hcju/isabelleps huggingface dataset:
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> synth H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>agt. P (msg.Agent agt)"   and "\<And>n. P (Number n)"   and "\<And>X. \<lbrakk>X \<in> synth H; P X\<rbrakk> \<Longrightarrow> P (Hash X)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P \<lbrace>X, Y\<rbrace>"   and "\<And>X K. \<lbrakk>X \<in> synth H; P X; Key K \<in> H\<rbrakk> \<Longrightarrow> P (Crypt K X)" shows "P x"
prems_2: shows "insert s S \<subseteq> carrier (union_ring C)"
p53: shows "\<exists>br\<in>allBlocksRead s p. bal (dblock s' p) = bal br"
ideal_restrict_indets: fixes G :: "(('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "ideal G = ideal (homogenize None ` extend_indets ` F)" shows "ideal (restrict_indets ` G) = ideal F"
psubstEnv_updEnv_idEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort" shows "idEnv [x \<leftarrow> X]_xs &[rho] = rho [x \<leftarrow> X #[rho]]_xs"
has_integral_eq: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and k :: "'d" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"   and "(f has_integral k) s" shows "(g has_integral k) s"
HR: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (x \<le> R p q)"
split_list: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = append ys (Cons x zs)"
embed_measure_eq_distr: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj f" shows "embed_measure M f = distr M (embed_measure M f) f"
conseq: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile> {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile> {P'} c {Q'}"
subst3Chain: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'a"   and Xs :: "name set" assumes "length xvec = length Tvec"   and "distinct xvec"   and "set xvec \<subseteq> supp T"   and "Xs \<sharp>* T[xvec::=Tvec]" shows "Xs \<sharp>* Tvec"
prod_dvd_intro: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k dvd m \<or> k dvd n" shows "k dvd m * n"
perm_char_def: fixes pi :: "('a \<times> 'a) list"   and c :: "char" shows "pi \<bullet> c = c"
chine_eq_apex: shows "\<rho>.chine = \<rho>.apex"
poly_y_x_const: fixes p :: "'a poly" shows "poly_y_x [:p:] = poly_lift p"
normalize_lcm_factorial: fixes a :: "'a"   and b :: "'a" shows "normalize (lcm_factorial a b) = lcm_factorial a b"
monotone_def: fixes orda :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ordb :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
INST: fixes x :: "variable"   and T :: "PDDL_STRIPS_Semantics.type" assumes "is_of_type Q x T" shows "is_of_type objT (f x) T"
procs_subset: shows "procs c \<subseteq> dom \<Gamma>"
cv_part: fixes X :: "V set"   and n :: "nat"   and i :: "nat"   and a :: "'a set" assumes "\<forall>x\<in>[X]\<^bsup>n\<^esup>. f (cv x) = i"   and "X \<subseteq> elts \<beta>"   and "a \<in> [cv X]\<^bsup>n\<^esup>" shows "f a = i"
phl_test_while: fixes p :: "'a"   and b :: "'a"   and x :: "'a" assumes "test p"   and "test b"   and "\<lbrace>p \<cdot> b\<rbrace>x\<lbrace>p\<rbrace>" shows "\<lbrace>p\<rbrace>while b do x od\<lbrace>p \<cdot> !b\<rbrace>"
tki: fixes xs :: "int list" shows "incr (take_incr xs)"
diamond_cube_is_only_horizontal_div_of_rot: shows "only_horizontal_division (boundary diamond_cube) {rot_diamond_cube}"
SeqE: fixes c1 :: "com"   and c2 :: "com"   and s1 :: "char list \<Rightarrow> int"   and s3 :: "char list \<Rightarrow> int" assumes "(c1;; c2, s1) \<Rightarrow> s3" obtains s\<^sub>2 :: "char list \<Rightarrow> int" where "(c1, s1) \<Rightarrow> s\<^sub>2" and "(c2, s\<^sub>2) \<Rightarrow> s3"
orbit_perm_rem_eq: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "bij f" shows "orbit (perm_rem y f) x = (if x = y then {y} else orbit f x - {y})"
