^sub>2 R\<rbrakk> \<Longrightarrow> P P (C\<^sub>1;; C\<^sub>2) R"   and "\<And>P b C. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s} C { (\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1}; P (\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s) C ((\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1)\<rbrakk> \<Longrightarrow> P ((\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1) (WHILE b DO C) (\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s)"   and "\<And>P c Q P' Q'. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {P} c { Q}; P P c Q; \<And>s n. P' (s, n) \<Longrightarrow> P (s, n); \<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n)\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
Min_in: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
in_quotient_imp_in_rel: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set"   and x :: "'b"   and y :: "'b" assumes "equiv A r"   and "X \<in> A // r"   and "{x, y} \<subseteq> X" shows "(x, y) \<in> r"
OAlist_tc_update_by_fun_def: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('a, 'b) oalist_tc" shows "OAlist_tc_update_by_fun k f xs = OAlist_tc (update_by_fun_pair_tc k f (list_of_oalist_tc xs))"
right_inverse_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and g :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "f \<circ> g = id" shows "Vector_Spaces.linear (*s) (*s) g"
mult_comp_cls_val_process_measurable: fixes Mkt2 :: "('a, 'c) discrete_market"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<forall>n. cls_val_process Mkt2 pf n \<in> borel_measurable (F n)"   and "portfolio pf"   and "\<forall>n. qty (Suc n) \<in> borel_measurable (F n)" shows "\<forall>n. cls_val_process Mkt2 (qty_mult_comp pf qty) n \<in> borel_measurable (F n)"
iMOD_geD: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "r \<le> x"
hom_add: fixes x :: "'a poly"   and y :: "'a poly" shows "map_poly hom (x + y) = map_poly hom x + map_poly hom y"
success_effectE: fixes c :: "'a Heap"   and h :: "heap" assumes "success c h" obtains r :: "'a" and h' :: "heap" where "effect c h h' r"
inorder_insert: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (RBT_Set.insert x t) = ins_list x (Tree2.inorder t)"
prv_imp_cnj_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp (cnj \<phi> \<chi>) \<psi>) (imp \<phi> (imp \<chi> \<psi>)))"
l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf \<equiv> type_wf = ShadowRootClass.type_wf"
AtCatchExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a2 c2 a" shows "atomicsR \<Gamma> \<Theta> (AnnComp a1 a2) (Catch c1 c2) a"
wpt_assertion: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.wpt one times inf top x = x"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case EV_FINISH of EV_PREP \<Rightarrow> f1 | EV_WAIT \<Rightarrow> f2 | EV_FINISH \<Rightarrow> f3) = f3"
A: shows "\<forall>n<Suc (length ws). P ((w # ws) ! n) (drop n ws @ xs)"
fst_image_times': shows "A = fst ` (A \<times> B)"
elims_3: fixes x :: "Core.term" assumes "\<not> is_app x" obtains v :: "String.literal" and va :: "typ" where "x = Ct v__ va__"   | v :: "variable" and va :: "typ" where "x = Fv vb__ vaa__"   | v :: "nat" where "x = Bv vc__"   | v :: "typ" and va :: "Core.term" where "x = Abs v va"
finite_Union: fixes X :: "'a set set"   and M :: "'a measure" assumes "finite X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
subst_var_eq_code_def: shows "subst_var_eq_code = SubstVar.subst_var_eq subst_var"
h2a: shows "\<forall>f. sum f {..0} = f 0"
exists_best_degenerate: shows "\<exists>x\<in>local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. x \<succeq>[\<R>] y"
lift_opt_simps_1: fixes e :: "'a" shows "lift_opt None e = Inl e"
zero_less_diff: fixes n :: "nat"   and m :: "nat" shows "m \<in># (-) n = (m < n)"
strong_exhaust: fixes y :: "tm" obtains "y = Zero"   | x2 :: "name" where "y = Var x2"   | x31 :: "tm" and x32 :: "tm" where "y = Eats x31 x32"
is_partition_of_def: fixes P :: "'b set set"   and A :: "'b set" shows "P partitions A = is_partition_of' P A"
Seq_is_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "(action.relation_of P ;; action.relation_of Q) \<in> {p. is_CSP_process p}"
incseq_le: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b"   and n :: "nat" assumes "incseq X"   and "X \<longlonglongrightarrow> L" shows "X n \<le> L"
