hcju/isabelleps huggingface dataset:
ProcModifyReturn_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/UNIV\<^esub> {\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P call init p return c Q,A"
AbsI: fixes t1 :: "typ"   and A :: "type_scheme list"   and e :: "expr"   and t2 :: "typ" assumes "mk_scheme t1 # A |- e :: t2" shows "A |- Abs e :: t1 -> t2"
refined_prog_secure: fixes cms\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and cms\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list" assumes "length cms\<^sub>C = length cms"   and "length cms\<^sub>A = length cms"   and "\<And>i mem\<^sub>C. i < length cms \<Longrightarrow> ((cms\<^sub>A ! i, mem\<^sub>A_of mem\<^sub>C), cms\<^sub>C ! i, mem\<^sub>C) \<in> \<R>_rel (cms ! i)"   and "\<And>i mem\<^sub>C mem\<^sub>C'. \<lbrakk>i < length cms; conc.low_mds_eq (snd (cms\<^sub>C ! i)) mem\<^sub>C mem\<^sub>C'\<rbrakk> \<Longrightarrow> ((cms\<^sub>A ! i, mem\<^sub>A_of mem\<^sub>C), cms\<^sub>A ! i, mem\<^sub>A_of mem\<^sub>C') \<in> \<R>\<^sub>A_rel (cms ! i)"   and "\<And>mem\<^sub>A. abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A)"   and "modes_respect_priv (map snd cms\<^sub>C)" shows "conc.prog_sifum_secure_cont cms\<^sub>C"
DIAndProof_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndProof vid1 pid1 pid2 pid3 pid4 = (ids.DIAndConcl vid1 pid1 pid2 pid3 pid4, [(0, Rrule ImplyR 0), (0, Lrule AndL 0), (0, Rrule ImplyR 0), (0, Cut (ids.DIAndCutP1 vid1 pid1)), (1, Cut (ids.DIAndSG1 vid1 pid1 pid3)), (0, Rrule CohideR 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc (Suc 0)), CloseId 1 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc 0), CloseId 0 0), (Suc (Suc 0), Cut (ids.DIAndCut34Elim1 vid1 pid3 pid4)), (0, Lrule ImplyL 0), (Suc (Suc (Suc 0)), Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Rrule CohideRR 0), (Suc 0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc (Suc 0)))), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc (Suc (Suc 0)))), Lrule AndL 0), (Suc (Suc (Suc (Suc (Suc 0)))), CloseId 0 0), (Suc (Suc (Suc 0)), AxSubst AK (ids.DIAndSubst341 pid1 pid2 pid3 pid4)), (Suc (Suc 0), CloseId 0 0), (Suc 0, CloseId 0 0), (0, Cut (ids.DIAndCut12Intro vid1 pid1 pid2)), (Suc 0, Rrule CohideRR 0), (Suc (Suc 0), AxSubst AK (ids.DIAndSubst12 pid1 pid2)), (0, Lrule ImplyL 0), (1, Lrule ImplyL 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, Cut (ids.DIAndCutP12 vid1 pid1 pid2)), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc 0))), AxSubst AK (ids.DIAndCurry12 pid1 pid2)), (Suc (Suc (Suc 0)), Rrule CohideRR 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc 0), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc (Suc 0))), Rrule ImplyR 0), (Suc (Suc (Suc (Suc 0))), Rrule AndR 0), (Suc (Suc (Suc (Suc (Suc 0)))), CloseId 0 0), (Suc (Suc (Suc (Suc 0))), CloseId 1 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, Cut (ids.DIAndCut34Elim2 vid1 pid3 pid4)), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc 0))), AxSubst AK (ids.DIAndSubst342 pid1 pid2 pid3 pid4)), (Suc (Suc (Suc 0)), Rrule CohideRR 0), (Suc (Suc (Suc 0)), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc 0)), Lrule AndL 0), (Suc (Suc (Suc 0)), CloseId 1 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc 0, CloseId 0 0), (1, Cut (ids.DIAndSG2 vid1 pid2 pid4)), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc 0)), CloseId 4 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc (Suc 0)), CloseId 0 0), (Suc (Suc (Suc 0)), CloseId 0 0), (1, CloseId 1 0)])"
2: shows "\<forall>r\<in>num. prv (neg (PPf r \<langle>\<phi>R\<rangle>))"
assms_2: shows "\<forall>\<^sub>F x in G. f x = g x"
