hcju/isabelleps huggingface dataset:
assms_5: fixes var :: "Vars.var"   and c :: "'a"   and exp :: "exp" assumes "{atom var} \<sharp>* c"   and "\<And>c. P c exp" shows "P c (Lam [var]. exp)"
erH_def: fixes R :: "('e, 'g) Ring_scheme"   and S :: "('e, 'h) Ring_scheme"   and X :: "'e"   and A :: "('f, 'i) Ring_scheme"   and B :: "('f, 'j) Ring_scheme"   and Y :: "'f"   and h :: "'e \<Rightarrow> 'f" shows "erH R S X A B Y h = (\<lambda>x\<in>carrier R. erh R S X A B Y h (fst (s_cf R S X x)) (s_cf R S X x))"
Z''_1: shows "Z'' = Closure\<^sub>\<alpha> \<Z>"
bisim1_xcp_\<tau>Red: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and P :: "'addr J1_prog"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'addr expr1 \<Rightarrow> 'a"   and t :: "'thread_id" assumes "typeof_addr h a = \<lfloor>Class_type D\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* Throwable"   and "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"   and "match_ex_table (compP f P) (cname_of h a) pc (compxE2 e 0 0) = None"   and "\<exists>n. n + max_vars e' \<le> length xs \<and> \<B> e n" shows "\<tau>red1r P t h (e', xs) (Throw a, loc) \<and> P,e,h \<turnstile> (Throw a, loc) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"
fields_table_SomeI: fixes G :: "prog"   and C :: "qtname"   and n :: "fspec"   and f :: "Decl.field"   and P :: "fspec \<Rightarrow> Decl.field \<Rightarrow> bool" assumes "table_of (DeclConcepts.fields G C) n = Some f"   and "P n f" shows "fields_table G C P n = Some (type f)"
convex_hull_Times: fixes s :: "'a set"   and t :: "'b set" shows "convex hull s \<times> t = (convex hull s) \<times> (convex hull t)"
sat_preserving_ground_sound_\<Gamma>: shows "sat_preserving_inference_system ground_sound_\<Gamma>"
trdsp_alt: fixes bs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" shows "trdsp bs (p, q) = trd bs (spoly (fst p) (fst q))"
prove'_Cons: fixes x :: "(nat \<times> form) list"   and l :: "(nat \<times> form) list list" shows "prove' (x # l) = prove' (subs x @ flatten (map subs l))"
\<gamma>'_def: shows "\<gamma>' = (\<lambda>t. \<i> * (\<gamma> t - z) + z)"
While: fixes P :: "state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "\<turnstile> {\<lambda>s. P s \<and> b s} c {P}" shows "\<turnstile> {P} WHILE b DO c {\<lambda>s. P s \<and> \<not> b s}"
diff_right_commute: fixes a :: "'q"   and c :: "'q"   and b :: "'q" shows "a - c - b = a - b - c"
bounded_incs_axioms: fixes b :: "nat list"   and ys :: "nat list" shows "bounded_incs (cond1 b ys) (Max (set b))"
subst_closed: fixes t :: "'a FOL_Fitting.term"   and i :: "nat"   and p :: "('a, 'b) form" assumes "closedt 0 t"   and "closed (Suc i) p" shows "closed i (p[t/i])"
ratfps_eqI: fixes x :: "'b ratfps"   and y :: "'b ratfps" assumes "fps_of_ratfps x = fps_of_ratfps y" shows "x = y"
Predict_elem_in_Gen: fixes I :: "item set"   and P :: "(symbol \<times> character list) list set"   and k :: "nat"   and x :: "item" assumes "I \<subseteq> Gen P"   and "k \<le> length Doc"   and "x \<in> Predict k I" shows "x \<in> Gen P"
leq_prodI: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" assumes "fst p \<sqsubseteq> fst q"   and "snd p \<sqsubseteq> snd q" shows "p \<sqsubseteq> q"
word_plus_mono_right2: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a \<le> a + b"   and "c \<le> b" shows "a \<le> a + c"
Assign: fixes a :: "aexp"   and s :: "char list \<Rightarrow> int option"   and i :: "int"   and x :: "char list" assumes "Def_Init_Exp.aval a s = Some i" shows "(x ::= a, s) \<rightarrow> (SKIP, s(x \<mapsto> i))"
