hcju/isabelleps huggingface dataset:
mult_less_cancel_right_disj: fixes 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 abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
intro: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder ord ord_strict"   and "ordered_powerprod_axioms ord" shows "ordered_powerprod ord ord_strict"
red_hext_incr: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "h \<unlhd> h'"
ics: shows "ics = Called [] \<or> ics = No_ics"
sums_iff_shift: fixes n :: "nat"   and s :: "'a" shows "(\<lambda>i. f (i + n)) sums s = f sums (s + sum f {..<n})"
mertens_bounded: shows "(\<lambda>x. \<MM> x - ln x) \<in> O(\<lambda>_. 1)"
homotopy_eqv_homotopic_triviality_null_imp: fixes S :: "'a set"   and T :: "'b set"   and U :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "S homotopy_eqv T"   and "continuous_on U f"   and "f ` U \<subseteq> T"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c)" shows "\<exists>c. homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)"
eval_gext_lemma: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')" shows "snd s\<le>|snd s' \<and> (case w of In2 vf \<Rightarrow> normal s \<longrightarrow> (\<forall>v x s. s\<le>|snd (assign (snd vf) v (x, s))) | _ \<Rightarrow> True)"
gt_iff_wt_unary_diff_same: 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 t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s = (tkbo_coefs.gt_wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t s \<or> tkbo_coefs.gt_unary ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s \<or> tkbo_coefs.gt_diff ground_heads_var gt_sym \<delta> arity_sym wt_sym coef_sym t s \<or> tkbo_coefs.gt_same ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s)"
induct: fixes P :: "fmla \<Rightarrow> bool"   and fmla :: "fmla" assumes "\<And>x1 x2. P (x1 EQ x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 OR x2)"   and "\<And>x. P x \<Longrightarrow> P (Robinson_Arithmetic.neg x)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (exi x1 x2)" shows "P fmla"
qty_empty_def: shows "qty_empty = (\<lambda>x n w. 0)"
span_list_def: fixes vs :: "'a vec list" shows "span_list vs \<equiv> {uu_. \<exists>c. uu_ = lincomb_list c vs}"
is_SDG_path_hd: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "ns \<noteq> []" shows "hd ns = n"
strict_trans1: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" assumes "b \<le>s a"   and "c <s b" shows "c <s a"
stable_on_allocation: fixes dsa :: "'b set"   and d :: "'b"   and ds :: "'b set"   and X :: "'a set" assumes "Singleton_for_d.stable_on dsa d ds X" shows "inj_on Xd X"
flip_fresh_fresh: fixes a :: "'d"   and x :: "'a"   and b :: "'d" assumes "atom a \<sharp> x"   and "atom b \<sharp> x" shows "(a \<leftrightarrow> b) \<bullet> x = x"
truncation_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and S :: "'b set" assumes "stellar slices W" shows "stellar.truncation slices W p S \<equiv> {p'. stellar.reachable_through slices W p S p'}"
sorted_wrt_linorder_of_list: fixes xs :: "'a list" assumes "distinct xs" shows "Linorder_Relations.sorted_wrt (linorder_of_list xs) xs"
set_iterator_map_rev_linord_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "('b \<times> 'c) set" shows "set_iterator_map_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k' \<le> k)"
prems_4: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
sifum_types_def: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool" shows "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B \<equiv> sifum_lang ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars \<and> sifum_security Stop (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) some_val some_val'"
LLL_inv_wD_1: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list" assumes "LLL.LLL_invariant_weak n m fs_init fs" shows "cof_vec_space.lin_indpt_list n (map of_int_hom.vec_hom fs)"
take_drop_take: fixes n :: "nat"   and xs :: "'b list"   and m :: "nat" shows "take n xs @ drop n (take m xs) = take (max n m) xs"
indets_monom_mult_subset: fixes c :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (MPoly_Type_Class.punit.monom_mult c t p) \<subseteq> keys t \<union> indets p"
geodesic_segment_param_in_geodesic_spaces_3: fixes t :: "real"   and x :: "'a"   and y :: "'a" assumes "t \<in> {0..dist x y}" shows "geodesic_segment_param {x--y} x t \<in> {x--y}"
