hcju/isabelleps huggingface dataset:
inverse_negative_imp_negative: 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 a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) zero"   and "a \<noteq> zero" shows "less a zero"
ASSERT_refine_right: fixes \<Phi> :: "bool"   and S :: "'a nres"   and R :: "('a \<times> 'b) set"   and S' :: "'b nres" assumes "\<Phi> \<Longrightarrow> S \<le> \<Down> R S'" shows "S \<le> \<Down> R (ASSERT \<Phi> \<bind> (\<lambda>_. S'))"
non_trivial_ker_imp_algebraic: fixes K :: "'b set"   and x :: "'b" assumes "a_kernel (K [X]\<^bsub>S\<^esub>) S (\<lambda>p. Hom.S.eval p x) \<noteq> {[]}" shows "(Hom.S.algebraic over K) x"
plus_adds_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus (plus a b) c" shows "plus.adds plus b c"
languageE: fixes w :: "'interpretation stream" assumes "w \<in> R.language" obtains p :: "'state" and v :: "'action stream" where "w = smap int (p ## trace v p)" and "init p" and "R.run v p"
hyps_2: fixes mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mds\<^sub>2' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "mds\<^sub>2 \<le> mds\<^sub>2''" shows "\<forall>mem. (\<langle>Stop, mds\<^sub>2', mem'\<rangle> \<in> loc_reach \<langle>c, mds\<^sub>2, mem\<rangle> \<longrightarrow> mds\<^sub>2' \<le> mds\<^sub>2'''__) \<and> locally_sound_mode_use \<langle>c, mds\<^sub>2, mem\<rangle>"
of_rat_divide: fixes a :: "rat"   and b :: "rat" shows "of_rat (a / b) = of_rat a / of_rat b"
assms_3: fixes k' :: "int"   and \<gamma>' :: "real \<Rightarrow> real \<times> real" assumes "(k', \<gamma>') \<in> set ((- 1, \<gamma>) # l)" shows "valid_path \<gamma>'"
finite_supp_map_bset_Not: fixes xset :: "('b, 'c, 'd) formula set['e]" assumes "finite (supp xset)" shows "finite (supp (map_bset Formula.Not xset))"
deg_p_nzero: shows "deg R p \<noteq> 0"
Conj\<^sub>\<alpha>_eq_iff: fixes tset1\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]"   and tset2\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" shows "(Conj\<^sub>\<alpha> tset1\<^sub>\<alpha> = Conj\<^sub>\<alpha> tset2\<^sub>\<alpha>) = (tset1\<^sub>\<alpha> = tset2\<^sub>\<alpha>)"
mndet_subset_FD: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b process" assumes "A \<noteq> {}" shows "(\<sqinter>xa\<in>A \<union> B \<rightarrow>  P) \<le> (\<sqinter>xa\<in>A \<rightarrow>  P)"
cases: fixes a1 :: "((nat \<times> nat) \<times> transition) fset"   and a2 :: "nat"   and a3 :: "nat \<Rightarrow>f value option"   and a4 :: "((nat \<times> nat) \<times> transition) fset"   and a5 :: "nat"   and a6 :: "nat \<Rightarrow>f value option"   and a7 :: "(String.literal \<times> value list) list" assumes "executionally_equivalent a1 a2 a3 a4 a5 a6 a7" obtains e1 :: "((nat \<times> nat) \<times> transition) fset" and s1 :: "nat" and r1 :: "nat \<Rightarrow>f value option" and e2 :: "((nat \<times> nat) \<times> transition) fset" and s2 :: "nat" and r2 :: "nat \<Rightarrow>f value option" where "a1 = e1__" and "a2 = s1__" and "a3 = r1__" and "a4 = e2__" and "a5 = s2__" and "a6 = r2__" and "a7 = []"   | e1 :: "((nat \<times> nat) \<times> transition) fset" and s1 :: "nat" and r1 :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and e2 :: "((nat \<times> nat) \<times> transition) fset" and s2 :: "nat" and r2 :: "nat \<Rightarrow>f value option" and es :: "(String.literal \<times> value list) list" where "a1 = e1" and "a2 = s1" and "a3 = r1" and "a4 = e2" and "a5 = s2" and "a6 = r2" and "a7 = (l, i) # es" and "\<forall>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es" and "\<forall>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. \<exists>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es"
