hcju/isabelleps huggingface dataset:
NSLIM_isNSCont: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a" shows "isNSCont f a"
rot_diamond_cube_boundary_to_subdiv_def: fixes gamma :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "rot_diamond_cube_boundary_to_subdiv gamma \<equiv> if gamma = diamond_left_edges then {diamond_bot_left_edge, diamond_top_left_edge} else if gamma = diamond_right_edges then {diamond_bot_right_edge, diamond_top_right_edge} else {}"
intro: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>x y. \<lbrakk>less x y; less y x; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> False" shows "asymmetric A less"
Carmichael_eq_totient_imp_primroot: fixes n :: "nat" assumes "0 < n"   and "Carmichael n = totient n" shows "\<exists>g. residue_primroot n g"
Inv: shows "\<forall>x\<in>P. x \<noteq> 0 \<longrightarrow> 1 / x \<in> P"
llast_LCons2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a llist" shows "llast (LCons x (LCons y xs)) = llast (LCons y xs)"
good_map_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'f" shows "good_map h = (surj h \<and> (\<forall>x y. fst (inv h (h (x, y))) = x))"
Done_eq_assert_gpv: fixes x :: "unit"   and b :: "bool" shows "(Generative_Probabilistic_Value.Done x = assert_gpv b) = b"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and S1 :: "'h \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'i \<Rightarrow> bool"   and S2 :: "'i \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_optionT' (R1 OO S1) (R2 OO S2) = rel_optionT' R1 R2 OO rel_optionT' S1 S2"
functor_at_arr_is_transformation: fixes F :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F"   and "A.arr a" shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at (A.dom a) F) (at (A.cod a) F) (at a F)"
fds_of_real_def: fixes f :: "real fds" shows "fds_of_real f = fds (\<lambda>n. of_real (fds_nth f n))"
neq_no_wait_locks_conv: fixes ln :: "'a \<Rightarrow>f nat" shows "(ln \<noteq> no_wait_locks) = (\<exists>l. 0 < ln $ l)"
graph_accs_2: fixes V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "finite V"   and "finite E" shows "edges (graph V E) = E \<union> E\<inverse> - Id"
rel_gpv''D: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')"
hom_mult: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "times_hom hom" shows "hom (x * y) = hom x * hom y"
sc: shows "h \<turnstile> get_scdom_component ptr' \<rightarrow>\<^sub>r sc"
Seq_le: fixes c1 :: "'a acom"   and c2 :: "'a acom"   and c :: "'a acom" shows "(c1;; c2 \<le> c) = (\<exists>c1' c2'. c = c1';; c2' \<and> c1 \<le> c1' \<and> c2 \<le> c2')"
length_univ_list_card: shows "length univ_list = CARD('a)"
assms: shows "\<forall>C\<in>AbstrLevel i. S \<notin> DSources i C"
in_sets: shows "g -` {x} \<inter> space M \<in> sets M"
rlex_not_sym: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x ys" shows "\<not> ys <\<^sub>r\<^sub>l\<^sub>e\<^sub>x xs"
refine_trans_using_invariants: fixes R :: "('a \<times> 'b) set"   and I :: "'a set"   and J :: "'b set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "{R \<inter> I \<times> J} TS.trans Ta, TS.trans Tc {> R}"   and "{I \<inter> Domain (R \<inter> UNIV \<times> J)} TS.trans Ta {> I}"   and "{J \<inter> Range (R \<inter> I \<times> UNIV)} TS.trans Tc {> J}" shows "{R \<inter> I \<times> J} TS.trans Ta, TS.trans Tc {> R \<inter> I \<times> J}"
subgame_finite: fixes V' :: "'a set" shows "finite (\<omega>\<^bsub>subgame V'\<^esub> ` V\<^bsub>subgame V'\<^esub>)"
constrains_UN: fixes I :: "'b set"   and F :: "'c program"   and A :: "'b \<Rightarrow> 'c set"   and A' :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i co A' i" shows "F \<in> \<Union> (A ` I) co \<Union> (A' ` I)"
