hcju/isabelleps huggingface dataset:
preorder_on_def: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Order_Predicates.preorder_on carrier le \<equiv> ((\<forall>x y. le x y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. le x y \<longrightarrow> y \<in> carrier)) \<and> (\<forall>x. x \<in> carrier \<longrightarrow> le x x) \<and> (\<forall>x y z. le x y \<longrightarrow> le y z \<longrightarrow> le x z)"
simps_2: shows "project_operation NONSAT_PUSH' = PUSH"
size_single: fixes b :: "'k" shows "size {#b#} = 1"
cl_mono: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<subseteq> Y"   and "Y \<subseteq> \<E>" shows "\<E>.cl X \<subseteq> \<E>.cl Y"
Q_FreeS: shows "order.greater_eq (FreeGroup S) \<langle>\<Union>h\<in>FreeGroup S. lconjby h ` Abs_freeword ` map (charpair S) ` R\<rangle>"
approx_hcmod_add_hcmod: fixes u :: "complex star"   and x :: "complex star" assumes "u \<approx> 0" shows "hcmod (x + u) \<approx> hcmod x"
tanh_def: fixes x :: "'a" shows "tanh x = sinh x / cosh x"
pow_mono: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq b a"   and "less_eq zero b" shows "less_eq (power.power one times b n) (power.power one times a n) \<and> less_eq zero (power.power one times b n)"
simps_1: fixes x :: "'a" shows "lookup_pair [] x = (0::'e)"
set_attribute_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_attribute_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
simps_1: fixes S :: "'a st"   and X :: "char list set" shows "top_on\<^sub>o (Some S) X = top_on\<^sub>s S X"
alt_state_def: fixes m1 :: "('s, 'm) stateT"   and m2 :: "('s, 'm) stateT" shows "local.alt_state m1 m2 = StateT (\<lambda>s. alt (run_state m1 s) (run_state m2 s))"
igWlsAbsIsInBar_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igWlsAbsIsInBar MOD \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
constructive_security_obsf'D: fixes \<eta> :: "nat" shows "constructive_security_obsf (real1 \<eta>) (ideal1 \<eta>) (sim1 \<eta>) (\<I>_real1 \<eta>) (\<I>_inner1 \<eta>) (\<I>_common1 \<eta>) (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>))))) (advantage (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>))))) (obsf_resource (sim1 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal1 \<eta>)) (obsf_resource (real1 \<eta>)))"
swap_row_to_front_result: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and I :: "nat" assumes "A \<in> carrier_mat n m"   and "I < n" shows "swap_row_to_front A I = mat n m (\<lambda>(i, j). if i = 0 then A $$ (I, j) else if i \<le> I then A $$ (i - 1, j) else A $$ (i, j))"
card_1_exists: fixes s :: "'b set" shows "(card s = 1) = (\<exists>!x. x \<in> s)"
openin_open_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin X S" shows "openin (subtopology X S) T = (openin X T \<and> T \<subseteq> S)"
arith_simps_44: shows "\<bar>1::'b\<bar> = (1::'b)"
USUP_Suc_shift: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> i \<in> {Suc 0..<Suc n} \<bullet> P i) = (\<Squnion> i \<in> {0..<n} \<bullet> P (Suc i))"
hull_redundant_eq: fixes a :: "'a"   and S :: "'a set \<Rightarrow> bool"   and s :: "'a set" shows "(a \<in> S hull s) = (S hull insert a s = S hull s)"
lookup_of_alist: fixes xs :: "('d \<times> 'c) list"   and k :: "'d" shows "Mapping.lookup (Mapping.of_alist xs) k = map_of xs k"
map_id: fixes t :: "'e llist" shows "lmap id t = t"
bl_or_bin: fixes n :: "nat"   and v :: "int"   and w :: "int" shows "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)"
higher_differentiable_on: fixes S :: "'a set"   and n :: "nat" shows "higher_differentiable_on S f n"
invalid_pure_recover: fixes R :: "('b \<times> 'a) set"   and x :: "'a"   and y :: "'b" shows "invalid_assn (pure R) x y = pure R x y * true"
borel_measurable_eq: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w = g w} \<in> sets M"
