hcju/isabelleps huggingface dataset:
illformed_theory_not_wf_theory: shows "\<not> wf_theory ((Map.empty, Map.empty, {}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]]), {})"
max_triangle_bound: fixes z :: "nat"   and e :: "nat" assumes "triangle z \<le> e" shows "z \<le> e"
arcs_del_arc: fixes a :: "'b" shows "arcs (G.del_arc a) = arcs G - {a}"
ts_ord_refl: fixes x :: "'a" shows "!x \<sqsubseteq> !x"
CompDiag_Inv_Can: fixes t :: "'a local.term" assumes "Can t"   and "Diag t" shows "Inv t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = Dom t"
smult_assoc1: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and a :: "'e"   and b :: "'e"   and x :: "'g" assumes "module R M"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M" shows "a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
FV_Call: fixes t :: "sterm"   and l :: "Label"   and a :: "sterm" shows "FV (Call t l a) = FV t \<union> FV a"
pq: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> p i (h i)" shows "q (\<lambda>x. \<lambda>i\<in>I. h i (x i))"
step_invariant_def: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool" shows "(A \<TTurnstile>\<^sub>A (I \<rightarrow>) P) = (\<forall>a. I a \<longrightarrow> (\<forall>s\<in>reachable A I. \<forall>s'. (s, a, s') \<in> automaton.trans A \<longrightarrow> P (s, a, s')))"
\<epsilon>_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf" shows "0 < \<epsilon>"
hcomp_def: fixes g :: "'d"   and f :: "'d" shows "D'.hcomp g f = (if D'.arr f \<and> D'.arr g \<and> trg\<^sub>D f = src\<^sub>D g then g \<star>\<^sub>D f else D'.null)"
xs: fixes i :: "nat" assumes "i < num_params p" shows "x i = xs ! i"
conj_bot_right: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "Conjunction.conj (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) \<bottom> = \<bottom>"
intensional_simps_9: fixes y :: "'b \<Rightarrow> bool" shows "y = #True \<equiv> y"
uhnfSum: fixes P :: "pi"   and Q :: "pi" assumes "uhnf P"   and "uhnf Q"   and "valid P"   and "valid Q" shows "\<exists>R. uhnf R \<and> valid R \<and> P \<oplus> Q \<equiv>\<^sub>e R \<and> depth R \<le> depth (P \<oplus> Q)"
mkM_kripke: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview" shows "kripke (PreEnvironmentJView.mkM envVal jview T)"
assms_2: shows "\<rho>\<^sub>2 \<equiv> run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w"
a_stone: fixes x :: "'a" shows "- x \<squnion> - - x = top"
forget: fixes a :: "name"   and P :: "pi"   and b :: "name" assumes "a \<sharp> P" shows "P[a::=b] = P"
HPhase1or2ReadThen_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv4c s p" shows "HInv4c s' p"
C\<^sub>1s: shows "\<^bold>\<forall>x xa xb. E (xa \<cdot> xb) \<^bold>\<leftarrow> E (x \<cdot> xa) \<^bold>\<and> E ((x \<cdot> xa) \<cdot> xb)"
djointish: shows "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
left_commute: fixes b :: "'a DBMEntry"   and a :: "'a DBMEntry"   and c :: "'a DBMEntry" shows "dbm_add b (dbm_add a c) = dbm_add a (dbm_add b c)"
assms: shows "anybranch T (\<lambda>p. red (ds @ p))"
App_def: fixes f :: "'e \<Rightarrow> 'f" shows "App f \<equiv> f"
borel_measurable_neq: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
asymp_equiv_at_infinity_transfer: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g"   and "filterlim f at_infinity F" shows "filterlim g at_infinity F"
hyps: shows "applied_rule_rev Cp x (a @ [b] @ xs) = \<lfloor>b\<rfloor> \<longrightarrow> r \<in> set xs \<longrightarrow> b \<notin> set xs \<longrightarrow> x \<notin> dom (Cp r)"
nat_pow_Suc2: fixes x :: "'c \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier F" shows "x [^]\<^bsub>F\<^esub> Suc n = x \<otimes>\<^bsub>F\<^esub> x [^]\<^bsub>F\<^esub> n"
max_v_ev_3: shows "norm1 local.max_v = 1"
length_butlast: fixes xs :: "'c list" shows "length (butlast xs) = length xs - 1"
adds_term_refl: fixes v :: "'a" shows "local.punit.adds_term v v"