pr0_in_hom': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>\<pp>\<^sub>0[a, b] : local.dom \<pp>\<^sub>0[a, b] \<rightarrow> b\<guillemotright>"
tagged_division_of: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" shows "(s tagged_division_of i) = (finite s \<and> (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> interior K1 \<inter> interior K2 = {}) \<and> \<Union> {K. \<exists>x. (x, K) \<in> s} = i)"
induct: fixes x1 :: "nat \<Rightarrow> pdf_type"   and x2 :: "cexpr"   and x3 :: "pdf_type"   and P :: "(nat \<Rightarrow> pdf_type) \<Rightarrow> cexpr \<Rightarrow> pdf_type \<Rightarrow> bool" assumes "x1 \<turnstile>\<^sub>c x2 : x3"   and "\<And>\<Gamma> v. P \<Gamma> (CVal v) (val_type v)"   and "\<And>\<Gamma> x. P \<Gamma> (CVar x) (\<Gamma> x)"   and "\<And>\<Gamma> e1 t1 e2 t2. \<lbrakk>\<Gamma> \<turnstile>\<^sub>c e1 : t1; P \<Gamma> e1 t1; \<Gamma> \<turnstile>\<^sub>c e2 : t2; P \<Gamma> e2 t2\<rbrakk> \<Longrightarrow> P \<Gamma> <e1, e2>\<^sub>c (PRODUCT t1 t2)"   and "\<And>\<Gamma> e t oper t'. \<lbrakk>\<Gamma> \<turnstile>\<^sub>c e : t; P \<Gamma> e t; op_type oper t = Some t'\<rbrakk> \<Longrightarrow> P \<Gamma> (oper $$\<^sub>c e) t'"   and "\<And>\<Gamma> b e1 t e2. \<lbrakk>\<Gamma> \<turnstile>\<^sub>c b : BOOL; P \<Gamma> b BOOL; \<Gamma> \<turnstile>\<^sub>c e1 : t; P \<Gamma> e1 t; \<Gamma> \<turnstile>\<^sub>c e2 : t; P \<Gamma> e2 t\<rbrakk> \<Longrightarrow> P \<Gamma> (IF\<^sub>c b THEN e1 ELSE e2) t"   and "\<And>t \<Gamma> e. \<lbrakk>case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL; P (case_nat t \<Gamma>) e REAL\<rbrakk> \<Longrightarrow> P \<Gamma> (\<integral>\<^sub>c e \<partial>t) REAL" shows "P x1 x2 x3"
set_ins_ops: fixes oper :: "'a operation"   and rects :: "'a rectangle list" shows "(oper \<in> set (ins_ops rects)) = (op_idx oper < length rects \<and> oper = ins_op rects (op_idx oper))"
msb_word_of_int: fixes x :: "int" shows "msb (word_of_int x) = bit x (LENGTH('b) - 1)"
traces_contain_events_def: fixes ES :: "'a ES_rec" shows "traces_contain_events ES \<equiv> \<forall>l\<in>Tr\<^bsub>ES\<^esub>. \<forall>e\<in>set l. e \<in> E\<^bsub>ES\<^esub>"
not_iFROM_iMOD_subset: fixes m :: "nat"   and n' :: "nat"   and r :: "nat" assumes "m \<noteq> Suc 0" shows "\<not> [n'\<dots>] \<subseteq> [ r, mod m ]"
suffix_first: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "first (s |\<^sub>s n) = s n"
summable_norm_cancel: fixes f :: "nat \<Rightarrow> 'd" assumes "summable (\<lambda>n. norm (f n))" shows "summable f"
exhaust: fixes y :: "int_point_t" obtains x11 :: "ipc_direction_t" and x12 :: "ipc_stage_t" and x13 :: "thread_id_t" and x14 :: "page_t" where "y = SK_IPC x11 x12 x13 x14"   | x21 :: "ev_wait_stage_t" and x22 :: "ev_consume_t" where "y = SK_EV_WAIT x21 x22"   | x31 :: "ev_signal_stage_t" and x32 :: "thread_id_t" where "y = SK_EV_SIGNAL x31 x32"   | "y = NONE"
p_shunting_swap: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(- z \<le> x \<squnion> y) = (- y \<le> x \<squnion> z)"
monad_fail_alt_axioms: shows "monad_fail_alt return bind fail alt"
meta_le_everything_if_top: fixes m :: "'b"   and x :: "'b" assumes "m = top" shows "x \<le> m"
HEndPhase0_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4d s p"   and "Inv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "HInv4b s' p"
eql4: shows "exH_k k = st2.ptensor_mat (1\<^sub>m 2) (st3.ptensor_mat (H_k k) (1\<^sub>m st4.d2))"
injective_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.injective f = (\<forall>i j k. f (j, i) \<noteq> bot \<and> f (k, i) \<noteq> bot \<longrightarrow> j = k)"
A: fixes a :: "'a"   and b :: "'a" assumes "r' a b" shows "(a \<in> A) = (a \<in> A')"
Im_mult_real: fixes a :: "complex"   and b :: "complex" assumes "is_real a" shows "Im (a * b) = Re a * Im b"
