hcju/isabelleps huggingface dataset:
iso_comp: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'d set set"   and g :: "'c \<Rightarrow> 'd" assumes "ChamberComplexIsomorphism X Y f"   and "ChamberComplexIsomorphism Y Z g" shows "ChamberComplexIsomorphism X Z (g \<circ> f)"
dist_transform: fixes p :: "int \<times> int"   and l :: "int"   and \<delta> :: "real" shows "(dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>) = (real_of_int l - \<delta> < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + \<delta>)"
peq: shows "Preds (USubst.adjoint (dotsubstt \<theta>) I \<omega>) = Preds (repc I dotid (term_sem I \<theta> \<omega>))"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_right_moving_1 ires rs (l, r) = (\<exists>ml mr rn. l = Bk \<up> ml @ Oc # ires \<and> r = Bk \<up> mr @ Oc \<up> Suc rs @ Bk \<up> rn \<and> Suc 0 < ml + mr)"
inj_comm_ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_comm_ring_hom hom \<equiv> comm_ring_hom hom \<and> zero_hom_0 hom"
reflection_of_binary_predicates_and_closures_2: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_reflects_binary_pred Rel Pred" shows "rel_reflects_binary_pred (Rel\<^sup>+) Pred"
DERIV_diff: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)" shows "((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
iso_inv_iso: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "category.iso C (category.inv C f)"
id: shows "poly_mult_rat r p = poly_mult_rat_main n d p"
predicate_eq_def: fixes P :: "'a Predicate.pred"   and Q :: "'a Predicate.pred" shows "local.predicate_eq P Q = (local.leq_pred P Q \<and> local.leq_pred Q P)"
rel_conversep: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_pmf R\<inverse>\<inverse> = (rel_pmf R)\<inverse>\<inverse>"
rel_gpv'_def: fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool" shows "spmf_to_gpv.rel_gpv' A = rel_gpv A (=)"
ex_\<pi>: shows "I \<noteq> UNIV \<and> admits_tupling I"
Mp_shrink_modulus: fixes k :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.Mp (m * k) f = poly_mod.Mp (m * k) g"   and "k \<noteq> 0" shows "f =m g"
lemma_R_for_Lim: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and i :: "nat"   and d :: "nat" assumes "learn_lim \<psi> U (\<phi> i)" shows "learn_lim \<psi> U (totalizer d i)"
np: assumes "w1 = NEG_INF \<and> w2 = POS_INF" shows "P w1 w2"
add_less_zeroD: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (plus x y) zero" shows "less x zero \<or> less y zero"
Derivation_dest_ex_unique: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "Derivation a D b" shows "\<exists>!x. Derivation a D x"
P1_def: shows "P1 \<equiv> \<lambda>tp. tp = ([], <code_tcontra>)"
length_repeat: fixes n :: "nat"   and a :: "'a" shows "length (repeat n a) = n"
backlinks_distinct: fixes E :: "('b \<times> 'b) list" assumes "distinct E" shows "distinct (backlinks E)"
ignores_old_vars_conc_meval: fixes P :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and n :: "nat"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "ignores_old_vars P"   and "conc.meval_abv gc\<^sub>C n gc\<^sub>C'"   and "length (fst gc\<^sub>C) = length cms" shows "P (map snd (fst gc\<^sub>C)) = P (map snd (fst gc\<^sub>C'))"
f: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "comp_fun_commute_on f A" shows "f \<in> A \<rightarrow> A \<rightarrow> A"
a1: shows "low_equal s1 s2 \<and> t1 = s1\<lparr>user_reg := new_user_reg\<rparr> \<and> t2 = s2\<lparr>user_reg := new_user_reg\<rparr>"
absorb1: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "min a b = a"
V_def: shows "V = arith_prog_fb (int_of_fbint x) b"
changes_itv_smods_0_1: fixes a :: "real"   and b :: "real"   and q :: "real poly" shows "changes_itv_smods a b 0 q = 0"
