hcju/isabelleps huggingface dataset:
denote_positive_trace_dim: fixes dims :: "nat list"   and S :: "com"   and \<rho> :: "complex mat" assumes "state_sig.well_com dims S"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>" shows "positive (state_sig.denote dims S \<rho>) \<and> trace (state_sig.denote dims S \<rho>) \<le> trace \<rho> \<and> state_sig.denote dims S \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
simps_6: shows "cospan_shape.comp cospan_shape.TT cospan_shape.BT = cospan_shape.BT"
D: shows "distributed M lborel X (\<lambda>x. ennreal (exponential_density l x))"
nat_eq_le: fixes m :: "nat"   and n :: "nat" assumes "m = n" shows "m \<le> n"
simps: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" shows "(P,a1 \<turnstile> a2 [::] a3) = ((\<exists>E. a1 = E \<and> a2 = [] \<and> a3 = []) \<or> (\<exists>E e T es Ts. a1 = E \<and> a2 = e # es \<and> a3 = T # Ts \<and> P,E \<turnstile> e :: T \<and> P,E \<turnstile> es [::] Ts))"
rcond_lift_false: shows "\<lceil>false\<rceil>\<^sub>\<leftarrow> = false"
right_neutral: fixes a :: "'b" shows "a XOR (0::'b) = a"
assms_2: shows "AE x in M. f (T x) = f x"
member_in_def: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" shows "G \<turnstile> m member_in C = (\<exists>provC. G\<turnstile>C\<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC)"
d: fixes g :: "'a mod_ring poly" assumes "g dvd f"   and "degree g = d" shows "irreducible g"
li_le_dim_2: fixes A :: "'c set" assumes "V.fin_dim"   and "A \<subseteq> carrier V"   and "V.module.lin_indpt A" shows "card A \<le> V.dim"
Transset_TC_eq: fixes x :: "V" assumes "Transset x" shows "TC x = x"
hyps_4: assumes "\<And>s. s \<in> ss \<Longrightarrow> \<exists>rs. s = s0 \<star> rs \<and> set rs \<subseteq> rset" shows "\<exists>rs. \<Sqinter> ss = s0 \<star> rs \<and> set rs \<subseteq> rset"
diff_add_cancel: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
indecomposable_is_\<omega>_power: fixes \<mu> :: "V" assumes "indecomposable \<mu>" obtains "\<mu> = 0"   | \<delta> :: "V" where "Ord \<delta>" and "\<mu> = \<omega> \<up> \<delta>"
mset_heap_def: fixes ts :: "'b tree list" shows "mset_heap ts = \<Sum>\<^sub># (image_mset mset_tree (mset ts))"
assms_1: shows "x \<in> state_set (state_list s as)"
simps_12: shows "ReleaseAcquire \<noteq> UnlockFail"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n A \<le> a"
p_def: fixes b :: "'b set" shows "p b \<equiv> b \<subseteq> U\<^sub>V\<^sub>S \<and> \<not> local.dependent b \<and> local.span b = local.span V"
pred_pmf_code: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c pmf" shows "pred_pmf P p = (\<forall>x\<in>set_pmf p. P x)"
measurable_countable_negligible_Union_bounded: fixes S :: "nat \<Rightarrow> 'c set" assumes "\<And>n. S n \<in> lmeasurable"   and "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"   and "bounded (\<Union> (range S))" shows "\<Union> (range S) \<in> lmeasurable"
distinct_2: fixes x2 :: "'b"   and x1 :: "'a" shows "Inr x2 \<noteq> Inl x1"
sup_absorb2: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "sup x y = y"
two_pred_below_arg: fixes f :: "one\<^sub>\<bottom>" shows "two_pred\<cdot>f \<sqsubseteq> f"
l3_inv7I: fixes x :: "l3_state" assumes "\<And>A B M. implConfid A B M \<in> synth (analz (ik x)) \<and> M \<in> payload \<Longrightarrow> implConfid A B M \<in> ik x \<or> M \<in> synth (analz (ik x))" shows "x \<in> l3_inv7"
noFaultnI': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Fault f \<Longrightarrow> False" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Fault f}"
concat_sing_list_pow: fixes ws :: "'b list list"   and u :: "'b list"   and k :: "nat" assumes "ws \<in> lists {u}"   and "\<^bold>|ws\<^bold>| = k" shows "concat ws = u \<^sup>@ k"
\<psi>_\<phi>: fixes f :: "'c"   and b :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : b \<rightarrow>\<^sub>C a\<guillemotright>" shows "HomC.\<psi> (b, a) (inC f) = f"