rell_G_mono_strong: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and x :: "('p, 'r, 't, 'u, 'v, 'w, 'x) G"   and y :: "('q, 's, 't, 'u, 'v, 'w, 'x) G"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2' :: "'r \<Rightarrow> 's \<Rightarrow> bool" assumes "rell_G L1 L2 x y"   and "\<And>a b. \<lbrakk>a \<in> set1_G x; b \<in> set1_G y; L1 a b\<rbrakk> \<Longrightarrow> L1' a b"   and "\<And>a b. \<lbrakk>a \<in> set2_G x; b \<in> set2_G y; L2 a b\<rbrakk> \<Longrightarrow> L2' a b" shows "rell_G L1' L2' x y"
nfoldli_alt_def: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli_alt l c f s \<equiv> REC\<^sub>T (\<lambda>D (l, s). case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> let b = c s in if b then f x s \<bind> (\<lambda>s'. D (ls, s')) else RETURN s) (l, s)"
env_empty_db_def: shows "env_empty_db \<equiv> \<bottom>"
HUN_iff: fixes b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" shows "(b \<^bold>\<in> \<Squnion>RepFun A B) = (\<exists>x\<^bold>\<in>A. b \<^bold>\<in> B x)"
bin_last_numeral_simps_1: shows "\<not> odd 0"
set_pmf_map: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" shows "set_pmf (map_pmf f M) = f ` set_pmf M"
equalityD2: fixes A :: "'c set"   and B :: "'c set" assumes "A = B" shows "B \<subseteq> A"
Sup_image_eadd2: fixes Y :: "enat set"   and x :: "enat" assumes "Y \<noteq> {}" shows "\<Squnion> ((+) x ` Y) = x + \<Squnion> Y"
H: shows "H \<equiv> Hermite_reduce_above M (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res"
flatten_verdicts_def: fixes Vs :: "('a \<times> 'b set) list" shows "flatten_verdicts Vs = \<Union> (set (map (\<lambda>(i, X). Pair i ` X) Vs))"
assms_1: shows "(u \<longlongrightarrow> l) F"
case_prodI: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
abs_eq: fixes xa :: "'d"   and x :: "'b" shows "Pred xa x = Abs_formula (Pred\<^sub>\<alpha> xa x)"
Pbij_extends_def: fixes \<gamma> :: "(Location \<times> Location) set"   and \<beta> :: "(Location \<times> Location) set" shows "Pbij_extends \<gamma> \<beta> = (\<beta> \<subseteq> \<gamma>)"
sterms_stermsl: fixes ps :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "ps \<in> sterms \<Gamma> p"   and "wellformed \<Gamma>" shows "ps \<in> stermsl p \<or> (\<exists>pn. ps \<in> stermsl (\<Gamma> pn))"
wf_prec_F: shows "minimal_element (\<prec>\<cdot>) UNIV"
llcp_ltake1: fixes n :: "enat"   and xs :: "'a llist"   and ys :: "'a llist" shows "llcp (ltake n xs) ys = min n (llcp xs ys)"
chamber_system_into: shows "order.greater_eq \<C> (f \<turnstile> \<C>)"
N_f': fixes n :: "nat" assumes "N_f' \<le> n" shows "\<bar>\<Sum>i. f' x0 (i + n)\<bar> < r / 3"
nths_vars2': shows "nths a.dims0 (- a.vars1') = a.dims2"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "ospec None uu = False"
simps_2: fixes X :: "nat set"   and n :: "nat" shows "witness X (Suc n) = witness X n @ [n \<in> X]"
ENFnr_to_ENFnrSR: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "sinvar_all_edges_normal_form_not_refl P" shows "sinvar_all_edges_normal_form_not_refl_SR (\<lambda>v1 _ v2 _. P v1 v2)"
intro: fixes r\<^sub>1 :: "V"   and r\<^sub>2 :: "V" assumes "fbrelation r\<^sub>1"   and "fbrelation r\<^sub>2" shows "fbrelation_pair r\<^sub>1 r\<^sub>2"
mask_eqs_7: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" shows "(a AND mask n) + (b AND mask n) AND mask n = a + b AND mask n"
map_option_eq_Some: fixes f :: "'b \<Rightarrow> 'a"   and xo :: "'b option"   and y :: "'a" shows "(map_option f xo = Some y) = (\<exists>z. xo = Some z \<and> f z = y)"
Im_half: fixes z :: "complex" shows "Im (z / 2) = Im z / 2"
int_ops_4: fixes a :: "nat" shows "int (Suc a) = int a + 1"
llength_eq_infty_conv_lfinite: fixes xs :: "'a llist" shows "(llength xs = \<infinity>) = (\<not> lfinite xs)"
set_spmf_I_N: fixes N :: "nat"   and e :: "nat"   and d :: "nat" assumes "((N, e), d) \<in> set_spmf I" obtains P :: "nat" and Q :: "nat" where "N = P * Q" and "P \<noteq> Q" and "prime P" and "prime Q" and "coprime e ((P - 1) * (Q - 1))" and "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))"