that: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> UNIV \<rightarrow> I"   and "pdevs_val e (pdevs_of_list xs) = (\<Sum>P\<in>set xs. f P *\<^sub>R P)" shows "thesis"
intros_1: fixes a :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" shows "(a, []) \<in> seq ars"
is_abs_def: fixes t :: "'a" shows "is_abs t = term_cases (\<lambda>_. False) (\<lambda>_. False) (\<lambda>_ _. False) True t"
word_msb_nth: fixes w :: "'b word" shows "msb w = bit (uint w) (LENGTH('b) - 1)"
finite_list: fixes A :: "'c set" assumes "finite A" shows "\<exists>xs. set xs = A"
in_arcs_eq: fixes w :: "'a" shows "in_arcs H w = rev_H ` out_arcs H w"
nu_par_aux1: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> \<tau> y = d (\<tau> y) \<cdot> x"
vimage_const: fixes c :: "'b"   and A :: "'b set" shows "(\<lambda>x. c) -` A = (if c \<in> A then UNIV else {})"
mat_times_vec_leq_def: fixes A :: "'b mat"   and x :: "'b vec"   and b :: "'b vec" shows "[A*\<^sub>vx]\<le>b = ((\<forall>i<dim_vec b. (A *\<^sub>v x) $ i \<le> b $ i) \<and> dim_row A = dim_vec b \<and> dim_col A = dim_vec x)"
Pareto_iff: fixes x :: "'alt"   and y :: "'alt" shows "Pareto R x y = (\<forall>i\<in>agents. R i x y)"
"3_2": fixes x :: "'d" assumes "comp kx ky = Lt"   and "x = f kx vx (0::'c)"   and "oalist_inv_raw xs"   and "oalist_inv_raw ((ky, vy) # ys)"   and "map2_val_compat g"   and "map2_val_compat h" shows "fst ` set (map2_val_pair f g h xs ((ky, vy) # ys)) \<subseteq> fst ` set xs \<union> fst ` set ((ky, vy) # ys)"
mcont_lfilter: fixes P :: "'a \<Rightarrow> bool" shows "mcont lSup lprefix lSup lprefix (lfilter P)"
simps_9: fixes v :: "event_data list"   and l :: "formula list"   and e :: "String.literal \<times> event_data list" shows "matches v (Ands l) e = (\<exists>\<phi>\<in>set l. matches v \<phi> e)"
bl_to_bin_inj: fixes bs :: "bool list"   and cs :: "bool list" assumes "bl_to_bin bs = bl_to_bin cs"   and "length bs = length cs" shows "bs = cs"
WhileF: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and c :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,s\<^sub>1\<rangle>"
id: shows "real_of_1 (p, l', r') = real_of_1 (p, l, r)"
isNth: shows "SG ! i = ([], [Fsubst ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>')) \<sigma>])"
HSem_reorder: fixes \<Gamma> :: "('d \<times> 'e) list"   and \<Delta> :: "('d \<times> 'e) list"   and ESem :: "'e \<Rightarrow> ('d \<Rightarrow> 'f) \<rightarrow> 'f"   and \<rho> :: "'d \<Rightarrow> 'f" assumes "map_of \<Gamma> = map_of \<Delta>" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> = has_ESem.HSem ESem \<Delta>\<cdot>\<rho>"
WCFG_CondThen: fixes c\<^sub>1 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and b :: "expr"   and c\<^sub>2 :: "cmd" assumes "c\<^sub>1 \<turnstile> n -et\<rightarrow> n'"   and "n \<noteq> (_Entry_)" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> n \<oplus> 1 -et\<rightarrow> n' \<oplus> 1"
u_map_up: fixes f :: "'f \<rightarrow> 'e"   and x :: "'f" shows "u_map\<cdot>f\<cdot>(up\<cdot>x) = up\<cdot>(f\<cdot>x)"
lunstream_simps_2: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d"   and s' :: "'d" assumes "g s = Skip s'" shows "lunstream g s = lunstream g s'"
subresultant_resultant: fixes f :: "'b poly"   and g :: "'b poly" shows "subresultant 0 f g = [:resultant f g:]"
assms_3: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0" shows "crit_pair_cbelow_on d m F p q"
tendsto_cadd_ereal: fixes y :: "ereal"   and x :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter" assumes "y \<noteq> - \<infinity>"   and "x \<noteq> - \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y) \<longlongrightarrow> x + y) F"
isCont_Im: fixes a :: "'a"   and g :: "'a \<Rightarrow> complex" assumes "isCont g a" shows "isCont (\<lambda>x. Im (g x)) a"
