hcju/isabelleps huggingface dataset:
set_fst_sunion_with: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" shows "set (map fst (sunion_with f xs ys)) = set (map fst xs) \<union> set (map fst ys)"
max_stepsize_def: fixes n :: "nat" shows "max_stepsize n = Max (stepsize ` {0..n})"
prime_natI: fixes p :: "nat" assumes "2 \<le> p"   and "\<And>m n. p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n" shows "prime p"
to_char_poly: fixes a :: "nat" assumes "a < n" shows "(\<Sum>p | p permutes {0..<n}. f p a) = char_poly (mat_delete A a a)"
monotone_w: fixes i :: "int"   and i' :: "int"   and j :: "int"   and j' :: "int" assumes "i \<le> i'"   and "i' \<le> j"   and "j \<le> j'" shows "w i' j \<le> w i j'"
FVc: fixes V :: "val" shows "fv (Val V) = {}"
assms: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))"
independent_bound_general: fixes S :: "'b set" assumes "vs1.independent S" shows "finite S \<and> card S \<le> vs1.dim S"
UnCI: fixes c :: "'k"   and B :: "'k set"   and A :: "'k set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
in_homE: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.in_hom f a b" obtains "S.VV.arr f" and "S.VV.dom f = a" and "S.VV.cod f = b"
var_X: shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = \<sigma>2"
as_good_asIff: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation" shows "(x \<in> as_good_as y carrier relation) = x \<approx>[relation] y"
C_def: fixes off :: "nat"   and k :: "nat" shows "C off k = c / real (K off k) ^ (m - 1)"
pp_dist_comp: fixes x :: "'a"   and y :: "'a" shows "- - (x * y) = - - x * - - y"
op_set_npick_remove_def: fixes X :: "'b set" shows "op_set_npick_remove X = SPEC (\<lambda>(x, X'). x \<in> X \<and> (X' = X - {x} \<or> X' = X))"
n_comm_star: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "!z \<cdot> x = x \<cdot> !z"   and "!z \<cdot> y = y \<cdot> !z" shows "!z \<cdot> x \<cdot> (!z \<cdot> y)\<^sup>\<star> = !z \<cdot> x \<cdot> y\<^sup>\<star>"
poly_add_hom': fixes p :: "'a list"   and q :: "'a list" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R" shows "S.normalize (map h (R.poly_add p q)) = S.poly_add (map h p) (map h q)"
l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_tag_name create_element known_ptr \<equiv> known_ptr = DocumentClass.known_ptr \<and> create_element = l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_element get_disconnected_nodes set_disconnected_nodes set_tag_name"
crename_OrR2: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = OrR2 <c>.N d"   and "c \<sharp> (R, a, b)"   and "d \<sharp> (a, b)" shows "\<exists>N'. R = OrR2 <c>.N' d \<and> N'[a\<turnstile>c>b] = N"
neg_less_iff_less: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus b) (uminus a) = less a b"
valid_frame: fixes s :: "recf \<times> nat list \<times> nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list" assumes "valid (s # ss)" shows "valid ss \<and> recfn (length (fst (snd s))) (fst s)"
simps_5: fixes f :: "Core.term"   and u :: "Core.term"   and f' :: "Core.term"   and u' :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" shows "raw_match_term (f $ u) (f' $ u') subs = Option.bind (raw_match_term f f' subs) (raw_match_term u u')"
lq_len: fixes r :: "'b list"   and s :: "'b list" assumes "r \<le>p s" shows "\<^bold>|r\<^bold>| + \<^bold>|r\<inverse>\<^sup>>s\<^bold>| = \<^bold>|s\<^bold>|"
isDiagonal_mat_def: fixes A :: "'b mat" shows "isDiagonal_mat A = (\<forall>i j. i \<noteq> j \<and> i < dim_row A \<and> j < dim_col A \<longrightarrow> A $$ (i, j) = (0::'b))"
