hcju/isabelleps huggingface dataset:
nfa_of_dfa_accepts: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and bss :: "bool list list" assumes "wf_dfa A n"   and "list_all (is_alph n) bss" shows "nfa_accepts (nfa_of_dfa A) bss = dfa_accepts A bss"
symmetry_rule_def: fixes t :: "'b" shows "symmetry_rule t = transl_rule (A_Cnv (A_Lbl t) \<sqsubseteq> A_Lbl t)"
parts_crypt_pubkey: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt (SesK SK) (PubKey C) \<in> parts (used s)" shows "C \<in> snd (snd SK) \<and> ((\<exists>n. (Owner n, SesKey SK) \<in> s) \<or> (\<exists>n B. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s)) \<or> SesKey SK \<in> spied s"
assms: assumes "\<Phi>" shows "m \<le>\<^sub>n m'"
Reals_minus: fixes a :: "'b" assumes "a \<in> \<real>" shows "- a \<in> \<real>"
appendl_correct_2: fixes s :: "'queue"   and x :: "'t" assumes "queue_invar s" shows "queue_invar (queue_push x s)"
c\<^sub>2'_props: shows "\<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<approx> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
card_1_singleton_iff: fixes A :: "'e set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
is_interval_translation: fixes x :: "'b"   and X :: "'b set" shows "is_interval ((+) x ` X) = is_interval X"
project_leadsTo_D_lemma: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> stable C"   and "F \<squnion> project h C G \<in> project_set h C \<inter> A \<longmapsto> B" shows "extend h F \<squnion> G \<in> C \<inter> extend_set h A \<longmapsto> extend_set h B"
REFC: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R" shows "c \<sigma> = c' \<sigma>'"
lengthLyx: shows "length (Lxy init {y, x}) = 2"
del_unEdge_valid: fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "valid_unMultigraph g" shows "valid_unMultigraph (del_unEdge v e v' g)"
id: shows "Suc (length r) - length d = Suc (length r - length d)"
expansion_level_real_def: fixes uu :: "real itself" shows "expansion_level uu = 0"
types_preserved_return: fixes n :: "nat"   and i :: "nat"   and vls :: "v list"   and LI :: "e list"   and ves :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and j :: "nat"   and lholed :: "Lholed" assumes "\<lparr>[Local n i vls LI]\<rparr> \<leadsto> \<lparr>ves\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [Local n i vls LI] : ts _> ts'"   and "const_list ves"   and "length ves = n"   and "Lfilled j lholed (ves @ [$Return]) LI" shows "\<S>\<bullet>\<C> \<turnstile> ves : ts _> ts'"
HeronConf_interp_stepwise_implies_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
uminus_set_def: fixes A :: "'e set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
sum_sum_mod_div: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and y :: "nat"   and x :: "nat" shows "(\<Sum>ia = 0..<x. sum (f ia) {0..<y}) = (\<Sum>ia = 0..<x * y. f (ia div y) (ia mod y))"
adj_msog_def: fixes xs :: "(accent \<times> 'a) list"   and zs :: "(accent \<times> 'a) list"   and l :: "(accent \<times> 'a) \<times> (accent \<times> 'a) list" shows "adj_msog xs zs l \<equiv> case l of (y, ys) \<Rightarrow> (y, case fst y of Acute \<Rightarrow> ys @ zs | Grave \<Rightarrow> xs @ ys | Macron \<Rightarrow> ys)"
pivot_positions_form_helper_2: fixes z :: "'b"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" shows "strict_sorted (map fst (pivot_positions_main_gen z A nr nc i j))"
id_rule: fixes c :: "'a"   and I :: "'b itself" assumes "c ::\<^sub>i I" shows "ID c c I"
language_coinduct: fixes R :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool"   and S :: "'a set"   and K :: "'a list set"   and L :: "'a list set" assumes "\<And>K L. R K L \<Longrightarrow> K \<subseteq> lists S \<and> L \<subseteq> lists S"   and "R K L"   and "\<And>K L. R K L \<Longrightarrow> ([] \<in> K) = ([] \<in> L)"   and "\<And>K L x. \<lbrakk>R K L; x \<in> S\<rbrakk> \<Longrightarrow> R (Deriv x K) (Deriv x L)" shows "K = L"
gt_trans: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and default :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "order_pair gt default"   and "gt x y"   and "gt y z" shows "gt x z"
