hcju/isabelleps huggingface dataset:
M: shows "M \<equiv> mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))"
eq_mult_str: fixes U :: "'b set" assumes "U \<subseteq> X" shows "ind_mult_str U = mult_str U"
unifier_sub1: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set"   and L' :: "fterm literal set" assumes "unifier\<^sub>l\<^sub>s \<sigma> L"   and "L' \<subseteq> L" shows "unifier\<^sub>l\<^sub>s \<sigma> L'"
submatrix_split: fixes A :: "'a mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = submatrix (submatrix A UNIV J) I UNIV"
assms: shows "\<bar>k\<bar> \<noteq> 1"
ring_axioms: shows "ring pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
fds_nth_uminus: fixes g :: "'b fds" shows "fds_nth (- g) = (\<lambda>n. - fds_nth g n)"
ideal_generated_repeated: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ring plus zero minus uminus times" shows "ring.ideal_generated plus zero uminus times {a, a} = ring.ideal_generated plus zero uminus times {a}"
image_mset_add_mset: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and M :: "'b multiset" shows "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
im_def: shows "l.im = (\<lambda>v. fst v \<oplus>\<^bsub>M\<^esub> snd v) ` carrier (direct_sum (md N1) (md N2))"
rel_set_transfer: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (rel_fun A (rel_fun B (=))) (rel_fun (rel_set A) (rel_fun (rel_set B) (=))) rel_set rel_set"
degree: fixes x :: "'b pdevs"   and j :: "nat" assumes "degree x \<le> j" shows "pdevs_apply x j = (0::'b)"
l_get_root_node_def: fixes get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_root_node get_root_node get_parent \<equiv> (\<forall>ptr h. pure (get_root_node ptr) h) \<and> (\<forall>h node_ptr. h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None \<longrightarrow> h \<turnstile> get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"
T_vrestr_meas_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "T --` A \<in> sets M"
append1_eq_conv: fixes xs :: "'f list"   and x :: "'f"   and ys :: "'f list"   and y :: "'f" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
a1: shows "(\<Sum>i\<le>p. (a i)\<^sup>2) = 1"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])"
rbt_map_def: fixes t :: "('a, 'b) rbt" shows "rbt_map t = map_of_alist (rbt_in_traverse_pairs t)"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signals :: "signal \<Rightarrow> nat"   and more :: "'a" shows "progress \<lparr>ik = ik, secret = secret, progress = progress, signals = signals, \<dots> = more\<rparr> = progress"
map_eq_imp_length_eq: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and g :: "'i \<Rightarrow> 'g"   and ys :: "'i list" assumes "map f xs = map g ys" shows "length xs = length ys"
univalent_composes: fixes R :: "('d \<times> 'e) set"   and S :: "('e \<times> 'f) set" assumes "univalent R"   and "univalent S" shows "univalent (R O S)"
IH_5: assumes "delete\<langle>t,s\<rangle> \<in> set S" shows "fv t \<union> fv s \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
induct: fixes x1 :: "'a formula multiset"   and x2 :: "'a formula multiset"   and P :: "'a formula multiset \<Rightarrow> 'a formula multiset \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2"   and "\<And>\<Gamma> \<Delta>. \<bottom> \<in># \<Gamma> \<Longrightarrow> P \<Gamma> \<Delta>"   and "\<And>k \<Gamma> \<Delta>. \<lbrakk>Atom k \<in># \<Gamma>; Atom k \<in># \<Delta>\<rbrakk> \<Longrightarrow> P \<Gamma> \<Delta>"   and "\<And>\<Gamma> F \<Delta>. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> F, \<Gamma>) \<Delta>"   and "\<And>F \<Gamma> \<Delta>. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta>; P (F, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P \<Gamma> (\<^bold>\<not> F, \<Delta>)"   and "\<And>F G \<Gamma> \<Delta>. \<lbrakk>F, G, \<Gamma> \<Rightarrow> \<Delta>; P (F, G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G, \<Gamma>) \<Delta>"   and "\<And>\<Gamma> F \<Delta> G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>); \<Gamma> \<Rightarrow> G, \<Delta>; P \<Gamma> (G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<and> G, \<Delta>)"   and "\<And>F \<Gamma> \<Delta> G. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta>; P (F, \<Gamma>) \<Delta>; G, \<Gamma> \<Rightarrow> \<Delta>; P (G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G, \<Gamma>) \<Delta>"   and "\<And>\<Gamma> F G \<Delta>. \<lbrakk>\<Gamma> \<Rightarrow> F, G, \<Delta>; P \<Gamma> (F, G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<or> G, \<Delta>)"   and "\<And>\<Gamma> F \<Delta> G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>); G, \<Gamma> \<Rightarrow> \<Delta>; P (G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G, \<Gamma>) \<Delta>"   and "\<And>F \<Gamma> G \<Delta>. \<lbrakk>F, \<Gamma> \<Rightarrow> G, \<Delta>; P (F, \<Gamma>) (G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<rightarrow> G, \<Delta>)" shows "P x1 x2"
MorphismsPreserved: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t"   and i :: "('o, 'm) IType" assumes "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i" shows "\<exists>g. i = IMor g \<and> g maps\<^bsub>iC\<^esub> Ty\<lbrakk>A\<rbrakk> to Ty\<lbrakk>B\<rbrakk>"
beh_prefix_closed: fixes c :: "'a list"   and b :: "'a list"   and T :: "('a, 'b) TS_scheme" assumes "c @ b \<in> beh T" shows "b \<in> beh T"
consistency_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "consistency_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. f B \<subseteq> C \<longrightarrow> f B = f C)"
loop_def: shows "loop \<equiv> while (dwa_cond (brc_det_algo (build_rqrm \<delta>) \<delta>)) (dwa_step (brc_det_algo (build_rqrm \<delta>) \<delta>)) (dwa_initial (brc_det_algo (build_rqrm \<delta>) \<delta>))"
PP_PPf: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (PP \<langle>\<phi>\<rangle>) = bprv (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>))"
MF_Rel_def: shows "MF_Rel \<equiv> rel_prod M_Rel (rel_mset MP_Rel)"
rep_eq: fixes x :: "'c pmf" shows "set_pmf x = {xa. measure_pmf.prob x {xa} \<noteq> 0}"
vec_to_iarray_basis_row_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec_to_iarray ` basis_row_space A = basis_row_space_iarrays (matrix_to_iarray A)"
simps_4: fixes p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and vb :: "int \<times> int"   and vc :: "(int \<times> int) list" shows "closest_pair_bf (p\<^sub>0 # v # vb # vc) = (let (c\<^sub>0, c\<^sub>1) = closest_pair_bf (v # vb # vc); p\<^sub>1 = find_closest_bf p\<^sub>0 (v # vb # vc) in if dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (c\<^sub>0, c\<^sub>1) else (p\<^sub>0, p\<^sub>1))"
abs_minus_commute: fixes abs :: "'y \<Rightarrow> 'y"   and plus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and zero :: "'y"   and minus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and uminus :: "'y \<Rightarrow> 'y"   and less_eq :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and less :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and a :: "'y"   and b :: "'y" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
exec_merge_guards_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>merge_guards c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
Icarr: fixes i :: "'a" assumes "i \<in> I" shows "i \<in> carrier R"
step_sound: fixes Cs :: "fterm literal set set"   and Cs' :: "fterm literal set set"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "resolution_step Cs Cs'"   and "eval\<^sub>c\<^sub>s F G Cs" shows "eval\<^sub>c\<^sub>s F G Cs'"
