hcju/isabelleps huggingface dataset:
sound_opr_alt: fixes opr :: "ground_action"   and f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" shows "sound_opr opr f = ((\<forall>s. close_eq s \<Turnstile> ground_action.precondition opr \<longrightarrow> (\<exists>s'. f s = Some s' \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set (dels (ground_action.effect opr)) \<and> close_eq s \<Turnstile> atm \<longrightarrow> close_eq s' \<Turnstile> atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set (adds (ground_action.effect opr)) \<and> close_eq s \<Turnstile> \<^bold>\<not> atm \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> atm) \<and> (\<forall>atm. atm \<in> set (adds (ground_action.effect opr)) \<longrightarrow> close_eq s' \<Turnstile> atm) \<and> (\<forall>fmla. fmla \<in> set (dels (ground_action.effect opr)) \<and> fmla \<notin> set (adds (ground_action.effect opr)) \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> fmla) \<and> (\<forall>a b. close_eq s \<Turnstile> Atom (Eq a b) \<longrightarrow> close_eq s' \<Turnstile> Atom (Eq a b)) \<and> (\<forall>a b. close_eq s \<Turnstile> \<^bold>\<not> (Atom (Eq a b)) \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> (Atom (Eq a b))))) \<and> (\<forall>fmla\<in>set (adds (ground_action.effect opr)). is_predAtom fmla))"
Helly_induct: fixes f :: "'b set set"   and n :: "nat" assumes "card f = n"   and "DIM('b) + 1 \<le> n"   and "\<forall>s\<in>f. convex s"   and "\<forall>t\<subseteq>f. card t = DIM('b) + 1 \<longrightarrow> \<Inter> t \<noteq> {}" shows "\<Inter> f \<noteq> {}"
simps_2: fixes V :: "'a list"   and v1 :: "'a"   and v2 :: "'a"   and Es :: "('a \<times> 'a) list" shows "wf_list_graph_impl V ((v1, v2) # Es) = (v1 \<in> set V \<and> v2 \<in> set V \<and> wf_list_graph_impl V Es)"
Abs_uint64_inverse: fixes y :: "64 word" assumes "y \<in> UNIV" shows "Rep_uint64 (Abs_uint64 y) = y"
solve_exec_def: fixes cs :: "('i \<times> constraint) list" shows "solve_exec cs \<equiv> let cs' = to_ns cs in case solve_exec_ns cs' of Inl is \<Rightarrow> Inl is | Inr v \<Rightarrow> Inr (from_ns v (map snd cs'))"
e: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> e) = (p = q)"
secrecy_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NA :: "msg"   and K :: "nat"   and NB :: "msg" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> otway" shows "Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs \<longrightarrow> Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
beukers_integral1_integrable': fixes r :: "nat"   and s :: "nat" shows "set_integrable lborel ({0<..<1} \<times> {0<..<1} \<times> {0<..<1}) (\<lambda>(z, x, y). x ^ r * y ^ s / (1 - (1 - x * y) * z))"
bfs_core_def: fixes src :: "nat"   and dst :: "nat" shows "bfs_core src dst \<equiv> while\<^sub>T (\<lambda>(f, P, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, P, C, N, d). (spec v. v \<in> C) \<bind> (\<lambda>v. let C = C - {v}; succ = E `` {v} in add_succ_spec dst succ v P N \<bind> (\<lambda>(f, P, N). if f then return (f, P, C, N, d + 1) else if C = {} then let C = N; N = {}; d = d + 1 in return (f, P, C, N, d) else return (f, P, C, N, d)))) (False, [src \<mapsto> src], {src}, {}, 0) \<bind> (\<lambda>(f, P, uu_, uu_, d). if f then return (Some (d, P)) else return None)"
nan_fcompare_double: fixes x :: "double"   and y :: "double" assumes "Code_Double.is_nan x \<or> Code_Double.is_nan y" shows "fcompare_double x y = Und"
fst_proj_mem: fixes A :: "'c"   and M :: "('c \<times> 'd) set" shows "(A \<in> fst_proj M) = (\<exists>B. (A, B) \<in> M)"
b: fixes t :: "real" assumes "t0 < t"   and "t \<le> t1" shows "v' t - f t (v t) < w' t - f t (w t)"
path_nret_ex_all_cd: fixes \<pi> :: "nat \<Rightarrow> 'n" assumes "is_path \<pi>"   and "\<forall>i. \<pi> i \<noteq> local.return" shows "\<forall>i. \<exists>j\<ge>i. \<forall>k>j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