string_rel_order_map: fixes x :: "String.literal"   and a :: "char list"   and y :: "String.literal"   and aa :: "char list" assumes "(x, a) \<in> string_rel"   and "(y, aa) \<in> string_rel" shows "(x \<le> y) = (a \<le> aa)"
phi_hom: fixes g1 :: "'a"   and g2 :: "'a" assumes "g1 \<in> carrier G"   and "g2 \<in> carrier G" shows "\<phi> (g1 \<otimes> g2) = \<phi> g1 \<otimes>\<^bsub>H\<^esub> \<phi> g2"
tab_def: shows "Tfg.\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1)"
eval\<^sub>\<sigma>: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "(cms\<^sub>1 ! k, subst \<sigma> (fst (mems ! k))) \<leadsto> (cms\<^sub>1' ! k, subst (g1 \<sigma>) mem\<^sub>1')"
from_list_correct_3: fixes l :: "(('v \<times> 'v) \<times> nat) list" assumes "valid_wgraph_repr l" shows "edges (\<alpha>g (from_list l)) = fst ` set l \<union> (fst ` set l)\<inverse>"
nsqn\<^sub>r_invalidate: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "dip \<in> dom dests" shows "nsqn\<^sub>r (the (invalidate rt dests dip)) = the (dests dip) - 1"
P'_def: shows "P' = (\<lambda>m. if prime (m + 1) \<and> m dvd 2 * n then 1 else 0)"
disjoint_insert_1: fixes B :: "'b set"   and a :: "'b"   and A :: "'b set" shows "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
thm_cont_propos_2: fixes v :: "i"   and p :: "\<o>" shows "[Contingent p \<^bold>\<equiv> \<^bold>\<diamond>p \<^bold>& \<^bold>\<diamond>\<^bold>\<not>p in v]"
of_nat_prod_mset: fixes A :: "nat multiset" shows "of_nat (\<Prod>\<^sub># A) = \<Prod>\<^sub># (image_mset of_nat A)"
assms_1: shows "is_arith_prog_on l start step a b"
deadlockD3: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'a" assumes "multithreaded_base.deadlock final r s"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" shows "\<forall>w. wset s t \<noteq> \<lfloor>PostWS w\<rfloor>"
simps: fixes args :: "Monitor.args"   and X :: "'a option list set"   and t :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping" shows "gc_join_mmsaux args X (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = (if right (args_ivl args) < enat (t - gc) then join_mmsaux args X (gc_mmsaux (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) else join_mmsaux args X (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since))"
pc: fixes k :: "real" shows "piecewise_continuous_on 0 k (I k) f"
ts_j: shows "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)"
ale_trans: fixes i :: "ant"   and j :: "ant"   and k :: "ant" assumes "i \<le> j"   and "j \<le> k" shows "i \<le> k"
A\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
wp_loop_step_sound: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "sound P" shows "sound (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P)"
assms: shows "(sig_red (\<prec>\<^sub>t) top_tail F)\<^sup>*\<^sup>* p q"
vector_meet_comp_x': fixes x :: "'a" shows "x ; 1 ; x = x ; 1 \<cdot> 1 ; x"
fls_times_nth_4: fixes f :: "'a fls"   and g :: "'a fls"   and n :: "int" shows "fls_nth (f * g) n = (\<Sum>i = 0..n - (fls_subdegree f + fls_subdegree g). fls_nth f (fls_subdegree f + i) * fls_nth g (n - fls_subdegree f - i))"
length_eq_distinct: fixes xs :: "'d list"   and ys :: "'d list" assumes "set xs = set ys"   and "distinct xs"   and "length xs = length ys" shows "distinct ys"
ltl_to_rabin_correct_exposed': fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_FG_to_rabin \<Sigma> \<G> w" shows "\<PP>\<^sub>\<infinity> \<phi> \<G> w = mojmir_def.accept \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q}"
fv\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('f, 'e) stateful_strand_step list"   and B :: "('f, 'e) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t (A @ B) = fv\<^sub>s\<^sub>s\<^sub>t A \<union> fv\<^sub>s\<^sub>s\<^sub>t B"