preserve: fixes 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"   and t :: "'t"   and tas :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   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" assumes "s -t\<triangleright>tas\<rightarrow> s'"   and "P s" shows "P s'"
sharing_consis: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent \<S> \<O> sb"
strict_prefix_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "strict_prefix [] (x # xs) = True"
HPhase1or2Write_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv3 s" shows "HInv3 s'"
canonical_fun_def: fixes U :: "'a set"   and x :: "'b" shows "stk2.canonical_fun U x = stk2.class_of U x"
hupd_del_locs: fixes a :: "loc"   and obj :: "obj"   and s :: "state" shows "hupd(a\<mapsto>obj) (del_locs s) = del_locs (hupd(a\<mapsto>obj) s)"
seq_conc_subdistr: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" shows "R \<parallel> S \<cdot> T \<subseteq> R \<cdot> T \<parallel> (S \<cdot> T)"
seqI': fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>B c\<guillemotright>" shows "B.seq g f"
assms_1: shows "\<forall>x\<in>A. \<not> P x x"
rel_def: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "stk2.rel x y \<equiv> (fst x \<in> I \<and> fst y \<in> I) \<and> (snd x \<in> \<FF>2 (fst x) \<and> snd y \<in> \<FF>2 (fst y)) \<and> (\<exists>W. W \<in> I \<and> W \<subseteq> fst x \<inter> fst y \<and> \<rho>2 (fst x) W (snd x) = \<rho>2 (fst y) W (snd y))"
constant_transformation_axioms: shows "constant_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) g"
valid_all_lift: fixes P :: "'a \<Rightarrow> (('b, 'c, 'd) response, char list, 'e process_name, 'e process_name \<times> ('b, 'e, 'c, 'd) request_op, ('b, 'e, 'c, 'd) Model.local_state) system_state \<Rightarrow> bool"   and p :: "'e process_name"   and Q :: "'a \<Rightarrow> (('b, 'c, 'd) response, char list, 'e process_name, 'e process_name \<times> ('b, 'e, 'c, 'd) request_op, ('b, 'e, 'c, 'd) Model.local_state) system_state \<Rightarrow> bool" assumes "\<And>x. \<lbrace>P x\<rbrace> p \<lbrace>Q x\<rbrace>" shows "\<lbrace>\<lambda>s. \<forall>x. P x s\<rbrace> p \<lbrace>\<lambda>s. \<forall>x. Q x s\<rbrace>"
terminates_mark_guards_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>mark_guards f c \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
vpair_app: fixes j :: "V"   and a :: "V"   and b :: "V" assumes "j = a" shows "ZFC_in_HOL.set {\<langle>a, b\<rangle>}\<lparr>j\<rparr> = b"
main_lemma: fixes A :: "'b list set" shows "\<exists>M. finite M \<and> SUPSEQ A = SUPSEQ M"
gcd_unique_nat: fixes d :: "nat"   and a :: "nat"   and b :: "nat" shows "(d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d)) = (d = gcd a b)"
alternating_list2: fixes s :: "'a"   and t :: "'a" shows "alternating_list 2 s t = [s, t]"
new_element_new_ptr: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 new_element_ptr|}"
distinct_19: fixes x4 :: "fm"   and x51 :: "name"   and x52 :: "fm" shows "Neg x4 \<noteq> SyntaxN.Ex x51 x52"
axioms_3: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "RModuleHom_axioms R smult M smult' T"
Class_left_inverse: fixes a :: "'a" assumes "a \<in> G" shows "Class (inverse a) [\<cdot>] Class a = Class \<one>"
word_nchotomy: shows "\<forall>w. \<exists>n. w = word_of_nat n \<and> n < 2 ^ LENGTH('a)"
intros_4: fixes \<O>s :: "nat set list"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and i :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" shows "\<O>s,\<R>s,i\<turnstile> (Fence # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
assms_4: shows "ground_clause (subst_cl (cl_ecl C) \<theta>)"