simps_1: fixes x :: "real_alg_2"   and xs :: "real_alg_2 list" shows "real_alg_2_list_hd_intern (x # xs) = x"
np_specD1: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "np_spec np" shows "snd ` set (np gs bs hs data) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
sorted_wrt_lower_sublist_still_sorted: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and lo' :: "nat"   and xs' :: "'c list" assumes "sorted_sublist_wrt R xs lo (lo' - Suc 0)"   and "lo \<le> lo'"   and "lo' < length xs"   and "\<forall>i. lo \<le> i \<and> i < lo' \<longrightarrow> xs' ! i = xs ! i"   and "length xs' = length xs" shows "sorted_sublist_wrt R xs' lo (lo' - Suc 0)"
nsgTr0: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<in> N \<bullet> a" shows "a \<cdot> \<rho> b \<in> N \<and> \<rho> a \<cdot> b \<in> N"
arr_MkArr: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'e" assumes "A \<in> {A. |A| <o \<AA>}"   and "B \<in> {A. |A| <o \<AA>}"   and "f \<in> extensional A \<inter> (A \<rightarrow> B)" shows "arr (MkArr A B f)"
distinct_85: fixes x4 :: "val"   and x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list" shows "Val x4 \<noteq> x81\<bullet>x82{x83}"
assms_7: shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"
mat_mult_index: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'a list list"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat"   and ze :: "'a"   and pl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ti :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Matrix_Legacy.mat nr n m1"   and "Matrix_Legacy.mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI ze pl ti nr m1 m2 ! j ! i = scalar_prodI ze pl ti (Matrix_Legacy.row m1 i) (Matrix_Legacy.col m2 j)"
elims_3: fixes x :: "'a Point"   and xa :: "'a Cone" assumes "\<not> outsideCone x xa" obtains p :: "'a Point" and cone :: "'a Cone" where "x = p" and "xa = cone" and "\<not> slope cone * slope cone * time2 (vertex cone) p < space2 (vertex cone) p"
uint_0_iff: fixes x :: "'b word" shows "(uint x = 0) = (x = 0)"
bot_unique: fixes a :: "'d upred" shows "(false \<sqsubseteq> a) = (a = false)"
is_non_volatile_Read\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_non_volatile_Read\<^sub>s\<^sub>b r = (case r of Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> \<not> volatile | _ \<Rightarrow> False)"
senior_states_initial: fixes q :: "'b" shows "senior_states q 0 = {}"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_step_subst: fixes \<delta> :: "'e \<Rightarrow> ('d, 'c) Term.term"   and F :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
get_nth_i_def: fixes ba :: "'a list \<times> 'a list" shows "get_nth_i ba = hd (snd ba)"
empty: shows "\<Sum>\<^sub># {#} = (0::'a)"
sp_generalize2: fixes F_empty :: "'k"   and F_get :: "'k \<Rightarrow> 'l \<Rightarrow> ('m, 'n, 'l, 'o, 'p, 'q, 'r, 's) instr fundef option"   and F_add :: "'k \<Rightarrow> 'l \<Rightarrow> ('m, 'n, 'l, 'o, 'p, 'q, 'r, 's) instr fundef \<Rightarrow> 'k"   and F_to_list :: "'k \<Rightarrow> ('l \<times> ('m, 'n, 'l, 'o, 'p, 'q, 'r, 's) instr fundef) list"   and heap_empty :: "'t"   and heap_get :: "'t \<Rightarrow> 'n \<times> 'm \<Rightarrow> 'm option"   and heap_add :: "'t \<Rightarrow> 'n \<times> 'm \<Rightarrow> 'm \<Rightarrow> 't"   and heap_to_list :: "'t \<Rightarrow> (('n \<times> 'm) \<times> 'm) list"   and is_true :: "'m \<Rightarrow> bool"   and is_false :: "'m \<Rightarrow> bool"   and box_ubx1 :: "'r \<Rightarrow> 'm"   and unbox_ubx1 :: "'m \<Rightarrow> 'r option"   and box_ubx2 :: "'s \<Rightarrow> 'm"   and unbox_ubx2 :: "'m \<Rightarrow> 's option"   and \<OO>\<pp> :: "'o \<Rightarrow> 'm list \<Rightarrow> 'm"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'o \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'p \<Rightarrow> 'm list \<Rightarrow> 'm"   and \<II>\<nn>\<ll> :: "'o \<Rightarrow> 'm list \<Rightarrow> 'p option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'p \<Rightarrow> 'm