hcju/isabelleps huggingface dataset:
mem_lcs: fixes H :: "'a set"   and a :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> a \<diamondsuit> H" shows "x \<in> carrier G"
SV_inv1_inductive_2: shows "{Vinv1} TS.trans sv_TS {> Vinv1}"
all_is_def: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and xs :: "nat set" shows "all_is p ps xs \<equiv> \<forall>v\<in>xs. ps !. v = Some p"
FRESH_binop_iff: fixes P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'c"   and binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite (supp P)"   and "finite (supp Q)" shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
iex_conjR1: fixes I1 :: "nat set"   and I2 :: "nat set"   and P2 :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and P1 :: "nat \<Rightarrow> bool" shows "(\<diamond> t1 I1. (\<diamond> t2 I2. P2 t1 t2) \<and> P1 t1) = (\<diamond> t1 I1. \<diamond> t2 I2. P2 t1 t2 \<and> P1 t1)"
SAStates_Chi_trans: fixes U :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and T :: "'a"   and S :: "'a"   and SA :: "('a, 'b, 'c) seqauto" assumes "U \<in> Chi A T"   and "S \<in> Chi A U"   and "T \<in> States SA"   and "SA \<in> SAs A"   and "U \<in> HAStates A" shows "S \<notin> States SA"
factorial_monoidI: assumes "\<And>a. \<lbrakk>a \<in> carrier G; a \<notin> Units G\<rbrakk> \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a"   and "\<And>a fs fs'. \<lbrakk>a \<in> carrier G; set fs \<subseteq> carrier G; set fs' \<subseteq> carrier G; wfactors G fs a; wfactors G fs' a\<rbrakk> \<Longrightarrow> essentially_equal G fs fs'" shows "factorial_monoid G"
ball_empty: fixes e :: "real"   and x :: "'a" assumes "e \<le> 0" shows "ball x e = {}"
dvd: shows "p ^ x - 1 dvd p ^ (x * Suc k) - 1"
sources_aux_append: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'d list"   and ys :: "'d list" shows "sources_aux I D U (xs @ ys) = sources_aux I D (sources_aux I D U ys) xs"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and tell :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad return bind"   and "monad_writer_axioms bind tell" shows "monad_writer return bind tell"
prv_imp_neg_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp \<phi> (neg (neg \<phi>)))"
eval_pareto_loser_aux: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt" assumes "is_pref_profile R" shows "(x \<in> pareto_losers R) = (\<exists>y\<in>{a, b, c}. x \<prec>[Pareto R] y)"
adjoint_pair_antipar_4: fixes f :: "'a"   and g :: "'a" assumes "adjoint_pair f g" shows "src g = trg f"
collapse: fixes writerT :: "('d, 'e, 'f) writerT" shows "WriterT (run_writer writerT) = writerT"
density_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "density M f = measure_of (space M) (sets M) (\<lambda>A. set_nn_integral M A f)"
get_attribute_reads: fixes element_ptr :: "(_) element_ptr"   and k :: "char list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h'"
slice_eq_bounds_empty: fixes i :: "nat"   and xs :: "'c list" shows "Misc.slice i i xs = []"
subclseq_cases: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D" obtains "C = D"   | "G\<turnstile>C\<prec>\<^sub>C D"
simps_1: shows "local.eval [] = (\<lambda>_. \<zero>)"
Diff_insert_absorb: fixes x :: "'c"   and A :: "'c set" assumes "x \<notin> A" shows "{x} \<union> A - {x} = A"
the_pure_def: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> assn" shows "the_pure P \<equiv> THE P'. \<forall>x x'. P x x' = \<up> ((x', x) \<in> P')"
simps: fixes \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and a1 :: "(status \<times> 'c) \<times> 'd"   and a2 :: "('e, 'f, status \<times> 'c, 'd, 'g, 'h action) thread_action"   and a3 :: "(status \<times> 'c) \<times> 'd" shows "\<tau>multithreaded.init_fin_\<tau>move \<tau>move a1 a2 a3 = (\<exists>x m ta x' m'. a1 = ((Running, x), m) \<and> a2 = convert_TA_initial (convert_obs_initial ta) \<and> a3 = ((Running, x'), m') \<and> \<tau>move (x, m) ta (x', m'))"