is_certified_associated_string_unique: fixes M :: "('c, 'd) message set"   and s :: "'e + ('c, 'd option) insert_message list"   and t :: "'e + ('c, 'd option) insert_message list" assumes "is_certified_associated_string M s"   and "is_certified_associated_string M t" shows "s = t"
comp_by_index_inj2: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "comp_by_index x1 y1 = comp_by_index x2 y2" shows "y1 = y2"
degree_pderiv: fixes p :: "'b poly" shows "degree (pderiv p) = degree p - 1"
A: shows "A = four_block_mat B (0\<^sub>m n m) C D"
set_def: fixes t :: "'b trie" shows "Trie_Fun.set t = {xs. Trie_Fun.isin t xs}"
less_multiset_less_multiset\<^sub>H\<^sub>O: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M < N) = less_multiset\<^sub>H\<^sub>O M N"
2: fixes \<Gamma> :: "'a formula multiset" assumes "is_nnf_mset \<Gamma>"   and "mset (concat (map (\<lambda>c. map (\<lambda>d. disj_of_clause c \<^bold>\<or> disj_of_clause d) (cnf_lists G)) (cnf_lists F))) + \<Gamma> \<Rightarrow>\<^sub>n" shows "image_mset disj_of_clause (mset (concat (map (\<lambda>f. map ((@) f) (cnf_lists G)) (cnf_lists F)))) + \<Gamma> \<Rightarrow>\<^sub>n"
return_bind: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'c + 'd" shows "Inr x \<bind> f = f x"
rjoin: fixes r :: "'a" shows "redex (Rjoin (VE (Rid r)))"
dominates_refl': fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "old.dominates g n n"
eval_poly_poly: fixes p :: "'c poly"   and x :: "'c" shows "eval_poly (\<lambda>a. [:a:]) p [:x:] = [:poly p x:]"
13: shows "i < row_length (A1 \<circ> A2) * row_length (B1 \<circ> B2)"
that_1: assumes "\<bar>y\<bar> = y" shows "thesis"
rep_eq: fixes x :: "'a"   and xa :: "'a fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
incomplete_imp_proper_subset: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "bl \<subset> \<V>"
inject_3: fixes x31 :: "'a convnet"   and x32 :: "'a convnet"   and y31 :: "'a convnet"   and y32 :: "'a convnet" shows "(Pool x31 x32 = Pool y31 y32) = (x31 = y31 \<and> x32 = y32)"
fds_nth_fds_moebius_mu: shows "($) (fds moebius_mu) = moebius_mu"
Aexp_Var_singleton: fixes x :: "Vars.var"   and n :: "Arity" shows "esing x\<cdot>(up\<cdot>n) \<sqsubseteq> \<A>\<^bsub>n\<^esub> (Var x)"
configuration_Max_in: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {}" shows "Max (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n) \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"
inj_on_iff_card_le: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (card A \<le> card B)"
tendsto_add_zero: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" assumes "(f \<longlongrightarrow> (0::'d)) F"   and "(g \<longlongrightarrow> (0::'d)) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> (0::'d)) F"
always_different_outputs_direct_subsumption_def: fixes m1 :: "((nat \<times> nat) \<times> transition) fset"   and m2 :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and s' :: "nat"   and t2 :: "'a transition_scheme" shows "always_different_outputs_direct_subsumption m1 m2 s s' t2 = (\<exists>p c1 c. obtains s c1 m1 0 <> p \<and> obtains s' c m2 0 <> p \<and> (\<exists>i. can_take_transition t2 i c))"
inner_verts_with_proj_def: fixes G :: "'b pair_pre_digraph"   and p :: "('b \<times> 'b) list" shows "pre_digraph.inner_verts (with_proj G) p = tl (map fst p)"
scale_scale_measure: fixes r :: "ennreal"   and r' :: "ennreal"   and M :: "'b measure" shows "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"
ResB: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "y \<sharp> a"   and "y \<noteq> x" shows "<\<nu>y>P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> <\<nu>y>P'"
iT_Plus_neg_0: fixes I :: "nat set" shows "I \<oplus>- 0 = I"
prv_imp_neg_allWI: fixes \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "bprv (neg (subst \<chi> t x))" shows "bprv (neg (all x \<chi>))"
