hcju/isabelleps huggingface dataset:
otherwith_actionD: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and P :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and a :: "'c" assumes "otherwith Q I P \<sigma> \<sigma>' a" shows "P \<sigma> a"
hide_un_D2: fixes A :: "'b set"   and P :: "'b process"   and B :: "'b set" assumes "finite A" shows "D ((P \ A) \ B) \<subseteq> D (P \ (A \<union> B))"
S5n_rels_eq: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and w' :: "'g"   and a :: "'e" assumes "S5n M"   and "(w, w') \<in> relations M a" shows "relations M a `` {w} = relations M a `` {w'}"
has_vector_derivative_eq_has_derivative_blinfun: fixes f :: "real \<Rightarrow> 'e"   and f' :: "'e"   and x :: "real"   and U :: "real set" shows "(f has_vector_derivative f') (at x within U) = (f has_derivative blinfun_apply (blinfun_scaleR_left f')) (at x within U)"
wp3_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3 c Q = (\<lambda>(s, n). \<exists>t m. m \<le> n \<and> (c, s) \<Rightarrow>\<^sub>A m \<Down> t \<and> Q (t, n - m))"
right_total_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "right_total R = ((=) \<le> R\<inverse>\<inverse> OO R)"
rep_eq: fixes x :: "'a fset" shows "fMin x = Min (fset x)"
m: shows "emeasure M (induced_map A --` W) = (\<Sum>n. emeasure M (DW (n + 1)))"
sel: fixes x :: "'b" shows "\<partial>\<^sup>- (\<partial> x) = x"
has_codomain_iff_arr: fixes f :: "'b" shows "(C'.codomains f \<noteq> {}) = C'.arr f"
atom_fresh_perm: fixes x :: "name"   and y :: "name" assumes "x \<in> V"   and "y \<in> V" shows "atom x \<sharp> p \<bullet> y"
closedin_compact_projection: fixes S :: "'a set"   and T :: "'b set"   and U :: "('a \<times> 'b) set" assumes "compact S"   and "closedin (top_of_set (S \<times> T)) U" shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
affine_hull_nonempty_interior: fixes S :: "'a set" assumes "interior S \<noteq> {}" shows "affine hull S = UNIV"
card_eq1_circline_type_zero: fixes H :: "circline" assumes "\<exists>z. circline_set H = {z}" shows "circline_type H = 0"
simps_1: fixes uu :: "'b"   and acc :: "'b list" shows "start_remove [] uu acc = rev acc"
Exp: shows "(R, expandSet P Q) \<in> sumComposeSet"
Zl_mI_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and I :: "'c set"   and j :: "nat" shows "Zl_mI K P I j = (SOME x. x \<in> I \<and> (\<nu>\<^bsub>K P j\<^esub>) x = LI K (\<nu>\<^bsub>K P j\<^esub>) I)"
p: shows "p \<in> by_length k P \<and> admissible (p @ [u])"
termMOD_igAbsIPresIGWls: shows "igAbsIPresIGWls termMOD"
hashcode_iarray_def: fixes a :: "'b iarray" shows "hashcode a = foldl (\<lambda>h v. h * 33 + hashcode v) 0 (IArray.list_of a)"
intros_3: fixes a :: "letter"   and b :: "letter"   and ws :: "letter list list"   and zs :: "letter list list"   and w :: "letter list" assumes "a \<noteq> b"   and "T a ws zs" shows "T a ws ((b # w) # zs)"
eq\<rho>E: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value"   and x :: "Vars.var"   and a :: "Arity" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "eq\<rho> ae \<rho>1 \<rho>2"   and "ae x = up\<cdot>a" shows "eq a (\<rho>1 x) (\<rho>2 x)"
sa_iso_triple_pair: fixes x :: "'c" shows "(Abs_regular (- - x), stone_phi (Abs_regular (- x)) \<squnion> up_filter (Abs_dense (x \<squnion> - x))) \<in> stone_phi.pairs"
simps_1: fixes a :: "('fsym, 'psym) atm" shows "wtL (Pos a) = wtA a"
subset_image_inj: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and T :: "'c set" shows "(S \<subseteq> f ` T) = (\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U)"
cod_simps_2: shows "\<pi>.cod.leg0 = \<pi>.leg0"
decompose_init_prod: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod f {0..n} = f 0 * prod f {1..n}"
converse_idem: shows "\<smile> \<circ> \<smile> = id"
f_base: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "0 \<le> f x"
in_quotient_imp_closed: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "X \<in> A // r"   and "x \<in> X"   and "(x, y) \<in> r" shows "y \<in> X"