rerename_noop: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and x :: "'a"   and "from" :: "nat"   and V :: "'a set"   and to :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "x \<notin> freshenLC from ` V" shows "Abstract_Formulas.rerename freshenLC V from to f x = f x"
subid_meet: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> Id"   and "S \<subseteq> Id" shows "R \<inter> S = R ; S"
Object_def: shows "Object \<equiv> STR ''java/lang/Object''"
permutation_lift_signed_lconjperm_eq_signed_list_lconjaction': fixes S :: "'b set"   and xs :: "('b \<times> bool) list" assumes "PreCoxeterSystem S"   and "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "(\<rightarrow>) (freeword_funlift signed_lconjpermutation (Abs_freeword xs)) = signed_list_lconjaction (map fst xs)"
UN_extend_simps_6: fixes A :: "'b \<Rightarrow> 'a set"   and C :: "'b set"   and B :: "'a set" shows "\<Union> (A ` C) - B = (\<Union>x\<in>C. A x - B)"
mult_coprime_aux: fixes a :: "nat"   and b :: "nat" assumes "1 < a"   and "1 < b"   and "coprime a b" shows "moebius_mu (a * b) = moebius_mu a * moebius_mu b"
simps_2: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and x :: "'a"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.subst_zpassign ground_heads_var \<delta> arity_sym wt_sym coef_sym \<rho> A (PCoef x i) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef coef_sym (\<rho> x) i))"
mat_eq: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'b list list"   and m2 :: "'b list list" assumes "mat nr nc m1"   and "mat nr nc m2" shows "(m1 = m2) = (\<forall>i<nc. \<forall>j<nr. m1 ! i ! j = m2 ! i ! j)"
Word_eq_word_of_int: shows "Word.Word = word_of_int"
simps_1: fixes p :: "nat" shows "showsp_bool p True = shows_string ''True''"
antipar_1: shows "trg\<^sub>D (map\<^sub>0 a) = src\<^sub>D (\<tau>\<^sub>0 a)"
smod_int_alt_def: fixes a :: "int"   and b :: "int" shows "a smod b = sgn a * (\<bar>a\<bar> mod \<bar>b\<bar>)"
finite_t1_space_imp_discrete_topology: fixes X :: "'c topology"   and U :: "'c set" assumes "topspace X = U"   and "finite U"   and "t1_space X" shows "X = discrete_topology U"
inverse_subgroupD: fixes H :: "('a \<Rightarrow> 'a) set" assumes "subgroup (transformation.symmetric.inverse ` H) transformation.Sym (compose G) (identity G)"   and "H \<subseteq> transformation.symmetric.Units" shows "subgroup H transformation.Sym (compose G) (identity G)"
assms_2: shows "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
is_set_category: shows "set_category (\<cdot>) \<AA>"
ereal_mult_zero: fixes a :: "ereal" shows "a * 0 = 0"
diff_fun_scaleR_left: fixes b :: "'a \<Rightarrow> 'd"   and c :: "real" assumes "diff_fun k charts b" shows "diff_fun k charts (c *\<^sub>R b)"
product_sigma_finite_def: fixes M :: "'d \<Rightarrow> 'e measure" shows "product_sigma_finite M \<equiv> \<forall>i. sigma_finite_measure (M i)"
simps_1: fixes S :: "'a"   and S' :: "'a" shows "(SKIP {S} \<le> SKIP {S'}) = (S \<le> S')"
a: shows "\<nexists>l. D \<in> N_of_state (lnth Sts l) \<and> D \<notin> N_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
fset_fmdom_fmrestrict_set: fixes xs :: "'c set"   and N :: "('c, 'd) fmap" shows "fset (fmdom (fmrestrict_set xs N)) = fset (fmdom N) \<inter> xs"
assms_3: shows "tt q \<prec>\<^sub>t tt p"
unsigned_of_bits_replicate_False: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) (replicate n False) = zero"
fin_digraph_axioms: shows "fin_digraph (app_iso iso_f G)"
d_gchain_pre: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "d_gchain G (Suc n) f" shows "d_gchain G n f"
finite_fun_UNIVD1: assumes "finite UNIV"   and "card UNIV \<noteq> Suc 0" shows "finite UNIV"