arrow_of_spans_axioms: shows "arrow_of_spans (\<odot>) (SPN r \<circ> SPN s)"
simps_7: fixes K\<^sub>1 :: "clock"   and n\<^sub>1 :: "nat"   and K\<^sub>2 :: "clock"   and n\<^sub>2 :: "nat"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool" shows "\<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n\<^sub>1), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n\<^sub>2)\<rfloor> \<in> R \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. R (time (Rep_run \<rho> n\<^sub>1 K\<^sub>1), time (Rep_run \<rho> n\<^sub>2 K\<^sub>2))}"
linordered_field_no_lb: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less y x"
normal_vector_span: fixes D :: "'a vec set" assumes "Suc (card D) = n"   and "D \<subseteq> carrier_vec n"   and "finite D"   and "lin_indpt D" shows "local.span D = {x \<in> carrier_vec n. x \<bullet> normal_vector D = (0::'a)}"
wf_ExtC: shows "wf_cdecl tprg (Ext, ExtCl)"
join_ir_double_exists: fixes v :: "vname"   and a :: "value"   and v' :: "vname" shows "\<exists>i r. join_ir i r v = Some a \<and> join_ir i r v' = Some a"
internal_iface_name_wildcard_longest_correct_None: assumes "internal_iface_name_wildcard_longest i1 i2 = None" shows "\<not> (internal_iface_name_match i1 p_i \<and> internal_iface_name_match i2 p_i)"
fls_residue_deriv_times_lr_inverse_eq_subdegree_4: fixes y :: "'a"   and f :: "'a fls" assumes "y * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "fls_residue (fls_right_inverse f y * fls_deriv f) = of_int (fls_subdegree f)"
in_countE: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" obtains n :: "nat" where "count M x = Suc n"
Rep_stone_phi_pair_inject: fixes x :: "'c stone_phi_pair"   and y :: "'c stone_phi_pair" shows "(Rep_stone_phi_pair x = Rep_stone_phi_pair y) = (x = y)"
distinct_251: shows "BinAnd \<noteq> ShiftRightSigned"
rreq_rrep_sn_simps_1: fixes hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "rreq_rrep_sn (Rreq hops rreqid dip dsn dsk oip osn sip handled) = (1 \<le> osn)"
override_idem_implies_vwb: fixes X :: "'a \<Longrightarrow> 'b" assumes "mwb_lens X"   and "\<And>s. s \<triangleleft>\<^bsub>X\<^esub> s = s" shows "vwb_lens X"
list_all2_ConsL: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and xs' :: "'c list"   and ys :: "'d list" assumes "list_all2 R (x # xs') ys" shows "\<exists>y ys'. ys = y # ys' \<and> R x y \<and> list_all2 R xs' ys'"
r'_def: shows "r' = rec_nat (r 0, 0) (\<lambda>i (q, n). (r (Suc i), if n \<in> acc q then (n + 1) mod num_acc else n))"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'b" shows "(a *s x = (0::'b)) = (a = (0::'a) \<or> x = (0::'b))"
StartBallot_HInv3_nL: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "StartBallot s s' t"   and "\<not> HInv3_L s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "\<not> HInv3_L s' p q d"
injI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
otherassms_4: shows "l' \<in> labels \<Gamma> q"
ord_isom_bij_to: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Order E"   and "ord_isom D E f" shows "bij_to f (carrier D) (carrier E)"
t_stop_pos: assumes "other.s' \<delta> \<noteq> 0" shows "0 < delayed_safe_distance.other.t_stop"
progress_L0: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and cs :: "e list"   and es_c :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "const_list cs" shows "\<lparr>s;vs;cs @ es @ es_c\<rparr> \<leadsto>_ i \<lparr>s';vs';cs @ es' @ es_c\<rparr>"
distinct_sorted_merge: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "class.linorder less_eq less"   and "distinct xs"   and "distinct ys"   and "linorder.sorted less_eq xs"   and "linorder.sorted less_eq ys" shows "distinct (linorder.merge less xs ys) \<and> linorder.sorted less_eq (linorder.merge less xs ys)"
a_comm: fixes G :: "('d, 'e) ring_scheme"   and x :: "'d"   and y :: "'d" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