d_delta_arctan_upper_31_pos: fixes x :: "real" assumes "x \<noteq> 0" shows "0 < diff_delta_arctan_upper_31 x"
atLs_def: fixes p :: "'g"   and labels :: "'h set set"   and s :: "('i, 'h, 'g, 'j, 'k) system_state" shows "atLs p labels s = (AT s p \<in> labels)"
h3g2: shows "(\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i) = isolate_variable_sparse p2 var 0 * Var var ^ 0 + isolate_variable_sparse p2 var 1 * Var var ^ 1"
lookup_update_unfold: fixes k :: "'e"   and v :: "'d"   and m :: "('e, 'd) mapping"   and k' :: "'e" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
Jacobi_neg_left: fixes n :: "int"   and a :: "int" assumes "odd n"   and "1 < n" shows "Jacobi (- a) n = (if n mod 4 = 1 then 1 else - 1) * Jacobi a n"
simps_1: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf" shows "T\<^sub>p_on_rand' A s [] = 0"
gfinprod_mono_neutral_cong_right: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> g i = \<one>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "g \<in> B \<rightarrow> carrier G" shows "gfinprod G g B = gfinprod G h A"
valid_lookup_cls_is_class: fixes G :: "prog"   and statT :: "ref_ty"   and dynC :: "qtname"   and static_membr :: "bool" assumes "G,statT \<turnstile> dynC valid_lookup_cls_for static_membr"   and "isrtype G statT"   and "wf_prog G" shows "is_class G dynC"
get_scdom_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok Shadow_DOM_SC_DOM_Components.get_scdom_component ptr"
g_add: fixes x :: "'a"   and y :: "'a" shows "g (x + y) = g x + g y"
has_integral_Beta_real': fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 < b" shows "((\<lambda>u. u powr (b - 1) / (1 + u) powr (a + b)) has_integral Beta a b) {0<..}"
val_pos: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and pos :: "'a \<Rightarrow> int"   and x :: "'a" assumes "class.spark_enum less_eq less pos" shows "spark_enum.val pos (pos x) = x"
n_def: shows "n = (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)"
sub_distribD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and s :: "'b" assumes "sub_distrib t"   and "sound P" shows "t P s \<ominus> 1 \<le> t (\<lambda>s'. P s' \<ominus> 1) s"
right_def: shows "right \<equiv> \<lambda>R. R (jumpF f (at_right a)) 0 \<or> continuous (at_right a) f \<and> R (f a) 0"
eval_S': fixes xs :: "nat list" assumes "length xs = 1" shows "eval S xs \<down>= Suc (hd xs)"
L_2: fixes x :: "real" assumes "x \<in> {a..b}" shows "dist (f x (\<phi> x)) (f x (\<psi> x)) \<le> L * dist (\<phi> x) (\<psi> x)"
mat_to_list_def: fixes A :: "'c mat" shows "mat_to_list A = map (\<lambda>i. map (\<lambda>j. A $$ (i, j)) [0..<dim_col A]) [0..<dim_row A]"
stable_FP_Int: fixes F :: "'b program"   and B :: "'b set" shows "F \<in> stable (FP F \<inter> B)"
add_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (add_instr instr s1)) \<and> t2 = snd (fst (add_instr instr s2))" shows "low_equal t1 t2"
rep_list_monom_mult: fixes c :: "'b"   and t :: "'a"   and r :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (monom_mult c t r) = local.punit.monom_mult c t (rep_list r)"
Complexs_mult: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<complex>"   and "b \<in> \<complex>" shows "a * b \<in> \<complex>"
Q1: fixes t :: "'trm" assumes "t \<in> atrm" shows "prv (LLq zer t)"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "PA a"   and "PB a'"   and "a \<sim> a'" shows "\<exists>b'. B a' b' \<and> b \<sim> b'"
a_heap_is_wellformed_def: fixes h :: "(_) heap" shows "local.a_heap_is_wellformed h = (local.CD.a_heap_is_wellformed h \<and> acyclic (local.CD.a_parent_child_rel h \<union> local.a_host_shadow_root_rel h) \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_shadow_root_valid h)"
1: shows "(k * l * t - t + a1 * k) * d' = a"
helper2: fixes v :: "'b vec"   and ss :: "'b vec list"   and ls :: "'b vec list"   and nr :: "nat" assumes "v \<in> carrier_vec (length ss)"   and "\<And>x. x \<in> set ls \<Longrightarrow> dim_vec x = nr" shows "mat_of_cols nr ss *\<^sub>v v = mat_of_cols nr (ls @ ss) *\<^sub>v (0\<^sub>v (length ls) @\<^sub>v v)"