list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'p \<Rightarrow> 'o"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'q \<Rightarrow> ('m, 'r, 's) unboxed list \<Rightarrow> ('m, 'r, 's) unboxed option"   and \<UU>\<bb>\<xx> :: "'p \<Rightarrow> Unboxed.type option list \<Rightarrow> 'q option"   and \<BB>\<oo>\<xx> :: "'q \<Rightarrow> 'p"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'q \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and F :: "'u \<Rightarrow> 'v fundef option"   and xs :: "('m, 'w, 'u, 'o, 'p, 'q, 'r, 's) instr list"   and n :: "nat"   and \<Sigma>' :: "Unboxed.type option list" assumes "ubx_sp F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"   and "ubx_sp.sp \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> F xs (replicate n None) = Ok \<Sigma>'" shows "ubx_sp.sp \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> F (map (ubx.generalize_instr box_ubx1 box_ubx2 \<BB>\<oo>\<xx>) xs) (replicate n None) = Ok (map Map.empty \<Sigma>')"
constant_degree: fixes p :: "'c poly" shows "constant (poly p) = (degree p = 0)"
ran_restrict_def: fixes p :: "'e \<Rightarrow> 'f decision option"   and S :: "'f decision set" shows "p \<triangleright> S \<equiv> \<lambda>x. if p x \<in> Some ` S then p x else \<bottom>"
extf: shows "\<forall>f\<in>local.ground_heads (head (App s t)). extf f (>\<^sub>t) (args (App s t)) (args s)"
split_at_def: fixes n :: "nat"   and xs :: "'b list" shows "split_at n xs = (take n xs, drop n xs)"
r1: assumes "nextElem vs c x = y" shows "is_sublist [x, y] vs"
B: shows "y = \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e"
UNIV: fixes S :: "'d set set" shows "generate_topology S UNIV"
inj_onI: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
concat_injective: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "concat xs = concat ys"   and "length xs = length ys"   and "\<forall>(x, y)\<in>list.set (zip xs ys). length x = length y" shows "xs = ys"
properties_existence: obtains lambda :: "nat" and mu :: "nat" where "local.properties lambda mu"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case init of init \<Rightarrow> f1 | loop \<Rightarrow> f2 | final \<Rightarrow> f3) = f1"
quicksort_ref_quicksort: fixes hi :: "nat"   and xs :: "'c list"   and lo :: "nat"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'e"   and x0 :: "nat \<times> nat \<times> 'd list" assumes "hi < length xs"   and "lo \<le> hi"   and "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. R (h x) (h y) \<or> R (h y) (h x)" shows "quicksort_ref R h x0 \<le> \<Down> Id (WB_Sort.quicksort R h x0)"
option_comp_def: fixes f :: "'e \<Rightarrow> 'd option"   and g :: "'f \<Rightarrow> 'e option"   and x :: "'f" shows "(f \<Lleftarrow> g) x \<equiv> Option.bind (g x) f"
assms: shows "\<sigma> \<in> space (state_measure V \<Gamma>)"
Prefixes_reflexive: fixes p :: "'a list" shows "p \<in> Prefixes p"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "Left_a.ide a"   and "Left_a.ide a'"   and "Left_a.in_hom g (H\<^sub>L a a') (H\<^sub>L a a)" shows "\<exists>f. Left_a.in_hom f a' a \<and> H\<^sub>L a f = g"
fps_XD'_0: fixes a :: "'a fps" shows "fps_XD' a 0 = 0"
compare_expansions_real: fixes c1 :: "real"   and c2 :: "real" shows "compare_expansions c1 c2 = (EQ, c1, c2)"
rcis_eq_zero_iff: fixes r :: "real"   and a :: "real" shows "(rcis r a = 0) = (r = 0)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
perm_cases: fixes pi :: "(name \<times> name) list"   and A :: "name set"   and B :: "name set" assumes "set pi \<subseteq> A \<times> A" shows "pi \<bullet> B \<subseteq> A \<union> B"
closure_of_subtopology_open: fixes X :: "'a topology"   and U :: "'a set"   and S :: "'a set" assumes "openin X U \<or> S \<subseteq> U" shows "subtopology X U closure_of S = U \<inter> X closure_of S"
a1: shows "low_equal s1 s2 \<and> \<not> snd (rett_instr instr s1) \<and> \<not> snd (rett_instr instr s2) \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (rett_instr instr s1)) \<and> t2 = snd (fst (rett_instr instr s2))"