ci_def: shows "ci \<equiv> \<lambda>i. real_interval (z ! i) - interval_of (real_of_float (mid (z ! i)))"
tgt_not_marked_imp: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and re :: "('d \<times> nat) edge" assumes "RedBlack prb"   and "re \<in> edges (red prb)"   and "\<not> marked prb (tgt re)" shows "\<not> marked prb (src re)"
bin_rep_aux_index: fixes n :: "nat"   and m :: "nat"   and i :: "nat" assumes "1 \<le> n"   and "m < 2 ^ n"   and "0 \<le> m"   and "i \<le> n" shows "bin_rep_aux n m ! i = m mod 2 ^ (n - i) div 2 ^ (n - 1 - i)"
U: shows "splits_at (\<alpha>' @ \<delta>) (i + ix) a1 U a2 \<and> splits_at \<gamma> j b1 U b2 \<and> n \<le> length D \<and> LeftDerivation a1 (take n D) b1 \<and> derivation_ge (drop n D) (Suc (length b1)) \<and> LeftDerivation a2 (derivation_shift (drop n D) (Suc (length b1)) 0) b2 \<and> (n = length D \<or> n < length D \<and> is_word (b1 @ [U]))"
all_not_in_conv: fixes A :: "'c set" shows "(\<forall>x. x \<notin> A) = (A = {})"
delete_cnj: fixes si :: "nat"   and x :: "complex"   and xis :: "(complex \<times> nat) list" assumes "si \<le> order x (\<Prod>(x, i)\<leftarrow>xis. [:- x, 1:] ^ Suc i)"   and "si \<noteq> 0" shows "(\<Prod>(x, i)\<leftarrow>xis. [:- x, 1:] ^ Suc i) = [:- x, 1:] ^ si * (\<Prod>(x, i)\<leftarrow>delete_cnj x si xis. [:- x, 1:] ^ Suc i)"
coeff_map_poly_hom: fixes p :: "'a mod_ring poly"   and i :: "nat" shows "poly.coeff (to_int_poly p) i = to_int_mod_ring (poly.coeff p i)"
inv_from_bot_step: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and i :: "nat" assumes "p A i"   and "local.inv_from_bot p A (Suc i)" shows "local.inv_from_bot p A i"
ssubst: fixes t :: "'d"   and s :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
class_of_eqI: fixes U1 :: "'a set"   and s1 :: "'b"   and U2 :: "'a set"   and s2 :: "'b"   and W :: "'a set" assumes "(U1, s1) \<in> Sigma I \<FF>"   and "(U2, s2) \<in> Sigma I \<FF>"   and "W \<in> I"   and "W \<subseteq> U1 \<inter> U2"   and "\<rho> U1 W s1 = \<rho> U2 W s2" shows "\<lfloor>U1, s1\<rfloor> = \<lfloor>U2, s2\<rfloor>"
IH: shows "fv\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<union> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q assign) S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q check) S)) \<union> \<Union> (set (map fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q S))"
l_neg: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
sel_14: fixes x21 :: "nat"   and x22 :: "'b"   and x23 :: "'b"   and x24 :: "'a"   and x25 :: "'a"   and x26 :: "'c" shows "un_Send4 (Send x21 x22 x23 x24 x25 x26) = x24"
box_eq_set_sum_Basis: fixes B :: "'b \<Rightarrow> real set" shows "{x. \<forall>i\<in>Basis. x \<bullet> i \<in> B i} = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` B i)"
substBinp_def2: fixes ys :: "'varSort"   and Y :: "('f, 'g, 'varSort, 'var, 'h) qTerm set"   and y :: "'var" shows "substBinp ys Y y = lift (substAbs ys Y y)"
of_rat_1: shows "of_rat 1 = (1::'c)"
map_fun_apply: fixes f :: "'r \<Rightarrow> 's"   and g :: "'t \<Rightarrow> 'q"   and h :: "'s \<Rightarrow> 't"   and x :: "'r" shows "map_fun f g h x = g (h (f x))"
nn_integral_count_compose_inj: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f :: "'c \<Rightarrow> ennreal" assumes "inj_on g A" shows "\<integral>\<^sup>+x\<in>A. f (g x)\<partial>count_space UNIV = set_nn_integral (count_space UNIV) (g ` A) f"
space_lim_stream: fixes M :: "'b measure"   and K :: "'b \<Rightarrow> 'b measure"   and x :: "'b" assumes "discrete_Markov_process M K" shows "space (discrete_Markov_process.lim_stream M K x) = streams (space M)"
prime_factorization_prod: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> (0::'a)" shows "prime_factorization (prod f A) = (\<Sum>n\<in>A. prime_factorization (f n))"