mapping_of_mpoly_of_poly: fixes i :: "nat"   and p :: "'b poly" shows "mapping_of (mpoly_of_poly i p) = mpoly_of_poly_aux i p"
left_commute: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" shows "signed.min b (signed.min a c) = signed.min a (signed.min b c)"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" where "x = [xa__]"   | xa :: "'a" and y :: "'a" and xs :: "'a list" where "x = xa \<Colon> y \<Colon> xs"
INF_lower2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
fbb_ffd_dual: fixes f :: "'c \<Rightarrow> 'd set" shows "\<partial> \<circ> bb\<^sub>\<F> f = fd\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
ground_resolution_with_selection_axioms: shows "ground_resolution_with_selection S_Q"
r_ifless_def: shows "r_ifless \<equiv> Cn 4 r_ifle [recf.Id 4 1, recf.Id 4 0, recf.Id 4 3, recf.Id 4 2]"
lb_set_def: fixes X :: "'a set" shows "lb_set X = {y. \<forall>x\<in>X. y \<le> x}"
partn_lst_fewer_colours: fixes \<beta> :: "'b set"   and n :: "nat"   and \<alpha> :: "nat list"   and \<gamma> :: "nat" assumes "partn_lst \<beta> (n # \<alpha>) \<gamma>"   and "\<gamma> \<le> n" shows "partn_lst \<beta> \<alpha> \<gamma>"
WT_gpv_absorb: fixes \<I>' :: "('i, 'j) \<I>"   and gpv :: "('i, 'j) distinguisher"   and \<I> :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter" assumes "\<I>' \<turnstile>g gpv \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "\<I> \<turnstile>g absorb gpv conv \<surd>"
wellformed: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "card {i \<in> A. i < x} \<in> {i. enat i < esize A}"
inorder_upd: fixes t :: "('c \<times> 'd) tree234"   and a :: "'c"   and b :: "'d" assumes "sorted1 (inorder t)" shows "inorder (tree\<^sub>i (upd a b t)) = upd_list a b (inorder t)"
exp_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite I" shows "exp (sum f I) = (\<Prod>x\<in>I. exp (f x))"
SUP_empty: fixes f :: "'c \<Rightarrow> 'b" shows "\<Sqinter> (f ` {}) = \<top>"
card_Diff1_le: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "card (A - {x}) \<le> card A"
mem: fixes a :: "'a poly"   and i :: "nat"   and x :: "'a" assumes "(a, i) \<in> set bs"   and "poly a x = (0::'a)" shows "order x p = Suc i"
alls_Cons_tl_conv: fixes A :: "nat"   and as :: "nat list" shows "alls A as = (zeroes (length as), 0) # tl (alls A as)"
poly_induct3: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>p q. \<lbrakk>q \<in> carrier P; p \<in> carrier P; Q p; Q q\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>P\<^esub> q)"   and "\<And>a n. a \<in> carrier R \<Longrightarrow> Q (monom P a n)" shows "Q p"
a_half_less_sum: shows "\<bar>real_of_float (Float m1 e1)\<bar> / 2 < \<bar>real_of_float (Float m1 e1) + real_of_float (Float m2 e2)\<bar>"
well_def_signs1: fixes j :: "rat list" assumes "j \<in> set (get_signs_R (calculate_data_R p qs1))" shows "length j = length qs1"
simps_1: fixes x :: "name" shows "height (Var x) = 1"
name_prm_name_def: fixes pi :: "(name \<times> name) list"   and a :: "name" shows "pi \<bullet> a \<equiv> name_prm_name pi a"
t_ins_out_def: fixes X :: "'b t_type" shows "t_ins_out X \<equiv> hd (subtrees X)"
add_pos_nonneg: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subset># a"   and "{#} \<subseteq># b" shows "{#} \<subset># a + b"
foreach_impl_correct_2: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"   and "prim_invar_impl Qi \<pi>i" shows "M_invar \<pi>i'"
Span_subgroup_props_2: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "\<zero> \<in> R.Span K Us"