vs'_def: shows "vs' = map (\<lambda>i. n - perm_vars_inv n vs (n - i - 1) - 1) [0..<n]"
fresh_fun_equiv: fixes h :: "'b \<Rightarrow> 'a"   and pi :: "('b \<times> 'b) list" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp h)"   and "\<exists>a. a \<sharp> (h, h a)" shows "pi \<bullet> fresh_fun h = fresh_fun (pi \<bullet> h)"
W4: shows "-l (1::'a) = -r (1::'a)"
from_nat_to_nat: fixes x :: "'a" shows "Countable.from_nat (Countable.to_nat x) = x"
rel_option_unfold: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b option"   and y :: "'c option" shows "rel_option R x y = (Option.is_none x = Option.is_none y \<and> (\<not> Option.is_none x \<longrightarrow> \<not> Option.is_none y \<longrightarrow> R (the x) (the y)))"
cblinfun_of_mat_plus: fixes M :: "complex mat"   and N :: "complex mat" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)"   and "N \<in> carrier_mat (length canonical_basis) (length canonical_basis)" shows "cblinfun_of_mat (M + N) = cblinfun_of_mat M + cblinfun_of_mat N"
plus_stream_def: fixes x :: "'a stream"   and y :: "'a stream" shows "x + y = pure (+) \<diamondop> x \<diamondop> y"
itrm_brackets_Nil: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and x :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket" shows "itrm_abstraction.itrm_brackets itrm_bracket [] x = Some x"
antisym_conv3: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> ord_strict (PP y) (PP x)" shows "(\<not> ord_strict (PP x) (PP y)) = (x = y)"
0: shows "integral {0..1} (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) = integral {0..1} (\<lambda>x. \<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + integral {0..1} (\<lambda>x. \<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))"
WN_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "WN_on r A = (\<forall>a\<in>A. \<exists>b. (a, b) \<in> r\<^sup>!)"
order_prop: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<exists>z. x \<squnion> z = y)"
All2_term_Iff: fixes i :: "name"   and t :: "tm"   and j :: "name"   and A :: "fm" assumes "atom i \<sharp> t"   and "atom j \<sharp> (i, t, A)" shows "{} \<turnstile> All2 i t A IFF SyntaxN.Ex j (Var j EQ t AND All2 i (Var j) A)"
eq: shows "msubset_of A F = image_mset f (mset_set A)"
linorder_le_cases: fixes x :: "'a"   and y :: "'a" obtains "x \<le> y"   | "y \<le> x"
f10: shows "(4294967296 + - 1 * uint w + - 1 * ((4294967296 + - 1 * uint w) mod 4294967296) \<le> 0) = (4294967296 \<le> uint w + (4294967296 + - 1 * uint w) mod 4294967296)"
assms: shows "(x, z) \<in> A ^^ Suc n"
blinfun_euclidean_eqI: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i" shows "x = y"
sum_wt_Par_sub_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "I \<subseteq> {..<brn (cl ! n)}" shows "sum (wt (Par cl) s) (local.shift cl n I) = 1 / real (length cl) * sum (wt (cl ! n) s) I"
productive_not_empty: fixes C :: "'a literal multiset" assumes "production C \<noteq> {}" shows "C \<noteq> {#}"
simps: fixes a1 :: "('d, 'e, 'f) com"   and a2 :: "('d, 'e, 'f) com" shows "com_relp a1 a2 = ((\<exists>c1 c2. a1 = c1 \<and> a2 = Seq c1 c2) \<or> (\<exists>c2 c1. a1 = c2 \<and> a2 = Seq c1 c2) \<or> (\<exists>c1 b c2. a1 = c1 \<and> a2 = Cond b c1 c2) \<or> (\<exists>c2 b c1. a1 = c2 \<and> a2 = Cond b c1 c2) \<or> (\<exists>c b. a1 = c \<and> a2 = While b c) \<or> (\<exists>c x. a1 = c x \<and> a2 = DynCom c) \<or> (\<exists>c f g. a1 = c \<and> a2 = Guard f g c) \<or> (\<exists>c1 c2. a1 = c1 \<and> a2 = Catch c1 c2) \<or> (\<exists>c2 c1. a1 = c2 \<and> a2 = Catch c1 c2))"
A: shows "integrable M (\<lambda>x. f1 x + C x)"
inner_minus_left: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (uminus x) y = - inner x y"