lookup_raw_inj: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list"   and ko :: "'a"   and ys :: "('b \<times> 'c) list" assumes "oalist_raw.oalist_inv rep_key_order (xs, ko)"   and "oalist_raw.oalist_inv rep_key_order (ys, ko)"   and "oalist_raw.lookup_raw rep_key_order (xs, ko) = oalist_raw.lookup_raw rep_key_order (ys, ko)" shows "xs = ys"
cases_2: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<rightarrow> y"   and "y \<rightarrow>\<^bsup>n\<^esup> v"   and "\<And>z m. \<lbrakk>m \<le> n; z \<rightarrow>\<^bsup>m\<^esup> v\<rbrakk> \<Longrightarrow> P m z" shows "P (Suc n) x"
size_list_estimation: fixes x :: "'b"   and xs :: "'b list"   and y :: "nat"   and f :: "'b \<Rightarrow> nat" assumes "x \<in> set xs"   and "y < f x" shows "y < size_list f xs"
qSwapInp_def: fixes xs :: "'f"   and x :: "'g"   and y :: "'g"   and qinp :: "'c \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option" shows "qSwapInp xs x y qinp \<equiv> lift (qSwap xs x y) qinp"
ex_distrib_star: fixes P :: "'a \<Rightarrow> assn"   and Q :: "assn" shows "(\<exists>\<^sub>Ax. P x * Q) = (\<exists>\<^sub>Ax. P x) * Q"
simps_13: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y21 :: "'a"   and y22 :: "'a list" shows "\<not> list_all2 R (y21 # y22) []"
br_rel_prod: fixes a :: "'d \<Rightarrow> 'f"   and I :: "'d \<Rightarrow> bool"   and b :: "'e \<Rightarrow> 'g"   and J :: "'e \<Rightarrow> bool" shows "br a I \<times>\<^sub>r br b J = br (\<lambda>(x, y). (a x, b y)) (\<lambda>(x, y). I x \<and> J y)"
invar_succ_step: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat"   and v :: "nat" assumes "nf_invar c src dst PRED C N d"   and "v \<in> C"   and "dst \<notin> Graph.E c `` {v} - dom PRED" shows "nf_invar' c src dst (map_mmupd PRED (Graph.E c `` {v} - dom PRED) v) (C - {v}) (N \<union> (Graph.E c `` {v} - dom PRED)) d"
y_def: shows "y \<equiv> getFr [x, z] [t] []"
EQ_def: fixes G :: "Var \<Rightarrow> L"   and p :: "L" shows "EQ G p = (\<lambda>s t. \<forall>x. LEQ (G x) p \<longrightarrow> s x = t x)"
wfUNIVI: fixes r :: "('a \<times> 'a) set" assumes "\<And>P x. \<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wf r"
preserves_reflects_arr: fixes f :: "'a" shows "arr (\<rr>.map f) = arr f"
RDirProd_list_truncate: fixes Rs :: "('a, 'b) ring_scheme list" shows "monoid.truncate (RDirProd_list Rs) = DirProd_list Rs"
col_mat_of_cols: fixes j :: "nat"   and vs :: "'a vec list"   and n :: "nat" assumes "j < length vs"   and "vs ! j \<in> carrier_vec n" shows "col (mat_of_cols n vs) j = vs ! j"
abs_pdevs_val_le_tdev': fixes e :: "nat \<Rightarrow> real"   and y :: "'a pdevs"   and p :: "nat" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "\<bar>pdevs_val e y\<bar> \<le> tdev' p y"
simps_22: fixes v1 :: "val"   and v2 :: "val" shows "eval_binop CondOr v1 v2 = Bool (the_Bool v1 \<or> the_Bool v2)"
set_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list" assumes "ShadowRootClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_tag_name element_ptr tag"
reg\<^sub>C_not_visible_abs: shows "reg\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d regex" shows "atms (map_regex f v) = f ` atms v"
ln_nth_prime_asymptotics: shows "(\<lambda>n. ln (real (nth_prime n)) - (ln (real n) + ln (ln (real n)))) \<in> o(\<lambda>_. 1)"
mem_imp_elem: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> carrier L"   and "x \<in> A" shows "x .\<in> A"
Infinitesimal_less_SReal2: fixes y :: "real star" assumes "y \<in> Infinitesimal" shows "\<forall>r\<in>\<real>. 0 < r \<longrightarrow> y < r"
in_one_stalk: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c"   and V :: "'c set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x"   and "V \<in> I" shows "(V, one_str V) \<in> stalk.one_stalk \<FF> \<rho> one_str I V"