C_def: shows "C = (\<lambda>x. (fst x, - snd x)) -` A"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "ToTC f \<equiv> if CCC.arr f then T (T (fst f, fst (snd f)), snd (snd f)) else C.null"
fmeasurable_measure_inner_outer_le: fixes S :: "'c set"   and M :: "'c measure"   and m :: "real" shows "(S \<in> fmeasurable (completion M) \<and> m = Sigma_Algebra.measure (completion M) S) = ((\<forall>e>0. \<exists>T\<in>fmeasurable (completion M). T \<subseteq> S \<and> m - e \<le> Sigma_Algebra.measure (completion M) T) \<and> (\<forall>e>0. \<exists>U\<in>fmeasurable (completion M). S \<subseteq> U \<and> Sigma_Algebra.measure (completion M) U \<le> m + e))"
pochhammer_poly_altdef: fixes n :: "nat" shows "pochhammer_poly n = (\<Prod>i<n. [:of_nat i, 1::'b:])"
usum_In0I: fixes M :: "('a, 'b) node set"   and A :: "('a, 'b) node set set"   and B :: "('a, 'b) node set set" assumes "M \<in> A" shows "In0 M \<in> usum A B"
assms: shows "\<forall>x xs. P x (filter Q xs) = P x xs"
map_poly_1: fixes f :: "'c \<Rightarrow> 'b" shows "map_poly f 1 = [:f (1::'c):]"
un_Skip_def: fixes step :: "('d, 'c) step" shows "un_Skip step = (case step of Skip x2 \<Rightarrow> x2)"
alg_dvd_triv_left: fixes y :: "'a"   and x :: "'a" assumes "algebraic_int y" shows "x alg_dvd x * y"
integrable_on_const: fixes S :: "'a set"   and c :: "'b" assumes "S \<in> lmeasurable" shows "(\<lambda>x. c) integrable_on S"
intros: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt t = wt s"   and "head t = head s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "gt_same t s"
M'_def: shows "M' \<equiv> reset M n (v c) d"
Class_equivalence: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "(Class a = Class b) = ((a, b) \<in> E)"
constrains_imp_Constrains: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> A co A'" shows "F \<in> A Co A'"
strand_vars_split_3: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list" shows "fv\<^sub>s\<^sub>t (S @ S') = fv\<^sub>s\<^sub>t S \<union> fv\<^sub>s\<^sub>t S'"
Group_addfunset: fixes M :: "'a set"   and R :: "'b set" assumes "Group M" shows "Group (addfunset R M)"
neval_induct: fixes x1 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and x2 :: "nat"   and x3 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and P :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val) \<Rightarrow> nat \<Rightarrow> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "abs.neval x1 x2 x3"   and "\<And>x y. x = y \<Longrightarrow> P x 0 y"   and "\<And>x y n z. \<lbrakk>x \<leadsto>\<^sub>A y; abs.neval y n z; P y n z\<rbrakk> \<Longrightarrow> P x (Suc n) z" shows "P x1 x2 x3"
sup_continuous_def: fixes F :: "'c \<Rightarrow> 'd" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (Sup (range M)) = (SUP i. F (M i)))"
convolution_erlang_density: fixes l :: "real"   and k\<^sub>1 :: "nat"   and k\<^sub>2 :: "nat" assumes "0 < l" shows "(\<lambda>x. \<integral>\<^sup>+ y. ennreal (erlang_density k\<^sub>1 l (x - y)) * ennreal (erlang_density k\<^sub>2 l y) \<partial>lborel) = (\<lambda>x. ennreal (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l x))"
map_scale_eq_monom_mult: fixes c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "c \<cdot> p = punit.monom_mult c (0::'c) p"
preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "S.dom (Cones.map f) = Cones.map (Cop.dom f)"
init: fixes a :: "'a" assumes "a \<in> A" shows "P (a ## LNil)"
symevalb_def: fixes P :: "(char list \<Rightarrow> int option) \<times> 'b \<Rightarrow> bool"   and e :: "bexp"   and v :: "bool" shows "symevalb P e v \<equiv> \<forall>s n. P (s, n) \<longrightarrow> pbval' e s = Some v"
p_in_Rats: fixes k :: "nat" assumes "\<And>k. k \<in> {1..3} \<Longrightarrow> p k \<in> \<rat>" shows "p k \<in> \<rat>"
mult_im_sheaf_def: shows "im0.mult_im_sheaf \<equiv> \<lambda>V x y. 0::'e"