PiE_I: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = undefined" shows "f \<in> Pi\<^sub>E A B"
int_to_bv_lt0: fixes n :: "int" assumes "n < 0" shows "int_to_bv n = norm_signed (bv_not (\<zero> # nat_to_bv (nat (- n - 1))))"
l1_l2_def: shows "N_Chop(lan v,i,j) \<and> |i| = x \<and> |j| = y"
assms_4: shows "local.dij i j \<le> x ! i"
size_merge: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "size (merge t1 t2) = size t1 + size t2"
state_succ_def: fixes s :: "('d, 'e) fmap"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" shows "state_succ s a \<equiv> if fst a \<subseteq>\<^sub>f s then snd a ++ s else s"
simps_2: fixes c :: "'b"   and c' :: "'b poly"   and n' :: "nat"   and p' :: "'b poly" shows "(c)\<^sub>p +\<^sub>p CN c' n' p' = CN ((c)\<^sub>p +\<^sub>p c') n' p'"
splitFace_empty_ram1_ram2_in_f21: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF []"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF []" shows "(ram1, ram2) \<in> \<E> f21"
division_of_def: fixes s :: "'a set set"   and i :: "'a set" shows "(s division_of i) = (finite s \<and> (\<forall>K\<in>s. K \<subseteq> i \<and> K \<noteq> {} \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>K1\<in>s. \<forall>K2\<in>s. K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}) \<and> \<Union> s = i)"
case_eq_if: fixes f :: "'b \<Rightarrow> 'b msstream \<Rightarrow> 'a"   and msstream :: "'b msstream" shows "(case msstream of MSSCons x xa \<Rightarrow> f x xa) = f (un_MSSCons1 msstream) (un_MSSCons2 msstream)"
parent_edge: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a" assumes "RootedTree G root"   and "v \<in> V\<^bsub>G\<^esub>"   and "v \<noteq> root" shows "v \<rightarrow>\<^bsub>G\<^esub> RootedTree.parent G root v"
invmode_Static_eq: fixes m :: "'a member_scheme"   and e :: "expr" shows "(invmode m e = Static) = is_static m"
inj_on_directional_derivative: fixes k :: "enat"   and a :: "'c"   and S :: "'c set" assumes "k \<noteq> 0" shows "inj_on (directional_derivative k a) S"
itop_sub_ttop_base: fixes A :: "'b set" shows "(\<Union>s\<in>A\<^sup>\<star>. {infsuff A s}) = (\<Union>t\<in>\<Union>s\<in>A\<^sup>\<star>. {suff A s}. {t \<inter> \<Union> (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s})})"
neq_shrK: fixes s :: "nat"   and evs :: "event list"   and B :: "agent" assumes "Key s \<notin> used evs" shows "s \<noteq> shrK B"
sel_9: fixes x61 :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and x62 :: "'a expr list" shows "un_Operation1 (Operation x61 x62) = x61"
wlsAll_inversion2: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs_s :: "'varSort \<times> 'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "(wls s X \<longrightarrow> True) \<and> (wlsAbs xs_s A \<longrightarrow> isInBar xs_s \<and> (\<exists>x X. wls (snd xs_s) X \<and> A = Abs (fst xs_s) x X)) \<and> (wlsInp delta inp \<longrightarrow> True) \<and> (wlsBinp delta binp \<longrightarrow> True)"
ifte_def: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "ifte a x y = a \<sqinter> x \<squnion> - a \<sqinter> y"
r_rpp: shows "bs_env \<rho> \<rho>''"
convolution_transfer: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "Abs_aezfun (convolution f g) = Abs_aezfun f * Abs_aezfun g"
append_constant_length_induct: fixes k :: "nat"   and P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>ys. k = 0 \<or> length ys < k \<Longrightarrow> P ys"   and "\<And>xs ys. \<lbrakk>length xs = k; P ys\<rbrakk> \<Longrightarrow> P (xs @ ys)" shows "P xs"