effect_to_assignments_iii_a: fixes v :: "'a"   and op :: "'a strips_operator"   and as :: "('a \<times> bool) list" assumes "v \<in> set (add_effects_of op)"   and "as = effect_to_assignments op" obtains a :: "'a \<times> bool" where "a \<in> set as" and "a = (v, True)"
agents_eq: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "agents = {i. R i \<noteq> (\<lambda>_ _. False)}"
lappend_code_1: fixes ys :: "'c llist" shows "lappend LNil ys = ys"
lowered_lotteries: shows "lowered.lotteries \<subseteq> lotteries"
map_cong: fixes x :: "'c llist"   and ya :: "'c llist"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> lset ya \<Longrightarrow> f z = g z" shows "lmap f x = lmap g ya"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "\<And>x y. hom (x + y) = hom x + hom y" shows "plus_hom hom"
CallRedsObj: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s,b\<rangle> \<rightarrow>* \<langle>e'\<bullet>M(es),s',b'\<rangle>"
A_def: shows "A = mat_of_rows n [vec_of_list xs]"
dvd_div_eq_mult: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> (0::'a)"   and "a dvd b" shows "(b div a = c) = (b = c * a)"
commutes_with_composition: fixes x :: "'b"   and y :: "'b" assumes "x \<in> \<FF> z'"   and "y \<in> \<FF> z'" shows "\<rho> z' w (\<cdot>\<^bsub>z'\<^esub> x y) = \<cdot>\<^bsub>w\<^esub> (\<rho> z' w x) (\<rho> z' w y)"
cinner_mult_left': fixes a :: "'c"   and m :: "complex"   and b :: "'c" shows "cinner (a * of_complex m) b = cnj m * cinner a b"
addcol_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and a :: "'a"   and l :: "nat" assumes "A \<in> carrier_mat nr n"   and "k < n" shows "addcol a l k A = A * addrow_mat n a k l"
prod_decode_triangle_add: fixes k :: "nat"   and m :: "nat" shows "prod_decode (triangle k + m) = prod_decode_aux k m"
rep_eq: fixes x :: "'a \<Rightarrow> 'a" shows "\<lbrakk>\<langle>x\<rangle>\<^sub>a\<rbrakk>\<^sub>e = (\<lambda>(b, b'). b' = x b)"
cases: fixes x :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" obtains name :: "char list" and preconds :: "(nat \<times> nat) list" and effects :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list" and cost :: "nat" where "x = (name, preconds, effects, cost)"
eq0: fixes n :: "nat" assumes "x [^] n = \<one>"   and "n < N" shows "n = 0"
noh_le: fixes z :: "complex" assumes "cmod z < 1" shows "cmod (h z) \<le> 1"
wrm_set1: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<and> sortedTree h t \<longrightarrow> setOf (wrm h t) \<subseteq> setOf t"
finite_measure_Union': fixes A :: "'b set"   and M :: "'b pmf"   and B :: "'b set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M (A \<union> B) = measure_pmf.prob M A + measure_pmf.prob M (B - A)"
Domainp_rel: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (rel_H R) = pred_H (Domainp R)"
igSwapIGAbs_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
exhaust: fixes y :: "Unboxed.type" obtains "y = Ubx1"   | "y = Ubx2"
assertion_inf_comp_eq: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf x y = times x y"
gen_g_impl_rel_ext_def: fixes Rm :: "('m \<times> 'o) set"   and Rv :: "('j \<times> 'n set) set"   and Re :: "('k \<times> ('n \<times> 'n) set) set"   and Rv0 :: "('l \<times> 'n set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
\<R>\<^sub>1_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma>' :: "'Var \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "\<turnstile> \<Gamma>' {c\<^sub>1} \<Gamma>" and "mds_consistent mds \<Gamma>'" and "mem\<^sub>1 =\<^bsub>\<Gamma>'\<^esub> mem\<^sub>2"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "state" assumes "a1 -a2\<rightarrow> a3" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = sa__" and "a2 = Do f__" and "a3 = ta__" and "ta__ \<in> f__ sa__"   | s0 :: "state" and c1 :: "com" and s1 :: "state" and c2 