converges_def: fixes T :: "'b set set"   and F :: "'b set set"   and x :: "'b" shows "F \<longlongrightarrow>\<^bsub>T\<^esub> x = (nhds\<^bsub>T\<^esub> x \<subseteq> F)"
nres_cases: fixes M :: "'b nres" obtains "M = FAIL"   | X :: "'b set" where "M = RES X"
coboundedI1: fixes a :: "'b \<times> 'c"   and c :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod a c" shows "less_eq_prod (inf_prod a b) c"
x_mat_eq_vec_sol: fixes b :: "rat vec"   and A :: "rat mat"   and x :: "nat \<Rightarrow> rat"   and i :: "nat" assumes "x \<Turnstile>\<^sub>c\<^sub>s set (x_mat_eq_vec b A)"   and "i < dim_vec b" shows "matrix_to_lpolies A ! i \<lbrace> x \<rbrace> = b $ i"
c_prod_comm: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y = y \<parallel> x"
elims_2: fixes x :: "'a \<times> nat" assumes "is_valid_event_id x" obtains i :: "'a" and j :: "nat" where "x = (i, j)" and "j < length (events i)"
src_in_sources: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src \<mu> \<in> sources \<mu>"
linear: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "x \<le>s y \<or> y \<le>s x"
Compl_partition: fixes A :: "'b set" shows "A \<union> - A = UNIV"
simple_match_valid_fbs_rlen: fixes rt :: "('b, 'c) routing_rule_scheme list"   and fw :: "'b simple_rule list"   and a :: "nat"   and aa :: "'b simple_match"   and ab :: "char list"   and b :: "simple_action"   and ifs :: "'d" assumes "valid_prefixes rt"   and "simple_fw_valid fw"   and "(a, aa, ab, b) \<in> set (annotate_rlen (lr_of_tran_fbs rt fw ifs))" shows "simple_match_valid aa"
h5: shows "\<forall>n. l2 ! n = m \<longrightarrow> (l1 @ l2) ! (n + length l1) = m"
fresh_map_of: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list"   and a :: "atom" assumes "x \<in> domA \<Gamma>"   and "a \<sharp> \<Gamma>" shows "a \<sharp> the (map_of \<Gamma> x)"
asymp_equiv_eventually_same_sign: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. sgn (f x) = sgn (g x)"
zz_def: shows "zz \<equiv> Variable (Suc (Suc n))"
impr_one_bl: fixes a :: "'a" shows "a r\<rightarrow> a = (1::'a)"
xsx1_xsx2: shows "\<langle>vcard xs\<^sub>1, x\<^sub>1\<rangle> = \<langle>vcard xs\<^sub>2, x\<^sub>2\<rangle>"
mset_subset_eq_add_mset_cancel: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(add_mset a A \<subseteq># add_mset a B) = (A \<subseteq># B)"
x_2: shows "\<forall>y\<in>convex hull S. norm (y - a) \<le> norm (x - a)"
assms_1: shows "AE x in M. f x \<le> f (T x)"
microstep_call_or_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp" assumes "p \<leadsto>\<^bsub>\<Gamma>\<^esub> q" shows "(\<exists>pn. p = call(pn)) \<or> (\<exists>p1 p2. p = p1 \<oplus> p2)"
divide_nonneg_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less y zero" shows "less_eq (divide x y) zero"
roifex_var_not_in_subtree_1: fixes b :: "'a ifex"   and v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" assumes "ro_ifex b"   and "b = IF v t e" shows "v \<notin> ifex_var_set t"
cases: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and a1 :: "(nat \<times> 'b) stream"   and a2 :: "nat \<times> 'b" assumes "grun A w V a1 a2" obtains a :: "nat \<times> 'b" and p :: "nat \<times> 'b" and r :: "(nat \<times> 'b) stream" where "a1 = a ## r" and "a2 = p" and "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" and "grun A w V r a"
assms: shows "pre_adjust \<langle>l, (x, lv), r\<rangle>"
assms_2: shows "x \<in> set (local.filter_min xs)"
det_swap_initial_cols: fixes A :: "'b mat"   and m :: "nat"   and k :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m m"   and "k + n \<le> m" shows "det A = (- (1::'b)) ^ (k * n) * det (mat m m (\<lambda>(i, j). A $$ (i, if j < n then j + k else if j < k + n then j - n else j)))"
Image_Rel_set: fixes succs :: "'c \<Rightarrow> 'c list"   and x :: "'c" shows "(Rel succs)\<^sup>* `` set (succs x) = (Rel succs)\<^sup>+ `` {x}"
