hcju/isabelleps huggingface dataset:
ftf_syn1: fixes a :: "'a event"   and u :: "'a event list"   and t :: "'a event list"   and s :: "'a event list"   and A :: "'a event set" shows "a \<notin> set u \<and> a \<notin> set t \<and> s setinterleaves ((t, u), A) \<longrightarrow> a \<notin> set s"
fps_nth_coeff_fps_poly_swap2: fixes p :: "'b poly fps"   and d :: "nat"   and m :: "nat"   and n :: "nat" assumes "\<And>n. degree (p $ n) \<le> d" shows "coeff (fps_poly_swap2 p) m $ n = coeff (p $ n) m"
PInfty_neq_ereal_1: fixes r :: "real" shows "ereal r \<noteq> \<infinity>"
kronecker_invertible: fixes A :: "'a mat"   and B :: "'a mat" assumes "invertible_mat A"   and "invertible_mat B" shows "invertible_mat (kronecker_product A B)"
zero_int_def: shows "0 \<equiv> abs_int (0, 0)"
set_dom: fixes U :: "'b set"   and f :: "'b set_arrow" shows "Dom\<^bsub>set_cat U\<^esub> f = set_dom f"
wf_exec_thread_start_actions_okD: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "P \<turnstile> (E, ws) \<surd>" shows "thread_start_actions_ok E"
elims: fixes x :: "'a"   and xa :: "('a \<times> 'b) list"   and y :: "('a \<times> 'b) list" assumes "delete_aux x xa = y" obtains k :: "'a" where "x = k__" and "xa = []" and "y = []"   | k :: "'a" and k' :: "'a" and v :: "'b" and xs :: "('a \<times> 'b) list" where "x = k" and "xa = (k', v) # xs" and "y = (if k = k' then xs else (k', v) # delete_aux k xs)"
twoCisTypeII_10: shows "(\<lambda>y. twoC (y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
simps_1: fixes x :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list, 'b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>m"   and ya :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list, 'b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>m" shows "(Tree\<^sub>m x = Tree\<^sub>m ya) = (x = ya)"
ide_char: fixes t :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.ide t = (t \<noteq> J_C.null \<and> functor (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map t) \<and> J_C.Dom t = J_C.Map t \<and> J_C.Cod t = J_C.Map t)"
Hausdorff_space_subtopology: fixes X :: "'c topology"   and S :: "'c set" assumes "Hausdorff_space X" shows "Hausdorff_space (subtopology X S)"
m1: assumes "i \<in> set xs" shows "m1 \<in> {k. k < length xs \<and> xs ! k = i}"
frag_cmul_zero2: fixes c :: "int" shows "frag_cmul c 0 = 0"
graph_ext_def: fixes nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b" shows "\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons graph_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple nodes edges) more"
hyps: shows "remaining_qty Mkt v pf asset (Suc n) \<in> borel_measurable (F n)"
Nil_not_valid: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a" shows "\<not> IO_Automaton.valid step []"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>) (\<cdot>) L local.map \<ll>"
mult_neg_pos: fixes a :: "'b"   and b :: "'b" assumes "a < (0::'b)"   and "(0::'b) < b" shows "a * b < (0::'b)"
prob_space_distr: fixes f :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'e"   and M' :: "'e measure" assumes "P.random_variable M' f" shows "prob_space (distr (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) M' f)"
B: shows "\<forall>x. poly p x \<noteq> 0"
cinsert_is_cUn: fixes a :: "'d"   and A :: "'d cset" shows "cinsert a A = cUn (csingle a) A"
ptrm_infer_type_alpha_equiv: fixes X :: "'b ptrm"   and Y :: "'b ptrm"   and \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option" assumes "X \<approx> Y" shows "ptrm_infer_type \<Gamma> X = ptrm_infer_type \<Gamma> Y"
is_regular_spairD1: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "rep_list p \<noteq> 0"
prems_4: shows "partial_density_operator \<rho>\<^sub>2"
UnE: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
k: shows "k \<noteq> 0 \<and> x \<in> elts (y * ord_of_nat k)"
exactly_one_monic_factorization: fixes f :: "'a poly" assumes "monic f" shows "\<exists>!fs. f = \<Prod>\<^sub># fs \<and> set_mset fs \<subseteq> {q. irreducible q \<and> monic q}"