elims: fixes x :: "nat"   and xa :: "real"   and xb :: "nat"   and y :: "real" assumes "power_down x xa xb = y" obtains p :: "nat" and xc :: "real" where "x = p__" and "xa = xc__" and "xb = 0" and "y = 1"   | p :: "nat" and xc :: "real" and n :: "nat" where "x = p" and "xa = xc" and "xb = Suc n" and "y = (if odd n then truncate_down (Suc p) ((power_down p xc (Suc n div 2))\<^sup>2) else truncate_down (Suc p) (xc * power_down p xc n))"
cnt_b: shows "count (image_mset f M) b < count (image_mset f N) b"
sing_len: fixes a :: "'b" shows "\<^bold>|[a]\<^bold>| = 1"
lex_half_segments_last': fixes d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "d \<in> set (map snd (half_segments_of_aform X))" shows "Counterclockwise_2D_Arbitrary.lex (snd (last (half_segments_of_aform X))) d"
kauff_mat_swingpos: shows "kauff_mat r_over_braid = kauff_mat l_over_braid"
imageEquivalence: fixes f :: "('c \<times> 'd set) set"   and x :: "'c" assumes "runiq f"   and "x \<in> Domain f" shows "f ,, x = f ,,, x"
simps: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "nat"   and a3 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "a1 \<leadsto>\<^bsup>a2\<^esup> a3 = ((\<exists>x y. a1 = x \<and> a2 = 0 \<and> a3 = y \<and> x = y) \<or> (\<exists>x y n z. a1 = x \<and> a2 = Suc n \<and> a3 = z \<and> x \<leadsto> y \<and> y \<leadsto>\<^bsup>n\<^esup> z))"
is_ring_morphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "ring_homomorphism (id.im_sheaf_morphisms U V) (id.im_sheaf U) (id.add_im_sheaf U) (id.mult_im_sheaf U) (id.zero_im_sheaf U) (id.one_im_sheaf U) (id.im_sheaf V) (id.add_im_sheaf V) (id.mult_im_sheaf V) (id.zero_im_sheaf V) (id.one_im_sheaf V)"
set_bits_aux_def: fixes f :: "nat \<Rightarrow> bool"   and w :: "'a word"   and n :: "nat" shows "set_bits_aux f w n = push_bit n w OR take_bit n (set_bits f)"
chamber_facet_is_chamber_facet: fixes C :: "'a set"   and D :: "'a set"   and z :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "z \<lhd> C"   and "order.greater_eq D z" shows "z \<lhd> D"
finite_psubset_induct: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite A"   and "\<And>A. \<lbrakk>finite A; \<And>B. B \<subset> A \<Longrightarrow> P B\<rbrakk> \<Longrightarrow> P A" shows "P A"
mono_sinvar: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and N :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and nP :: "'b \<Rightarrow> 'c" assumes "SecurityInvariant_preliminaries sinvar"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "sinvar \<lparr>nodes = N, edges = E\<rparr> nP" shows "sinvar \<lparr>nodes = N, edges = E'\<rparr> nP"
omega_simulation: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> x \<le> y \<cdot> z" shows "z \<cdot> x\<^sup>\<omega> \<le> y\<^sup>\<omega>"
add_succ_spec_def: fixes dst :: "'c"   and succ :: "'c set"   and v :: "'d"   and PRED :: "'c \<Rightarrow> 'd option"   and N :: "'c set" shows "add_succ_spec dst succ v PRED N \<equiv> assert (N \<subseteq> dom PRED) \<bind> (\<lambda>_. SPEC (\<lambda>(f, PRED', N'). case f of True \<Rightarrow> dst \<in> succ - dom PRED \<and> PRED \<subseteq>\<^sub>m PRED' \<and> PRED' \<subseteq>\<^sub>m map_mmupd PRED (succ - dom PRED) v \<and> dst \<in> dom PRED' | False \<Rightarrow> dst \<notin> succ - dom PRED \<and> PRED' = map_mmupd PRED (succ - dom PRED) v \<and> N' = N \<union> (succ - dom PRED)))"
assms: shows "run_one_step d i (s, vs, ves, $b_e.If x71 x72 x73) = (s', vs', res)"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (map\<^sub>0 a) = map\<^sub>0 a"
above_Above_singl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "above r a = Above r {a}"
neg_diag_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
ACI_norm_deriv: fixes as :: "'b"   and r :: "'b rexp" shows "\<guillemotleft>deriv as \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>deriv as r\<guillemotright>"