Prev: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and first :: "bool"   and buf :: "'a option list set list"   and nts :: "nat list"   and I :: "\<I>" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "first = (j = 0)"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>')) [min (Monitor.progress \<sigma> \<phi>' j) (j - 1)..<Monitor.progress \<sigma> \<phi>' j] buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi>' j) (j - 1)..<j] nts" shows "wf_mformula \<sigma> j n R (MPrev I \<phi> first buf nts) (formula.Prev I \<phi>')"
quasi_inverse_transpose_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<cong> h"   and "equivalence_map g" shows "f \<cong> h \<star> g\<^sup>~"
set_spmf_def: fixes p :: "'b spmf" shows "set_spmf p = set_pmf p \<bind> set_option"
L_sim_delta: fixes s :: "'s" assumes "wf_state s" shows "map L (sim_delta s) = map (\<lambda>a. lQuot a (L s)) (\<sigma> n)"
foundation18: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<upsilon> X) = (X \<tau> \<noteq> invalid \<tau>)"
inf_sup_aci_5: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (y::'b) = sup y x"
ih_aux1: fixes a' :: "'e"   and b :: "'f"   and R :: "('e \<times> 'f) set"   and a :: "'e" assumes "(a', b) \<in> R"   and "a' = a" shows "(a, b) \<in> R"
dg_prod_2_Cod_vdomain: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "\<D>\<^sub>\<circ> ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr>) = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
currentI: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "\<And>x. d_OUT f x \<le> weight \<Gamma> x"   and "\<And>x. d_IN f x \<le> weight \<Gamma> x"   and "\<And>x. \<lbrakk>x \<notin> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> d_OUT f x \<le> d_IN f x"   and "\<And>e. e \<notin> \<^bold>E \<Longrightarrow> f e = 0" shows "current \<Gamma> f"
inject: fixes x3 :: "'b"   and y3 :: "'b" shows "(Inner x3 = Inner y3) = (x3 = y3)"
module_hom_axioms: fixes A :: "(('a, 'b) vec, 'c) vec" shows "module_hom (*s) (*s) ((*v) A)"
StableI: fixes F :: "'d program"   and A :: "'d set" assumes "F \<in> A Co A" shows "F \<in> Stable A"
intros: fixes S :: "'procS" shows "indRelRp (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"
subset_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
image_ident: fixes Y :: "'g set" shows "(\<lambda>x. x) ` Y = Y"
ivp_solsD_2: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "X \<in> Sols f U S t\<^sub>0 s" shows "X t\<^sub>0 = s"
usolves_ode_from_def: fixes y :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and t0 :: "real"   and T :: "real set"   and X :: "'d set" shows "(y usolves_ode f from t0) T X = ((y solves_ode f) T X \<and> t0 \<in> T \<and> is_interval T \<and> (\<forall>z T'. t0 \<in> T' \<and> is_interval T' \<and> T' \<subseteq> T \<and> (z solves_ode f) T' X \<longrightarrow> z t0 = y t0 \<longrightarrow> (\<forall>t\<in>T'. z t = y t)))"
emeasure_lborel_cbox_eq: fixes l :: "'a"   and u :: "'a" shows "emeasure lborel (cbox l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
max_flow_def: shows "max_flow = while (\<lambda>f. \<exists>p. NPreflow.isAugmentingPath c s t f p) (\<lambda>f. NFlow.augment_with_path c f (shortest_augmenting_path f)) (\<lambda>_. 0::'capacity)"
map_idx_eq_map: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'b"   and xs :: "'c list"   and n :: "nat" shows "map_idx f xs n = map (\<lambda>i. f (xs ! i) (i + n)) [0..<length xs]"