code: fixes a :: "'c" shows "lconst a = LCons a (lconst a)"
simps: fixes a1 :: "('a, 'b, 'c, 'd) Tree"   and a2 :: "('a, 'b, 'c, 'd) Tree" shows "((a1, a2) \<in> Tree_wf) = ((\<exists>t tset. a1 = t \<and> a2 = tConj tset \<and> t \<in> set_bset tset) \<or> (\<exists>t. a1 = t \<and> a2 = tNot t) \<or> (\<exists>t f \<alpha>. a1 = t \<and> a2 = tAct f \<alpha> t))"
is_ground_cls_mono: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "C \<subseteq># D"   and "substitution_ops.is_ground_cls subst_atm D" shows "substitution_ops.is_ground_cls subst_atm C"
simps_1: fixes x21 :: "color"   and x22 :: "('a, 'b) rbt"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt"   and y21 :: "color"   and y22 :: "('a, 'b) rbt"   and y23 :: "'a"   and y24 :: "'b"   and y25 :: "('a, 'b) rbt" shows "(Branch x21 x22 x23 x24 x25 = Branch y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
MP: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> M" shows "t \<in> SMP M"
add_poly_p_add_to_result: assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#(xs, n)#} + {#}) ({#}, {#}, {#(xs, n)#} + r)"
dom_simp: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.dom f = \<F>C.mkarr (\<F>C.DOM f)"
keys_def: shows "keys \<equiv> map_fun lookup id (\<lambda>f. {k. f k \<noteq> (0::'c)})"
Arg2pi_add: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "Arg2pi w + Arg2pi z = (if Arg2pi w + Arg2pi z < 2 * pi then Arg2pi (w * z) else Arg2pi (w * z) + 2 * pi)"
get_tag_name_locs_impl: shows "get_tag_name_locs = a_get_tag_name_locs"
rep_eq: fixes x :: "'b fpxs"   and xa :: "'b fpxs" shows "fpxs_nth (x * xa) = (\<lambda>xb. \<Sum>(y, z)\<in>{(y, z). y \<in> Formal_Puiseux_Series.supp (fpxs_nth x) \<and> z \<in> Formal_Puiseux_Series.supp (fpxs_nth xa) \<and> xb = y + z}. fpxs_nth x y * fpxs_nth xa z)"
introR: fixes a :: "'c"   and a' :: "'d"   and R :: "('c \<times> 'd) set" assumes "(a, a') \<in> R" shows "(a, a') \<in> R"
inf_aci_1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
yun: shows "yun_factorization gcd p = (d, pis)"
elims_2: fixes x :: "'a list"   and xa :: "'a list"   and xb :: "nat" assumes "sublist_at x xa xb" obtains xc :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = xc # xs__" and "xa = y__ # ys__" and "xb = 0" and "xc = y__ \<and> sublist_at xs__ ys__ 0"   | xs :: "'a list" and y :: "'a" and ys :: "'a list" and i :: "nat" where "x = xs" and "xa = y # ysa__" and "xb = Suc i" and "sublist_at xs ysa__ i"   | ys :: "'a list" where "x = []" and "xa = ys" and "xb = 0"
Un_UNIV_left: fixes B :: "'d set" shows "UNIV \<union> B = UNIV"
commutes_with_composition: fixes x :: "'b"   and y :: "'b" assumes "x \<in> A"   and "y \<in> A" shows "f (addA x y) = addB (f x) (f y)"
abs_eq: fixes x :: "int" shows "of_int_mod_ring x = Abs_mod_ring (x mod int CARD('a))"
generate_H_elim: fixes emBits :: "nat"   and x :: "bit list"   and y :: "bit list"   and z :: "bit list" assumes "roundup emBits 8 * 8 - emBits \<le> length x"   and "length (maskedDB_zero x emBits) = roundup emBits 8 * 8 - 168"   and "length y = 160" shows "generate_H (maskedDB_zero x emBits @ y @ z) emBits 160 = y"
b2T: shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle> \<rightarrow>* \<langle>C\<bullet>\<^sub>sM(map Val vs),(h\<^sub>2, l\<^sub>2, sh\<^sub>2),True\<rangle>"
mult_cancel_left1: fixes c :: "'a"   and b :: "'a" shows "(c = c * b) = (c = (0::'a) \<or> b = (1::'a))"
add_closed: fixes S :: "'d set"   and pls :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "semigroup_add_ow S pls"   and "a \<in> S"   and "b \<in> S" shows "pls a b \<in> S"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs"   and "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_set_mode_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
same_prefix_prefix: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
prj_chine_assoc_1: shows "Prj\<^sub>1 \<cdot> chine_assoc = Prj\<^sub>1\<^sub>1"
