hcju/isabelleps huggingface dataset:
weakPsiCongParComm: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<doteq> Q \<parallel> P"
vfsequence_of_vlist_vlist_of_vfsequence: fixes xs :: "V" assumes "vfsequence xs" shows "vfsequence_of_vlist (vlist_of_vfsequence xs) = xs"
list_comb: fixes f :: "'a"   and xs :: "'a list" shows "P (list_comb f xs) = (P f \<and> list_all P xs)"
assms_4: shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
tag_name_eq_h: fixes ptr' :: "(_) element_ptr"   and disc_nodes :: "char list" shows "h \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes = h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
approx_diff: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star"   and d :: "'a star" assumes "a \<approx> b"   and "c \<approx> d" shows "a - c \<approx> b - d"
set_4: fixes x21 :: "('b, 'a) tm"   and x22 :: "('b, 'a) tm" shows "vars (App x21 x22) = vars x21 \<union> vars x22"
subseq_to_smaller: fixes j :: "nat"   and i :: "nat"   and w :: "nat \<Rightarrow> 'a" assumes "j \<le> i" shows "w [i \<rightarrow> j] = []"
ordinal_0_exp: fixes x :: "ordinal" shows "0 ** x = (if x = 0 then 1 else 0)"
simps_2: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d" shows "add_effects_of \<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = add_effects_of"
elims: fixes x :: "'c net_state"   and y :: "nat set" assumes "net_ips x = y" obtains i :: "nat" and s :: "'c" and R :: "nat set" where "x = NodeS i s R" and "y = {i}"   | n1 :: "'c net_state" and n2 :: "'c net_state" where "x = SubnetS n1 n2" and "y = net_ips n1 \<union> net_ips n2"
rewrite_negated_primitives_not_has_disc_negated: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and negate_f :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "has_disc_negated disc False m \<Longrightarrow> \<forall>pts. \<not> has_disc_negated disc False (negate_f C pts)" shows "\<not> has_disc_negated disc False (rewrite_negated_primitives (disc, sel) C negate_f m)"
InitSeqThrowReds: fixes C :: "char list"   and b :: "bool"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp" assumes "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> unit,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
subset_val_process': fixes A :: "'c set"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and n :: "nat"   and w :: "'d" assumes "finite A"   and "support_set p \<subseteq> A" shows "val_process Mkt p n w = (\<Sum>x\<in>A. prices Mkt x n w * p x (Suc n) w)"
INF_lower: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A" shows "Inf (f ` A) \<le> f i"
jordan_nf_smult: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and k :: "'b" assumes "jordan_nf A n_as"   and "k \<noteq> (0::'b)" shows "jordan_nf (k \<cdot>\<^sub>m A) (map (\<lambda>(n, a). (n, k * a)) n_as)"
ASSt: fixes x :: "char list"   and e :: "aexp"   and s :: "char list \<Rightarrow> int" shows "\<down>\<^sub>s (x ::= e, s) = s[e/x]"
chine_hcomp_ide_src: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "chine_hcomp f (src f) = Leg0 (Dom f) \<down>\<down> C.cod (Leg0 (Dom f))"
indep_sets_def: fixes F :: "'b \<Rightarrow> 'a set set"   and I :: "'b set" shows "indep_sets F I = ((\<forall>i\<in>I. F i \<subseteq> events) \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j)))))"
f_Equiv_Exec_Stream_expand_shrink_output_set_eq: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "'h list" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "\<And>input0 m. \<lbrakk>set input0 \<subseteq> A; m \<in> A\<rbrakk> \<Longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input0 \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input0 \<odot> k2) c2)"   and "set input \<subseteq> A" shows "map output_fun1 (f_Exec_Comp_Stream trans_fun1 (map input_fun1 input \<odot> k1) c1) \<div> k1 = map output_fun2 (f_Exec_Comp_Stream trans_fun2 (map input_fun2 input \<odot> k2) c2) \<div> k2"
proper_interval_set_Compl_aux_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and n :: "nat"   and s1 :: "'d"   and s2 :: "'e" shows "proper_interval_set_Compl_aux_fusion g1 g2 ao n s1 s2 = proper_interval_set_Compl_aux ao n (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
fork_sfoldl: fixes f1 :: "'s \<rightarrow> 'r \<rightarrow> 's"   and z1 :: "'s"   and f2 :: "'t \<rightarrow> 'r \<rightarrow> 't"   and z2 :: "'t" shows "sfoldl\<cdot>f1\<cdot>z1 && sfoldl\<cdot>f2\<cdot>z2 = sfoldl\<cdot>(\<Lambda> (a, b) z. (f1\<cdot>a\<cdot>z, f2\<cdot>b\<cdot>z))\<cdot>(z1, z2)"