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c1__; c2__)" and "a3 = s2" and "s0 -c1__\<rightarrow> s1" and "s1 -c2__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and t :: "state" and c2 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c1a__ ELSE c2a__" and "a3 = tb__" and "ba__ sb__" and "sb__ -c1a__\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and t :: "state" and c1 :: "com" where "a1 = sc__" and "a2 = IF bb__ THEN c1 ELSE c2" and "a3 = tc__" and "\<not> bb__ sc__" and "sc__ -c2\<rightarrow> tc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = sd__" and "a2 = WHILE bc__ DO ca__" and "a3 = sd__" and "\<not> bc__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and t :: "state" and u :: "state" where "a1 = se__" and "a2 = WHILE b DO cb__" and "a3 = u" and "b se__" and "se__ -cb__\<rightarrow> td__" and "td__ -WHILE b DO cb__\<rightarrow> u"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = g s t" and "f s -c\<rightarrow> t"
is_lub_bigger1: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" shows "is_lub (r\<^sup>*) x y y = ((x, y) \<in> r\<^sup>*)"
simps: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f" shows "Rangep r a = (\<exists>aa b. a = b \<and> r aa b)"
conn: fixes y :: "'a" shows "connected ({0..1} \<inter> f -` {y})"
compact_sphere: fixes a :: "'a"   and r :: "real" shows "compact (sphere a r)"
set_sort: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "set (Sorting_Algorithms.sort cmp xs) = set xs"
SWF_def: fixes swf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and Pcond :: "'b set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> bool" shows "SWF swf A Is Pcond \<equiv> \<forall>P. Pcond A Is P \<longrightarrow> rpr A (swf P)"
asubg_pOp_closed: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "A +> H"   and "x \<in> H"   and "y \<in> H" shows "x \<plusminus> y \<in> H"
leg0_composite: shows "SPN_fgh.\<mu>\<nu>.leg0 = SPN_fgh.\<nu>.leg0 \<odot> SPN_fgh.\<mu>\<nu>.prj\<^sub>0"
LeftDerives1_propagate_prefix: fixes \<delta> :: "symbol list"   and \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and \<beta> :: "symbol list" assumes "LeftDerives1 (\<delta> @ \<alpha>) i r \<beta>"   and "length \<delta> \<le> i" shows "is_prefix \<delta> \<beta>"
red_preserves_hconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and T :: "ty" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E,h \<turnstile> e : T"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
intros: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x2"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x' :: "'x2"   and ln' :: "'l \<Rightarrow>f nat"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "thr s' t = \<lfloor>(x', ln')\<rfloor>"   and "\<tau>move2 (x, shr s) ta (x', shr s')" shows "m\<tau>move2 s (t, ta) s'"
closedin_Hausdorff_singleton: fixes X :: "'c topology"   and x :: "'c" assumes "Hausdorff_space X"   and "x \<in> topspace X" shows "closedin X {x}"
div_eq_sgn_abs: fixes k :: "int"   and l :: "int" assumes "sgn k = sgn l" shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
sst_vars_disj_cons: fixes a :: "('b, 'a) stateful_strand_step"   and A :: "('b, 'a) stateful_strand_step list" assumes "fv\<^sub>s\<^sub>s\<^sub>t (a # A) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = {}" shows "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"
card_cartesian_product_singleton: fixes x :: "'a"   and A :: "'b set" shows "card ({x} \<times> A) = card A"
methd_declC: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "methd G C sig = Some m"   and "ws_prog G"   and "is_class G C" shows "(\<exists>d. class G (declclass m) = Some d \<and> table_of (methods d) sig = Some (mthd m)) \<and> G\<turnstile>C\<preceq>\<^sub>C declclass m \<and> methd G (declclass m) sig = Some m"