homeomorphic_ball_UNIV: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "ball a r homeomorphic UNIV"
trivial_same: fixes X :: "'c set" shows "trivial X = (\<forall>x1\<in>X. \<forall>x2\<in>X. x1 = x2)"
choose_u_member: fixes xs :: "int poly list" assumes "xs \<noteq> []" shows "choose_u xs \<in> set xs"
INF_seq_distrib: fixes f :: "'d::type \<Rightarrow> 'a::refinement_lattice"   and C :: "'d::type set"   and d :: "'a::refinement_lattice" shows "\<Sqinter> ((f::'d::type \<Rightarrow> 'a::refinement_lattice) ` (C::'d::type set)) ; (d::'a::refinement_lattice) = (\<Sqinter>c::'d::type\<in>C. f c ; d)"
cmp_in_hom_2: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b"   and "src\<^sub>C a = trg\<^sub>C b" shows "\<guillemotleft>\<Phi>\<^sub>F (a, b) : F a \<star>\<^sub>D F b \<Rightarrow>\<^sub>D F (a \<star>\<^sub>C b)\<guillemotright>"
exists_trace_for_any_i_j: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat" assumes "distributed_system channel"   and "\<exists>c'. distributed_system.trace channel trans send recv c t c'"   and "i \<le> j" shows "distributed_system.trace channel trans send recv (distributed_system.s channel trans send recv c t i) (take (j - i) (drop i t)) (distributed_system.s channel trans send recv c t j)"
AKcryptSKI: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "AKcryptSK authK servK evs"
bfs_reachability: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and r :: "'b"   and s :: "'b" assumes "class.stone_kleene_relation_algebra_arc star one times sup less_eq less bot inf top conv uminus"   and "finite {x. x = uminus (uminus x)}" shows "[r = uminus (uminus r) \<and> s = uminus (uminus s) \<and> times s top = s]   t := bot; q := s; p := inf (uminus s) (times (conv r) s); WHILE p \<noteq> bot INV {r = uminus (uminus r) \<and> q = uminus (uminus q) \<and> times q top = q \<and> inf t (conv t) = bot \<and> less_eq t r \<and> less_eq t q \<and> q = times (star (conv t)) s \<and> p = inf (uminus q) (times (conv r) q)}  VAR {\<lambda>(p, q, t). card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus (inf (uminus q) (times (star (conv r)) s))))}}  DO t := sup t (inf r (times q (conv p))); q := sup q p; p := inf (uminus q) (times (conv r) p) OD   [inf t (conv t) = bot \<and> less_eq t r \<and> q = times (star (conv t)) s \<and> q = times (star (conv r)) s]"
less_eq_nd_fun_def: shows "(\<le>) \<equiv> id (\<lambda>f g. f\<^sub>\<bullet> \<le> g\<^sub>\<bullet>)"
empty: fixes g :: "'d \<Rightarrow> 'c" shows "bot_homomorphism (sup_monoid.sum g)"
assms_2: shows "f \<in> carrier L \<rightarrow> carrier L"
simps_6: fixes c :: "int"   and e :: "MIR.num" shows "minusinf (Le (CN 0 c e)) = T"
ring_hom_ring_axioms_def: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "ring_hom_ring_axioms R S h \<equiv> h \<in> ring_hom R S"
finite_quotient: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "finite A"   and "r \<subseteq> A \<times> A" shows "finite (A // r)"
geom_rand_walk_borel_adapted: shows "borel_adapt_stoch_proc nat_filtration geom_proc"
that: fixes as :: "'edge list" assumes "S,slice_kind S \<turnstile> (m # msx',s) =as\<Rightarrow>\<^sub>\<tau> (m' # msx',s)" shows "thesis"
approximation_preproc_nat_3: fixes i :: "nat"   and j :: "nat" shows "real (i + j) = real i + real j"
with_def: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and d :: "'c" shows "with P f d = (if \<exists>x. P x then f (SOME x. P x) else d)"
mult_inj_if_coprime_nat: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set"   and g :: "'b \<Rightarrow> nat"   and B :: "'b set" assumes "inj_on f A"   and "inj_on g B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> coprime (f a) (g b)" shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)"
not_on_circlepathI: fixes z :: "complex"   and z0 :: "complex"   and r :: "real"   and st :: "real"   and tt :: "real" assumes "cmod (z - z0) \<noteq> \<bar>r\<bar>" shows "z \<notin> path_image (part_circlepath z0 r st tt)"