l_comb_add1: fixes A :: "'b set"   and H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "t \<in> {j. j \<le> n} \<rightarrow> A" shows "l_comb R M n (\<lambda>x\<in>{j. j \<le> n}. s x \<plusminus>\<^bsub>R\<^esub> t x) f = l_comb R M n s f \<plusminus> l_comb R M n t f"
simps_4: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and S :: "'a"   and b' :: "bexp"   and c1' :: "'a acom"   and c2' :: "'a acom"   and S' :: "'a" shows "(IF b THEN c1 ELSE c2 {S} \<sqsubseteq> IF b' THEN c1' ELSE c2' {S'}) = (b = b' \<and> c1 \<sqsubseteq> c1' \<and> c2 \<sqsubseteq> c2' \<and> S \<sqsubseteq> S')"
tuple_prj_spc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "elementary_category_with_pullbacks.tuple C prj0 prj1 (prj1 f g) f g (prj0 f g) = elementary_category_with_pullbacks.pbdom C prj0 f g"
Gamma_altdef: fixes z :: "'a" shows "Gamma z = lim (Gamma_series z)"
k: shows "\<forall>n\<ge>k. b \<le> s n"
insert_before_def: shows "Shadow_DOM.insert_before \<equiv> l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before Shadow_DOM.get_parent Shadow_DOM.get_child_nodes Shadow_DOM.set_child_nodes get_ancestors_si Shadow_DOM.adopt_node set_disconnected_nodes get_disconnected_nodes Shadow_DOM.get_owner_document"
env_restr_belowI: fixes S :: "'a set"   and m\<^sub>1 :: "'a \<Rightarrow> 'b"   and m\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> m\<^sub>1 x \<sqsubseteq> m\<^sub>2 x" shows "m\<^sub>1 f|` S \<sqsubseteq> m\<^sub>2 f|` S"
sets_measure_of_conv: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})"
deg_def: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a" shows "deg R p = (LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p))"
fimage_is_fempty: fixes f :: "'n \<Rightarrow> 'm"   and A :: "'n fset" shows "(f |`| A = {||}) = (A = {||})"
\<sigma>f_def: fixes D :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "\<sigma>f D \<equiv> \<lambda>(A, A'). D (A, A'\<lparr>ok := False\<rparr>)"
rel_interior_nonempty_interior: fixes S :: "'c set" assumes "interior S \<noteq> {}" shows "rel_interior S = interior S"
homh: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring_hom_ring R S h" shows "h \<in> ring_hom R S"
zero_one_infty_not_equal_4: shows "1\<^sub>h \<noteq> 0\<^sub>h"
T_deter_rand: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s0 :: "'a"   and qs :: "'b list"   and A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)"   and x :: "'d" shows "real (On_Off.T step t s0 qs (On_Off.off2 step A (s0, x) qs)) = On_Off.T_on_rand' step t (On_Off.embed A) (return_pmf (s0, x)) qs"
1: shows "s_TS init h (as @ [x] @ bs @ [x]) (length (as @ [x] @ bs @ [x])) = fst (Partial_Cost_Model.Step (rTS h) (TSdet init h (as @ [x] @ bs @ [x]) (length (as @ [x] @ bs))) ((as @ [x] @ bs @ [x]) ! length (as @ [x] @ bs)))"
sort_oalist_raw_id: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" assumes "ko_ntm.oalist_inv xs" shows "sort_oalist_ko_ntm xs = xs"
axioms_3: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" shows "subdigraph_axioms \<BB> \<CC>"
mat_geI: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> B $$ (i, j) \<le> A $$ (i, j)" shows "A \<ge>\<^sub>m B"
uv0: shows "u \<noteq> 0 \<and> v \<noteq> 0"
assms_1: shows "G \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"
set_valued_transformation_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "set_valued_transformation C S F G \<tau> \<equiv> (category C \<and> replete_set_category S) \<and> set_valued_functor C S F \<and> set_valued_functor C S G \<and> natural_transformation C S F G \<tau>"
Err_le_conv: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" shows "(Err \<sqsubseteq>\<^bsub>le r\<^esub> e) = (e = Err)"
eq_numeral_simps_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus n = one) = (n = num.One)"
antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "less_eq x y"   and "less_eq y x" shows "x = y"