lb_arctan_bound': fixes x :: "float"   and prec :: "nat" assumes "0 \<le> real_of_float x" shows "real_of_float (lb_arctan prec x) \<le> arctan (real_of_float x)"
fix_clock_alt_def: fixes x1ba :: "'f state"   and xa :: "'d state \<times> 'e" shows "fix_clock x1ba xa = (case xa of (x3a, x2ba) \<Rightarrow> (update_clock (\<lambda>_. if clock x3a \<le> clock x1ba then clock x3a else clock x1ba) x3a, x2ba))"
accepts_def: fixes tr :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> 'a"   and s :: "'b"   and as :: "'c list" shows "accepts tr P s as = P (foldl tr s as)"
a1: shows "s' = snd (fst (mul_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r)"
simps: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "argmax f A = {x \<in> A. f x = Max (f ` A)}"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_fail_alt return bind fail alt" shows "monad_alt return bind alt"
hyps: fixes \<S> :: "nat \<Rightarrow> bool option" shows "dom (share sb \<S>) \<subseteq> dom \<S> \<union> all_shared sb"
Nmlize_Trg_2: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Trg t\<^bold>\<rfloor> = Trg t"
converse_def: shows "\<smile> \<equiv> \<lambda>r. {(xa, x). (\<lambda>x xa. s2p r (x, xa))\<inverse>\<inverse> xa x}"
vcg_name_pre_state: fixes P :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<And>s. P s \<Longrightarrow> \<lbrace>(=) s\<rbrace> c \<lbrace>Q\<rbrace>" shows "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
elims_2: fixes x :: "'at Literal"   and xa :: "'at Literal" assumes "literal_ordering x xa" obtains L1 :: "'at Literal" and L2 :: "'at Literal" where "x = L1" and "xa = L2" and "(atom L1, atom L2) \<in> atom_ordering"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "ord x y"
simps_3: fixes v :: "game" shows "Diamondo (Agame v) undeff = undeff"
lookup_add: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and g :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "poly_mapping.lookup (f + g) k = poly_mapping.lookup f k + poly_mapping.lookup g k"
subscene_implies_sublens': fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y" shows "\<lbrakk>X\<rbrakk>\<^sub>\<sim> \<subseteq>\<^sub>S \<lbrakk>Y\<rbrakk>\<^sub>\<sim> = X \<subseteq>\<^sub>L' Y"
morphism_sheaves_of_rings_def: fixes X :: "'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 \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" shows "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
vector_3_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "vector [x, y, z] $ 1 = x"
reachable_empty_stack: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and y :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "fst x = []"   and "reachable x y" shows "fst y = []"
advantage_def: fixes \<G> :: "'d cyclic_group"   and \<A> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool spmf" shows "Diffie_Hellman.ddh.advantage \<G> \<A> = \<bar>spmf (Diffie_Hellman.ddh.ddh_0 \<G> \<A>) True - spmf (Diffie_Hellman.ddh.ddh_1 \<G> \<A>) True\<bar>"
elims_3: fixes x :: "'c \<times> 'd"   and xa :: "'c \<Rightarrow> 'd option" assumes "\<not> x \<in>\<^sub>m xa" obtains k :: "'c" and v :: "'d" and m :: "'c \<Rightarrow> 'd option" where "x = (k, v)" and "xa = m" and "m k \<noteq> Some v"
diff_indexed_Ints_vec: fixes x :: "'b vec"   and n :: "nat"   and y :: "'b vec"   and I :: "nat set" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "x \<in> indexed_Ints_vec I"   and "y \<in> indexed_Ints_vec I" shows "x - y \<in> indexed_Ints_vec I"
bounded_linear_blinfun_mult_left: shows "bounded_linear blinfun_mult_left"
ord_resolve_reductive: fixes M :: "'a literal multiset set"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset" assumes "gr.ord_resolve M CAs DA AAs As E" shows "E < DA"