simps_7: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> X \<phi> = suffix 1 w \<Turnstile> \<phi>"
prefix_code_1: fixes xs :: "'a list" shows "prefix [] xs = True"
cring_simprules_10: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
pl_3: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> \<^bold>\<rightarrow> (\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> \<phi>)]]"
nummul_nz: fixes i :: "int"   and a :: "Ferrack.num" assumes "i \<noteq> 0"   and "nozerocoeff a" shows "nozerocoeff (nummul a i)"
proj2_incident_Col: fixes p :: "proj2"   and l :: "proj2_line"   and q :: "proj2"   and r :: "proj2" assumes "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident r l" shows "proj2_Col p q r"
statically_complete_calculus_axioms: shows "statically_complete_calculus Bot_FL Inf_FL entails_\<G> std.Red_I std.Red_F"
existence_ivl_maximal_interval: fixes x :: "real \<Rightarrow> 'a"   and S :: "real set"   and x0 :: "'a" assumes "(x solves_ode f) S X"   and "x t0 = x0"   and "t0 \<in> S"   and "is_interval S"   and "S \<subseteq> T" shows "S \<subseteq> existence_ivl t0 x0"
w: fixes \<eta> :: "nat" shows "wiring (\<I>_ideal \<eta>) (\<I>_real \<eta>) (cnv \<eta>) (w \<eta>)"
v0_v1_Digraph_axioms: shows "v0_v1_Digraph (remove_vertex v) v0 v1"
h1c3: shows "isolate_variable_sparse p x (n + 1) \<noteq> 0"
yun_factorization_induct: fixes as :: "('c poly \<times> nat) set"   and p :: "'c poly"   and P :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> bool"   and bn :: "'c poly"   and cn :: "'c poly" assumes "monic_factorization as p"   and "\<And>bn cn. bn = 1 \<Longrightarrow> P bn cn"   and "\<And>bn cn. \<lbrakk>bn \<noteq> 1; P (bn div gcd bn (cn - pderiv bn)) ((cn - pderiv bn) div gcd bn (cn - pderiv bn))\<rbrakk> \<Longrightarrow> P bn cn"   and "bn = p div gcd p (pderiv p)"   and "cn = pderiv p div gcd p (pderiv p)" shows "P bn cn"
H2: shows "geodesic_segment_between H2 (h a) (h (dist x z))"
formSubst: fixes y :: "NominalSequents.var"   and x :: "NominalSequents.var"   and A :: "form"   and F :: "char list" assumes "y \<sharp> x \<and> y \<sharp> A" shows "F \<nabla> [x].A = F \<nabla> [y].[y,x]A"
pe_p: shows "set (vwalk_arcs p) \<subseteq> arcs_ends G"
units_of_one: fixes G :: "('b, 'c) monoid_scheme" shows "\<one>\<^bsub>units_of G\<^esub> = \<one>\<^bsub>G\<^esub>"
abs_idempotent: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (abs a) = abs a"
le_cases: fixes x :: "'c word"   and y :: "'c word" obtains "x \<le>s y"   | "y \<le>s x"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" shows "satC \<xi> \<equiv> list_ex (satL \<xi>)"
absolutely_integrable_approximate_continuous: fixes S :: "real set"   and f :: "real \<Rightarrow> real"   and e :: "real" assumes "f absolutely_integrable_on S"   and "S \<in> lmeasurable"   and "0 < e" obtains g :: "real \<Rightarrow> real" where "continuous_on UNIV g" and "g absolutely_integrable_on S" and "LINT x|lebesgue_on S. \<bar>f x - g x\<bar> < e"
cl_in_subset_carrier: fixes X :: "'a set" shows "cl_in \<E> X \<subseteq> \<E>"
Fun_pr0: shows "Fun (pr0 a b) = (\<lambda>x\<^bold>\<in>local.set (local.prod a b). UP (hsnd (DOWN x)))"
qAFresh_qSwap_exchange1: fixes zs :: "'a"   and z2 :: "'b"   and z1 :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qAFresh zs z2 (X #[[z1 \<and> z2]]_zs) = qAFresh zs z1 X"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
span_is_subset2: fixes S :: "rat vec set" assumes "S \<subseteq> Rn" shows "gs.span S \<subseteq> Rn"
e: shows "epath steps \<and> fst (shd steps) \<in> S"
disjoint_list_disjoint_list_rec: fixes ts :: "'b set list" assumes "disjoint_list ts" shows "disjoint_list_rec ts"
reachable_snapshot_invI: fixes m :: "'f"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "\<And>y. mut_m.reachable m y s \<Longrightarrow> in_snapshot y s" shows "mut_m.reachable_snapshot_inv m s"