op_Union_coll_def: shows "op_Union_coll = \<Union>"
matchC_C_sym: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.matchC_C aval tval cval indis theta c d" shows "PL_Indis.matchC_C aval tval cval indis (theta\<inverse>) d c"
row_space_eq_col_space_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_space A = col_space (Finite_Cartesian_Product.transpose A)"
flatten_pi_1_sf_subset: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1) \<subseteq> (\<Union>\<phi>\<in>sf \<phi>. sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1))"
pause_writer_def: fixes pause :: "'n \<Rightarrow> ('o \<Rightarrow> 'm) \<Rightarrow> 'm"   and out :: "'n"   and c :: "'o \<Rightarrow> ('k, 'l, 'm) writerT" shows "pause_writer pause out c = WriterT (pause out (\<lambda>input. run_writer (c input)))"
option_eq_def: fixes eq :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and v1 :: "'c option"   and v2 :: "'d option" shows "option_eq eq v1 v2 \<equiv> case (v1, v2) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x1, None) \<Rightarrow> False | (Some x1, Some x2) \<Rightarrow> eq x1 x2"
elims_1: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "'f"   and xb :: "'d list"   and xc :: "'f"   and y :: "bool" assumes "d_reaches x xa xb xc = y" obtains M :: "('d, 'e, 'f) FSM" and p :: "'f" and xs :: "'d list" and q :: "'f" where "x = M" and "xa = p" and "xb = xs" and "xc = q" and "y = (\<exists>tr ys. d_reached_by M p xs q tr ys)"
homeomorphism_chart_basis: shows "homeomorphism (Collect (chart_basis_domainP b)) UNIV (chart_basis b) local.chart_basis_inv"
has_bochner_integral_integral_eq: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "has_bochner_integral M f x" shows "integral\<^sup>L M f = x"
assms: shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
pair_by_index_main: fixes n1 :: "nat"   and n2 :: "nat" shows "nat_to_pr (pair_by_index n1 n2) = c_f_pair (nat_to_pr n1) (nat_to_pr n2)"
simulation_this: shows "simulation ((R1 O R2)\<inverse>) C A"
b_e_check_single_sound_unop_testop_cvtop: fixes \<C> :: "t_context"   and e :: "b_e"   and tn' :: "checker_type"   and tm' :: "checker_type"   and t :: "t"   and uu :: "unop_i"   and uv :: "testop"   and uw :: "unop_f"   and t1 :: "t"   and sx :: "sx option"   and tm :: "t list" assumes "check_single \<C> e tn' = tm'"   and "(e = Unop_i t uu \<or> e = Testop t uv) \<and> is_int_t t \<or> e = Unop_f t uw \<and> is_float_t t \<or> e = Cvtop t1 Convert t sx \<and> convert_cond t1 t sx \<or> e = Cvtop t1 Reinterpret t sx \<and> t1 \<noteq> t \<and> t_length t1 = t_length t \<and> sx = None"   and "c_types_agree tm' tm"   and "tn' \<noteq> Bot"   and "tm' \<noteq> Bot" shows "\<exists>tn. c_types_agree tn' tn \<and> \<C> \<turnstile> [e] : tn _> tm"
NatTransCompCompDef: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and X :: "'g" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2"   and "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" shows "\<eta>1 $$ X \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> \<eta>2 $$ X"
interpret_Atoms_2: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list" shows "QEdlo_ex.interpret (Atom (Less i j)) xs = (xs ! i < xs ! j)"
gen_ideal_ideal: fixes x :: "'a" assumes "x \<in> R" shows "ideal \<langle>x\<rangle> R (+) (\<cdot>) \<zero> \<one>"
word_of_int_eq_iff: fixes k :: "int"   and l :: "int" shows "(word_of_int k = word_of_int l) = (take_bit LENGTH('k) k = take_bit LENGTH('k) l)"