mod_eq_0_iff_dvd: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "(modulo a b = zero) = dvd.dvd times b a"
vars_apps: fixes s :: "('d, 'c) tm"   and ss :: "('d, 'c) tm list" shows "vars (apps s ss) = vars s \<union> \<Union> (vars ` set ss)"
rsplit_aux_0_simps_1: fixes m :: "nat"   and n :: "nat"   and c :: "int"   and x :: "int list" assumes "m = 0" shows "bin_rsplit_aux n m c x = x"
nat_seg_image_imp_finite: fixes A :: "'c set"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "A = f ` {i. i < n}" shows "finite A"
suminf_split_initial_segment': fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k + 1)) + sum f {..k}"
assms_3: shows "x \<in> B\<^sub>1 - B\<^sub>2"
CACH_RET: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "ins_is C m l vreturn"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "Anno\<down>l = None" shows "deriv G C m l (Cachera 0)"
order_le_prod: fixes r\<^sub>A :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "order (Product.le r\<^sub>A r\<^sub>B) = (order r\<^sub>A \<and> order r\<^sub>B)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and err :: "'c err" shows "P (case err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = ((err = Err \<longrightarrow> P f1) \<and> (\<forall>x2. err = OK x2 \<longrightarrow> P (f2 x2)))"
continuous_on_closed_Un: fixes s :: "'d set"   and t :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
compl_bot_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<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" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus bot = top"
assms: fixes x :: "'a"   and y :: "'a"   and u :: "real"   and v :: "real" assumes "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
simps_5: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp"   and \<beta> :: "('b, 'c, 'a) hp" shows "prog_sem I (\<alpha> \<union>\<union> \<beta>) = prog_sem I \<alpha> \<union> prog_sem I \<beta>"
map_prod_simp: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "map_prod f g (a, b) = (f a, g b)"
shift_1: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "valid_decomp X (local.shift k m qs)"
LIM_eq: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a" shows "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r)"
simps: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a list"   and a3 :: "'a" shows "rtrancl_path r a1 a2 a3 = ((\<exists>x. a1 = x \<and> a2 = [] \<and> a3 = x) \<or> (\<exists>x y ys z. a1 = x \<and> a2 = y # ys \<and> a3 = z \<and> r x y \<and> rtrancl_path r y ys z))"
sep_list_conj_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Ps :: "('b \<Rightarrow> bool) list" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_list_conj plus zero sep_disj Ps \<equiv> foldl (sep_algebra.sep_conj plus sep_disj) (sep_algebra.sep_empty zero) Ps"
pcr_cr_eq: shows "pcr_nonzero (=) = cr_nonzero"
fixp_above_Field: fixes a :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "a \<in> Field leq" shows "fixp_above a \<in> Field leq"
image_Union: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set set" shows "f ` \<Union> S = \<Union> (f \<turnstile> S)"
valid_vmuaux_def: shows "valid_vmuaux = (\<lambda>_ cur (t, aux) auxlist. t = cur \<and> aux = auxlist)"
subset_mem_compact_implies_subset_existence_interval: fixes T' :: "real set"   and x0 :: "'a"   and K :: "'a set" assumes "0 \<in> T'"   and "is_interval T'"   and "T' \<subseteq> UNIV"   and "x0 \<in> X"   and "\<And>t. \<lbrakk>t \<in> T'; t \<in> existence_ivl0 x0\<rbrakk> \<Longrightarrow> flow0 x0 t \<in> K"   and "compact K"   and "K \<subseteq> X" shows "T' \<subseteq> existence_ivl0 x0"
binary_functor_Prod_1: shows "binary_functor (\<cdot>) (\<cdot>) (\<cdot>) CCC.Prod"