comp_def: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'f" shows "f \<circ> g = (\<lambda>x. f (g x))"
igWlsBinp_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'i option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
semiring_norm_90: fixes m :: "Num.num"   and n :: "Num.num" shows "(num.Bit1 m = num.Bit1 n) = (m = n)"
mult_comm: fixes x :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and y :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "x \<in> \<O> U"   and "y \<in> \<O> U" shows "mult_sheaf_spec U x y = mult_sheaf_spec U y x"
lists_of_len_fin1: fixes P :: "'a set"   and n :: "nat" assumes "finite P" shows "finite (lists P \<inter> {l. length l = n})"
top_unclocked: fixes env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and count1 :: "nat"   and count2 :: "nat" shows "(evaluate_top False env s tp (s', r) \<longrightarrow> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s') \<and> evaluate_top False env (update_clock (\<lambda>_. count1) s) tp (update_clock (\<lambda>_. count1) s', r) = evaluate_top False env (update_clock (\<lambda>_. count2) s) tp (update_clock (\<lambda>_. count2) s', r)"
a_Pi: shows "a \<in> B \<rightarrow> carrier R"
absolutely_integrable_Un: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'a set" assumes "f absolutely_integrable_on S"   and "f absolutely_integrable_on T" shows "f absolutely_integrable_on S \<union> T"
bdd_below1: shows "bdd_below {\<parallel>x\<parallel>\<^sup>2 |x. x \<in> M}"
distance_refl: fixes s :: "'v" shows "\<delta> s s = 0"
seq: shows "\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l)"
f_succ_def: shows "f_succ \<equiv> \<lambda>\<delta>. {(xa, x). f_succp (\<lambda>x. x \<in> \<delta>) xa x}"
nothrows_\<Gamma>: shows "\<forall>p\<in>dom \<Gamma>. nothrows (the (\<Gamma> p))"
rel_fun_def: fixes A :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
monomial_0D: fixes t :: "'d"   and c :: "'e" assumes "monomial c t = 0" shows "c = (0::'e)"
classical_P: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "let PP = \<lambda>t. subst P t xx in prv (neg (neg (PP \<langle>\<phi>\<rangle>)))" shows "prv (PP \<langle>\<phi>\<rangle>)"
fd_iso1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<le> d y" shows " |z\<rangle> x \<le>  |z\<rangle> y"
is_natural_1: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd"   and \<tau> :: "'b \<Rightarrow> 'd"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
apsnd_compose: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'a \<times> 'd" shows "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
mult_right_isotone: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<preceq> y" shows "z \<odot> x \<preceq> z \<odot> y"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VxVxV.seq g f = (seq (fst g) (fst f) \<and> VxV.seq (snd g) (snd f))"
one_subst_automatic: fixes M :: "lam"   and M' :: "lam"   and N :: "lam"   and N' :: "lam"   and x :: "name" assumes " M \<longrightarrow>\<^sub>1 M'"   and " N \<longrightarrow>\<^sub>1 N'" shows " M[x::=N] \<longrightarrow>\<^sub>1 M'[x::=N']"
star_n_minus: fixes X :: "nat \<Rightarrow> 'a" shows "- star_n X = star_n (\<lambda>n. - X n)"
attach_shadow_root_is_weakly_dom_component_safe: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'n" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>h h'"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 |h \<turnstile> attach_shadow_root element_ptr shadow_root_mode|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_dom_component (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 element_ptr)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
nstut_fun2: fixes X :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and Y :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "NSTUTINV X"   and "NSTUTINV Y" shows "NSTUTINV f<X, Y>"