wf_interp_for_formula_FAnd: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi>1 :: "'a formula"   and \<phi>2 :: "'a formula" shows "wf_interp_for_formula (w, I) (FAnd \<phi>1 \<phi>2) = (wf_interp_for_formula (w, I) \<phi>1 \<and> wf_interp_for_formula (w, I) \<phi>2)"
cut_ge_subset: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t \<subseteq> I"
induct: fixes P :: "('b \<Rightarrow> bool) \<Rightarrow> 'b negation_type list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> bool"   and a1 :: "'b negation_type list" assumes "\<And>uu_. P uu_ []"   and "\<And>f a as. P f as \<Longrightarrow> P f (Pos a # as)"   and "\<And>f a as. P f as \<Longrightarrow> P f (Neg a # as)" shows "P a0 a1"
seqr_right_unit: fixes P :: "('a, 'b) urel" shows "P ;; II = P"
simps: fixes a :: "rat"   and b :: "rat" shows "qdfst (QDelta a b) = a"
agree_refl: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and A :: "('d + 'd) set" shows "Vagree \<nu> \<nu> A"
measurable_eq_EXP: shows "(=) \<in> POW \<rightarrow>\<^sub>M EXP"
mpoly_map_vars_Var: fixes f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "bij f" shows "mpoly_map_vars f (Var i) = Var (f i)"
transitive_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "transitive A r = transitive A r'"
mergesort_by_rel_split_length: fixes xs1 :: "'a list"   and xs2 :: "'a list"   and xs :: "'a list" shows "length (fst (mergesort_by_rel_split (xs1, xs2) xs)) = length xs1 + length xs div 2 + length xs mod 2 \<and> length (snd (mergesort_by_rel_split (xs1, xs2) xs)) = length xs2 + length xs div 2"
assms_1: shows "i < min \<^bold>|u\<^bold>| \<^bold>|v\<^bold>|"
2: fixes v :: "'a" shows "higher_differentiable_on S (\<lambda>x. frechet_derivative g (at x) v) n"
product_def: fixes A :: "'b set"   and B :: "'c set" shows "Product_Type.product A B = A \<times> B"
jmpl_instr_low_equal_sub1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1))))))) PC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1))))))))))) \<and> t2 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2))))))) PC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2)))))))))))" shows "low_equal t1 t2"
OAlist_tc_lookup_eq_valueI: fixes k :: "'b"   and v :: "'c"   and xs :: "('b, 'c) oalist_tc" assumes "(k, v) \<in> set (list_of_oalist_tc xs)" shows "OAlist_tc_lookup xs k = v"
resumption_ord_abort: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "Resumption.resumption.is_Done r"   and "\<not> Resumption.resumption.is_Done r'" shows "Resumption.resumption.result r = None"
split_list_first_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>y\<in>set ys. \<not> P y)"
compress_normalize_primitive_not_introduces_C: fixes disc :: "'c \<Rightarrow> bool"   and m :: "'c match_expr"   and sel :: "'c \<Rightarrow> 'd"   and C' :: "'d \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr" assumes "\<not> has_disc disc m"   and "wf_disc_sel (disc, sel) C'"   and "normalized_nnf_match m"   and "compress_normalize_primitive (disc, sel) C f m = Some m'"   and "\<And>as_pos as_neg. f [] = Some (as_pos, as_neg) \<Longrightarrow> as_pos = [] \<and> as_neg = []" shows "\<not> has_disc disc m'"
f_last_message_hold_le: fixes xs :: "'b message_af list"   and k :: "nat" assumes "length xs \<le> k" shows "xs \<longmapsto> k = last_message_hold xs"
add_mset_replicate_mset_safe: fixes M :: "'b multiset"   and a :: "'b" assumes "NO_MATCH {#} M" shows "add_mset a M = {#a#} + M"
stutter_selection_lproject_visible: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a stream"   and p :: "'b" assumes "transition_system_interpreted_traces ex en int ind"   and "transition_system.run ex en u p" shows "stutter_selection (lift (liset (transition_system_interpreted.visible ex en int) (llist_of_stream u))) (llist_of_stream (smap int (p ## sscan ex u p)))"
norm_bound_Basis_le: fixes b :: "'c"   and x :: "'c"   and e :: "real" assumes "b \<in> Basis"   and "norm x \<le> e" shows "\<bar>x \<bullet> b\<bar> \<le> e"