to_fun_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "x \<in> carrier R" shows "UP_cring.to_fun R f x \<in> carrier R"
qty_single_def: fixes asset :: "'b"   and qt_proc :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "qty_single asset qt_proc = qty_empty(asset := qt_proc)"
rep_eq: fixes x :: "nat"   and xa :: "'b trace" shows "Rep_prefix (take_prefix x xa) = stake x (Rep_trace xa)"
vec_to_lpoly_eval_dot_prod: fixes v :: "rat vec"   and x :: "nat \<Rightarrow> rat" shows "vec_to_lpoly v \<lbrace> x \<rbrace> = v \<bullet> vec (dim_vec v) x"
proves_eta_steps: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "t \<rightarrow>\<^sub>\<eta>\<^sup>* u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
1: shows "interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f = integral {g a..g b} f"
interaction_bound_bind_step: fixes interaction_bound' :: "('f, 'out, 'e) gpv \<Rightarrow> enat"   and f :: "'d \<Rightarrow> ('f, 'out, 'e) gpv"   and p :: "('d, 'out, 'e) gpv" assumes "\<And>p. interaction_bound' (p \<bind> f) \<le> local.interaction_bound p + (\<Squnion>x\<in>results'_gpv p. interaction_bound' (f x))"   and "\<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "(\<Squnion>generat\<in>set_spmf (the_gpv (p \<bind> f)). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input)) \<le> local.interaction_bound p + (\<Squnion>x\<in>results'_gpv p. \<Squnion>generat\<in>set_spmf (the_gpv (f x)). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))"
tendsto_at_within_iff_tendsto_nhds: fixes g :: "'a \<Rightarrow> 'b"   and l :: "'a"   and S :: "'a set" shows "(g \<longlongrightarrow> g l) (at l within S) = (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
scale_spmf_neg: fixes r :: "real"   and p :: "'c spmf" assumes "r \<le> 0" shows "scale_spmf r p = return_pmf None"
E: shows "\<forall>X\<in>set PS. ref_union_closed X \<and> sequential X"
apply_infmatrix_def: fixes f :: "'b \<Rightarrow> 'c" shows "apply_infmatrix f \<equiv> \<lambda>A j i. f (A j i)"
compact_sums: fixes s :: "'b set"   and t :: "'b set" assumes "compact s"   and "compact t" shows "compact {x + y |x y. x \<in> s \<and> y \<in> t}"
Collect_empty_eq_bot: fixes P :: "'b \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot)"
frontier_interiors: fixes s :: "'a set" shows "frontier s = - interior s - interior (- s)"
lowner_le_P': shows "P' \<le>\<^sub>L tensor_P post (1\<^sub>m K)"
po_subset_po: fixes cl :: "'b potype"   and S :: "'b set" assumes "PO cl"   and "S \<subseteq> pset cl" shows "\<lparr>pset = S, order = induced S (order cl)\<rparr> \<in> PartialOrder"
CollectionRangeTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and b :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>P CollectionRange a b : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "\<tau> = Integer[1]" and "\<Gamma> \<turnstile>\<^sub>E a : \<tau>'" and "\<Gamma> \<turnstile>\<^sub>E b : \<sigma>" and "UnlimitedNatural[1] \<le> \<tau>'" and "\<tau>' \<le> Integer[1]" and "UnlimitedNatural[1] \<le> \<sigma>" and "\<sigma> \<le> Integer[1]"
OK_check: fixes eX :: "'a withERR" assumes "eX \<noteq> ERR" shows "OK (check eX) = eX"
NSLIM_Re: fixes f :: "'a \<Rightarrow> complex"   and a :: "'a"   and L :: "complex" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "(\<lambda>x. Re (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S Re L"
honest_never_says_newer_timestamp_in_auth: fixes evs :: "event list"   and T :: "nat"   and A :: "agent"   and X :: "msg" assumes "CT evs \<le> T"   and "A \<notin> bad"   and "Number T \<in> parts {X}"   and "evs \<in> kerbIV" shows "\<forall>B Y. Says A B \<lbrace>Y, X\<rbrace> \<notin> set evs"