set_8: fixes x8 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Rjoin x8) = RID\<^sub>E x8"
Int_left_absorb: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
interval_integral_Ioo: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<bar>a\<bar> < \<infinity>"   and "\<bar>b\<bar> < \<infinity>" shows "interval_lebesgue_integral lborel a b f = set_lebesgue_integral lborel {real_of_ereal a<..<real_of_ereal b} f"
rep_eq: shows "(*\<^sub>v) 0 = (\<lambda>x. 0::'d)"
Infinitesimal_add_cancel: fixes y :: "'a star"   and x :: "'a star"   and z :: "'a star" assumes "y \<in> Infinitesimal"   and "x + y \<approx> z" shows "x \<approx> z"
Gauss_Jordan_upt_k_det_P_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k_det_P A k = (let foldl = foldl Gauss_Jordan_column_k_det_P (1::'b, 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
bal_up\<^sub>i_tree: fixes t :: "'a up\<^sub>i" shows "bal_up\<^sub>i t = bal (tree\<^sub>i t)"
abs_eq: fixes xa :: "'e \<Rightarrow> 'c"   and x :: "'d \<Longrightarrow> 'e" shows "mk\<^sub>e xa \<restriction>\<^sub>e x = mk\<^sub>e (\<lambda>b. xa (create\<^bsub>x\<^esub> b))"
mult2_alt_ns_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_alt_ns ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
equivclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "equivclp r a b"   and "P a"   and "\<And>y z. \<lbrakk>equivclp r a y; r y z \<or> r z y; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
med0sk1iar_def: fixes t :: "skl1_state" shows "med0sk1iar t \<equiv> \<lparr>signals = signalsResp t\<rparr>"
distinct_1: fixes x11 :: "nat"   and x12 :: "'c"   and x13 :: "nat set"   and x21 :: "'c net_state"   and x22 :: "'c net_state" shows "NodeS x11 x12 x13 \<noteq> SubnetS x21 x22"
seq_def: fixes P :: "'c process"   and Q :: "'c process" shows "P `;` Q \<equiv> Abs_process ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)"
eval_member: fixes xq :: "'b Predicate.seq" shows "Predicate.member xq = pred.eval (Predicate.pred_of_seq xq)"
finite_\<R>: shows "finite \<R>\<^sub>\<beta>"
Int_Inter_eq_2: fixes \<B> :: "'c set set"   and A :: "'c set" shows "\<Inter> \<B> \<inter> A = (if \<B> = {} then A else \<Inter>B\<in>\<B>. B \<inter> A)"
imp: fixes g :: "'c \<Rightarrow> 'd"   and Q :: "'d \<Rightarrow> bool" assumes "bij g"   and "\<exists>!x. Q (g x)" shows "\<exists>!y. Q y"
intro: fixes x :: "'b \<Longrightarrow> 'a" assumes "\<And>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u" shows "mwb_lens_axioms x"
sign_def: fixes K :: "nat"   and X :: "msg" shows "sign K X = \<lbrace>X, Crypt K (Hash X)\<rbrace>"
q_def: shows "q = drop (Suc (length (butlast (prio_emb_pos prio (prio_emb_step prio t s) s)))) (butlast (prio_emb_pos prio t s))"
always_imp_trans: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and R :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<box>(\<lambda>s. P s \<longrightarrow> Q s)) \<sigma>"   and "(\<box>(\<lambda>s. Q s \<longrightarrow> R s)) \<sigma>" shows "(\<box>(\<lambda>s. P s \<longrightarrow> R s)) \<sigma>"
bt_dfs_distinct_children: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and x :: "('d, 'e) bintree" assumes "distinct (bt_dfs f x)" shows "distinct (bts_dfs f (children x))"
cong_less_imp_eq_int: fixes a :: "int"   and m :: "int"   and b :: "int" assumes "0 \<le> a"   and "a < m"   and "0 \<le> b"   and "b < m"   and "[a = b] (mod m)" shows "a = b"
simps_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.Derivation \<NN> \<TT> \<RR> a (d # D) b = (\<exists>x. CFG.Derives1 \<NN> \<TT> \<RR> a (fst d) (snd d) x \<and> CFG.Derivation \<NN> \<TT> \<RR> x D b)"
assms: shows "x \<in> configuration q\<^sub>0 (Suc n)"
loose_bvar1_eta_norm: fixes t :: "Core.term"   and lev :: "nat" assumes "loose_bvar1 t lev" shows "loose_bvar1 (eta_norm t) lev"