eq_residue_mod: fixes f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and k :: "nat" assumes "f \<in> carrier Zp"   and "g \<in> carrier Zp"   and "f k = g k" shows "\<exists>h. h \<in> carrier Zp \<and> f = g \<oplus> [p] \<cdot> \<one> [^] k \<otimes> h"
integrable_uniform_limit_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" shows "f integrable_on {a..b}"
unique_session_keys: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and NA' :: "msg"   and B' :: "agent"   and X' :: "msg" assumes "Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Says Server A' (Crypt (shrK A') \<lbrace>NA', Agent B', Key K, X'\<rbrace>) \<in> set evs"   and "evs \<in> ns_shared" shows "A = A' \<and> NA = NA' \<and> B = B' \<and> X = X'"
dep2_def: fixes u :: "'b"   and v :: "'b" shows "dep2 u v \<equiv> \<exists>w r s. u = r *\<^sub>R w \<and> v = s *\<^sub>R w"
assms_5: shows "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'"
leg_simps_3: shows "C.arr src_\<mu>.cod.leg1"
assms_1: shows "\<guillemotleft>f : a' \<rightarrow>\<^sub>A a\<guillemotright>"
AnalBinds_delete_lookup: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "(FBinds (delete x \<Gamma>)\<cdot>ae) x = \<bottom>"
lisetI: fixes i :: "nat"   and w :: "'b llist"   and A :: "'b set" assumes "enat i < llength w"   and "w ?! i \<in> A" shows "i \<in> liset A w"
that: fixes p1 :: "('v \<times> 'v) list"   and u :: "'v"   and u' :: "'v"   and p2 :: "('v \<times> 'v) list" assumes "p = p1 @ (u, u') # p2"   and "int_vertices p1 \<subseteq> W"   and "u \<in> W"   and "u' \<notin> W" shows "thesis"
read_actions_actions: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> read_actions E" shows "a \<in> actions E"
Det_def: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P \<box> Q \<equiv> action_of (R (\<not> action.relation_of P\<^sup>f\<^sub>f \<and> \<not> action.relation_of Q\<^sup>f\<^sub>f \<turnstile> action.relation_of P\<^sup>t\<^sub>f \<and> action.relation_of Q\<^sup>t\<^sub>f \<triangleleft> \<lambda>(A, A'). tr A = tr A' \<and> wait A' \<triangleright>  action.relation_of P\<^sup>t\<^sub>f \<or> action.relation_of Q\<^sup>t\<^sub>f))"
poly_subst_trivial: fixes p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst (\<lambda>_. 0) p = monomial (lookup p 0) 0"
value_flow_cleanup: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Delta> :: "('k, 'l) network_scheme" assumes "\<And>x. f (x, source \<Delta>) = 0" shows "value_flow \<Delta> (cleanup f) = value_flow \<Delta> f"
rel_inject_2: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x21 :: "'i"   and x22 :: "'k"   and y21 :: "'j"   and y22 :: "'l" shows "rel_generat R1 R2 R3 (IO x21 x22) (IO y21 y22) = (R2 x21 y21 \<and> R3 x22 y22)"
Q_def: shows "Q \<equiv> distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
red_n_Base: fixes c :: "com"   and s :: "char list \<Rightarrow> val option" shows "\<langle>c,s\<rangle> \<rightarrow>\<^bsup>0\<^esup> \<langle>c,s\<rangle>"
impl_of_cases: fixes y :: "('a \<times> 'b) list" assumes "y \<in> {xs. (distinct \<circ> list.map fst) xs}" obtains x :: "('a, 'b) alist" where "y = impl_of x"
sign_r_pos_minus: fixes p :: "real poly"   and x :: "real" assumes "p \<noteq> 0" shows "sign_r_pos p x = (\<not> sign_r_pos (- p) x)"
real_add: fixes a :: "'a"   and b :: "'a" shows "real_of (a + b) = real_of a + real_of b"
simps: fixes k :: "'c"   and v :: "'d"   and m :: "'c \<Rightarrow> 'd option" shows "((k, v) \<in>\<^sub>m m) = (m k = Some v)"
dvd_div_mult: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c dvd b" shows "b div c * a = b * a div c"
GatewaySystem_L3: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "ServiceCenter i a"   and "GatewayReq req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "dt (Suc t) \<noteq> [] \<or> dt (Suc (Suc t)) \<noteq> []"   and "ack t = [init_state]"   and "req (Suc t) = [init]"   and "\<forall>t1\<le>t. req t1 = []"   and "\<forall>m\<le>k + 2. req (t + m) \<noteq> [send]"   and "req (t + 3 + k) = [send]"   and "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" shows "vc (2 * d + (t + (4 + k))) = [vc_com]"