kcomp_def: fixes f :: "'b \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'c set" shows "f \<circ>\<^sub>K g = \<mu> \<circ> \<P> g \<circ> f"
A4_DAcc_level0: shows "DAcc level0 sA4 = {sA2, sA5}"
f_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S" shows "f (x + y) = f x + f y"
step: fixes m :: "nat"   and M :: "nat set"   and l :: "nat" assumes "\<Phi> m M"   and "0 < l" shows "Ex (\<lambda>(n, N, j). \<Phi> n N \<and> \<Psi> l m n M N j)"
assms_1: shows "y \<notin> fst ` fv t"
assms_2: shows "P'' \<longmapsto>\<alpha> \<prec> P'''"
MittenTauschS: fixes \<Gamma> :: "'b formula list"   and \<Delta> :: "'b formula list"   and \<DD> :: "'b formula"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<Delta> @ \<DD> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<DD> # \<Delta> @ \<Theta>"
wf_residual_networkD_1: fixes x :: "'v"   and y :: "'v" assumes "res.wf_residual_network"   and "edge (residual_network f) x y" shows "\<not> edge (residual_network f) y x"
dot_lmul_matrix: fixes x :: "('a, 'rows) vec"   and A :: "(('a, 'cols) vec, 'rows) vec"   and y :: "('a, 'cols) vec" shows "inner_cols (x v* A) y = inner_rows x ((\<chi>i j. cnj_class.cnj (A $ i $ j)) *v y)"
assms_1: shows "hn_ctxt A (fst x) (fst y) \<or>\<^sub>A hn_ctxt A' (fst x) (fst y) \<Longrightarrow>\<^sub>t hn_ctxt Am (fst x) (fst y)"
redT_updW_total: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action" shows "\<exists>ws'. redT_updW t ws wa ws'"
Compl_atMost: fixes k :: "'a" shows "- {..k} = {k<..}"
a: shows "A \<and> (SOME x. A \<and> x \<or> \<not> x) \<or> \<not> (SOME x. A \<and> x \<or> \<not> x)"
insert_fset_left_idem: fixes x :: "'b"   and S :: "'b fset" shows "insert_fset x (insert_fset x S) = insert_fset x S"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
list_all2_Cons2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
inputAlpha: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "xvec \<sharp>* P"   and "(p \<bullet> xvec) \<sharp>* P" shows "\<Psi> \<rhd> P \<longmapsto> M\<lparr>(p \<bullet> N)\<rparr> \<prec> p \<bullet> P'"
floor_conv_div_nat: fixes m :: "nat"   and n :: "nat" shows "real_of_int \<lfloor>real m / real n\<rfloor> = real (m div n)"
ground_imp_subst_iden: fixes s :: "('b, 'a) tm"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm" assumes "ground s" shows "subst \<rho> s = s"
tdghm_NTMap_app_in_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
imp_disjL: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))"
mono_indspace: fixes G :: "'g set"   and FH :: "('f, 'g) aezfun set" shows "mono (indspace G FH)"
r_def: shows "r = [:- a / of_nat n, 1::'a:]"
distinct_7: fixes x1 :: "'a"   and x51 :: "'a"   and x52 :: "bexp"   and x53 :: "'a acom"   and x54 :: "'a" shows "SKIP {x1} \<noteq> {x51} WHILE x52 DO x53 {x54}"
intros_1: fixes \<sigma> :: "'tp"   and c :: "('a, 'b) lit list" assumes "infTp \<sigma>" shows "\<sigma> \<turnstile> c"
max_less_iff_conj: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" shows "(signed.max x y <s z) = (x <s z \<and> y <s z)"
set_map: fixes f :: "'g \<Rightarrow> 'e"   and v :: "'g set['f]" shows "set_bset (map_bset f v) = f ` set_bset v"
cbelow_on_induct: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "cbelow_on A ord z rel a b"   and "\<lbrakk>a \<in> A; ord a z\<rbrakk> \<Longrightarrow> P a"   and "\<And>b c. \<lbrakk>cbelow_on A ord z rel a b; rel b c; c \<in> A; ord c z; P b\<rbrakk> \<Longrightarrow> P c" shows "P b"