possible_single: fixes x :: "'a"   and x' :: "'a" shows "possible (single x) x' = (x = x')"
map_gpv_Done: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and x :: "'g" shows "map_gpv f g (Generative_Probabilistic_Value.Done x) = Generative_Probabilistic_Value.Done (f x)"
wf_lexn: fixes r :: "('a \<times> 'a) set"   and n :: "nat" assumes "wf r" shows "wf (lexn r n)"
pw_ords_iff_1: fixes S :: "'c nres"   and S' :: "'c nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
real_poly_power: fixes p :: "'b poly"   and n :: "nat" assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (p ^ n)) \<subseteq> \<real>"
adm_disj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "adm P"   and "adm Q" shows "adm (\<lambda>x. P x \<or> Q x)"
smult_eq_0_iff: fixes a :: "'a"   and p :: "'a poly" shows "(Polynomial.smult a p = 0) = (a = (0::'a) \<or> p = 0)"
rel_fun_comp: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and R2 :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and h :: "'k \<Rightarrow> 'l" assumes "(R1 ===> S1) f g"   and "(R2 ===> S2) g h" shows "(R1 OO R2 ===> S1 OO S2) f h"
iprev_cut_greater_fix: fixes n :: "nat"   and I :: "nat set"   and t :: "nat" assumes "iprev n I \<le> t" shows "iprev n (I \<down>> t) = n"
learn_consE_3: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "learn_cons \<psi> U s"   and "f \<in> U" shows "\<forall>k\<le>n. \<psi> (the (s (f \<triangleright> n))) k = f k"
if_true_eval\<^sub>w\<^sub>C: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat"   and x :: "var\<^sub>C"   and c\<^sub>C_then :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_else :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" shows "mem\<^sub>C x = 0 \<longrightarrow> eval_abv\<^sub>C \<langle>Stmt.If (bexp\<^sub>C.Eq x 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>C_then ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C"
eq_linorder_comp2eq_eq: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "eq_linorder cmp" shows "comp2eq cmp = (=)"
order_less_irrefl: fixes x :: "'c" shows "\<not> x < x"
downclosed_mono: fixes f :: "'d \<Rightarrow> 'c" assumes "\<forall>Y. downclosed_set Y \<longrightarrow> downclosed_set (f -` Y)" shows "mono f"
SFAccRedsNonStatic: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool" assumes "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s,False\<rangle>"
simps_3: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 or\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<mu> = \<psi>\<^sub>1[Y]\<^sub>\<mu> or\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>"
sgn_vec_def: fixes qs :: "real poly list"   and x :: "real" shows "sgn_vec qs x \<equiv> map (Matrix_Equation_Construction.sgn \<circ> (\<lambda>q. poly q x)) qs"
abs_le_mult: fixes a :: "'b"   and b :: "'b" shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
lossless_funct_OT_12: fixes msgs :: "'g \<times> 'g"   and \<sigma> :: "bool" shows "lossless_spmf (funct_OT_12 msgs \<sigma>)"
lookup_tensor_prod: fixes is1 :: "nat list"   and A :: "'b tensor"   and is2 :: "nat list"   and B :: "'b tensor" assumes "is1 \<lhd> dims A"   and "is2 \<lhd> dims B" shows "lookup (A \<otimes> B) (is1 @ is2) = lookup A is1 * lookup B is2"
subg: shows "graph_union (map_graph (\<Union> (range g)) (chain_sup S)) C = C"
rep_eq: fixes x :: "circline_mat" shows "circline_type_clmat x = circline_type_cmat (Rep_circline_mat x)"
ftoh: shows "(red (insert q (G - {p})))\<^sup>*\<^sup>* f h"
A: fixes n :: "nat" shows "dist c ((f ^^ n) basepoint) \<le> e"
sum_upto_def: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "sum_upto f x = (\<Sum>i | 0 < i \<and> real i \<le> x. f i)"
pair_return_pmf2: fixes x :: "'c pmf"   and y :: "'d" shows "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
effect_returnI: fixes h :: "heap"   and h' :: "heap"   and x :: "'b" assumes "h = h'" shows "effect (return x) h h' x"