ActualNotI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<not>\<^bold>\<A>\<phi> in v]" shows "[\<^bold>\<A>\<^bold>\<not>\<phi> in v]"
eval_simps_3: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "cod (hfsetcat.eval b c) = c"
fst: shows "pseudo_proj_True (Suc n) -` {w} \<subseteq> pseudo_proj_True n -` {w}"
ptrm_prm_apply_compose: fixes \<pi> :: "'b prm"   and \<sigma> :: "'b prm"   and X :: "'b ptrm" shows "\<pi> \<bullet> \<sigma> \<bullet> X = (\<pi> \<diamondop> \<sigma>) \<bullet> X"
gen_llength_def: fixes n :: "nat"   and xs :: "'c llist" shows "gen_llength n xs = enat n + llength xs"
REF0: shows "(\<sigma>0, \<sigma>0') \<in> R"
intros_2: fixes A :: "'b tensor"   and B :: "'b tensor"   and j :: "nat" assumes "dims A = dims B"   and "cprank_max1 A"   and "cprank_max j B" shows "cprank_max (Suc j) (A + B)"
AbstrLevels_S4opt_A22: fixes i :: "AbstrLevelsID" assumes "sS4opt \<in> AbstrLevel i" shows "sA22 \<notin> AbstrLevel i"
NetModel_node_props_def: fixes P :: "('a, SINVAR_BLPtrusted.node_config, 'b) TopoS_Params_scheme" shows "SINVAR_BLPtrusted_impl.NetModel_node_props P = (\<lambda>i. case node_properties P i of None \<Rightarrow> SINVAR_BLPtrusted.default_node_properties | Some property \<Rightarrow> property)"
One_fract_def: shows "1 = Fract (1::'a) (1::'a)"
bounded_clinear_mult_left: fixes y :: "'b" shows "bounded_clinear (\<lambda>x. x * y)"
fps_integral0_linear: fixes a :: "'a"   and f :: "'a fps"   and b :: "'a"   and g :: "'a fps" shows "fps_integral0 (fps_const a * f + fps_const b * g) = fps_const a * fps_integral0 f + fps_const b * fps_integral0 g"
Elem_nat2Nat_Nat: fixes n :: "nat" shows "Elem (nat2Nat n) HOLZF.Nat"
lnull_lzip: fixes xs :: "'c llist"   and ys :: "'d llist" shows "lnull (lzip xs ys) = (lnull xs \<or> lnull ys)"
pair_leq_def: shows "pair_leq = pair_less\<^sup>="
B: fixes x :: "'a" assumes "x \<in> X" shows "small (B x)"
before_first_time: fixes \<rho> :: "'b run"   and K :: "clock"   and n :: "nat"   and \<tau> :: "'b tag_const"   and m :: "nat" assumes "first_time \<rho> K n \<tau>"   and "m < n" shows "time (Rep_run \<rho> m K) < \<tau>"
cosh_add: fixes y :: "'a" shows "cosh (x + y) = cosh x * cosh y + sinh x * sinh y"
partial_function_lift: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lb :: "'c set \<Rightarrow> 'c" assumes "partial_function_definitions ord lb" shows "partial_function_definitions (fun_ord ord) (fun_lub lb)"
resolution_class_sum_card: fixes S :: "'a set set" assumes "resolution_class S" shows "int (sum card S) = \<v>"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>\<tau> f : F' a \<rightarrow>\<^sub>C F b\<guillemotright>"
span_scale: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and S :: "'d set"   and c :: "'c" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
asms_3: shows "\<not> list_all (\<lambda>x. x < i) xs"
v: shows "v \<in> vs1.span (vs1.extend_basis B - B)"
wlp_measure_expand: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and P :: "complex mat" assumes "well_com (Measure n M S)" shows "wlp (Measure n M S) P = matrix_sum d (\<lambda>k. adjoint (M k) * wlp (S ! k) P * M k) n"
middle: shows "arr_mset mod_arr (m - 1) m = {#Min_mset (arr_mset arr l m)#}"
simps_2: fixes p :: "perm"   and t :: "('b, 'c, 'd) Tree" shows "p \<bullet> tNot t = tNot (p \<bullet> t)"
conv_radius_nonneg: fixes f :: "nat \<Rightarrow> 'a" shows "0 \<le> conv_radius f"
ipresFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'd, 'i, 'j) model_scheme" shows "ipresFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> fresh ys y X \<longrightarrow> igFresh MOD ys y (h X)"
fun_dual5: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f = \<partial> \<circ> g \<circ> \<partial>) = (\<partial> \<circ> f \<circ> \<partial> = g)"
map_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "C.\<alpha>' f = local.inv (\<alpha>.map (CCC.cod f)) \<cdot> T.ToCT f"
hn_refine_alt: fixes Fpre :: "assn"   and c :: "'a Heap"   and Fpost :: "assn"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" shows "hn_refine Fpre c Fpost P m \<equiv> nofail m \<longrightarrow> <Fpre> c <\<lambda>r. hn_rel P m r * Fpost>\<^sub>t"