finished_inf: fixes s :: "'state" assumes "s \<rightarrow>\<^sup>\<infinity>" shows "\<not> finished (\<rightarrow>) s"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat list \<Rightarrow> 'a mat \<Rightarrow> 'a mat \<times> 'a mat \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "nat list"   and a2 :: "'a mat"   and a3 :: "'a mat \<times> 'a mat" assumes "\<And>div_op H P K. P div_op [] H (P, K)"   and "\<And>div_op i xs H P K. (\<And>x xa xb. \<lbrakk>x = div_op (H $$ (i, 0)) (H $$ (0, 0)); xa = addrow_mat (dim_row H) (- x) i 0; xb = addrow (- x) i 0 K\<rbrakk> \<Longrightarrow> P div_op xs H (xa * P, xb)) \<Longrightarrow> P div_op (i # xs) H (P, K)" shows "P a0 a1 a2 a3"
Arg2pi_lt_pi: fixes z :: "complex" shows "(0 < Arg2pi z \<and> Arg2pi z < pi) = (0 < Im z)"
Act\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and \<alpha> :: "'g"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "p \<bullet> Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> (p \<bullet> \<alpha>) (p \<bullet> t\<^sub>\<alpha>)"
sup_eq_bot_iff: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x \<union># y = {#}) = (x = {#} \<and> y = {#})"
iso_verts_eq_iff: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'c"   and v :: "'c" assumes "iG.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom G)"   and "v \<in> verts (app_iso hom G)" shows "(iso_verts hom u = iso_verts hom v) = (u = v)"
floor_correct: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and floor :: "'a \<Rightarrow> int"   and x :: "'a" assumes "class.floor_ceiling divide inverse times one plus zero minus uminus less_eq less sgn abs floor" shows "less_eq (ring_1.of_int one plus zero minus (floor x)) x \<and> less x (ring_1.of_int one plus zero minus (floor x + 1))"
exhaust_sel: fixes llist :: "'b llist" obtains "llist = <>"   | "llist = lhd llist % ltl llist"
Smith_normal_form_upt_k_imp_Suc_k: fixes A :: "(('a, 'd) vec, 'e) vec"   and k :: "nat" assumes "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"   and "k < min (nrows A) (ncols A)" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<Suc k] bezout) k"
beta_coerce_cfun: fixes f :: "'e \<rightarrow> 'f"   and x :: "'d" shows "COERCE('e \<rightarrow> 'f, 'd \<rightarrow> 'c)\<cdot>f\<cdot>x = COERCE('f, 'c)\<cdot>(f\<cdot>(COERCE('d, 'e)\<cdot>x))"
inf_prefix_length_leL: fixes xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length xs ys \<le> length xs"
simps_1: fixes ys :: "'e list" shows "[] @ ys = ys"
sum_closed: fixes G :: "'a set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "AbGroup G"   and "finite A"   and "f ` A \<subseteq> G" shows "sum f A \<in> G"
take_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "(x # xs) \<down> Suc n = x # xs \<down> n"
cases: fixes x :: "symbol list" obtains "x = []"   | xa :: "symbol" and xs :: "symbol list" where "x = xa # xs"
distinct_107: fixes x7 :: "jump"   and x101 :: "stmt"   and x102 :: "stmt" shows "Jmp x7 \<noteq> x101 Finally x102"
k1_def: shows "k1 = (1 - s1 - s2) / (1 - s1)"
sum_list_Basis_list: fixes f :: "'b \<Rightarrow> 'a" shows "sum_list (map f Basis_list) = sum f Basis"
vec_of_basis_enum_minus: fixes b1 :: "'a"   and b2 :: "'a" shows "vec_of_basis_enum (b1 - b2) = vec_of_basis_enum b1 - vec_of_basis_enum b2"
f_aggregate_append_mult: fixes xs :: "'b list"   and m :: "nat"   and k :: "nat"   and ys :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "length xs = m * k" shows "f_aggregate (xs @ ys) k ag = f_aggregate xs k ag @ f_aggregate ys k ag"
exists_f_PAQ_Aii: fixes A :: "'b mat"   and P :: "'b mat"   and n :: "nat"   and Q :: "'b mat"   and i :: "nat" assumes "isDiagonal_mat A"   and "P \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "Q \<in> carrier_mat n n"   and "i < n" shows "\<exists>f. (P * A * Q) $$ (i, i) = (\<Sum>i\<in>set (diag_mat A). f i * i)"
hom_one: shows "rupture_surj K p \<one>\<^bsub>K [X]\<^esub> = \<one>\<^bsub>Rupt K p\<^esub>"