valid_Tree\<^sub>\<alpha>_Not\<^sub>\<alpha>: fixes P :: "'state"   and t\<^sub>\<alpha> :: "('b, 'pred, 'act) Tree\<^sub>\<alpha>" shows "valid_Tree\<^sub>\<alpha> P (Not\<^sub>\<alpha> t\<^sub>\<alpha>) = (\<not> valid_Tree\<^sub>\<alpha> P t\<^sub>\<alpha>)"
r_one_unique: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and x :: "'e" assumes "Group G"   and "a \<in> carrier G"   and "x \<in> carrier G"   and "a \<cdot>\<^bsub>G\<^esub> x = a" shows "x = \<one>\<^bsub>G\<^esub>"
inv_right_eq_iff: fixes f :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "bijection f" shows "(b = inv f a) = (f b = a)"
identity: shows "map_sum (\<lambda>x. x) (\<lambda>x. x) = id"
unique_limit': fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp"   and "Zp_converges_to s a" shows "a = res_lim s"
ind: shows "\<forall>ys. length ys < length xs \<longrightarrow> norm_unsigned ys = [] \<or> bv_msb (norm_unsigned ys) = \<one>"
finite_k_words: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and k :: "nat" assumes "source_code M b fi X enc dec" shows "finite {w. length w = k \<and> set w \<subseteq> source_code.L M X}"
invK_symK: fixes n :: "nat" shows "invK (symK n) = symK n"
strand_sem_split_4: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d \<theta>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>d \<theta>"
rel_pmf_return_pmf1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and M :: "'d pmf" shows "rel_pmf R (return_pmf x) M = (\<forall>a\<in>set_pmf M. R x a)"
SFAssThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
simps_4: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "monoidal_language C" shows "monoidal_language.Diagonalize C (t \<^bold>\<cdot> u) = monoidal_language.CompDiag C (monoidal_language.Diagonalize C t) (monoidal_language.Diagonalize C u)"
star_slide1: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<cdot> x \<le> x \<cdot> (y \<cdot> x)\<^sup>\<star>"
None_in_results'_gpv_catch_gpv: fixes gpv :: "('d, 'e, 'f) gpv" shows "(None \<in> results'_gpv (catch_gpv gpv)) = (\<not> colossless_gpv \<I>_full gpv)"
measurable_pair_swap_iff: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and M2 :: "'a measure"   and M1 :: "'b measure"   and M :: "'c measure" shows "(f \<in> M2 \<Otimes>\<^sub>M M1 \<rightarrow>\<^sub>M M) = ((\<lambda>(x, y). f (y, x)) \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M)"
le_comp_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq x y" shows "less_eq (times x z) (times y z)"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < length fs" shows "fs ! j $ i \<in> \<int>"
reachable_adj_trans: fixes G :: "('c, 'd) pre_digraph"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "wf_digraph G"   and "a \<rightarrow>\<^sup>*\<^bsub>G\<^esub> b"   and "b \<rightarrow>\<^bsub>G\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>G\<^esub> c"
obj_arr_loc: fixes r :: "Value"   and i :: "nat" assumes "isArrV r" shows "ref (r.[i]) = r"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_compat_list_axioms ext" shows "ext_compat_list ext"
take_is_prefix: fixes n :: "nat"   and xs :: "'d list" shows "prefix (take n xs) xs"
lookup_minus_fun: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (s - t) = lookup s - lookup t"
Neg_Melem_subst_atm_subst_cls: fixes A :: "'a"   and C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "Neg A \<in># C" shows "Neg (A \<cdot>a \<sigma>) \<in># C \<cdot> \<sigma>"
cycle_free_cycle_free_up_to: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "cycle_free m n"   and "k \<le> n" shows "cycle_free_up_to m k n"