simps_1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ys :: "'b list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_vec_Tensor f [] ys = []"
assms: fixes x :: "'a" assumes "x |\<in>| S" shows "rel_option P (fmlookup m x) (fmlookup n x)"
Zconst: fixes k :: "nat"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>Z {\<lambda>s. enat k * P s} c {\<lambda>s. enat k * Q s}"   and "0 < k" shows "\<turnstile>\<^sub>Z {P} c {Q}"
Rep_filter_inverse: fixes x :: "'c Filter.filter" shows "Filter.filter.Abs_filter (Filter.filter.Rep_filter x) = x"
inpt_st_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "'a"   and u' :: "'b"   and x :: "'c" shows "inpt_st r u u' x = (\<exists>y. r u u' x y)"
results_gpv_map_gpv_id: fixes \<I> :: "('b, 'c) \<I>"   and f :: "'d \<Rightarrow> 'a"   and gpv :: "('d, 'b, 'c) gpv" shows "results_gpv \<I> (map_gpv f id gpv) = f ` results_gpv \<I> gpv"
exhaust: fixes y :: "('d, 'e) stateful_strand_step" obtains x1 :: "('d, 'e) Term.term" where "y = send\<langle>x1\<rangle>"   | x2 :: "('d, 'e) Term.term" where "y = receive\<langle>x2\<rangle>"   | x31 :: "poscheckvariant" and x32 :: "('d, 'e) Term.term" and x33 :: "('d, 'e) Term.term" where "y = \<langle>x31: x32 \<doteq> x33\<rangle>"   | x41 :: "('d, 'e) Term.term" and x42 :: "('d, 'e) Term.term" where "y = insert\<langle>x41,x42\<rangle>"   | x51 :: "('d, 'e) Term.term" and x52 :: "('d, 'e) Term.term" where "y = delete\<langle>x51,x52\<rangle>"   | x61 :: "poscheckvariant" and x62 :: "('d, 'e) Term.term" and x63 :: "('d, 'e) Term.term" where "y = \<langle>x61: x62 \<in> x63\<rangle>"   | x71 :: "'e list" and x72 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" and x73 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" where "y = \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
negligible_locally_Lipschitz_image: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "DIM('c) \<le> DIM('d)"   and "negligible S"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>T B. open T \<and> x \<in> T \<and> (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> B * norm (y - x))" shows "negligible (f ` S)"
condition2_part4: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f" assumes "echelon_form_upt_k A k"   and "a < b"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) = nrows A" shows "(LEAST n. A $ a $ n \<noteq> (0::'d)) < (LEAST n. A $ b $ n \<noteq> (0::'d))"
assms_1: shows "LS M1 q1 \<subseteq> LS M2 q2"
assign_eval\<^sub>w_const\<^sub>A: fixes x :: "Eg1.var"   and c :: "nat"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat" shows "eval_abv\<^sub>A \<langle>x \<leftarrow> aexp.Const c, mds, mem\<rangle>\<^sub>A \<langle>Stop, mds, mem(x := c)\<rangle>\<^sub>A"
implicit_module_ow: fixes U\<^sub>M :: "'e set" shows "module_ow U\<^sub>M (+) (0::'e) (-) uminus = module_on U\<^sub>M"
pos_real_map: shows "{x. 1 / x \<in> (\<lambda>x. x + 1) ` {x. 0 < Re x}} = circle_01_diam"
ordLess_def: shows "ordLess = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
lift_Suc_mono_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f n) (f (Suc n))"   and "n \<le> n'" shows "less_eq (f n) (f n')"
infinity_evalUni_NeqUni: fixes p :: "real \<times> real \<times> real"   and x :: "real" shows "(\<exists>x. \<forall>y<x. aEvalUni (NeqUni p) y) = evalUni (substNegInfinityUni (NeqUni p)) x"
prems_4: shows "(A, y) \<in> foldSetD D (\<cdot>) e"
insertI2: fixes a :: "'e"   and B :: "'e set"   and b :: "'e" assumes "a \<in> B" shows "a \<in> insert b B"
gseq_iff: fixes f :: "nat \<Rightarrow> 'a list"   and g :: "nat \<Rightarrow> 'a list" shows "((f, g) \<in> gseq) = (f \<in> SEQ (lists A) \<and> g \<in> SEQ (lists A) \<and> (\<exists>i. length (g i) < length (f i) \<and> (\<forall>j<i. f j = g j)))"
scaleR: fixes x :: "'a"   and r :: "real"   and b :: "'b" shows "f' y x (r *\<^sub>R b) = r *\<^sub>R f' y x b"
fsps_chain_boundTr2: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set"   and x :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a"   and "x \<in> fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C})"   and "C \<noteq> {}" shows "(THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x)) \<in> carrier N \<and> (\<exists>a1 b1. (a1, b1) \<in> C \<and> x \<in> fgs R M a1 \<and> (THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x)) = b1 x)"