i_shrink_last_i_append_mod: fixes xs :: "'b list"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "length xs mod k = 0" shows "(xs \<frown> f) \<div>\<^sub>l k = xs \<div>\<^sub>l k \<frown> f \<div>\<^sub>l k"
convergent_const: fixes c :: "'d" shows "convergent (\<lambda>n. c)"
softSubst: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "softSubst \<phi> t x \<in> fmla"
rep_eq: fixes x :: "('a, 'b) lifted_distrib_lattice_top"   and xa :: "('a, 'b) lifted_distrib_lattice_top" shows "Rep_lifted_distrib_lattice_top (x \<sqinter> xa) = (\<lambda>f. Rep_lifted_distrib_lattice_top x f \<sqinter> Rep_lifted_distrib_lattice_top xa f)"
opt: fixes C' :: "'a set" assumes "finite C'"   and "vertex_cover E C'" shows "card C \<le> k * card C'"
zero_ext_gh: fixes y :: "'a" assumes "y \<in> GH - supp gh" shows "f (x - y) * gh y = (0::'b)"
EX: fixes v :: "'a"   and rr :: "'a list" assumes "s' = v # rr @ r"   and "((u # s, c), ee, v # rr, c') \<in> ntrp fg" shows "P"
initial_state_1: shows "LLL_impl_inv (initial_state m fs_init) 0 fs_init"
det_permute_columns: fixes p :: "'b \<Rightarrow> 'b"   and A :: "(('c, 'b) vec, 'b) vec" assumes "p permutes UNIV" shows "det (\<chi>i j. A $ i $ p j) = of_int (sign p) * det A"
oexp_def: shows "rs.oexp = {(f, g). f \<in> rs.FINFUNC \<and> g \<in> rs.FINFUNC \<and> ((let m = s.max_fun_diff f g in (f m, g m) \<in> r) \<or> f = g)}"
DynPDG_axioms: shows "DynPDG sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_) dyn_control_dependence"
elims_3: fixes x :: "'a word \<times> nat"   and xa :: "'a word" assumes "\<not> simple_match_ip x xa" obtains base :: "'a word" and len :: "nat" and p_ip :: "'a word" where "x = (base, len)" and "xa = p_ip" and "p_ip \<notin> ipset_from_cidr base len"
hyps: fixes s :: "nat set" assumes "enat i < llength (lselect s w)" shows "lselect s w ?! i = w ?! nth_least s i"
semilattice_order_set_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "semilattice_order_set f less_eq less \<equiv> semilattice_order f less_eq less \<and> semilattice_set f"
simple_integral_indicator_only: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "integral\<^sup>S M (indicator A) = emeasure M A"
contra_subsetD: fixes A :: "'g set"   and B :: "'g set"   and c :: "'g" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
list_all2_iff: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
generalized_fw_join_def: fixes l1 :: "('d simple_match \<times> 'e) list"   and l2 :: "('d simple_match \<times> 'f) list" shows "generalized_fw_join l1 l2 \<equiv> concat (map (\<lambda>(m1, a). concat (map (\<lambda>(m2, b). map (\<lambda>u. (u, a, b)) (option2list (simple_match_and m1 m2))) l2)) l1)"
image_poly_eval_indexed_var: fixes L :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and i :: "'c"   and q :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = M.indexed_const \<zero>\<^bsub>M\<^esub>"   and "M.indexed_const \<in> ring_hom M L"   and "q \<in> carrier (poly_ring L)"   and "1 < degree q"   and "pirreducible\<^bsub>L\<^esub> (carrier L) q" shows "ring.eval (image_ring (M.eval_pmod L i q) (poly_ring L)) q (M.indexed_var i) = \<zero>\<^bsub>image_ring (M.eval_pmod L i q) (poly_ring L)\<^esub>"
apos: fixes n :: "nat" shows "real n * T \<in> existence_ivl0 x \<and> flow0 x (real n * T) = x"
when_diff_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and P :: "bool" assumes "class.cancel_comm_monoid_add plus minus zero" shows "zero.when zero (minus a b) P = minus (zero.when zero a P) (zero.when zero b P)"
orderE: fixes a :: "nat"   and b :: "nat" assumes "a dvd b" obtains "a = gcd a b"
rel_fun_def_butlast: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and T :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g \<Rightarrow> 'h" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