secureI_secureFillI_Aux: fixes n :: "nat"   and k :: "nat"   and m :: "nat"   and s :: "Var \<Rightarrow> Val"   and d :: "IMP"   and t :: "Var \<Rightarrow> Val"   and ss :: "Var \<Rightarrow> Val"   and tt :: "Var \<Rightarrow> Val"   and C :: "CtxtProg"   and c :: "IMP"   and X :: "Var set" assumes "n \<le> k"   and "m \<le> k"   and " s , d \<rightarrow>\<^sub>n  t "   and " ss , d \<rightarrow>\<^sub>m  tt "   and " s \<approx> ss "   and "d = Fill C c"   and "LOW X C"   and "LOW X Ctxt_Body"   and "body = Fill Ctxt_Body c"   and "secure c" shows " t \<approx> tt "
witness_4: fixes b :: "'d" assumes "b \<in> T" shows "i b \<in> S"
closer_points_lemma: fixes y :: "'b"   and z :: "'b" assumes "0 < y \<bullet> z" shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R z - y) < norm y"
assert_spmf_def: fixes b :: "bool" shows "assert_spmf b = (if b then return_spmf () else return_pmf None)"
D: shows "(xs @ y # ipurge_tr I D (D y) zs, {}) \<in> failures P"
simps_5: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cTemperatureF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTemperatureF"
child_ancestors: shows "h \<turnstile> get_ancestors_si (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r child_ancestors"
c1c2_5: shows "k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
mon_conv_real_fun_def: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real" shows "u\<up>f \<equiv> (\<forall>n. u n \<le> u (Suc n)) \<and> (\<forall>w. (\<lambda>n. u n w) \<longlonglongrightarrow> f w)"
PO_m1_inv3_ifresh: shows "reach m1 \<subseteq> m1_inv3_ifresh"
dg_GRPH_components_3: fixes \<alpha> :: "V" shows "dg_GRPH \<alpha>\<lparr>Dom\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_dghms \<alpha>. \<FF>\<lparr>HomDom\<rparr>)"
complex_roots_of_int_poly_all_def: fixes p :: "int poly" shows "complex_roots_of_int_poly_all p = (let n = degree p in if 3 \<le> n then complex_roots_of_int_poly3 p else if n = 1 then [roots1 (complex_of_int_poly p)] else if n = 2 then croots2 (complex_of_int_poly p) else [])"
evalF_equiv: fixes A :: "formula"   and f :: "vbl \<Rightarrow> object"   and g :: "vbl \<Rightarrow> object"   and M :: "model" assumes "equalOn (freeVarsF A) f g" shows "evalF M f A = evalF M g A"
greaterThan_Int_greaterThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "ord.lessThan less a \<inter> ord.lessThan less b = ord.lessThan less (ord.min less_eq a b)"
holomorphic_on_imp_differentiable_on: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s" shows "f differentiable_on s"
IH: fixes s :: "('a, 'b) fmap" assumes "no_effectless_act as" shows "no_effectless_act (rem_condless_act s [] as)"
vcg_pre: fixes Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and R :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and P :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<lbrace>Q\<rbrace> c \<lbrace>R\<rbrace>"   and "\<And>s. P s \<Longrightarrow> Q s" shows "\<lbrace>P\<rbrace> c \<lbrace>R\<rbrace>"
perron_frobenius_spectral_radius_complex: fixes A :: "complex mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "real_nonneg_mat A"   and "\<And>x. poly (char_poly (map_mat Re A)) x = 0 \<Longrightarrow> x \<le> 1"   and "\<And>x. cmod x = 1 \<Longrightarrow> order x (char_poly A) \<le> d" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (d - 1))"
simps: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp t = (case unapp t of None \<Rightarrow> (t, []) | Some (t, u) \<Rightarrow> let (f, args) = term.strip_comb unapp t in (f, args @ [u]))"
apps_preserves_beta: fixes r :: "lam"   and s :: "lam"   and ss :: "lam list" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "r \<degree>\<degree> ss \<rightarrow>\<^sub>\<beta> s \<degree>\<degree> ss"
3: shows "src \<lparr>Chn = fgh.chine_assoc, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> = src h"
assms_2: shows "hom_component p (deg_pm t) = hom_component p n"
c_max_ext_special_cf: fixes n :: "nat" shows "c_max S (ext_cf S n (C\<^sub>0 S)) = n"