hom_group_mul_rev: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a"   and y :: "'a" assumes "(\<lambda>(x, y). x \<otimes> y) \<in> hom (subgroup_generated G A \<times>\<times> subgroup_generated G B) G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<in> A"   and "y \<in> B" shows "x \<otimes> y = y \<otimes> x"
bind_mono: fixes B :: "('b \<Rightarrow> 'c option) \<Rightarrow> 'd option"   and C :: "'d \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'e option" assumes "monotone option.le_fun option_ord B"   and "\<And>y. monotone option.le_fun option_ord (C y)" shows "monotone option.le_fun option_ord (\<lambda>f. Option.bind (B f) (\<lambda>y. C y f))"
crel_vs1_execute_Some: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap"   and heap :: "heap" assumes "crel_vs1 R a b"   and "P heap" obtains x :: "'f" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'"
Try: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and tn :: "qtname"   and vn :: "vname"   and c2 :: "stmt" assumes "E,dt\<Turnstile>c1\<Colon>\<surd>"   and "prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable"   and "lcl E (VName vn) = None"   and "E\<lparr>lcl := lcl E(VName vn \<mapsto> Class tn)\<rparr>,dt\<Turnstile>c2\<Colon>\<surd>" shows "E,dt\<Turnstile>Try c1 Catch(tn vn) c2\<Colon>\<surd>"
q_inter: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and I :: "'b set"   and Q :: "'b set"   and Q' :: "'b set" assumes "intact slices W I"   and "stellar.quorum (project slices I) W Q"   and "stellar.quorum (project slices I) W Q'"   and "Q \<inter> I \<noteq> {}"   and "Q' \<inter> I \<noteq> {}" shows "Q \<inter> Q' \<inter> I \<noteq> {}"
constructive_security_obsf_def: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and \<A> :: "('y + 'z, 'aa + 'ab) distinguisher_obsf"   and adv :: "real" shows "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv \<equiv> constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv \<and> constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv"
JVM_conf_read_axioms: shows "JVM_conf_read addr2thread_id thread_id2addr empty_heap allocate typeof_addr heap_read heap_write hconf P"
assms_2: fixes x :: "variable \<times> String.literal set" assumes "x \<in> tvsT_Set (snd ` set instT)" shows "x \<notin> fst ` set instT"
Ring1_def: fixes R :: "'a set" shows "Ring1 R \<equiv> Group R \<and> Ring1_axioms R"
of_nat_neq_0: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc n) \<noteq> zero"
del_list_sorted: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted_less (xs @ a # ys)" shows "del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
is_type_simps: 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 C :: "char list" shows "is_type P Void \<and> is_type P Boolean \<and> is_type P Integer \<and> is_type P NT \<and> is_type P (Class C) = is_class P C"
simps_2: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and p :: "String.literal"   and \<phi> :: "formula"   and \<psi> :: "formula" shows "sat \<sigma> V v i (formula.Let p \<phi> \<psi>) = sat \<sigma> (V(p \<mapsto> \<lambda>i. {v. length v = nfv \<phi> \<and> sat \<sigma> V v i \<phi>})) v i \<psi>"
LFO_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and R :: "('d \<times> 'c) set"   and fi :: "'a \<Rightarrow> 'd \<Rightarrow> 'd nres"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s0i :: "'d"   and s0 :: "'c" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li (\<lambda>_. True) fi s0i \<le> \<Down> R (foreach l f s0)"
prems: shows "k < length (trace_parallel_plan_sas_plus I (a # \<pi>)) - 1"
B: fixes cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "ccomp c\<^sub>1 \<Turnstile> cfs\<box>" shows "cpred c\<^sub>1 (cfs ! 0) (cfs ! (length cfs - Suc 0))"
roofed_circE: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and T :: "'a set" assumes "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> T" obtains "x \<in> RF\<^bsub>\<Gamma>\<^esub> T" and "\<not> essential \<Gamma> (B \<Gamma>) T x"
Arctan_def: shows "Arctan \<equiv> \<lambda>z. \<i> / 2 * Ln ((1 - \<i> * z) / (1 + \<i> * z))"
isSPath_sg_outgoing: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and u1 :: "nat"   and v1 :: "nat"   and v2 :: "nat" assumes "isSimplePath u p v"   and "(u1, v1) \<in> set p"   and "v1 \<noteq> v2" shows "(u1, v2) \<notin> set p"
self: fixes t :: "Lambda_Z.term" shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* t\<^sup>\<bullet>"