bind_return_distr': fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and N :: "'b measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> (\<lambda>x. return N (f x)) = distr M N f"
homologous_rel_add: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and a' :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b' :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a a'"   and "homologous_rel p X S b b'" shows "homologous_rel p X S (a + b) (a' + b')"
extend_map_lemma_cofinite0: fixes \<F> :: "'a set set"   and K :: "'a set"   and U :: "'a set"   and T :: "'b set"   and h :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and> continuous_on (\<Union> \<F> - C) g \<and> g ` (\<Union> \<F> - C) \<subseteq> T \<and> (\<forall>x\<in>(\<Union> \<F> - C) \<inter> K. g x = h x)"
lookup_lower: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t"   and v :: "'t" shows "lookup (lower p u) v = (if v \<prec>\<^sub>t u then lookup p v else (0::'c))"
fds_liouville_lambda_times_zeta: shows "fds liouville_lambda * fds_zeta = fds_ind is_square"
map_nth: fixes xs :: "'b list" shows "map ((!) xs) [0..<length xs] = xs"
\<psi>_def: fixes f :: "'a" shows "\<psi> f \<equiv> (\<sigma>.map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>.map\<^sub>0 (trg\<^sub>C f), \<tau>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D (\<sigma>.unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
ohearn_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "ohearn P c Q \<equiv> \<forall>t. Q t \<longrightarrow> (\<exists>s. P s \<and> (c, s) \<Rightarrow> t)"
unrest_uop: fixes x :: "'a \<Longrightarrow> 'b"   and e :: "('c, 'b) uexpr"   and f :: "'c \<Rightarrow> 'd" assumes "x \<sharp> e" shows "x \<sharp> uop f e"
lookup_restrict_indets_pp: fixes t :: "'c option \<Rightarrow>\<^sub>0 nat" shows "lookup (restrict_indets_pp t) = (\<lambda>x. lookup t (Some x))"
ah_def: shows "ah \<equiv> {h. \<forall>m. h m = {} \<or> m \<in> h m}"
leq_gen_cong: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R \<le> gen_cong R"
subspace_dim_equal: fixes S :: "'a set"   and T :: "'a set" assumes "subspace S"   and "subspace T"   and "S \<subseteq> T"   and "dim T \<le> dim S" shows "S = T"
intros_2: fixes l :: "int"   and d :: "int"   and u :: "int" assumes "- l \<le> d"   and "d \<le> u" shows "valid_intv' l u (Const' d)"
assms_2: shows "wp \<pi> (WHILE b DO c) Q s\<^sub>0"
red_not_final_thread: 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 ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   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" assumes "s -2-t\<triangleright>ta\<rightarrow> s'" shows "r2.not_final_thread s t"
card_T_bound: shows "card ((t \<circ> OB) ` msgs) \<le> (n + 1) ^ card observations"
col_cong_bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B D"   and "Cong A B C D"   and "Bet A C B" shows "Bet C A D \<or> Bet C B D"
elims_2: fixes x :: "ivl"   and xa :: "ivl" assumes "le_aux x xa" obtains l1 :: "int option" and h1 :: "int option" and l2 :: "int option" and h2 :: "int option" where "x = I l1 h1" and "xa = I l2 h2" and "le_option False l2 l1 \<and> le_option True h1 h2"
stable_sort_key_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sk :: "('h \<Rightarrow> 'g) \<Rightarrow> 'h list \<Rightarrow> 'h list" assumes "class.linorder less_eq less" shows "linorder.stable_sort_key sk = (\<forall>f xs k. filter (\<lambda>y. f y = k) (sk f xs) = filter (\<lambda>y. f y = k) xs)"
terminates_sequence_app: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and s :: "'e"   and ys :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile>sequence Seq xs \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>sequence Seq ys \<down> s'" shows "\<Gamma>\<turnstile>sequence Seq (xs @ ys) \<down> Normal s"