COND: fixes \<sigma> :: "'a"   and \<sigma>' :: "'a" assumes "\<sigma> \<le> \<sigma>'"   and "c \<sigma> \<noteq> c \<sigma>'" shows "\<sigma> = bot \<or> \<sigma>' = top"
continuous_on_sing: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on {x} f"
rank_Some_sink: fixes x :: "nat"   and n :: "nat"   and j :: "nat" assumes "rank x n = Some j" shows "\<not> sink (token_run x n)"
even_bit_succ_iff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "bit (plus one a) n = (bit a n \<or> n = 0)"
simply_connected_eq_Borsukian: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> Borsukian S)"
JOIN_VARS_def: fixes f :: "'e"   and g :: "'f"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" shows "JOIN_VARS f g P \<equiv> P f g"
SPN_is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>) (\<star>) \<a> \<i> src trg (\<bullet>) (\<circ>) Span.assoc Span.unit Span.src Span.trg SPN \<Phi>"
strict_sorted_equal: fixes xs :: "'a word list"   and ys :: "'a word list" assumes "signed.strict_sorted xs"   and "signed.strict_sorted ys"   and "set ys = set xs" shows "ys = xs"
isomorphicI': fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.iso_in_hom f a b" shows "D.isomorphic a b"
ae: shows "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
of_int_div: fixes y :: "int"   and x :: "int" assumes "y dvd x" shows "of_int (x div y) = of_int x / of_int y"
rel_spmf_reflI: fixes p :: "'e spmf"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
sat_dualD_2: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "y \<in> sat_dual A c" shows "\<forall>i<dim_vec y. (0::'a) \<le> y $ i"
FL_equivalent_iff_not_distinguished: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" shows "FL_logically_equivalent F P Q = (\<nexists>x. x \<in> \<A>[F] \<and> x distinguishes P from Q)"
dghm_ArrMap_vdomain: shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
power_minus': fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "NO_MATCH one x" shows "power.power one times (uminus x) n = times (power.power one times (uminus one) n) (power.power one times x n)"
wcount_add_mset: fixes f :: "'k \<Rightarrow> nat"   and x :: "'k"   and M :: "'k multiset"   and a :: "'k" shows "wcount f (add_mset x M) a = (if x = a then Suc (f a) else 0) + wcount f M a"
strongly_connected_def: fixes G :: "('e, 'f) pre_digraph" shows "strongly_connected G \<equiv> verts G \<noteq> {} \<and> (\<forall>u\<in>verts G. \<forall>v\<in>verts G. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v)"
neg_less_pos_iff: fixes A :: "'a"   and B :: "'a" shows "(Neg A < Pos B) = (A < B)"
bin_nth_1: fixes n :: "nat" shows "bit 1 n = (n = 0)"
dominated_convergence_at_top_1: fixes f :: "real \<Rightarrow> 'a \<Rightarrow> 'b"   and s :: "'a set"   and h :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>k. f k integrable_on s"   and "h integrable_on s"   and "\<And>k x. x \<in> s \<Longrightarrow> norm (f k x) \<le> h x"   and "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top" shows "g integrable_on s"
\<C>_vars_correct: fixes x :: "'Var" shows "\<C>_vars x = \<Union> (bexp_vars ` dma_type x)"
is_empty_def: shows "is_empty \<equiv> \<lambda>x. x = {}"
total': fixes x :: "'a"   and y :: "'a" assumes "\<not> le x y"   and "x \<in> carrier"   and "y \<in> carrier" shows "le y x"
count_add_mset: fixes b :: "'k"   and A :: "'k multiset"   and a :: "'k" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
mod_Suc_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "Suc (Suc (m mod n)) mod n = Suc (Suc m) mod n"
strict_sorted_sorted_wrt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less = sorted_wrt less"
fully_faithful_functor_axioms: shows "fully_faithful_functor Right_a.comp Right_a.comp (H\<^sub>R a)"