keys_sum: fixes b :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"   and I :: "'d set" shows "Poly_Mapping.keys (sum b I) \<subseteq> (\<Union>i\<in>I. Poly_Mapping.keys (b i))"
isIdom_def: fixes g :: "'a"   and n :: "'node"   and m :: "'node" shows "gE2.isIdom g n m = ((m \<noteq> n \<and> gE2.dominates g m n) \<and> (\<forall>m'\<in>set (\<alpha>n2 g). m' \<noteq> n \<and> gE2.dominates g m' n \<longrightarrow> gE2.dominates g m' m))"
comm_add_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s)"
g: fixes e :: "(char list \<Rightarrow> int) \<Rightarrow> 'a" shows "e \<circ> S = (\<lambda>s. e (S s))"
correct_state_def: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id" shows "correct_state \<Phi> t \<equiv> \<lambda>(xp, h, frs). P,h \<turnstile> t \<surd>t \<and> hconf h \<and> preallocated h \<and> (case frs of [] \<Rightarrow> True | f # fs \<Rightarrow> let (stk, loc, C, M, pc) = f in \<exists>Ts T mxs mxl\<^sub>0 is xt \<tau>. P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>(mxs, mxl\<^sub>0, is, xt)\<rfloor> in C \<and> \<Phi> C M ! pc = \<lfloor>\<tau>\<rfloor> \<and> conf_f P h \<tau> is f \<and> conf_fs P h \<Phi> M (length Ts) T fs \<and> conf_xcp P h xp (is ! pc))"
chine_assoc'_def: shows "doms.chine_assoc' \<equiv> \<langle>\<langle>doms.Prj\<^sub>1 \<lbrakk>doms.\<mu>.leg0, doms.\<nu>.leg1\<rbrakk> doms.Prj\<^sub>1\<^sub>0\<rangle> \<lbrakk>doms.\<nu>.leg0 \<cdot> doms.\<mu>\<nu>.prj\<^sub>0, doms.\<pi>.leg1\<rbrakk> doms.Prj\<^sub>0\<^sub>0\<rangle>"
arc_to_ends_def: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" shows "arc_to_ends G e \<equiv> (tail G e, head G e)"
L_transform_Not: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "L_transform (FL_Formula.Not x) = Formula.Not (L_transform x)"
univalent_conv_injective: fixes x :: "'a" shows "coreflexive (x[[x]]) = coreflexive (x[[x\<^sup>T\<^sup>T]])"
disjoint_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
rquot_accepting: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and q :: "nat" assumes "rquot_DFS A n"   and "dfa_is_node (rquot A n) q" shows "dfa_accepting (rquot A n) q = (\<exists>m. dfa_accepting A (dfa_steps A q (zeros m n)))"
t_stop_def: shows "ego.t_stop = - v\<^sub>e / a\<^sub>e"
Class_left_inverse: fixes a :: "'a" assumes "a \<in> G" shows "contained.Class (inverse a) [\<cdot>] contained.Class a = contained.Class \<one>"
ran: shows "(\<forall>n. coeff f n \<in> {0..<m}) = (range (coeff f) \<subseteq> {0..<m})"
ta_prod_def: fixes TA1 :: "('a, 'c, 'd) tree_automaton_rec_scheme"   and TA2 :: "('b, 'c, 'e) tree_automaton_rec_scheme" shows "ta_prod TA1 TA2 \<equiv> \<lparr>ta_initial = ta_initial TA1 \<times> ta_initial TA2, ta_rules = \<delta>_prod (ta_rules TA1) (ta_rules TA2)\<rparr>"
one_fun_apply: fixes x :: "'d" shows "1 x = (1::'c)"
cod_simp: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.VV.arr f" shows "Span.VV.cod f = Span.VxV.cod f"
perm_6: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "('a, 'b, 'c) psi" shows "pi \<bullet> \<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>(pi \<bullet> x1)\<rparr>(pi \<bullet> x2)"
uf_2: shows "uf \<longlonglongrightarrow> \<Sqinter> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. f = f1 + f2}"
generate_one: shows "generate G {\<one>} = {\<one>}"
distinct_23: fixes x2 :: "nat"   and x3 :: "nonce_t" shows "cNumber x2 \<noteq> cNonce x3"
connected_component_1_gen: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "DIM('a) = 1" shows "connected_component S a b = (closed_segment a b \<subseteq> S)"
intro: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "subgroup H (add_monoid G)" shows "additive_subgroup H G"
closed_segment: fixes a :: "'a"   and b :: "'a" shows "closed (closed_segment a b)"
zero_in_gen_ideal: fixes x :: "'a" assumes "x \<in> R" shows "\<zero> \<in> \<langle>x\<rangle>"
finite_atLeastAtMost_int: fixes l :: "int"   and u :: "int" shows "finite {l..u}"