rChoice2: fixes \<beta> :: "prog"   and \<nu> :: "fin_string \<Rightarrow> real"   and \<omega> :: "fin_string \<Rightarrow> real"   and \<alpha> :: "prog" assumes "[\<beta>]\<nu> \<downharpoonleft> \<omega>" shows "[\<alpha> \<union>\<union> \<beta>]\<nu> \<downharpoonleft> \<omega>"
fe_def: fixes x :: "'a" shows "fe x = ennreal (f x)"
IH: shows "(r' = 0 \<or> degree r' < degree d) \<and> smult (lc ^ n) (d * (smult lc q + b) + (smult lc r - b * d)) = d * q' + r'"
eq3: shows "[(a, c), (b, c), (a, c)] \<bullet> x = x"
distinct_mset_dom: fixes N :: "('c, 'd) fmap" shows "distinct_mset (dom_m N)"
filter_tab_SomeD: fixes c :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "filter_tab c t k = Some x" shows "t k = Some x \<and> c k x"
div_add: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
injI: fixes f :: "'k \<Rightarrow> 'l" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
rfd_rbb_galois: fixes R :: "('c \<times> 'd) set"   and X :: "'d set"   and Y :: "'c set" shows "(fd\<^sub>\<R> R X \<subseteq> Y) = (X \<subseteq> bb\<^sub>\<R> R Y)"
degree_mult_not_eq: fixes f :: "'b poly"   and g :: "'b poly" assumes "degree (f * g) \<noteq> degree f + degree g" shows "lead_coeff f * lead_coeff g = (0::'b)"
abs_le_D2: fixes abs :: "'a \<Rightarrow> '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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq (uminus a) b"
subst_name_def: fixes a :: "name"   and b :: "name"   and c :: "name" shows "a[b::=c] \<equiv> if a = b then c else a"
b_2: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p b"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_plus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vrat_plus" shows "b = c"
mod_exp_aux_correct: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and m :: "'b"   and y :: "'b"   and x :: "'b"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (euclidean_semiring_cancel.mod_exp_aux times modulo m y x n) m = modulo (times (power.power one times x n) y) m"
divide_le_eq: 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 b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide b c) a = (if less zero c then less_eq b (times a c) else if less c zero then less_eq (times a c) b else less_eq zero a)"
sum_over_fun_eq: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in># A \<Longrightarrow> f x = g x" shows "\<Sum>\<^sub># (image_mset f A) = \<Sum>\<^sub># (image_mset g A)"
not_different_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and authK' :: "nat" assumes "AKcryptSK authK servK evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK authK' servK evs \<and> servK \<in> symKeys"
g_ball_def: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_ball m P \<equiv> iteratei m id (\<lambda>kv \<sigma>. P kv) True"
cring_simprules_9: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
gallery_double_cross_not_minimal_Cons1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and C :: "'b set"   and D :: "'b set"   and Cs :: "'b set list" assumes "ThinishChamberComplexFolding X f"   and "B \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "C \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (B # C # Cs @ [D])" shows "\<not> SimplicialComplex.min_maxsimpchain X (B # C # Cs @ [D])"
next_sendupdD_5: fixes c0 :: "('b, 'loc \<times> 't) Exchange.configuration"   and c1 :: "('b, 'loc \<times> 't) Exchange.configuration"   and p :: "'b"   and tt :: "('loc \<times> 't) set" assumes "cri.next_sendupd' c0 c1 p tt" shows "c_glob c1 = c_glob c0"