valid_reusables: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "local.valid t"   and "l t \<le> fst w"   and "r t \<le> snd w" shows "Ball (set (reusables t w)) local.valid"
wf_tuple_cong: fixes n :: "nat"   and A :: "nat set"   and v :: "'c option list"   and w :: "'c option list" assumes "wf_tuple n A v"   and "wf_tuple n A w"   and "\<forall>x\<in>A. map the v ! x = map the w ! x" shows "v = w"
bullet_app_mono: fixes t :: "CL"   and t' :: "CL"   and u :: "CL"   and u' :: "CL" assumes "(t, t') \<in> red\<^sup>*"   and "(u, u') \<in> red\<^sup>*" shows "(bullet_app t u, bullet_app t' u') \<in> red\<^sup>*"
thin: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "\<exists>!D. D \<in> X - {C} \<and> z \<lhd> D"
finite_product: fixes A :: "('c \<times> 'd) set" assumes "finite (fst ` A)"   and "finite (snd ` A)" shows "finite A"
distinct_87: fixes x5 :: "ephkey"   and x11 :: "tag" shows "cEphK x5 \<noteq> cTag x11"
fin: fixes j :: "'a" assumes "j \<in> I" shows "finite {i \<in> I. U i \<inter> U j \<noteq> {}}"
unitor_coincidence_2: shows "\<r>[\<I>] = \<iota>"
permutations_of_list_impl_aux_correct': fixes acc :: "'a list"   and xs :: "'a list" shows "permutations_of_list_impl_aux acc xs = map (\<lambda>xs. rev xs @ acc) (permutations_of_list_impl xs)"
map\<^sub>0_simps_1: fixes a :: "'b" assumes "B.obj a" shows "C.obj (H.map\<^sub>0 a)"
has_domain_iff_has_codomain: fixes f :: "'c \<times> 'c" shows "(CopxC.domains f \<noteq> {}) = (CopxC.codomains f \<noteq> {})"
neval_0: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and x :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and y :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval"   and "x = y" shows "sifum_security_init.neval eval x 0 y"
equiv: shows "\<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<approx> \<epsilon>(r' \<mapsto> (\<epsilon>, \<epsilon>, e))"
get_add_eq: fixes e :: "'fenv_ubx"   and x :: "'fun"   and v :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" shows "Fubx_get (Fubx_add e x v) x = Some v"
pwD1: fixes S :: "'b nres"   and S' :: "'b nres" assumes "S \<le> S'"   and "nofail S'" shows "nofail S"
unrestrictn_path: fixes g' :: "'b ugraph"   and V :: "'b set"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path (restrict_nodes g' V) u p v" shows "path g' u p v"
suffix_def: fixes s :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "s |\<^sub>s i \<equiv> %n. s (n + i)"
bind_UNION: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd set" shows "Set.bind A f = \<Union> (f ` A)"
thetaSeqTT_def: shows "thetaSeqTT \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>T d1 \<and> c2 \<approx>T d2}"
bij_betw_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" shows "bij_betw f A B = (inj_on f A \<and> (\<forall>y\<in>B. \<exists>x\<in>A. f x = y) \<and> (\<forall>x\<in>A. f x \<in> B))"
aeq_ale: fixes a :: "ant"   and b :: "ant" assumes "a = b" shows "a \<le> b"
composite_functor_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and F :: "'d \<Rightarrow> 'e"   and G :: "'e \<Rightarrow> 'f" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
CFG: shows "CFG src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry"
twiddleStore_inverse: fixes \<beta> :: "(Location \<times> Location) set"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "s  \<approx>\<^sub>\<beta> t" shows "t  \<approx>\<^sub>Pbij_inverse \<beta> s"
simps_2: fixes k :: "nat" shows "make_vert_equiv (Suc k) = mat1 2 \<otimes> make_vert_equiv k"
simps_2: fixes h :: "'a \<times> bool"   and t :: "('a \<times> bool) list"   and l :: "nat"   and cl :: "nat" shows "prefixToLevel_aux (h # t) l cl = (if isDecision h then if l \<le> cl then [] else h # prefixToLevel_aux t l (cl + 1) else h # prefixToLevel_aux t l cl)"