perm_rem_simps_2: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "bij f"   and "f y = x" shows "perm_rem x f y = f x"
tiTable_ack_st_splitten: fixes lose :: "nat \<Rightarrow> bool list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and t :: "nat"   and req :: "nat \<Rightarrow> reqType list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> 'a list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> 'a list" assumes "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop"   and "st_in t = init_state \<and> req t = [init] \<longrightarrow> ack t = [call] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = call"   and "st_in t = init_state \<and> req t \<noteq> [init] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state"   and "(st_in t = call \<or> st_in t = connection_ok \<and> req t \<noteq> [send]) \<and> lose t = [False] \<longrightarrow> ack t = [connection_ok] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = connection_ok"   and "(st_in t = call \<or> st_in t = connection_ok \<or> st_in t = sending_data) \<and> lose t = [True] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state"   and "st_in t = connection_ok \<and> req t = [send] \<and> lose t = [False] \<longrightarrow> ack t = [sending_data] \<and> i1 t = b t \<and> vc t = [] \<and> st_out t = sending_data"   and "st_in t = sending_data \<and> a1 t = [] \<and> lose t = [False] \<longrightarrow> ack t = [sending_data] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = sending_data"   and "st_in t = sending_data \<and> a1 t = [sc_ack] \<and> lose t = [False] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = voice_com"   and "st_in t = voice_com \<and> stop t = [] \<and> lose t = [False] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = voice_com"   and "st_in t = voice_com \<and> stop t = [] \<and> lose t = [True] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = voice_com"   and "st_in t = voice_com \<and> stop t = [stop_vc] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state" shows "ack t = [st_out t]"
at_bij2: fixes pi :: "('a \<times> 'a) list"   and x :: "'a"   and y :: "'a" assumes "at TYPE('a)"   and "rev pi \<bullet> x = y" shows "x = pi \<bullet> y"
assms_3: shows "r\<lparr>a\<rparr> = b"
ET_def: fixes ST :: "('d, 'e, 'f) status" shows "ET ST \<equiv> \<Union>SA\<in>SAs (HA ST). (EnabledTrans `` {ST}) `` {SA}"
image_mono: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
tat: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], x # y # h) (v @ r) = length v + Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev qs @ h) r"
fds_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>n. 0 < n \<Longrightarrow> f n = g n" shows "fds f = fds g"
subst_subst: fixes t :: "('c, 'f) Term.term"   and \<sigma> :: "'f \<Rightarrow> ('c, 'e) Term.term"   and \<tau> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "t \<cdot> \<sigma> \<cdot> \<tau> = t \<cdot> \<sigma> \<circ>\<^sub>s \<tau>"
fac_fac_body_eq: shows "fac = fix\<cdot>fac_body"
antimonoE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" obtains "f y \<le> f x"
finite_dimensional_real_vector_space_pair_on_def: fixes S1 :: "'c set"   and S2 :: "'d set"   and Basis1 :: "'c set"   and Basis2 :: "'d set" shows "finite_dimensional_real_vector_space_pair_on S1 S2 Basis1 Basis2 \<equiv> finite_dimensional_real_vector_space_on S1 Basis1 \<and> finite_dimensional_real_vector_space_on S2 Basis2"
keys_map_key_PP: fixes p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "keys (Poly_Mapping.map_key PP p) = pp.mapping_of ` keys p"
natLess_def: shows "natLess \<equiv> {(x, y). x < y}"
rel_setD2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set"   and y :: "'b" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
