hcju/isabelleps huggingface dataset:
fbox_g_ode_subset: fixes f :: "'a \<Rightarrow> 'a"   and T :: "real set"   and S :: "'a set"   and \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and U :: "'a \<Rightarrow> real set"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "local_flow f T S \<phi>"   and "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "|x\<acute>=\<lambda>t. f & G on U S @ 0] Q = (\<lambda>s. s \<in> S \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"
append_simpl: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and i :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "Matrix_Tensor.mult id f" shows "i < length xs \<longrightarrow> (xs @ ys) ! i = xs ! i"
tendsto_ident_at: fixes a :: "'a"   and s :: "'a set" shows "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
invariant_red: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and i :: "'i" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P i t x m" shows "P i t x' m'"
tr1_def: shows "path M1 (io_init @ [io_last] || tr1_init @ [tr1_last]) (initial M1) \<and> length (tr1_init @ [tr1_last]) = length (io_init @ [io_last])"
sel_1: fixes x2 :: "'e" shows "un_Skip (Skip x2) = x2"
assms_3: shows "h \<in> C.hom (C.cod g) (C.dom f)"
simps_11: fixes I :: "\<I>"   and r :: "formula Regex.regex" shows "convert_multiway (MatchF I r) = MatchF I (convert_multiway_regex r)"
list_all2_Nil2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list" shows "list_all2 P xs [] = (xs = [])"
assms_2: shows "constructive_security_obsf' real middle sim_outer \<I>_real \<I>_middle \<I>_common \<A>"
prems_3: shows "history_consistent \<theta> p ((x # xs) @ ys)"
has_integral_real: fixes f :: "real \<Rightarrow> 'a"   and y :: "'a"   and a :: "real"   and b :: "real" shows "(f has_integral y) {a..b} = (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of {a..b} \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e))"
hd: shows "hd \<GG> = {\<one>\<^bsub>G\<^esub>}"
homologous_rel_sym: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "homologous_rel p X S a b = homologous_rel p X S b a"
conseq1: fixes G :: "'a triple set"   and P' :: "'a assn"   and c :: "com"   and Q :: "'a assn"   and P :: "'a assn" assumes "G|-{P'}. c .{Q}"   and "\<forall>Z s. P Z s \<longrightarrow> P' Z s" shows "G|-{P}. c .{Q}"
iso_tuple_update_accessor_cong_assist_id: fixes upd :: "('c \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'd"   and ac :: "'d \<Rightarrow> 'c" assumes "Record.iso_tuple_update_accessor_cong_assist upd ac" shows "upd id = id"
phiUses_exI': fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q"   and "p \<in> allVars g" obtains m :: "'node" where "q \<in> phiUses g m" and "m \<in> set (predecessors g (defNode g p))"
degree_less_iff: fixes p :: "'a mpoly"   and x :: "nat"   and k :: "nat" shows "(MPoly_Type.degree p x < k) = (0 < k \<and> (\<forall>m\<in>monomials p. lookup m x < k))"
T_App: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and t\<^sub>2 :: "trm" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2"   and "\<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<cdot> t\<^sub>2 : T\<^sub>2"
merge_t_def: fixes i :: "nat" shows "merge_t i = {n. \<exists>q q' j. state_rank q n = Some j \<and> j < i \<and> q' = \<delta> q (w n) \<and> q' \<notin> F \<and> ((\<exists>q''. q'' \<noteq> q \<and> q' = \<delta> q'' (w n) \<and> state_rank q'' n \<noteq> None) \<or> q' = q\<^sub>0)}"