sdiv_int_range: fixes a :: "int"   and b :: "int" shows "a sdiv b \<in> {- \<bar>a\<bar>..\<bar>a\<bar>}"
ardenable_def: fixes rhs :: "'b trm set" shows "ardenable rhs \<equiv> \<forall>Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> lang r"
assms_1: fixes f :: "'a" assumes "f \<in># F" shows "irreducible f"
diff_def: fixes x :: "'a"   and y :: "'a" shows "x - y = Abs ((Rep x - Rep y) mod n)"
static_bounds_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" shows "static_bounds a b x y = (let mx = maxne0_impl y b; my = maxne0_impl x a in (\<forall>x\<in>set x. x \<le> mx) \<and> (\<forall>y\<in>set y. y \<le> my))"
wf_red: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "\<not> waiting (wset s t)" shows "\<exists>ta' x' m'. t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle> \<and> (actions_ok s t ta' \<or> actions_ok' s t ta' \<and> actions_subset ta' ta)"
nrename_OrR2: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[x\<turnstile>n>y] = OrR2 <c>.N d"   and "c \<sharp> (R, d)" shows "\<exists>N'. R = OrR2 <c>.N' d \<and> N'[x\<turnstile>n>y] = N"
alog_spec: fixes fs :: "('a \<times> 'a) list"   and keeps :: "('a \<times> 'a) list" shows "TopoS_Interface_impl.minimalize_offending_overapprox (\<lambda>G. nm_sinvar m G nP) fs keeps G = TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. sinvar_spec G nP) fs keeps (list_graph_to_graph G)"
incomp_lcp_len: fixes u :: "'b list"   and v :: "'b list" assumes "\<not> u \<bowtie> v" shows "\<^bold>|u \<and>\<^sub>p v\<^bold>| < min \<^bold>|u\<^bold>| \<^bold>|v\<^bold>|"
set_baliL: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.set_tree (baliL l a r) = Tree2.set_tree l \<union> {a} \<union> Tree2.set_tree r"
inf_selective: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = x \<or> x \<sqinter> y = y"
esize_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "esize (f ` A) = esize A"
rhs_def: shows "rhs \<equiv> nth_cons (flatten (\<lambda>s'. s') (upto_prod 17) (upto_prod z)) (2, None) 8"
that: fixes f :: "'a \<Rightarrow> nat" assumes "inj_on f S" shows "thesis"
step_3: shows "\<I>' \<turnstile>g gpv \<surd>"
less_imp_le: fixes y :: "'d"   and x :: "'d" assumes "y < x" shows "y \<le> x"
subst_apply_strand_step_def: fixes s :: "('c, 'd) strand_step"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "s \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = (case s of send\<langle>t\<rangle>\<^sub>s\<^sub>t \<Rightarrow> send\<langle>t \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t | receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<Rightarrow> receive\<langle>t \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t | \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t \<Rightarrow> \<langle>a: (t \<cdot> \<theta>) \<doteq> t' \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t | \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<Rightarrow> \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>)\<rangle>\<^sub>s\<^sub>t)"
numVals_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and x :: "nat" shows "ast_problem.numVals problem x \<equiv> length (snd (snd (ast_problem.astDom problem ! x)))"
select_convs_3: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_src \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_src"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs"   and "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"
LID\<^sub>L_def: fixes s :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" shows "LID\<^sub>L s \<equiv> case s of (\<sigma>, \<tau>, e) \<Rightarrow> LID\<^sub>S \<sigma> \<union> LID\<^sub>S \<tau> \<union> LID\<^sub>E e"
Suc: shows "(\<Sum>k\<le>n. of_nat (stirling n k) * x ^ k) = pochhammer x n"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "\<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
llist_le_def: fixes s :: "'b llist"   and t :: "'b llist" shows "(s \<le> t) = (\<exists>d. t = s @@ d)"
add_match_def: fixes m :: "'b match_expr"   and rs :: "'b rule list" shows "add_match m rs = map (\<lambda>r. case r of Rule m' a' \<Rightarrow> Rule (MatchAnd m m') a') rs"