InvariantGetReasonIsReasonAfterNotifyWatches: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and M :: "(Literal \<times> bool) list"   and decision :: "bool"   and Q :: "Literal set"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "uniq Wl"   and "getM state = M @ [(opposite literal, decision)]"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) Q" shows "let state' = notifyWatches_loop literal Wl newWl state in Let (Q \<union> (set (getQ state') - set (getQ state))) (InvariantGetReasonIsReason (getReason state') (getF state') (getM state'))"
fdom_res_empty: fixes f :: "('a, 'b) ffun" shows "{} \<lhd>\<^sub>f f = {}\<^sub>f"
no_vs_change_obtain_snapshot_bound_2nd_step: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap" assumes "finite PROB"   and "vars_change as vs s = []"   and "sat_precond_as s as"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as = exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as' \<and> subseq as' as \<and> sat_precond_as s as' \<and> length as' \<le> problem_plan_bound (snapshot PROB (fmrestrict_set vs s))"
norm_sum_lemma: fixes e :: "real"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "e \<le> norm (a + b + c + d)" shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
ua_vb_0: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'d"   and b :: "'d" assumes "is_bezout_ext bezout"   and "snd (snd (snd (snd (bezout a b)))) \<noteq> (0::'d)" shows "fst (snd (snd (bezout a b))) * a + fst (snd (snd (snd (bezout a b)))) * b = (0::'d)"
subsetI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "L F f \<subseteq> L F g"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_with_State_Idle_conv2: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a"   and t1 :: "nat" assumes "0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "t1 \<in> [0\<dots>,k - Suc 0] \<oplus> t0"   and "State_Idle localState output_fun trans_fun (localState (s t1))"   and "output_fun (s t1) \<noteq> NoMsg" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<diamond> t1 [0\<dots>,k - Suc 0] \<oplus> t0. output_fun (s t1) = m \<and> State_Idle localState output_fun trans_fun (localState (s t1)))"
simply_connected_eq_contractible_path: fixes S :: "'b set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))))"
is_OK_returns_result_I: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and y :: "'n" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y" shows "h \<turnstile> ok f"
simps: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and a1 :: "('b, 'c) wait_set_action"   and a2 :: "'b \<Rightarrow> 'c wait_set_status option" shows "redT_updW t ws a1 a2 = ((\<exists>t' w. a1 = Notify w \<and> a2 = ws(t' \<mapsto> PostWS WSNotified) \<and> ws t' = \<lfloor>InWS w\<rfloor>) \<or> (\<exists>w. a1 = Notify w \<and> a2 = ws \<and> (\<forall>x. ws x \<noteq> \<lfloor>InWS w\<rfloor>)) \<or> (\<exists>w. a1 = NotifyAll w \<and> a2 = (\<lambda>t. if ws t = \<lfloor>InWS w\<rfloor> then \<lfloor>PostWS WSNotified\<rfloor> else ws t)) \<or> (\<exists>w. a1 = Suspend w \<and> a2 = ws(t \<mapsto> InWS w)) \<or> (\<exists>t' w WSInterrupted. a1 = WakeUp t' \<and> a2 = ws(t' \<mapsto> PostWS WSInterrupted) \<and> ws t' = \<lfloor>InWS w\<rfloor>) \<or> (\<exists>t'. a1 = WakeUp t' \<and> a2 = ws \<and> (\<forall>x. ws t' \<noteq> \<lfloor>InWS x\<rfloor>)) \<or> a1 = Notified \<and> a2 = ws(t := None) \<or> a1 = WokenUp \<and> a2 = ws(t := None))"
sup_bot_right: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a" assumes "bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
card_of_Fpow: fixes A :: "'b set" shows "|A| \<le>o |Fpow A|"
x_in_csv: shows "x \<in> set (map (consistent_sign_vec (qs1 @ qs2)) (characterize_root_list_p p))"
inv_eq_1_iff: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "(\<ominus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>) = (x = \<zero>\<^bsub>K [X]\<^esub>)"