fixp_induct: fixes P :: "(enat \<Rightarrow> 'c llist \<Rightarrow> 'c llist) \<Rightarrow> bool" assumes "llist.admissible (\<lambda>ldrop. P (curry ldrop))"   and "P (\<lambda>ldrop n. LNil)"   and "\<And>xs. \<lbrakk>P xs; llist.le_fun (\<lambda>(x, y). xs x y) (\<lambda>(b, a). ldrop b a)\<rbrakk> \<Longrightarrow> P (\<lambda>a aa. case a of 0 \<Rightarrow> aa | eSuc n' \<Rightarrow> case aa of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> xs n' xs')" shows "P ldrop"
c1_on_open_euclidean_def: fixes f :: "'d \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'd"   and X :: "'d set" shows "c1_on_open_euclidean f f' X \<equiv> c1_on_open f f' X"
zero_power2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero 2 = zero"
EpriKF_synth: fixes K :: "fresh_t"   and H :: "msg set" assumes "epriKF K \<in> synth H" obtains "epriKF K \<in> H"
simps_1: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.nlcons a Empty = Single a"
OUT_Edge: fixes x :: "'v"   and y :: "'v" shows "\<^bold>O\<^bold>U\<^bold>T\<^bsub>\<Delta>''\<^esub> (Edge x y) = (if edge \<Delta> x y then {Vertex y} else {})"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Cod t) (Cod t') (Map t')"
intros: fixes classes :: "'a fset"   and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and \<C> :: "'a"   and "from" :: "'c"   and role :: "'c"   and end\<^sub>1 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool"   and end\<^sub>2 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "association_ends' classes associations \<C> from role end\<^sub>1"   and "association_ends' classes associations \<C> from role end\<^sub>2"   and "end\<^sub>1 \<noteq> end\<^sub>2" shows "association_ends_not_unique' classes associations"
image_Collect_subsetI: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
dirichlet_inverse_cong: fixes f :: "nat \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'b"   and i :: "'b"   and i' :: "'b"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. 0 < n \<Longrightarrow> f n = f' n"   and "i = i'"   and "n = n'" shows "dirichlet_inverse f i n = dirichlet_inverse f' i' n'"
RedInitError: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Error)\<rfloor>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>RI (C,THROW NoClassDefFoundError) ; Cs \<leftarrow> e,(h, l, sh),b\<rangle>"
simps_2: fixes g :: "(node \<times> node) list"   and x :: "node"   and xs :: "node list"   and ys :: "node list" shows "dfs g (x # xs) ys = (if List.member ys x then dfs g xs ys else dfs g (nexts g x @ xs) (x # ys))"
ball_one_point2: fixes A :: "'c set"   and a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>A. a = x \<longrightarrow> P x) = (a \<in> A \<longrightarrow> P a)"
splitAt_distinct_snd: fixes vs :: "'c list"   and ram1 :: "'c" assumes "distinct vs" shows "distinct (snd (splitAt ram1 vs))"
poly_mapping_induct: fixes P :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and f :: "'b \<Rightarrow>\<^sub>0 'c" assumes "\<And>k v. P (Poly_Mapping.single k v)"   and "\<And>f g k v. \<lbrakk>P f; P g; g = Poly_Mapping.single k v; k \<notin> keys f\<rbrakk> \<Longrightarrow> P (f + g)" shows "P f"
tl_replicate: fixes n :: "nat"   and x :: "'a" shows "tl (x \<up> n) = x \<up> (n - 1)"
intros_1: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set"   and l :: "'b"   and t' :: "'a"   and t :: "'a"   and Ts :: "'a set"   and final :: "'c \<Rightarrow> bool" assumes "has_lock (locks s $ l) t'"   and "t' \<noteq> t"   and "t' \<in> Ts" shows "final_thread.must_wait final s t (Inl l) Ts"
Parallel_AllNone_lemma: fixes Ss :: "('a ann_com option \<times> 'a set) list"   and s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a"   and n :: "nat" assumes "All_None Ss"   and "(Parallel Ss, s) -Pn\<rightarrow> (Parallel Ts, t)" shows "Ts = Ss \<and> n = 0 \<and> s = t"