eq_key_imp_eq_value: fixes xs :: "('b \<times> 'c) list"   and k :: "'b"   and v1 :: "'c"   and v2 :: "'c" assumes "distinct (map fst xs)"   and "(k, v1) \<in> set xs"   and "(k, v2) \<in> set xs" shows "v1 = v2"
closed_glob_consistent_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and \<R> :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.closed_glob_consistent dma \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> \<R> \<longrightarrow> (\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (((c\<^sub>1, mds), mem\<^sub>1(x := v\<^sub>1)), (c\<^sub>2, mds), mem\<^sub>2(x := v\<^sub>2)) \<in> \<R>)) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (((c\<^sub>1, mds), mem\<^sub>1(x := v)), (c\<^sub>2, mds), mem\<^sub>2(x := v)) \<in> \<R>))))"
DisjointPaths_extend: fixes P :: "'a list" assumes "v0 \<leadsto>P\<leadsto> v1"   and "\<And>xs v. \<lbrakk>xs \<in> Q; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths G v0 v1 (insert P Q)"
set_upt_Suc: fixes i :: "nat" shows "{0..<Suc i} = insert i {0..<i}"
dd_lt_d: shows "\<forall>D'\<in>#DD. D' < D"
der: fixes x :: "'a" assumes "x \<in> path_image g" shows "f field_differentiable at x"
hpsafe_Evolve: fixes ODE :: "('d, 'e) ODE"   and P :: "('d, 'f, 'e) formula" assumes "osafe ODE"   and "fsafe P" shows "hpsafe (EvolveODE ODE P)"
step_CFG_SSA_Transformed_notriv: shows "CFG_SSA_Transformed_notriv \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs u_g p_g var chooseNext_all"
opnet_node_init: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and i :: "nat"   and R :: "nat set" assumes "(\<sigma>, s) \<in> init (opnet onp \<langle>i; R\<rangle>)" shows "(\<sigma>, s) \<in> {(\<sigma>, NodeS i ns R) |\<sigma> ns. (\<sigma>, ns) \<in> init (onp i)}"
conjugate_transpose_rank: fixes A :: "'b mat" shows "vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) A\<^sup>H"
ueq: shows "u = S \<inter> {x. a' \<bullet> x = b'}"
Polys_closed_monomial: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and c :: "'b" assumes "t \<in> .[X]" shows "monomial c t \<in> P[X]"
elims: fixes x :: "'a \<Rightarrow> 'b option"   and xa :: "('a, 'b) sas_plus_operator list list"   and y :: "('a \<Rightarrow> 'b option) list" assumes "trace_parallel_plan_sas_plus x xa = y" obtains s :: "'a \<Rightarrow> 'b option" where "x = s__" and "xa = []" and "y = [s__]"   | s :: "'a \<Rightarrow> 'b option" and ops :: "('a, 'b) sas_plus_operator list" and opss :: "('a, 'b) sas_plus_operator list list" where "x = s" and "xa = ops # opss" and "y = s # (if are_all_operators_applicable_in s ops \<and> are_all_operator_effects_consistent ops then trace_parallel_plan_sas_plus (execute_parallel_operator_sas_plus s ops) opss else [])"
spikes: shows "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
simps: fixes a :: "('fenv_ubx, 'a, ('fun, 'b) frame) state" shows "Subx.final a = (\<exists>F f fd pc H \<Sigma>. a = State F H [Frame f pc \<Sigma>] \<and> Fubx_get F f = Some fd \<and> pc = length (body fd))"
ext: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "C g f \<noteq> partial_magma.null C" shows "partial_magma.arr C (C g f)"
hyps_1: fixes i :: "nat" assumes "n = j - i"   and "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<in> set (msgs (S t j) cid)"   and "channel cid = Some (p, q)" shows "\<forall>k. i \<le> k \<and> k \<le> j \<longrightarrow> Marker \<in> set (msgs (S t k) cid)"
m_comm: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "x \<otimes>\<^bsub>poly_ring R\<^esub> y = y \<otimes>\<^bsub>poly_ring R\<^esub> x"