LT21: fixes F :: "'b \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> ((\<exists>x. F x) \<leadsto> G) = (\<forall>x. F x \<leadsto> G)"
\<Delta>_simps_1: shows "seq (\<mu> \<star> r\<^sub>0) \<rho>"
gcd_0_int: fixes x :: "int" shows "gcd x 0 = \<bar>x\<bar>"
wf_fdecl_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "wf_fdecl P \<equiv> \<lambda>(F, T). is_type P T"
vangle_0_right: fixes u :: "'b" shows "vangle u (0::'b) = pi / 2"
mark_guards_guards: fixes f :: "'f"   and gs :: "('g \<times> 'd set) list"   and c :: "('d, 'e, 'g) com" shows "mark_guards f (guards gs c) = guards (map (\<lambda>(f', g). (f, g)) gs) (mark_guards f c)"
rep_eq: shows "Rep_pfun {}\<^sub>p = Map.empty"
simps_4: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm"   and v :: "(real, 'd) vec" shows "frechet I (Times t1 t2) v = (\<lambda>v'. sterm_sem I t1 v * frechet I t2 v v' + frechet I t1 v v' * sterm_sem I t2 v)"
uniqRemdups: fixes x :: "'a list" shows "uniq (remdups x)"
fmpred_restrict_set: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and A :: "'c set" assumes "fmpred P m" shows "fmpred P (fmrestrict_set A m)"
leadsETo_Union: fixes S :: "'d set set"   and F :: "'d program"   and CC :: "'d set set"   and B :: "'d set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A leadsTo[CC] B" shows "F \<in> \<Union> S leadsTo[CC] B"
split_list_last_prop: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z)"
RECT_def: fixes B :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'d" shows "REC\<^sub>T B x \<equiv> if trimono B then flatf_gfp B x else top"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "[] \<noteq> Cons x21 x22"
era: shows "cs ! index (take (Suc i) cs) z = z"
fg_invariant1: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and p :: "'f"   and r :: "nat"   and q :: "'f"   and m :: "'h"   and n :: "nat" assumes "fg_run A rho HOs SHOs coords"   and "(p, r, q, m) \<in> network (rho n)" shows "m = sendMsg A r p q (state (rho (fg_start_round rho p r)) p)"
assms_2: fixes b :: "'a" assumes "b dvd a"   and "normalize b = b" shows "\<not> prime_elem b"
neg_mult_resr: fixes y :: "'a"   and x :: "'a" shows "inv_resl x (-' y) = inv_resr (- x) y"
S_V: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" assumes "Separation G v0 v1 S" shows "S \<subseteq> V\<^bsub>G\<^esub>"
state_update_vs_allInstances_generic_tc: fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and oid :: "nat"   and \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and Object :: "'d"   and P :: "('d state \<times> 'd state \<Rightarrow> Set(\<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a"   and "oid \<notin> dom \<sigma>'"   and "Type Object \<noteq> None"   and "cp P"   and "\<And>X. const X \<Longrightarrow> const (P X)" shows "(mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr> \<Turnstile> P (OclAllInstances_generic pre_post Type)) = (mk \<lparr>heap = \<sigma>', assocs = A\<rparr> \<Turnstile> P OclAllInstances_generic pre_post Type->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>Type Object\<rfloor>))"
apartment_vertex_set_int: fixes X :: "'a set set"   and Y :: "'a set set" assumes "X \<in> \<A>"   and "Y \<in> \<A>" shows "\<Union> (X \<inter> Y) = \<Union> X \<inter> \<Union> Y"
ps: shows "ps \<subseteq> bl \<and> card ps = 2"
simps_6: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Impl p q[s/k] = Impl (p[s/k]) (q[s/k])"
rcoset_rel_sym: fixes G :: "'a set" assumes "Group G" shows "sym (rcoset_rel G)"
distinct_58: fixes name' :: "name"   and pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "<\<nu>name'>pi' \<noteq> name1<name2>.pi"
count_roots_below_correct: fixes p :: "real poly"   and a :: "real" shows "count_roots_below p a = card {x. x \<le> a \<and> poly p x = 0}"