count_image_mset_le_count_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c multiset"   and y :: "'d" assumes "inj_on f (set_mset M)" shows "count (image_mset f M) y \<le> count M (inv_into (set_mset M) f y)"
times_Measurement_System_ext_def: fixes x :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>"   and y :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>" shows "x \<cdot> y = \<lparr>unit_sys = unit, \<dots> = mmore x \<cdot> mmore y\<rparr>"
is_zero_row_utp_0: fixes m :: "'d"   and A :: "(('e, 'f) vec, 'd) vec" shows "is_zero_row_upt_k m 0 A"
intros: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set"   and c :: "('a, 'c, 'd) Stmt"   and \<Gamma>' :: "'a \<Rightarrow> 'd set option"   and \<S>' :: "'a set \<times> 'a set"   and P' :: "'d set"   and mds :: "Mode \<Rightarrow> 'a set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P c \<Gamma>' \<S>' P'"   and "sifum_types.tyenv_wellformed bexp_vars \<C>_vars mds \<Gamma> \<S> P"   and "sifum_types.tyenv_eq ev\<^sub>B dma_type \<Gamma> mem\<^sub>1 mem\<^sub>2"   and "sifum_lang_no_dma.pred ev\<^sub>B P mem\<^sub>1"   and "sifum_lang_no_dma.pred ev\<^sub>B P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> sifum_types.type_max ev\<^sub>B (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "(((c, mds), mem\<^sub>1), (c, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>1 ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma>' \<S>' P'"
diff_f': shows "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0::'a})"
lossless_response: fixes init :: "'l \<Rightarrow> 'm \<Rightarrow> ('n \<times> 'o) spmf"   and response :: "'n \<Rightarrow> 'm \<Rightarrow> 'p \<Rightarrow> 'q spmf"   and check :: "'l \<Rightarrow> 'o \<Rightarrow> 'p \<Rightarrow> 'q \<Rightarrow> bool"   and Rel :: "('l \<times> 'm) set"   and S_raw :: "'l \<Rightarrow> 'p \<Rightarrow> ('o \<times> 'q) spmf"   and \<A>ss :: "'l \<Rightarrow> 'o \<times> 'p \<times> 'q \<Rightarrow> 'o \<times> 'p \<times> 'q \<Rightarrow> 'm spmf"   and challenge_space :: "'p set"   and valid_pub :: "'l set"   and G :: "('l \<times> 'm) spmf"   and r :: "'n"   and w :: "'m"   and e :: "'p" assumes "\<Sigma>_protocols_to_commitments init response check Rel S_raw \<A>ss challenge_space valid_pub G" shows "lossless_spmf (response r w e)"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "partial_order L"   and "complete_lattice_axioms L" shows "complete_lattice L"
valid_1: shows "valid_ownership \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b'"
l_one: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "\<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x = x"
validFrom_validTrans: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and s :: "'a"   and a :: "'b"   and ou :: "'c"   and s' :: "'a"   and tr :: "('a, 'b, 'c) trans list" assumes "IO_Automaton.validTrans step (Trans s a ou s')"   and "IO_Automaton.validFrom step s' tr" shows "IO_Automaton.validFrom step s (Trans s a ou s' # tr)"
\<Z>_srngD_3: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"
times_poly_def: fixes p :: "'b poly"   and q :: "'b poly" shows "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons (0::'b) p) p 0"
idem: fixes a :: "'d multiset" shows "a \<union># a = a"
callee_invariant: fixes y :: "'b"   and s' :: "'s2"   and s :: "'s2"   and x :: "'a" assumes "(y, s') \<in> set_spmf (oracle2 s x)"   and "I2 s"   and "x \<in> outs_\<I> \<I>" shows "I2 s'"