PPs_mono: fixes X :: "'c set"   and Y :: "'c set" assumes "X \<subseteq> Y" shows ".[X] \<subseteq> .[Y]"
chain: shows "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
catch_gpv_Fail: shows "catch_gpv Fail = Generative_Probabilistic_Value.Done None"
laplace_integrand_integrable: fixes M :: "real"   and c :: "real"   and B :: "real \<Rightarrow> real"   and s :: "complex" assumes "exponential_order M c f"   and "\<And>k. AE x\<in>{0..k} in lebesgue. cmod (f x) \<le> B k"   and "\<And>k. f integrable_on {0..k}"   and "c < Re s" shows "local.laplace_integrand s integrable_on {0..}"
Bot_in_lattice: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "CLF cl f" shows "Bot cl \<in> pset cl"
assoc_mult_mat_vec: fixes A :: "'b mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'b mat"   and n\<^sub>3 :: "nat"   and v :: "'b Matrix.vec" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "v \<in> carrier_vec n\<^sub>3" shows "A * B *\<^sub>v v = A *\<^sub>v (B *\<^sub>v v)"
dord_fun_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" assumes "class.linorder less_eq less" shows "linorder.dord_fun ord s t \<equiv> let d1 = deg_fun s; d2 = deg_fun t in d1 < d2 \<or> d1 = d2 \<and> ord s t"
gauss_jordan_row_echelon: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and A' :: "'b mat"   and B' :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan A B = (A', B')" shows "row_echelon_form A'"
step_invariant_induction: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and I :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool"   and P :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "c \<Rightarrow>\<^sup>* c'"   and "\<not> boring_step c'"   and "invariant (\<Rightarrow>) I"   and "I c"   and "\<And>\<Gamma> e x S. I (\<Gamma>, App e x, S) \<Longrightarrow> P (\<Gamma>, App e x, S) (\<Gamma>, e, Arg x # S)"   and "\<And>\<Gamma> y e x S. I (\<Gamma>, Lam [y]. e, Arg x # S) \<Longrightarrow> P (\<Gamma>, Lam [y]. e, Arg x # S) (\<Gamma>, e[y::=x], S)"   and "\<And>\<Gamma> x e S. \<lbrakk>map_of \<Gamma> x = Some e; \<not> isVal e; I (\<Gamma>, Var x, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Var x, S) (delete x \<Gamma>, e, Upd x # S)"   and "\<And>\<Gamma> x e S. \<lbrakk>map_of \<Gamma> x = Some e; isVal e; I (\<Gamma>, Var x, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Var x, S) ((x, e) # delete x \<Gamma>, e, S)"   and "\<And>\<Gamma> x e S. \<lbrakk>x \<notin> domA \<Gamma>; isVal e; I (\<Gamma>, e, Upd x # S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, e, Upd x # S) ((x, e) # \<Gamma>, e, S)"   and "\<And>\<Delta> \<Gamma> e S. \<lbrakk>atom ` domA \<Delta> \<sharp>* \<Gamma>; atom ` domA \<Delta> \<sharp>* S; I (\<Gamma>, Terms.Let \<Delta> e, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Terms.Let \<Delta> e, S) (\<Delta> @ \<Gamma>, e, S)"   and "\<And>\<Gamma> scrut e1 e2 S. I (\<Gamma>, scrut ? e1 : e2, S) \<Longrightarrow> P (\<Gamma>, scrut ? e1 : e2, S) (\<Gamma>, scrut, Alts e1 e2 # S)"   and "\<And>\<Gamma> b e1 e2 S. I (\<Gamma>, Bool b, Alts e1 e2 # S) \<Longrightarrow> P (\<Gamma>, Bool b, Alts e1 e2 # S) (\<Gamma>, if b then e1 else e2, S)"   and "\<And>c. P c c"   and "\<And>c c' c''. \<lbrakk>c \<Rightarrow>\<^sup>* c'; c' \<Rightarrow>\<^sup>* c''; P c c'; P c' c''\<rbrakk> \<Longrightarrow> P c c''" shows "P c c'"
only_horizontal_division: shows "only_horizontal_division \<gamma> two_chain"
obtain_complete: obtains TAC :: "(nat set option, 'L) tree_automaton_rec" where "ta_lang TAC = ta_lang TA" and "complete_tree_automaton TAC A"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P"
distinct_2: fixes x2 :: "name"   and x1 :: "name" shows "BoundOutputS x2 \<noteq> InputS x1"
wf_subclass_loc_axioms: shows "wf_subclass_loc cs"