ncloses_extend: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and a :: "coname"   and P :: "trm"   and B :: "ty" assumes "\<theta>_n ncloses \<Gamma>"   and "x \<sharp> \<Gamma>"   and "x \<sharp> \<theta>_n"   and "<a>:P \<in> \<parallel><B>\<parallel>" shows "(x, a, P) # \<theta>_n ncloses (x, B) # \<Gamma>"
weak_ranking_altdef: shows "weak_ranking le = weak_ranking_aux carrier"
\<alpha>\<beta>2\<gamma>_lt_\<alpha>\<beta>1\<gamma>: shows "\<alpha>2 + \<beta>2 * \<gamma> < \<alpha>1 + \<beta>1 * \<gamma>"
less_restrictive_correct: fixes atom_ordering :: "('a \<times> 'a) set"   and R1 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and R2 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.less_restrictive R1 R2"   and "propositional_atoms.Sound R1" shows "propositional_atoms.Sound R2"
nth_subdegree_mult_left: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_nth (f * g) (subdegree f) = fps_nth f (subdegree f) * fps_nth g 0"
elims_2: fixes x :: "('b, 'c) Combinators"   and xa :: "('b, 'c) Combinators list" assumes "memberP x xa" obtains r :: "('b, 'c) Combinators" and xb :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = r" and "xa = xb # xs" and "member r xb \<or> memberP r xs"
eq1: shows "w \<cdot> z = [a] \<^sup>@ m \<cdot> [a] \<^sup>@ k \<cdot> [b] \<cdot> z'"
dis_log_\<A>_def: fixes \<A> :: "'a \<Rightarrow> ('a \<times> nat \<times> nat \<times> nat \<times> nat) spmf"   and h :: "'a" shows "dis_log_\<A> \<A> h = \<A> h \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. assert_spmf (c = \<^bold>g [^] d \<otimes> h [^] m \<and> c = \<^bold>g [^] d' \<otimes> h [^] m') \<bind> (\<lambda>_. return_spmf (if m' < m then nat ((int d' - int d) * Number_Theory_Aux.inverse (m - m') (order \<G>) mod int (order \<G>)) else nat ((int d - int d') * Number_Theory_Aux.inverse (m' - m) (order \<G>) mod int (order \<G>))))))"
assms_2: shows "InvariantCEntailed (getConflictFlag state) F0 (getC state)"
cartesianTr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and x :: "'f \<times> 'h" assumes "R module M"   and "R module N"   and "x \<in> M \<times>\<^sub>c N" shows "\<exists>m n. m \<in> carrier M \<and> n \<in> carrier N \<and> x = (m, n)"
bind_left_identity: fixes a :: "'e"   and f :: "'e \<Rightarrow> ('c, 'd) state" shows "return a \<bind> f = f a"
perm_exits_Vs: fixes i :: "name" assumes "i \<in> Vs" shows "p \<bullet> i \<notin> Vs"
rprD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "rpr A r" shows "complete A r \<and> refl_on A r \<and> trans r"
d_def_split_var: fixes x :: "'a" shows "d (x \<sqinter> nc) + x \<cdot> (0::'a) \<parallel> 1\<^sub>\<sigma> = d x"
sup_state_trans: fixes P :: "'b prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list"   and c :: "ty list \<times> ty err list" assumes "P \<turnstile> a \<le>\<^sub>i b"   and "P \<turnstile> b \<le>\<^sub>i c" shows "P \<turnstile> a \<le>\<^sub>i c"
rtrancl3p_converse_induct: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and bs :: "'e list"   and a'' :: "'d"   and P :: "'d \<Rightarrow> 'e list \<Rightarrow> 'd \<Rightarrow> bool" assumes "rtrancl3p r a bs a''"   and "\<And>a. P a [] a"   and "\<And>a b a' bs a''. \<lbrakk>rtrancl3p r a' bs a''; r a b a'; P a' bs a''\<rbrakk> \<Longrightarrow> P a (b # bs) a''" shows "P a bs a''"
i_set0_mult_subset_i_set0: fixes k :: "nat" shows "i_set0_mult k \<subseteq> i_set0"
dist_pos_lt: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open"   and "x \<noteq> y" shows "0 < dist x y"
fraction_not_in_nats: fixes n :: "int"   and m :: "int" assumes "\<not> n dvd m"   and "n \<noteq> 0" shows "of_int m / of_int n \<notin> \<nat>"