Succ_Shift: fixes Kl :: "'d list set"   and k :: "'d"   and kl :: "'d list" shows "BNF_Greatest_Fixpoint.Succ (BNF_Greatest_Fixpoint.Shift Kl k) kl = BNF_Greatest_Fixpoint.Succ Kl (k # kl)"
carr_prodag_mem_eq: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and X :: "'d \<Rightarrow> 'e"   and Y :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A"   and "\<forall>l\<in>I. X l = Y l" shows "X = Y"
me_K': shows "measurable_envelope M (K \<inter> {x \<in> space M. b n \<le> f x}) K'"
foldr_fun_no_upd: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and y :: "'c"   and g :: "'b \<Rightarrow> 'd"   and \<A> :: "'c \<Rightarrow> 'd" assumes "inj_on f (set xs)"   and "y \<notin> f ` set xs" shows "foldr (\<lambda>x \<A>. \<A>(f x := g x)) xs \<A> y = \<A> y"
differentiable_chain_at: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and g :: "'c \<Rightarrow> 'd" assumes "f differentiable at x"   and "g differentiable at (f x)" shows "g \<circ> f differentiable at x"
all_rules: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> pterm) fset" assumes "prules C_info rs" shows "fBall rs prule"
smult_distribs_2: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "Polynomial.smult a (p + q) = Polynomial.smult a p + Polynomial.smult a q"
ComplE: fixes c :: "'b"   and A :: "'b set" assumes "c \<in> - A" obtains "c \<notin> A"
Red_F_\<G>_empty_L_q_def: fixes q :: "'q"   and N :: "('f \<times> 'l) set" shows "Red_F_\<G>_empty_L_q q N = {C. \<forall>D\<in>\<G>_F_L_q q C. D \<in> Red_F_q q (\<Union> (\<G>_F_L_q q ` N)) \<or> (\<exists>E\<in>N. False \<and> D \<in> \<G>_F_L_q q E)}"
assms_1: fixes ns :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and gs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" assumes "ns \<noteq> []" shows "set (ab gs bs ns data) = set bs \<union> set ns"
quorum_of_set_def: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'b set" assumes "with_w quorum_of" shows "with_w.quorum_of_set quorum_of S Q \<equiv> \<exists>p\<in>S. quorum_of p Q"
filterlim_pow_at_bot_even: fixes n :: "nat"   and f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "0 < n"   and "filterlim f at_bot F"   and "even n" shows "LIM x F. f x ^ n :> at_top"
preList_Seq: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList upds (C1;; C2) l s = preList (map (\<lambda>(x, e). (x, preT C2 e)) upds) C1 l s"
head_is_vert2: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> (g_E G)\<^sup>+" shows "v \<in> set (g_V G)"
intros_12: 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 c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Catch c\<^sub>1 c\<^sub>2 Q,A"
min_diff: fixes m :: "nat"   and i :: "nat"   and n :: "nat" shows "min (m - i) (n - i) = min m n - i"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e (pdevs_of_list (inl X)) = pdevs_val e' X"   and "e' \<in> UNIV \<rightarrow> I" shows "thesis"
B22: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'a" shows "mono f = (\<forall>n. f n \<le> f (Suc n))"
eSuc_inject: fixes m :: "enat"   and n :: "enat" shows "(eSuc m = eSuc n) = (m = n)"
up_filter_dense_antitone: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "up_filter (Abs_dense (x \<squnion> - x \<squnion> y \<squnion> z)) \<le> up_filter (Abs_dense (x \<squnion> - x \<squnion> y))"
join_above1: fixes x :: "'a"   and y :: "'a" assumes "compatible x y" shows "x \<sqsubseteq> x \<squnion> y"
slp_of_fa_un_def: fixes Unop :: "floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M1 :: "(floatarith, nat) mapping"   and slp1 :: "floatarith list" shows "slp_of_fa_un Unop a M slp M1 slp1 = (case Mapping.lookup M (Unop a) of None \<Rightarrow> (Mapping.update (Unop a) (length slp1) M1, slp1 @ [Unop (Var (slp_index_lookup slp1 M1 a))]) | Some i \<Rightarrow> (Mapping.update (Unop a) (length slp) M, slp @ [Var (slp_index slp i)]))"
basis_ex1_supset_circuit: fixes B :: "'a set"   and x :: "'a" assumes "basis B"   and "x \<in> carrier - B" shows "\<exists>!C. circuit C \<and> C \<subseteq> insert x B"