BuildSeq2_def: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf"   and y' :: "hf" shows "BuildSeq2 B C s k y y' \<equiv> BuildSeq (\<lambda>p. \<exists>x x'. p = \<langle>x, x'\<rangle> \<and> B x x') (\<lambda>p q r. \<exists>x x' y y' z z'. p = \<langle>x, x'\<rangle> \<and> q = \<langle>y, y'\<rangle> \<and> r = \<langle>z, z'\<rangle> \<and> C x x' y y' z z') s k \<langle>y, y'\<rangle>"
listsum2_lemma: fixes xs :: "nat list"   and vl :: "nat"   and n :: "nat" assumes "length xs = vl"   and "n \<le> vl" shows "rec_exec (rec_listsum2 vl n) xs = listsum2 xs n"
lo_lemma''': fixes y :: "nat"   and x :: "nat" assumes "\<not> Suc 0 < y" shows "rec_exec rec_lo [x, y] = lo x y"
sum_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) zero" shows "less_eq (comm_monoid_add.sum plus zero f A) zero"
continuous_on_cong: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "s = t"   and "\<And>x. x \<in> t \<Longrightarrow> f x = g x" shows "continuous_on s f = continuous_on t g"
i_take_length: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "length (f \<Down> n) = n"
cbox_translation: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "cbox (c + a) (c + b) = (+) c ` cbox a b"
comp_arr_dom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "VV.arr f"   and "VV.dom f = a" shows "VV.comp f a = f"
leftderivations1_def: shows "leftderivations1 = {(u, v) |u v. leftderives1 u v}"
ln_eq_minus_one: fixes x :: "real" assumes "0 < x"   and "ln x = x - 1" shows "x = 1"
livePhi: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" assumes "liveVal g v"   and "phiArg g v v'" shows "liveVal g v'"
vsv_is_VLambda: shows "VLambda (\<D>\<^sub>\<circ> f) (app f) = f"
iEx_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iEx I P \<equiv> \<exists>t\<in>I. P t"
known_dists_def: fixes G :: "graph"   and V :: "nat set" shows "known_dists G V = (V \<subseteq> verts G \<and> 0 \<in> V \<and> (\<forall>i\<in>verts G. has_dist_on G 0 i V) \<and> (\<forall>i\<in>V. has_dist G 0 i \<and> dist G 0 i = dist_on G 0 i V))"
succ_def: fixes v :: "'a" shows "succ v = (SOME w. v \<rightarrow> w \<and> (v \<in> W1 \<or> w \<notin> W1))"
iso_inv_iso: fixes f :: "'a" assumes "C.iso f" shows "C.iso (C.inv f)"
ide_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.ide (src\<^sub>C \<mu>)"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel"
preserves_section_retraction: fixes e :: "'d"   and m :: "'d" assumes "D.ide (e \<cdot>\<^sub>D m)" shows "C.ide (F e \<cdot>\<^sub>C F m)"
subst_cls_mset_id_subst: fixes CC :: "'a literal multiset multiset" shows "CC \<cdot>cm id_subst = CC"
nat_pow_closed: fixes x :: "nat"   and n :: "nat" assumes "x \<in> carrier (units_of R)" shows "x [^]\<^bsub>units_of R\<^esub> n \<in> carrier (units_of R)"
prop_atoms_subfrmlsn: fixes \<phi> :: "'a ltln" shows "prop_atoms \<phi> \<subseteq> subfrmlsn \<phi>"
imp_cong: fixes P :: "bool"   and P' :: "bool"   and Q :: "bool"   and Q' :: "bool" assumes "P = P'"   and "P' \<Longrightarrow> Q = Q'" shows "(P \<longrightarrow> Q) = (P' \<longrightarrow> Q')"
measurable_cong_sets: fixes M :: "'d measure"   and M' :: "'d measure"   and N :: "'e measure"   and N' :: "'e measure" assumes "sets M = sets M'"   and "sets N = sets N'" shows "M \<rightarrow>\<^sub>M N = M' \<rightarrow>\<^sub>M N'"
ran_maplets: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys"   and "distinct xs" shows "ran [xs [\<mapsto>] ys] = set ys"
assms_3: shows "set (labels_conv ss1) \<subseteq> S"
matcher_correctness: fixes r :: "'b rexp"   and s :: "'b list" shows "matcher r s = (s \<in> lang r)"
is_hpair_def: fixes z :: "hf" shows "is_hpair z = (\<exists>x y. z = \<langle>x, y\<rangle>)"