is_fieldI: fixes C :: "qtname"   and f :: "fdecl" shows "is_field (C, fdecl f)"
1: shows "(A * C) $$ (i div dim_row B, j div dim_col D) = row A (i div dim_row B) \<bullet> col C (j div dim_col D)"
weight_le_stable: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'c) Term.term" assumes "weight_le s t" shows "weight_le (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
lprefix_llist_ofI: fixes ys :: "'c list"   and xs :: "'c list" assumes "\<exists>zs. ys = xs @ zs" shows "llist_of xs \<sqsubseteq> llist_of ys"
kauff_mat_swingneg: shows "kauff_mat r_under_braid = kauff_mat l_under_braid"
symp_symclp_eq: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "symp r" shows "symclp r = r"
lset_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lset (iterates f x) = {(f ^^ n) x |n. True}"
Proj_idempotent: fixes M :: "'a ccsubspace" shows "Proj M o\<^sub>C\<^sub>L Proj M = Proj M"
Nat_case_zero: fixes z :: "'a"   and s :: "nat discr Box \<rightarrow> 'a" shows "Nat_case\<cdot>z\<cdot>s\<cdot>0 = z"
affine_dependent_subset: fixes s :: "'a set"   and t :: "'a set" assumes "affine_dependent s"   and "s \<subseteq> t" shows "affine_dependent t"
assms_4: fixes x :: "'a"   and y :: "'a" shows "R (h x) (h y) \<or> R (h y) (h x)"
lpd: shows "\<pi> l pd\<rightarrow> \<pi> (Suc k)"
restrict_map_sub_add: fixes h :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "h |` S ++ h `- S = h"
assms_1: shows "a_consistent (ae, a, as) (\<Gamma>, Var x, S)"
r_result_diverg: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and t :: "nat" assumes "i = encode f"   and "recfn (e_length x) f"   and "eval f (Partial_Recursive.list_decode x) \<up>" shows "eval r_result [t, i, x] \<down>= 0"
cases: fixes x :: "ternaryvalue" obtains "x = TernaryTrue"   | "x = TernaryFalse"   | "x = TernaryUnknown"
map_to_set_inj: fixes k :: "'g"   and v :: "'h"   and m :: "'g \<Rightarrow> 'h option"   and v' :: "'h" assumes "(k, v) \<in> map_to_set m"   and "(k, v') \<in> map_to_set m" shows "v = v'"
abs_eq: fixes x :: "'b ptrm" shows "depth (abs_trm x) = size x"
abs_inj: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat"   and "abs_nat m = abs_nat n" shows "m = n"
arcs1: shows "arcs a a xs = arcs a i zs @ arcs i a (j # ys)"
R': fixes x :: "'b"   and y :: "'a" shows "R x y \<Longrightarrow>\<^sub>t R' x y"
R1_def: fixes msgs :: "'grp \<times> 'grp"   and \<sigma> :: "bool" shows "R1 msgs \<sigma> = (let (m0, m1) = msgs in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>\<sigma> = a * b in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>\<sigma>'. return_spmf (msgs, \<^bold>g [^] a, \<^bold>g [^] b, if \<sigma> then \<^bold>g [^] c\<^sub>\<sigma>' else \<^bold>g [^] c\<^sub>\<sigma>, if \<sigma> then \<^bold>g [^] c\<^sub>\<sigma> else \<^bold>g [^] c\<^sub>\<sigma>')))))"
w': shows "weight (Fun f (ss1 @ t # ss2)) \<le> weight (Fun f (ss1 @ s # ss2))"
SUBSEQ_def: fixes A :: "'b list set" shows "SUBSEQ A \<equiv> {x. \<exists>y\<in>A. subseq x y}"
of_int_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w + z) = plus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
adjacentchain_card: fixes x :: "'b set"   and xs :: "'b set list"   and y :: "'b set" assumes "adjacentchain (x # xs @ [y])" shows "card x = card y"
rep_eq: fixes x :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a pdevs" shows "pdevs_apply (filter_pdevs x xa) = filter_pdevs_raw x (pdevs_apply xa)"
zero_least3: fixes x :: "'a" shows "!x \<cdot> o = o"
if_smult: fixes P :: "bool"   and x :: "real"   and y :: "real"   and v :: "'c" shows "(if P then x else y) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
bind_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and g :: "'d \<Rightarrow> 'c set" shows "f ` A \<bind> g = A \<bind> g \<circ> f"
fmadd_idempotent: fixes m :: "('a, 'b) fmap" shows "m ++\<^sub>f m = m"
oreachablep_oreachable_eq: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "oreachablep A S U = (\<lambda>x. x \<in> oreachable A S U)"
di_2: shows "case di of None \<Rightarrow> True | Some ddi \<Rightarrow> ddi \<in> set (wordinterval_CIDR_split_prefixmatch (uncurry WordInterval (dports r)))"
weakly_reduced_def: fixes \<alpha> :: "rat"   and k :: "nat" shows "gs.weakly_reduced \<alpha> k = (\<forall>i. Suc i < k \<longrightarrow> \<parallel>gs.gso i\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gs.gso (Suc i)\<parallel>\<^sup>2)"