Lp_infinity_zero_space: fixes p :: "ennreal"   and M :: "'b measure" assumes "0 < p" shows "zero_space\<^sub>N (\<LL> p M) = {f \<in> borel_measurable M. AE x in M. f x = 0}"
succ_chain_vars_id: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < length l"   and "j < length l" shows "tvars (\<T> (l ! i)) = tvars (\<T> (l ! j))"
ASSERT_leI: fixes \<Phi> :: "bool"   and M :: "'a nres"   and M' :: "'a nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "ASSERT \<Phi> \<bind> (\<lambda>_. M) \<le> M'"
check_mult_l_dom_err_def: fixes p_notin :: "bool"   and p :: "nat"   and i_already :: "bool"   and i :: "nat" shows "check_mult_l_dom_err p_notin p i_already i = SPEC (\<lambda>_. True)"
surj_nat_to_rat_surj: shows "surj nat_to_rat_surj"
lookupb_csubst: fixes a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and c :: "coname"   and N :: "trm"   and x :: "name"   and P :: "trm"   and y :: "name" assumes "a \<sharp> (\<theta>_c, c, N)" shows "Cut <c>.N x.P = lookupb y a \<theta>_c c N{a:=x.P}"
solve_cubic_depressed_Cardano_all_real_roots: fixes e :: "real"   and f :: "real"   and y :: "complex" assumes "0 \<le> discriminant_cubic_depressed e f"   and "y ^ 3 + cor e * y + cor f = 0" shows "y \<in> \<real>"
restr_to_subalg_def: fixes M :: "'b measure"   and F :: "'b measure" shows "restr_to_subalg M F = measure_of (space M) (sets F) (emeasure M)"
cod_def: fixes f :: "'a arrow_of_spans_data" shows "V.cod f = (if V.codomains f \<noteq> {} then SOME b. b \<in> V.codomains f else V.null)"
Suc_3: shows "\<mu>_small_row i fs (Suc j)"
PP_def: shows "PP \<equiv> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> B\<^sub>1 ===> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> (B\<^sub>1 ===> B\<^sub>1) ===> ((=) ===> B\<^sub>1 ===> B\<^sub>1) ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> B\<^sub>2 ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> (B\<^sub>2 ===> B\<^sub>2) ===> ((=) ===> B\<^sub>2 ===> B\<^sub>2) ===> (=)"
UP_smult_l_distr: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "p \<in> carrier (UP R)" shows "(a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>UP R\<^esub> p = a \<odot>\<^bsub>UP R\<^esub> p \<oplus>\<^bsub>UP R\<^esub> b \<odot>\<^bsub>UP R\<^esub> p"
help_rtrancl_Range: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set" assumes "a \<noteq> b"   and "(a, b) \<in> R\<^sup>*" shows "b \<in> Range R"
inf_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "inf x y = y"
finprod_relations_triv_harder_better_stronger: fixes A :: "'a set" assumes "A \<subseteq> carrier G"   and "relations A = {\<lambda>_\<in>A. 0}" shows "\<forall>f\<in>\<Pi>\<^sub>E a\<in>A. generate G {a}. finprod G f A = \<one> \<longrightarrow> (\<forall>a\<in>A. f a = \<one>)"
dord_pm_plus_monotone: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "ord s t \<Longrightarrow> ord (s + u) (t + u)"   and "linorder.dord_pm ord s t" shows "linorder.dord_pm ord (s + u) (t + u)"
simps: fixes xs :: "'b list" shows "list_of (IArray xs) = xs"
combine_states_def: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and n :: "char list" shows "<s|t> n = (if is_local n then s n else t n)"
mult_poly_add_left: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "(p + q) * r = p * r + q * r"
simps_5: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({I} WHILE b DO C) Q = I"
a_inv_closed: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<in> carrier G"
SDom_def: fixes \<sigma> :: "('g, 'h, 'i) subst" shows "SDom \<sigma> = {Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}"
a0i_agreementD: fixes x :: "a0n_state"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "x \<in> a0i_agreement" shows "signals x (Commit A B M) \<le> signals x (Running A B M)"
nstut_fun1: fixes X :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "NSTUTINV X" shows "NSTUTINV f<X>"
simps_27: fixes x2 :: "'a" shows "set_extended \<lbrakk>x2\<rbrakk> = {x2}"
ASSFMT_1: shows "Q' = Q \<union> br_dsq \<delta> q (Q, W)"