has_vector_derivative: fixes f :: "'b \<Rightarrow> 'c"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b"   and F :: "real filter" assumes "bounded_linear f"   and "(g has_vector_derivative g') F" shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
trans: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "P i j"   and "P j k"   and "i < j"   and "j \<le> k" shows "P i k"
column_add_code_nth: fixes A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and a :: "'d"   and b :: "'d"   and q :: "'c" shows "($h) (column_add A a b q) = column_add_row A a b q"
fps_of_int: fixes c :: "int" shows "fps_const (of_int c) = of_int c"
terminate_halt_lemma: fixes m :: "nat"   and r :: "nat"   and t :: "nat" assumes "rec_exec rec_nonstop ([m, r] @ [t]) = 0"   and "\<forall>i<t. 0 < rec_exec rec_nonstop ([m, r] @ [i])" shows "terminate rec_halt [m, r]"
integral_fst': fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) f" shows "LBINT x. LBINT y. f (x, y) = integral\<^sup>L (lborel \<Otimes>\<^sub>M lborel) f"
dfs_while_algo: fixes R :: "('a \<times> 'a) set"   and \<Sigma>i :: "'a set" assumes "finite (R\<^sup>* `` \<Sigma>i)" shows "while_algo (dfs_algo \<Sigma>i R)"
neg_closed: fixes G :: "'a set"   and g :: "'a" assumes "AbGroup G"   and "g \<in> G" shows "- g \<in> G"
strip_eq_Assign: fixes C :: "'a acom"   and x :: "char list"   and e :: "aexp" shows "(strip C = x ::= e) = (\<exists>P. C = x ::= e {P})"
fmdom'_def: shows "fmdom' \<equiv> map_fun fmlookup id dom"
node_ptr_kinds_def: fixes heap :: "(_) heap" shows "node_ptr_kinds heap = the |`| cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| ffilter is_node_ptr_kind (object_ptr_kinds heap)"
simps: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "\<alpha>.round_robin_reschedule t0 queue n0 s = (let t = hd queue; queue' = tl queue in if t = t0 then None else case \<alpha>.round_robin_step n0 (t # queue', n0) s t of None \<Rightarrow> \<alpha>.round_robin_reschedule t0 (queue' @ [t]) n0 s | \<lfloor>ttaxm\<sigma>\<rfloor> \<Rightarrow> \<lfloor>ttaxm\<sigma>\<rfloor>)"
TESL_interp_stepwise_timedelayed_coind_unfold: fixes master :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and measuring :: "clock"   and slave :: "clock"   and n :: "nat" shows "\<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = (\<lbrakk> master \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<union> \<lbrakk> master \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> measuring @ n \<oplus> \<delta>\<tau> \<Rightarrow> slave \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m) \<inter> \<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
mono: fixes x :: "'a"   and y :: "'a" assumes "I x"   and "I y"   and "x \<le> y" shows "f x \<le> f y"
d_comm_ax: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d y \<cdot> d x"
L_idx: fixes w :: "letter list"   and ws :: "letter list list"   and f :: "nat \<Rightarrow> letter list" assumes "L w ws"   and "is_prefix ws f" shows "\<exists>i. emb (f i) w \<and> i < length ws"
r_cancel: fixes a :: "'a list"   and c :: "'a list"   and b :: "'a list" assumes "a \<oplus>\<^bsub>K [X]\<^esub> c = b \<oplus>\<^bsub>K [X]\<^esub> c"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "c \<in> carrier (K [X])" shows "a = b"
circ_circ_circ: fixes x :: "'a" shows "x\<^sup>\<circ>\<^sup>\<circ>\<^sup>\<circ> = x\<^sup>\<circ>\<^sup>\<circ>"
hyp: shows "\<forall>l\<le>k. read_only_unowned (snd (snd (c (n + l)))) (fst (c (n + l)))"
less_induct: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "\<And>x. (\<And>y. x < y \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