right_inv_of_unitary_transpose: fixes U :: "complex Matrix.mat" assumes "unitary U" shows "U\<^sup>t * U\<^sup>t\<^sup>\<dagger> = 1\<^sub>m (dim_col U)"
Image_INT_eq: fixes r :: "('c \<times> 'b) set"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'c set" assumes "single_valued (r\<inverse>)"   and "A \<noteq> {}" shows "r `` \<Inter> (B ` A) = (\<Inter>x\<in>A. r `` B x)"
comm_monoid_add_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "comm_monoid_add_hom hom \<equiv> monoid_add_hom hom"
face_cycle_succ_inI: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and x :: "'d"   and y :: "'d" assumes "digraph_map G M"   and "x \<in> pre_digraph_map.face_cycle_set M y" shows "pre_digraph_map.face_cycle_succ M x \<in> pre_digraph_map.face_cycle_set M y"
rel_refl_strong: fixes x :: "'e set['f]"   and Ra :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> Ra z z" shows "rel_bset Ra x x"
src_dom: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>C (C.dom \<mu>) = src\<^sub>C \<mu>"
same_level_path_def: fixes as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" shows "CFGExit.same_level_path as \<equiv> CFGExit.same_level_path_aux [] as \<and> CFGExit.upd_cs [] as = []"
matches_simps_2: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d"   and m' :: "'c match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m' m) a p = matches \<gamma> m' a p"
carr2_2: shows "gs2.gso i \<in> Rn"
tc: fixes x :: "'a" assumes "x \<cdot> x \<le> x" shows "x\<^sup>\<star> \<cdot> x = x"
f_nxt_empty': fixes f :: "'a \<Rightarrow> 'a ttree"   and x :: "'a"   and T :: "'a set" assumes "f x = TTree.empty" shows "f_nxt f T x = f"
intros: fixes ta' :: "('f, 'g, 'x, 'h, 'i, 'j) thread_action"   and ta :: "('f, 'g, 'k, 'h, 'i, 'j) thread_action" assumes "collect_locks' \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<subseteq> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"   and "collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<subseteq> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"   and "collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<subseteq> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>" shows "actions_subset ta' ta"
Sup_set_def: fixes A :: "'g set set" shows "\<Union> A = {x. \<Squnion> ((\<in>) x ` A)}"
ExcessAt_def: fixes g :: "graph"   and v :: "nat" shows "ExcessAt g v \<equiv> if \<not> finalVertex g v then 0 else excessAtType (tri g v) (quad g v) (except g v)"
ring_finite_field_ops64: fixes pp :: "uint64" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "ring_ops (finite_field_ops64 pp) mod_ring_rel64"
DVDJ_l: fixes i :: "int"   and n :: "int"   and s :: "MIR.num" assumes "0 < i"   and "0 < n"   and "numbound0 s" shows "isrlfm (DVDJ i n s)"
simps: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "mergesort_by_rel R xs = (if length xs < 2 then xs else mergesort_by_rel_merge R (mergesort_by_rel R (fst (mergesort_by_rel_split ([], []) xs))) (mergesort_by_rel R (snd (mergesort_by_rel_split ([], []) xs))))"
listsE: fixes x :: "'g"   and l :: "'g list"   and A :: "'g set" assumes "x # l \<in> lists A" obtains "x \<in> A" and "l \<in> lists A"
simps_10: fixes x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree"   and x1 :: "('e, 'f, 'g) Tree set['e]" shows "tAct x41 x42 \<noteq> tConj x1"
alphaAbs_qAbs_imp_all_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "xs = xs' \<and> (\<forall>y. qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
abs_leI: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq (uminus a) b" shows "less_eq (abs a) b"
"2_3": assumes "\<And>x. x \<in> X \<Longrightarrow> sf x \<subseteq> X" shows "card (\<Union>\<phi>\<in>X. sf (\<phi>[N]\<^sub>\<Sigma>\<^sub>1)) \<le> card X \<and> card (\<Union>\<phi>\<in>X. sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1)) \<le> card X \<and> card (\<Union>\<phi>\<in>X. sf \<phi>[M]\<^sub>\<Sigma>\<^sub>2 \<union> sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1)) \<le> 3 * card X"