effect_to_assignments_simp: fixes op :: "'a strips_operator" shows "effect_to_assignments op = map (\<lambda>v. (v, True)) (add_effects_of op) @ map (\<lambda>v. (v, False)) (delete_effects_of op)"
open_Var_close_Var: fixes t :: "expr"   and i :: "nat"   and xT :: "char list \<times> Unary_PCF.type" assumes "lc t" shows "open_Var i xT (close_Var i xT t) = t"
simps_2: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "lhd (derivation_from St) = St"
P_k_1: shows "P_k \<in> paths_with_new"
INFM_sort_of_eq: fixes s :: "atom_sort" shows "\<exists>\<^sub>\<infinity>a. sort_of a = s"
inject_3: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a"   and y31 :: "nat set"   and y32 :: "nat set"   and y33 :: "'a" shows "(x31\<not>x32:arrive(x33) = y31\<not>y32:arrive(y33)) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
sum_list_0: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (0::'b)" shows "sum_list xs = (0::'b)"
one_mono_ordered_semiring_1_axioms: shows "one_mono_ordered_semiring_1 \<delta> (delta_gt \<delta>)"
assertion_conjunctive: 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" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "x \<in> conjunctive.conjunctive inf inf times"
functional_converse_relation: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "left_unique R"   and "right_total R" shows "\<forall>y. \<exists>!x. R x y"
inject_1: fixes x11 :: "char list"   and x12 :: "spec list option"   and x13 :: "dec list"   and y11 :: "char list"   and y12 :: "spec list option"   and y13 :: "dec list" shows "(Tmod x11 x12 x13 = Tmod y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
better_ex_eval_fm: fixes e :: "(name, hf) finfun"   and k :: "name"   and A :: "fm" shows "eval_fm e (SyntaxN.Ex k A) = (\<exists>x. eval_fm (finfun_update e k x) A)"
divide_right_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (divide a c) (divide b c)"
simps_10: fixes i :: "nat"   and T :: "ty"   and e :: "nat exp" shows "compE\<^sub>2 {i:T; e} = compE\<^sub>2 e"
invar: fixes s' :: "'s"   and s :: "'s" assumes "s' \<in> set_spmf (body s)"   and "I s"   and "guard s" shows "I s'"
D: shows "ExtAgrK (S (User m, n, run)) = Some f"
assms: shows "termFSwSbImorph h hA (fromMOD MOD)"
IH: assumes "spec_ops xs deps"   and "oid \<notin> set (map fst xs)" shows "\<exists>pre suf. xs = pre @ suf \<and> (\<forall>i\<in>set (map fst pre). i < oid) \<and> (\<forall>a\<in>set (map fst suf). oid < a)"
ereal_minus_le_iff: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" shows "(x - y \<le> z) = ((y = - \<infinity> \<longrightarrow> z = \<infinity>) \<and> (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y))"
res_c2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> y" shows "residual f (f x) \<le> residual f (f y)"
subprob_measurableD_1: fixes N :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and S :: "'b measure"   and x :: "'a" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "space (N x) = space S"
distinct_15: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> x41 ;; x42"
hd_drop_is_nth: fixes n :: "nat"   and x :: "'a list" assumes "n < length x" shows "hd (drop n x) = x ! n"
hsize_pos: fixes t :: "('a, 'b) tm" shows "0 < hsize t"
rep_number: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int"   and x :: "'b" assumes "constant_rep_design point_set block_collection design_rep_number"   and "x \<in> point_set" shows "block_collection rep x = design_rep_number"
assms: shows "\<theta> \<sim>[at_top] (\<lambda>x. x)"
prod_cont_iff: fixes f :: "'d \<times> 'e \<Rightarrow> 'f" shows "cont f = ((\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y))))"