match_type: fixes p :: "pat"   and U :: "ty"   and \<Delta> :: "(name \<times> ty) list"   and \<Gamma>\<^sub>2 :: "(name \<times> ty) list"   and u :: "trm"   and t :: "trm"   and T :: "ty"   and \<theta> :: "(name \<times> trm) list" assumes "\<turnstile> p : U \<Rightarrow> \<Delta>"   and "\<Gamma>\<^sub>2 \<turnstile> u : U"   and "\<Delta> @ \<Gamma>\<^sub>2 \<turnstile> t : T"   and "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>"   and "supp p \<sharp>* u" shows "\<Gamma>\<^sub>2 \<turnstile> \<theta>\<lparr>t\<rparr> : T"
lincomb_index: fixes i :: "nat"   and X :: "rat vec set"   and a :: "rat vec \<Rightarrow> rat" assumes "i < n"   and "X \<subseteq> Rn" shows "gs.lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
distribute_fixpoint_over_setmonotone_comp: fixes f :: "'b set \<Rightarrow> 'b set"   and g :: "'b set \<Rightarrow> 'b set"   and I :: "'b set" assumes "setmonotone f"   and "setmonotone g"   and "(f \<circ> g) I = I" shows "f I = I \<and> g I = I"
od_fm_fun_Chain: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "Chain (od_fm_fun R M N f H) C"   and "C \<noteq> {}" shows "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a"
prems: shows "set (x # Vs) \<subseteq> carrier_vec n"
if_SE_def: fixes c :: "'b \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<times> 'b) option"   and F :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "_if_SE c E F = (\<lambda>\<sigma>. if c \<sigma> then E \<sigma> else F \<sigma>)"
Sum_pmf_def: fixes p :: "real"   and Da :: "'a pmf"   and Db :: "'b pmf" shows "Sum_pmf p Da Db = bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)"
mkClockAuto_def: shows "mkClockAuto \<equiv> \<lambda>agents jkbp envInit envAction envTrans envVal envObs. mkAlgAuto acts_MapOps trans_MapOps (ClockView.clock_simObs envObs) (ClockView.clock_simInit envInit envObs) (ClockView.clock_simTrans agents jkbp envAction envTrans envVal envObs) (ClockView.clock_simAction jkbp envVal envObs) (\<lambda>a. map (ClockView.clock_simInit envInit envObs a \<circ> envObs a) envInit)"
fls_regpart_def: shows "fls_regpart \<equiv> map_fun fls_nth id (\<lambda>f. Abs_fps (\<lambda>n. f (int n)))"
validCE_act: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. active cid (t i)"   and "dynamic_component.eval tCMP active cid t t' n \<gamma>" shows "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (the_enat (dynamic_component.nAct tCMP active cid (enat n) (inf_llist t)))"
rep_eq: fixes x :: "'c \<Rightarrow> bool"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmfilter x xa) = map_filter x (fmlookup xa)"
finsum_smult: fixes g :: "'b \<Rightarrow> 'a vec"   and A :: "'b set"   and c :: "'a" assumes "True"   and "g \<in> A \<rightarrow> carrier_vec n" shows "c \<cdot>\<^sub>v finsum V g A = (\<Oplus>\<^bsub>V\<^esub>x\<in>A. c \<cdot>\<^sub>v g x)"
unitary11_gen_iff': fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary11_gen M = (\<exists>a b k. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> (M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a) \<or> M = k *\<^sub>s\<^sub>m (- 1, 0, 0, 1) *\<^sub>m\<^sub>m (a, b, cnj b, cnj a)))"
suffix_Nil: fixes xs :: "'a list" shows "suffix xs [] = (xs = [])"
j_less_k: shows "mod_type_class.to_nat j < k"
simps: fixes p :: "nat"   and m :: "int"   and e :: "int" shows "showsp_float10 p (m \<e> e) = (let ds = digits (nat \<bar>m\<bar>) []; d = int (length ds); e = e + d - 1; mp = take 1 ds; ms = drop 1 ds; ms = rev (dropWhile ((=) 0) (rev ms)); show_digits = shows_list_gen (showsp_nat p) ''0'' [] [] [] in (if m < 0 then shows_string ''-'' else (\<lambda>x. x)) \<circ> show_digits mp \<circ> (if ms = [] then \<lambda>x. x else shows_string ''.'' \<circ> show_digits ms) \<circ> (if e = 0 then \<lambda>x. x else shows_string ''e'' \<circ> showsp_int p e))"
map_nth_eq_self: fixes fas :: "'c list"   and l :: "nat" assumes "length fas = l" shows "map ((!) fas) [0..<l] = fas"
RecvMarker_implies_Marker_in_set: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "ev = RecvMarker cid p q" shows "Marker \<in> set (msgs c cid)"
bind_output_cong: fixes x :: "nat"   and b :: "nat"   and ab :: "nat"   and aa :: "nat"   and ac :: "nat" assumes "x < order \<G>" shows "(x = nat ((int b - int ab) * Number_Theory_Aux.inverse (aa - ac) (order \<G>) mod int (order \<G>))) = [x = nat ((int b - int ab) * Number_Theory_Aux.inverse (aa - ac) (order \<G>) mod int (order \<G>))] (mod order \<G>)"
triangle_permutes_2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "MinkowskiPrimitive.kinematic_triangle \<E> \<P> b a c"