filter_tab_SomeI: fixes t :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and x :: "'c"   and C :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "t k = Some x"   and "C k x" shows "filter_tab C t k = Some x"
l2norm_def: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" shows "l2norm S f \<equiv> sqrt (l2product S f f)"
simps_2: fixes A :: "'a" shows "atom (Neg A) = A"
remdups_eq_nil_iff: fixes x :: "'b list" shows "(remdups x = []) = (x = [])"
Cauchy_integral_formula_convex_simple: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "convex S"   and "f holomorphic_on S"   and "z \<in> interior S"   and "valid_path \<gamma>"   and "path_image \<gamma> \<subseteq> S - {z}"   and "pathfinish \<gamma> = pathstart \<gamma>" shows "((\<lambda>w. f w / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
at_map_atom_pres: fixes x :: "'a" assumes "atom x" shows "at_map x = Abs_atoms (\<eta> x)"
Mainj: fixes j :: "nat" assumes "0 < j" shows "\<forall>\<^sub>F n in sequentially. ln (B n) / real n \<le> (1 + ln L) / real j"
sub_2: fixes mds :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" assumes "mds_consistent mds \<Gamma>\<^sub>1"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^sub>1\<^esub> mem\<^sub>2"   and "\<langle>c, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>\<^sub>1'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
open_subopen: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "open S = (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
conjI2: fixes A :: "bool"   and B :: "bool" assumes "A"   and "A \<Longrightarrow> B" shows "A \<and> B"
nth_subdegree_mult_right: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_nth (f * g) (subdegree g) = fps_nth f 0 * fps_nth g (subdegree g)"
hcomp_in_vhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<nu> \<star>\<^sub>B \<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>" and "src\<^sub>B \<mu> = src\<^sub>B f" and "src\<^sub>B \<mu> = src\<^sub>B g" and "trg\<^sub>B \<nu> = trg\<^sub>B f" and "trg\<^sub>B \<nu> = trg\<^sub>B g"
content_eq_0: fixes a :: "'b"   and b :: "'b" shows "(content (cbox a b) = 0) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i)"
flush_step: shows "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m', sb', \<O>', \<R>', \<S>')"
circ_right_unfold: fixes x :: "'a" shows "Rf (x\<^sup>\<circ>) x = x\<^sup>\<circ>"
hyps_7: fixes b :: "(name \<times> trm) list" assumes "b closes (x, \<sigma>) # \<Gamma>" shows "b<t> \<in> RED (T \<tau>)"
simps_4: fixes v :: "'b"   and va :: "'b"   and vb :: "'c" shows "removeShadowRules1_alternative_rev [AllowPortFromTo v va vb] = [AllowPortFromTo v va vb]"
list_all_simps_2: fixes P :: "'b \<Rightarrow> bool" shows "list_all P [] = True"
image_empty: fixes f :: "'c::type \<Rightarrow> 'b::type" shows "(f::'c::type \<Rightarrow> 'b::type) ` {} = {}"
stutter_sampler_def: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'a" shows "stutter_sampler f \<sigma> \<equiv> f 0 = 0 \<and> strict_mono f \<and> (\<forall>k n. f k < n \<and> n < f (Suc k) \<longrightarrow> \<sigma> n = \<sigma> (f k))"
times_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x \<cdot> y \<equiv> pnat_plus x y"
prems: shows "\<exists>a\<in>set (x # xs). P a"
pos: shows "{r. 0 < r \<and> r < q} \<subset> {0<..}"
Abs_ttree_inverse: fixes y :: "'b list set" assumes "y \<in> {xss. [] \<in> xss \<and> downset xss}" shows "Rep_ttree (Abs_ttree y) = y"
det_wf_nfa: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" assumes "wf_nfa A n" shows "wf_dfa (det_nfa A) n"
gcd_nat_induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "\<And>m. P m 0"   and "\<And>m n. \<lbrakk>0 < n; P n (m mod n)\<rbrakk> \<Longrightarrow> P m n" shows "P m n"