stable_pair_on_fix_F: fixes ds :: "'b set"   and X :: "'a set \<times> 'a set" assumes "stable_pair_on ds X" shows "map_prod id dual X = F ds (map_prod id dual X)"
mem_diff_3: shows "\<forall>x. var_asm_not_written mds\<^sub>C' x \<longrightarrow> mem\<^sub>C' x = mem\<^sub>C'' x \<and> dma\<^sub>C mem\<^sub>C' x = dma\<^sub>C mem\<^sub>C'' x"
Says_K6: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and A :: "agent"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
newton_seq_fact7: fixes n :: "nat" shows "ns (Suc n) \<ominus> ns n = \<ominus> local.divide (f \<bullet> ns n) (f' \<bullet> ns n)"
DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> =n\<Rightarrow> t"
Nonce_Number: fixes X :: "nonce_t"   and X' :: "nat" shows "Nonce X \<noteq> Number X'"
ANR_imp_neighbourhood_retract: fixes S :: "'a set"   and U :: "'a set" assumes "ANR S"   and "closedin (top_of_set U) S" obtains V :: "'a set" where "openin (top_of_set U) V" and "S retract_of V"
head_app_iso: fixes hom :: "('g, 'h, 'f, 'e) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "head (app_iso hom G) = iso_head hom"
zero_nin_W: shows "0\<^sub>v n \<notin> W"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<CC> = vcard \<CC>"
lcm_proj2_if_dvd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a b" shows "lcm a b = normalize b"
card_eq_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "card B = vs1.dim V"   and "finite B" shows "vs1.independent B = (V \<subseteq> vs1.span B)"
f12_fdg: shows "f12 = fst (splitFace g v u f ws)"
more_temp_simps3_9: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>\<box>\<diamond>F \<equiv> \<box>\<diamond>F"
atLeastAtMost_singleton: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.order less_eq less" shows "ord.atLeastAtMost less_eq a a = {a}"
supp_rel_tAct: fixes t :: "('a, 'b, 'c, 'd) Tree"   and f :: "'d"   and \<alpha> :: "'c" assumes "finite (supp_rel (=\<^sub>\<alpha>) t)" shows "supp_rel (=\<^sub>\<alpha>) (tAct f \<alpha> t) = supp f \<union> (supp \<alpha> \<union> supp_rel (=\<^sub>\<alpha>) t - bn \<alpha>)"
irreducible_dvd_prod_list: fixes p :: "'a poly"   and as :: "'a poly list" assumes "irreducible p"   and "p dvd prod_list as" shows "Bex (set as) ((dvd) p)"
union_diff2: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "sum_0 g (A \<union> B) = sum_0 g (A - B) + sum_0 g (B - A) + sum_0 g (A \<inter> B)"
ocondition_def: fixes c :: "'d \<Rightarrow> bool"   and L :: "'d \<Rightarrow> 'e"   and R :: "'d \<Rightarrow> 'e" shows "ocondition c L R \<equiv> \<lambda>s. if c s then L s else R s"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> [$C v, $e] : ts _> ts'"
nyinitcls_def: fixes G :: "prog"   and s :: "state" shows "nyinitcls G s = {C. is_class G C \<and> \<not> initd C s}"
valid_countings_def: fixes a :: "nat"   and b :: "nat" shows "valid_countings a b = card {f \<in> {1..a + b} \<rightarrow>\<^sub>E {A, B}. card {x \<in> {1..a + b}. f x = A} = a \<and> card {x \<in> {1..a + b}. f x = B} = b \<and> (\<forall>m\<in>{1..a + b}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A})}"
takeWhile_append2: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "takeWhile P (xs @ ys) = xs @ takeWhile P ys"
Atm_transT_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c"   and s :: "'b"   and s' :: "'b" assumes "PL.transT tval aval (Atm atm, s) s'" obtains "s' = aval atm s"
pre_Trans: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language"   and z :: "'a language" assumes "Plus_Times_pre_cong R x y"   and "Plus_Times_pre_cong R y z" shows "Plus_Times_pre_cong R x z"
nv_OfFax_1: fixes x :: "Preliminaries.var"   and f :: "'fsym" shows "x \<notin> nvT (lOfFax f)"