nth_equalityI: fixes xs :: "'d list"   and ys :: "'d list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
res_lim_pushforward': fixes p :: "int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_continuous p f"   and "padic_integers.is_Zp_cauchy p s"   and "t = alt_seq s" shows "res_lim (f \<circ> s) = res_lim (f \<circ> t)"
closure_Un: fixes S :: "'b set"   and T :: "'b set" shows "closure (S \<union> T) = closure S \<union> closure T"
rel_mapping_map_upd: fixes Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and P :: "'c \<Rightarrow> 'a option"   and P' :: "'c \<Rightarrow> 'b option"   and p :: "'c" assumes "Q x y"   and "rel_mapping Q P P'" shows "rel_mapping Q (P(p \<mapsto> x)) (P'(p \<mapsto> y))"
simps_2: fixes m :: "'b SecurityInvariant_configured"   and Ms :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "generate_valid_topology_SOME (m # Ms) G = (if c_sinvar m G then generate_valid_topology_SOME Ms G else delete_edges (generate_valid_topology_SOME Ms G) (SOME F. F \<in> c_offending_flows m G))"
surj: shows "\<forall>y. \<exists>n. y = f n"
lift_p_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "('c, 'd) state" shows "lift_p P f = (\<forall>heap. P heap \<longrightarrow> (case run_state f heap of (x, heap) \<Rightarrow> P heap))"
map_2: fixes f1 :: "'f \<Rightarrow> 'b"   and f2 :: "'g \<Rightarrow> 'd"   and f3 :: "'h \<Rightarrow> 'e"   and x21 :: "'g"   and x22 :: "'h" shows "map_generat f1 f2 f3 (IO x21 x22) = IO (f2 x21) (f3 x22)"
select_convs_2: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "discovered \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = discovered"
subgraphI: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "verts H \<subseteq> verts G"   and "arcs H \<subseteq> arcs G"   and "compatible G H"   and "wf_digraph H"   and "wf_digraph G" shows "subgraph H G"
tl_append: fixes xs :: "'g list"   and ys :: "'g list" shows "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z # zs \<Rightarrow> zs @ ys)"
assms_5: shows "(G ===> (=)) inEdges' inEdges2"
limpt_sequential_inj: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (\<exists>f. (\<forall>n. f n \<in> S - {x}) \<and> inj f \<and> f \<longlonglongrightarrow> x)"
prems_3: shows "subst_domain (\<sigma>1(v := t)) \<inter> range_vars \<theta>2 = {}"
actionI: fixes A :: "'c \<times> 'd \<Rightarrow> bool" assumes "\<And>s t. A (s, t)" shows "\<turnstile> A"
rel_witness_converter1: fixes A :: "'a \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'i \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and R' :: "'j \<Rightarrow> 'h \<Rightarrow> bool"   and conv :: "('a, 'c, 'e, 'g) converter"   and conv' :: "('b, 'd, 'f, 'h) converter" assumes "rel_converter (A OO A') B C (R OO R') conv conv'"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'"   and "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_converter A (\<lambda>b (b', c). b = b' \<and> B b' c) (\<lambda>c (c', d). c = c' \<and> C c' d) R conv (rel_witness_converter A A' B C R R' (conv, conv'))"
mm_equiv_\<C>_eq: fixes a :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and b :: "'Var \<Rightarrow> 'Val"   and a' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and b' :: "'Var \<Rightarrow> 'Val" assumes "(a, b) \<approx> (a', b')"   and "snd a = snd a'" shows "\<forall>x\<in>\<C>. b x = b' x"
simps: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and a :: "('i \<Rightarrow> 'k) \<times> ('l \<Rightarrow> 'j)" shows "wiring \<I> \<I>' cnv a = (\<exists>f g. a = (f, g) \<and> \<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C \<and> \<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>)"
float_plus_comm_eq: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b" shows "a + b = b + a"
last_list_update: fixes xs :: "'b list"   and k :: "nat"   and x :: "'b" assumes "xs \<noteq> []" shows "last (xs[k := x]) = (if k = length xs - 1 then x else last xs)"
3: shows "Min (PLS s as) \<le> length p - 1"
gState_progress_rel_trans: fixes x :: "'a gState_scheme"   and y :: "'a gState_scheme"   and prog :: "program"   and z :: "'a gState_scheme" assumes "(x, y) \<in> gState_progress_rel prog"   and "(y, z) \<in> gState_progress_rel prog" shows "(x, z) \<in> gState_progress_rel prog"
nth_solution_complete: fixes D :: "nat"   and z :: "nat \<times> nat" assumes "pell D"   and "pell.solution D z" shows "z \<in> range (pell.nth_solution D)"