init_InvIncomingInfoUpright: fixes c :: "('c, 'd) configuration" assumes "init_config c" shows "InvIncomingInfoUpright c"
mat_diag_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "mat_diag n f \<in> carrier_mat n n"
cases: fixes a :: "'node SDG_node"   and n :: "'node SDG_node" assumes "a \<in> sum_SDG_slice1 n" obtains "a = n" and "valid_SDG_node n"   | n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a = n''__" and "n''__ s\<longrightarrow>\<^bsub>cd\<^esub> n'__" and "n'__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a = n''a__" and "n''a__ s-V__\<rightarrow>\<^sub>d\<^sub>d n'a__" and "n'a__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''b__" and "n''b__ s-p__\<rightarrow>\<^bsub>call\<^esub> n'b__" and "n'b__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and V :: "'var" and n' :: "'node SDG_node" where "a = n''c__" and "n''c__ s-pa__:V\<rightarrow>\<^bsub>in\<^esub> n'c__" and "n'c__ \<in> sum_SDG_slice1 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" and "n' \<in> sum_SDG_slice1 n"
intensional_simps_21: fixes P :: "'b \<Rightarrow> bool" shows "P \<and> \<not> P \<equiv> #False"
pderiv_of_int_poly: fixes p :: "int poly" shows "pderiv (of_int_poly p) = of_int_poly (pderiv p)"
step_Normal_elim_cases_11: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Catch c1 c2, Normal s) \<rightarrow> u" obtains c\<^sub>1' :: "('b, 'a, 'c) com" and s' :: "('b, 'c) xstate" where "u = (Catch c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, Normal s) \<rightarrow> (c\<^sub>1', s')"   | "u = (c2, Normal s)" and "c1 = Throw"   | "c1 = Skip" and "u = (Skip, Normal s)"
gen_common_subdiv_imp_common_subdiv: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_sudiv_exists one_chain1 one_chain2 = (\<exists>ps1 ps2. common_boundary_sudivision_exists (one_chain1 - ps1) (one_chain2 - ps2) \<and> (\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>))"
prv_cnjI: fixes \<phi> :: "fm"   and \<chi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV"   and "{} \<turnstile> \<phi>"   and "{} \<turnstile> \<chi>" shows "{} \<turnstile> \<phi> AND \<chi>"
eq: shows "B = K x \<bind> (\<lambda>y. B' y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>)))"
prv_imp_ldsj_in: fixes \<phi>s :: "'fmla list"   and \<phi> :: "'fmla" assumes "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s" shows "prv (imp \<phi> (ldsj \<phi>s))"
ref_dom_events_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list" shows "ref_dom_events P D u xs \<equiv> {x. u = D x \<and> {x} \<in> refusals P xs}"
assms: shows "((P, Map.empty), {#}, \<sigma>) \<in> matchrel\<^sup>*"
unambigous_prefix_routing_strong_mono: fixes rr :: "('b, 'c) routing_rule_scheme"   and rtbl :: "('b, 'c) routing_rule_scheme list"   and rr' :: "('b, 'c) routing_rule_scheme" assumes "is_longest_prefix_routing (rr # rtbl)"   and "unambiguous_routing (rr # rtbl)"   and "rr' \<in> set rtbl"   and "routing_match rr' = routing_match rr" shows "routing_rule_sort_key rr < routing_rule_sort_key rr'"
of_seq_code_2: fixes x :: "'b"   and P :: "'b Predicate.pred" shows "set_of_seq (seq.Insert x P) = insert x (set_of_pred P)"
R: fixes P :: "assn"   and s :: "nat \<times> nat"   and bi :: "nat \<times> nat \<Rightarrow> bool Heap"   and f :: "nat \<times> nat \<Rightarrow> (nat \<times> nat) Heap" assumes "wf (measure (\<lambda>(l, h). h - l))"   and "P \<Longrightarrow>\<^sub>A (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h)))"   and "\<And>s. <(case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h))) * true> bi s <\<lambda>r. (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h))) * \<up> (r = (case s of (l, h) \<Rightarrow> l < h))>\<^sub>t"   and "\<And>s. case s of (l, h) \<Rightarrow> l < h \<Longrightarrow> <(case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h))) * true> f s <\<lambda>r. (case r of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h))) * \<up> ((r, s) \<in> measure (\<lambda>(l, h). h - l))>\<^sub>t"   and "\<And>s. \<not> (case s of (l, h) \<Rightarrow> l < h) \<Longrightarrow> (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> h \<and> h \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (h < n \<longrightarrow> p \<le> xs ! h))) \<Longrightarrow>\<^sub>A (case s of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (l < n \<longrightarrow> p \<le> xs ! l)))" shows "<P> heap_WHILET bi f s <\<lambda>r. case r of (l, h) \<Rightarrow> is_pfa c xs (a, n) * \<up> (l \<le> n \<and> (\<forall>j<l. xs ! j < p) \<and> (l < n \<longrightarrow> p \<le> xs ! l))>\<^sub>t"
rel_resource_simps: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> ('e \<times> ('c, 'e) resource) spmf"   and g :: "'d \<Rightarrow> ('f \<times> ('d, 'f) resource) spmf" shows "rel_resource A B (Resource f) (Resource g) = rel_fun A (rel_spmf (rel_prod B (rel_resource A B))) f g"
assms: shows "run_one_step d i (s, vs, ves, $Load x191 x192 x193 x194) = (s', vs', res)"
BoxDmd_simple: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(F \<and> G)"
sorted_wrt_Cons: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and ys :: "'b list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
mediant_times_F: shows "mediant \<circ> (\<lambda>s. s \<otimes> F) = mediant"
simps_2: fixes x :: "'g"   and xs :: "'g list" shows "rotate1 (x # xs) = xs @ [x]"
comp_null_2: fixes f :: "'b" shows "B f B.null = B.null"
c1c2_4: shows "f ` domain c1 \<subseteq> domain c2"
Reals_mult: fixes a :: "'c"   and b :: "'c" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a * b \<in> \<real>"
always_imp_mp_suffix: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "(\<box>(\<lambda>s. P s \<longrightarrow> Q s)) \<sigma>"   and "P (\<sigma> |\<^sub>s i)" shows "Q (\<sigma> |\<^sub>s i)"
continuous_on_interior: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "continuous_on s f"   and "x \<in> interior s" shows "isCont f x"
polar_rep_non_zero: fixes p :: "proj2" shows "M *v proj2_rep p \<noteq> 0"
parabolic_isometry_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "parabolic_isometry (inv f)"
local_lipschitz_compose1: fixes g :: "'e \<Rightarrow> 'b"   and T :: "'e set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "local_lipschitz (g ` T) X f"   and "continuous_on T g" shows "local_lipschitz T X (\<lambda>t. f (g t))"
Ya_arr: fixes g :: "'c" assumes "C.arr g" shows "Y a g = S.mkArr (Hom.set (C.cod g, a)) (Hom.set (C.dom g, a)) (S.UP \<circ> inl \<circ> (\<cdot>\<^sup>o\<^sup>p) g \<circ> \<psi> (C.cod g, a))"
step_invariant_pairI: fixes A :: "('c \<times> 'd, 'e) automaton"   and I :: "'e \<Rightarrow> bool"   and P :: "('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd \<Rightarrow> bool" assumes "\<And>\<xi> p \<xi>' p' a. \<lbrakk>(\<xi>, p) \<in> reachable A I; ((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P ((\<xi>, p), a, \<xi>', p')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
is_red_addsE: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd" assumes "is_red F p" obtains f :: "'t \<Rightarrow>\<^sub>0 'd" and v :: "'t" where "f \<in> F" and "v \<in> keys p" and "f \<noteq> 0" and "lt f adds\<^sub>t v"
le_replicateI: fixes xs :: "nat list"   and b :: "nat" assumes "\<forall>x\<in>set xs. x \<le> b" shows "xs \<le>\<^sub>v replicate (length xs) b"
continuous_map_iff_continuous2: fixes g :: "'a \<Rightarrow> 'b" shows "continuous_map euclidean euclidean g = continuous_on UNIV g"
compact_sphere: fixes a :: "'d"   and r :: "real" shows "compact (sphere a r)"
map_indets_id: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and f :: "'c \<Rightarrow> 'c" assumes "\<And>x. x \<in> indets p \<Longrightarrow> f x = x" shows "map_indets f p = p"
fun_comp_eq_conv: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and fg :: "'d \<Rightarrow> 'e" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"