pr_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.cod C (pr0 a b) = b"
listall_nil_eq: fixes P :: "'a \<Rightarrow> bool" shows "listall P [] = True"
vec_lambda_eq_sum: fixes f :: "'a \<Rightarrow> real \<Rightarrow> real"   and x :: "(real, 'a) vec" shows "(\<chi>k. f k (x $ k)) = (\<Sum>k\<in>Basis. f (axis_index k) (x \<bullet> k) *\<^sub>R k)"
nat_to_list_def: shows "nat_to_list = (\<lambda>u. if u = 0 then [] else c_unfold (c_len u) (c_snd (u - 1)))"
axioms: fixes S1 :: "'b set"   and S2 :: "'c set"   and s1 :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "linear_on S1 S2 s1 s2 f" shows "module_hom_on S1 S2 s1 s2 f"
Complete_def: fixes Rule :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" shows "Complete Rule = (\<forall>S. saturated_binary_rule Rule S \<longrightarrow> all_fulfill finite S \<longrightarrow> {} \<notin> S \<longrightarrow> satisfiable S)"
intro: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d" assumes "\<And>G f nP. \<lbrakk>wf_graph G; f \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP\<rbrakk> \<Longrightarrow> \<forall>i\<in>fst ` f. \<not> sinvar G (nP(i := default_node_properties))"   and "\<And>otherbot. \<forall>G f nP i. wf_graph G \<and> f \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> i \<in> fst ` f \<longrightarrow> \<not> sinvar G (nP(i := otherbot)) \<Longrightarrow> otherbot = default_node_properties" shows "SecurityInvariant_ACS_axioms sinvar default_node_properties"
sum_distinct_of_odd: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "\<And>i. p i \<noteq> 0 \<Longrightarrow> 1 \<le> i \<and> i \<le> n"   and "\<And>i. p i * i \<le> n"   and "\<And>i. p i \<noteq> 0 \<Longrightarrow> odd i" shows "(\<Sum>i\<le>n. distinct_of_odd p i * i) = (\<Sum>i\<le>n. p i * i)"
inv\<^sub>3E_1: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and P\<^sub>1 :: "'b set set"   and P\<^sub>2 :: "'b set set"   and B\<^sub>1 :: "'b set"   and B\<^sub>2 :: "'b set"   and S :: "'b set"   and L :: "'b set" assumes "BinPacking_Complete U w c"   and "BinPacking_Complete.inv\<^sub>3 U w c P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "BinPacking_Complete.inv\<^sub>2 U w c P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
continuous_on_ext_cont: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_on (cbox a b) f" shows "continuous_on S (ext_cont f a b)"
Field_square: fixes x :: "'b set" shows "Field (x \<times> x) = x"
intro: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order" assumes "gd_nat_term (\<lambda>x. x) (\<lambda>x. x) to" shows "qpm_nat_inf_term to"
eucl_of_list_take_DIM: fixes d :: "nat"   and xs :: "real list" assumes "d = DIM('a)" shows "eucl_of_list (take d xs) = eucl_of_list xs"
UnE: fixes c :: "'x"   and A :: "'x set"   and B :: "'x set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
leg_simps_3: shows "C.arr src_\<mu>.dom.leg1"
lattice_partial_function_definition: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
test_de_morgan: fixes x :: "'a"   and y :: "'a" shows "n (t x \<cdot> t y) = n x + n y"
init_fin_bisim_wait_simps: fixes bisim_wait :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and status1 :: "status"   and x1 :: "'a"   and status2 :: "status"   and x2 :: "'b" shows "FWbisimulation_base.init_fin_bisim_wait bisim_wait (status1, x1) (status2, x2) = (status1 = Running \<and> status2 = Running \<and> bisim_wait x1 x2)"
mt: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "mono_trans (t P)"
equiv_utransD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "equiv_utrans t u"   and "unitary P" shows "t P = u P"
in_mono: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "order.greater_eq B A" shows "x \<in> A \<longrightarrow> x \<in> B"