klift_eta_inv1: fixes f :: "'a \<Rightarrow> 'b set" shows "f\<^sup>\<dagger> \<circ> \<eta> = f"
renders_commutative: shows "D.is_rendered_commutative_by p0 p1"
correct: fixes x0 :: "'a"   and x1 :: "'a"   and \<sigma> :: "bool" assumes "x0 \<in> carrier \<G>"   and "x1 \<in> carrier \<G>" shows "mal_def.trusted_party (x0, x1) \<sigma> = protocol_ot (x0, x1) \<sigma>"
cases: fixes a1 :: "point"   and a2 :: "point" assumes "swp_graph a1 a2" obtains swp_sum :: "point \<Rightarrow> point" and x :: "int" and y :: "int" where "a1 = \<lparr>Xcoord = x, Ycoord = y\<rparr>" and "a2 = \<lparr>Xcoord = y, Ycoord = x\<rparr>"
M5_lattice_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "M5_lattice a b c = (a \<sqinter> b = b \<sqinter> c \<and> c \<sqinter> a = b \<sqinter> c \<and> a \<squnion> b = b \<squnion> c \<and> c \<squnion> a = b \<squnion> c \<and> a \<sqinter> b < a \<squnion> b)"
nz: fixes i :: "nat" assumes "i < m + n" shows "M $$ (i, i) \<noteq> 0"
fds_nth_scale_nonpower: fixes c :: "nat"   and n :: "nat"   and f :: "'b fds" assumes "\<not> is_nth_power c n" shows "fds_nth (fds_scale c f) n = (0::'b)"
assms_2: shows "orient_lit_inst l u v polarity \<theta>"
Struct_axioms: shows "CU.Struct wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) D intF intP"
add_mono_thms_linordered_field_2: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i = j \<and> k < l" shows "i + k < j + l"
x_y_in_carrier: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "ordinal_utility carrier relation u"   and "x \<succeq>[relation] y" shows "x \<in> carrier \<and> y \<in> carrier"
fmpred_iff: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('d, 'e) fmap" shows "fmpred P m = (\<forall>x y. fmlookup m x = Some y \<longrightarrow> P x y)"
coeffs_map_poly: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly" shows "coeffs (map_poly f p) = strip_while ((=) (0::'c)) (map f (coeffs p))"
l_set_val_get_tag_name_axioms_def: fixes set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_tag_name_axioms set_val_locs get_tag_name_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
ln_fact_conv_mangoldt: fixes n :: "nat" shows "ln (fact n) = (\<Sum>d = 1..n. mangoldt d * real_of_int \<lfloor>real n / real d\<rfloor>)"
asm: shows "\<forall>i j. i < j \<and> j < length l \<longrightarrow> l ! i \<noteq> a \<or> l ! j = a"
frontier_of_triangle: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "DIM('b) = 2" shows "frontier (convex hull {a, b, c}) = closed_segment a b \<union> closed_segment b c \<union> closed_segment c a"
vector_inf_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqinter> top = x" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
multiplicity_lcm: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and p :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "x \<noteq> zero"   and "y \<noteq> zero"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.multiplicity times one p (lcm x y) = max (factorial_semiring.multiplicity times one p x) (factorial_semiring.multiplicity times one p y)"
HInfinite_add_le_zero: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HInfinite"   and "y \<le> 0"   and "x \<le> 0" shows "x + y \<in> HInfinite"
pa1_disj_in2: shows "path_image c1 \<inter> inside (path_image c2 \<union> path_image c) = {}"
flow_lift: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "flow \<Delta> f" shows "flow \<Delta>' (lift f)"
mat_det_cnj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (mat_cnj A) = cnj (mat_det A)"
dim_def: fixes p :: "'a"   and V :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set" shows "cotangent_space.dim p V = (if \<exists>b\<subseteq>cotangent_space p. independent b \<and> span b = span V then card (SOME b. b \<subseteq> cotangent_space p \<and> independent b \<and> span b = span V) else 0)"
univ_ce_lm_1: fixes n :: "nat"   and x :: "nat" shows "(c_pair n x \<in> univ_ce) = (x \<in> nat_to_ce_set n)"
eq: shows "\<sigma> \<circ> f = \<tau> \<circ> g"
simps_11: fixes p :: "fm"   and q :: "fm" shows "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
2: shows "v \<notin> range_vars (\<sigma>1 \<circ>\<^sub>s \<theta>2)"
ring_id_eval: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "U_function_ring R"   and "a \<in> carrier R" shows "U_function_ring.ring_id R a = a"