v0_conforms: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>"   and "v0 \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)" shows "\<sigma> v0 = vmc_path_no_deadend.w0 P"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) (\<cdot>) replete_setcat.comp Adj.\<phi>C Adj.\<phi>D \<Delta>.map G \<phi> \<psi> Adj.\<eta> Adj.\<epsilon> Adj.\<Phi> Adj.\<Psi>"
geseq_iff: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and A :: "'b set" shows "((f, g) \<in> mbs.geseq A) = (f \<in> SEQ A \<and> g \<in> SEQ A \<and> (f = g \<or> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j))))"
induct: fixes x1 :: "nat"   and x2 :: "Lholed"   and x3 :: "e list"   and x4 :: "e list"   and P :: "nat \<Rightarrow> Lholed \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> bool" assumes "Lfilled_exact x1 x2 x3 x4"   and "\<And>lholed es. lholed = LBase [] [] \<Longrightarrow> P 0 lholed es es"   and "\<And>vs lholed n es' l es'' k es lfilledk. \<lbrakk>const_list vs; lholed = LRec vs n es' l es''; Lfilled_exact k l es lfilledk; P k l es lfilledk\<rbrakk> \<Longrightarrow> P (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')" shows "P x1 x2 x3 x4"
iprod_left_add_distrib: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "\<langle>xs + ys,zs\<rangle> = \<langle>xs,zs\<rangle> + \<langle>ys,zs\<rangle>"
Abs_contour_inverse: fixes y :: "nat list" assumes "y \<in> contour" shows "Rep_contour (Abs_contour y) = y"
prop_equiv_finite: fixes P :: "'a ltln set" assumes "finite P" shows "finite {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P}"
LCons: fixes P :: "'a prog"   and obs :: "('b \<times> ('c, 'b) obs_event action) list"   and ob :: "'b \<times> ('c, 'b) obs_event action"   and obs' :: "('b \<times> ('c, 'b) obs_event action) llist" assumes "ta_hb_consistent P (obs @ [ob]) obs'"   and "case ob of (t, NormalAction (ReadMem ad al v)) \<Rightarrow> \<exists>w. w \<in> write_actions (llist_of (obs @ [ob])) \<and> (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w \<and> value_written P (llist_of (obs @ [ob])) w (ad, al) = v \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb length obs \<and> (\<forall>w'\<in>write_actions (llist_of (obs @ [ob])). (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w' \<longrightarrow> P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>hb length obs \<or> is_volatile P al \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>so w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>so length obs \<longrightarrow> w' = w) | (t, NormalAction _) \<Rightarrow> True | (t, _) \<Rightarrow> True" shows "ta_hb_consistent P obs (LCons ob obs')"
streamValue3: fixes D :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and y :: "nat"   and x :: "'a"   and j :: "nat" assumes "\<forall>j\<le>D. str (t + j + Suc y) = x"   and "j \<le> D"   and "str (t + y) = x" shows "str (t + j + y) = x"
Chernoff_on_iia_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Chernoff_on A f = iia_on A f"
contractible_space_upper_hemisphere: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "contractible_space (subtopology (nsphere n) {x. 0 \<le> x k})"
concat_eq_Nil_conv: fixes xss :: "'a list list" shows "(concat xss = []) = (\<forall>xs\<in>list.set xss. xs = [])"
iT_Plus_neg_Plus_ge_cut_eq: fixes b :: "nat"   and a :: "nat"   and I :: "nat set" assumes "b \<le> a" shows "I \<oplus>- a \<oplus> b = I \<down>\<ge> a \<oplus>- a - b"
ts_inv_const_into_ts_ok: fixes P :: "'h \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool"   and I :: "'h \<Rightarrow> 'g option"   and ts :: "'h \<Rightarrow> ('i \<times> 'k \<Rightarrow>f nat) option"   and m :: "'j" assumes "ts_inv (\<lambda>_. P) I ts m" shows "ts_ok P ts m"
simps_2: fixes x :: "nat"   and xs :: "nat list" shows "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))"
Ax: shows "Ax = MPoly_Type.monom (monomial (Suc 0) x') 1 * Axx + Ax'"
size_multiset_union: fixes f :: "'b \<Rightarrow> nat"   and M :: "'b multiset"   and N :: "'b multiset" shows "size_multiset f (M + N) = size_multiset f M + size_multiset f N"
last_linear_mul_lemma: fixes a :: "'a"   and p :: "'a list"   and x :: "'a"   and b :: "'a" shows "last (a %* p +++ (x # b %* p)) = (if p = [] then x else b * last p)"