iso_defs_2: shows "iso \<equiv> \<lambda>r r' f. embed r r' f \<and> bij_betw f (Field r) (Field r')"
igSwapIPresIGWls_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Iteration.model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<sqsubseteq> a \<squnion> b"
ect: shows "simAbs (runJP k_mkAlgAuto t a) = sim_equiv_class a t"
Fun_corestr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "set_category_given_img.Fun S (set_category.some_img S \<AA>) (set_category.corestr S \<AA> f) = set_category_given_img.Fun S (set_category.some_img S \<AA>) f"
image_image: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
a': shows "\<forall>y\<in>S - {a}. norm (v - a) < norm (v - y)"
iterate_add_simps_1: fixes a :: "'c" shows "iterate_add 0 a = (0::'c)"
False: shows "\<not> (A = {} \<or> B = {} \<or> \<not> bounded A \<or> \<not> bounded B)"
divide_nonpos_pos: 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 x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less zero y" shows "less_eq (divide x y) zero"
w: shows "(WHILE b DO c, s') \<rightarrow>* (SKIP, t)"
alw_ev_stl: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" shows "alw (ev P) (stl \<omega>) = alw (ev P) \<omega>"
lower_Interval: fixes x :: "'a \<times> 'a" assumes "fst x \<le> snd x" shows "lower (Interval x) = fst x"
fill_length: fixes xs :: "'c list"   and ns :: "nat list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and mi :: "'d"   and ma :: "'d" shows "length (fill xs ns index key n mi ma) = n"
prv_expl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "prv fls" shows "prv \<phi>"
resolutionD1: shows "partition_on_mset \<B> \<P>"
sig_regular_reduced_unique': fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "lt p = lt q"   and "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G p"   and "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G q" shows "local.punit.monom_mult (lc q) (0::'a) (rep_list p) = local.punit.monom_mult (lc p) (0::'a) (rep_list q)"
bounded_linear_divide: fixes y :: "'b" shows "bounded_linear (\<lambda>x. x / y)"
comp_fun_commute_axioms: shows "comp_fun_commute (\<lambda>x yo. yo \<bind> m x)"
real_inv_sqrt_pow2: fixes x :: "real" assumes "0 < x" shows "(inverse (sqrt x))\<^sup>2 = inverse x"
limit_mat_unique: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat"   and B :: "complex mat" assumes "limit_mat X A m"   and "limit_mat X B m" shows "A = B"
simps: fixes r :: "nat \<times> char list list" shows "the_path (Ref r) = snd r"
in_subterms_subset_Union: fixes t :: "('c, 'd) Term.term"   and M :: "('c, 'd) Term.term set" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "subterms t \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t M"
main: fixes pa :: "rat poly"   and i :: "nat" assumes "(pa, i) \<in> set pis" shows "set (roots_of_rat_poly_main pa) = {x. poly pa x = 0}"
nat_diff_add_eq2: fixes i :: "nat"   and j :: "nat"   and u :: "nat"   and m :: "nat"   and n :: "nat" assumes "i \<le> j" shows "i * u + m - (j * u + n) = m - ((j - i) * u + n)"
effect_for_induct: fixes i :: "nat"   and j :: "nat"   and f :: "nat \<Rightarrow> 'b Heap"   and h :: "heap"   and h' :: "heap"   and u :: "unit"   and I :: "nat \<Rightarrow> heap \<Rightarrow> bool" assumes "i \<le> j"   and "effect (for [i..<j] f) h h' u"   and "I i h"   and "\<And>k h h' r. \<lbrakk>i \<le> k; k < j; I k h; effect (f k) h h' r\<rbrakk> \<Longrightarrow> I (Suc k) h'" shows "I j h'"
check_poly_strict_mono_discrete: fixes p :: "('c monom \<times> 'a) list"   and v :: "'c" assumes "discrete"   and "check_poly_strict_mono_discrete (\<succ>) p v" shows "poly_strict_mono p v"