hcju/isabelleps huggingface dataset:
if_eval\<^sub>w: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and b :: "'d"   and t :: "('a, 'c, 'd) Stmt"   and e :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "(((Stmt.If b t e, mds), mem), (if eval\<^sub>B mem b then t else e, mds), mem) \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
intensional_simps_6: fixes P :: "'b \<Rightarrow> bool" shows "P = \<not> P \<equiv> #False"
eq: shows "A * pderiv B - pderiv A * B = pderiv C * B - C * pderiv B"
IH_1: assumes "opaque x = []" shows "x \<simeq> Pure (unlift_vars 0 x)"
f1: shows "(get_trap_set s = {} \<or> reset_trap_val s \<noteq> False \<or> get_ET (cpu_reg_val PSR s) \<noteq> 0) \<and> ((\<exists>e. fetch_instruction (delayed_pool_write s) = Inl e) \<or> (\<exists>v1 v2. fetch_instruction (delayed_pool_write s) = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> (annul_val (delayed_pool_write s) = True \<or> annul_val (delayed_pool_write s) = False \<and> (\<forall>v1' v2'. fetch_instruction (delayed_pool_write s) = Inr v1' \<and> decode_instruction v1' = Inr v2' \<longrightarrow> supported_instruction (fst v2') = True) \<and> (fst v2 \<noteq> ctrl_type RETT \<or> fst v2 = ctrl_type RETT \<and> (get_ET (cpu_reg_val PSR (delayed_pool_write s)) = 1 \<or> get_ET (cpu_reg_val PSR (delayed_pool_write s)) \<noteq> 1 \<and> get_S (cpu_reg_val PSR (delayed_pool_write s)) \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR (delayed_pool_write s))) + 1) mod NWINDOWS)) (cpu_reg_val WIM (delayed_pool_write s)) = 0 \<and> get_addr (snd v2) (delayed_pool_write s) AND 3 = 0)))))"
not_produces_imp_notin_production: fixes C :: "'a literal multiset"   and A :: "'a" assumes "production C \<noteq> {A}" shows "A \<notin> production C"
Star_1: fixes mr :: "mregex"   and ms :: "mregex" assumes "r = from_mregex mr \<phi>s"   and "safe_regex Futu g (from_mregex mr \<phi>s)"   and "ms \<in> LPD mr" shows "safe_regex Futu g (from_mregex ms \<phi>s)"
less_irrefl: fixes x :: "'b list" shows "\<not> strict_prefix x x"
col2_orth__orth: fixes U :: "'p"   and V :: "'p"   and P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "U \<noteq> V"   and "Col P Q U"   and "Col P Q V"   and "A B C Orth P Q" shows "A B C Orth U V"
path_defs_4: fixes g :: "real \<Rightarrow> 'b" shows "path_image g = g ` {0..1}"
snd_pseudo_divmod_main: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat"   and q' :: "'b poly" shows "snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)"
execute_return: fixes x :: "'a" shows "execute (return x) = Some \<circ> Pair x"
path_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a" shows "CFG.path sourcenode targetnode valid_edge (sourcenode a) [a] (targetnode a)"
CommuteWithIntervalCut4_def: fixes "fun" :: "'a set \<Rightarrow> 'a set" shows "CommuteWithIntervalCut4 fun \<equiv> \<forall>t fun2 I. fun2 = (\<lambda>I. I \<down>< t) \<or> fun2 = (\<lambda>I. I \<down>\<le> t) \<or> fun2 = (\<lambda>I. I \<down>> t) \<or> fun2 = (\<lambda>I. I \<down>\<ge> t) \<longrightarrow> fun (fun2 I) = fun2 (fun I)"
monotone_times_ennreal2: fixes y :: "ennreal" shows "monotone (\<le>) (\<le>) ((*) y)"
int_or_zero: fixes x :: "int" shows "0 OR x = x"
lnull_ldropWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lnull (ldropWhile P xs) = (\<forall>x\<in>lset xs. P x)"
inverse_1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse one = one"
linear_poly_eqI: fixes p :: "linear_poly"   and q :: "linear_poly" assumes "\<And>v. p \<lbrace> v \<rbrace> = q \<lbrace> v \<rbrace>" shows "p = q"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and event :: "event" shows "P (case event of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = ((\<forall>x11 x12 x13. event = Says x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x21 x22. event = Gets x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. event = Notes x31 x32 \<longrightarrow> P (f3 x31 x32)))"
finite_j_OUT: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "(\<Sum>\<^sup>+ y\<in>\<^bold>O\<^bold>U\<^bold>T x. j (\<langle>x\<rangle>, \<langle>y\<rangle>)) \<noteq> \<top>"
induct: fixes P :: "(lit \<times> lit) list \<Rightarrow> bool"   and a0 :: "(lit \<times> lit) list" assumes "P []"   and "\<And>cl cls. P cls \<Longrightarrow> P (cl # cls)" shows "P a0"
m_closed: fixes x :: "Vertex \<Rightarrow> Vertex"   and y :: "Vertex \<Rightarrow> Vertex" assumes "x \<in> carrier tetrahedral_group"   and "y \<in> carrier tetrahedral_group" shows "x \<otimes>\<^bsub>tetrahedral_group\<^esub> y \<in> carrier tetrahedral_group"
edges_\<alpha>g_aux: fixes g :: "'m" assumes "invar g" shows "edges (\<alpha>g g) = \<alpha>edges_aux g"
commute: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
h7: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set (get_signs (reduce_system p (qs, matrix, subsets, signs)))"
lift_stop_oracle_def: fixes "oracle" :: "'b \<Rightarrow> 'c \<Rightarrow> ('a \<times> 'b) spmf"   and s :: "'b"   and x :: "'c" shows "lift_stop_oracle oracle s x = map_spmf (apfst Some) (oracle s x)"
binop: shows "binop \<real>\<^sub>\<circ> vreal_plus"
is_processT5_S3: fixes s :: "'c event list"   and P :: "'c process"   and c :: "'c event" assumes "(s, {}) \<in> F P"   and "(s @ [c], {}) \<notin> F P" shows "(s, {c}) \<in> F P"
irreducibleD': fixes a :: "'c"   and b :: "'c" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> b dvd (1::'c)"
distinct_91: fixes x4 :: "val"   and x111 :: "char list"   and x112 :: "ty"   and x113 :: "expr" shows "Val x4 \<noteq> {x111:x112; x113}"
termFSwImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'b, 'c, 'd) model_scheme" shows "termFSwImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSwapAll h hA MOD"
subset_of_opens: fixes U :: "'a set" assumes "U \<in> {U. is_open U \<and> x \<in> U}" shows "is_open U"
set_integral_space: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "set_lebesgue_integral M (space M) f = integral\<^sup>L M f"
points_index_sum: fixes B :: "'b set multiset multiset"   and ps :: "'b set" shows "\<Sum>\<^sub># B index ps = (\<Sum>b\<in>#B. b index ps)"
nb\<^sub>1\<^sub>2: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> set (effect_of op)" shows "(\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) (v, a) = Some True"
that: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b \<in> B"   and "b \<noteq> 0"   and "red {b} p q" shows "thesis"
3: shows "LBINT x:{a<..}. ln (d * x + b) * (d * x + b) powr c = C"
subset_imageE: fixes B :: "'e set"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "B \<subseteq> f ` A" obtains C :: "'f set" where "C \<subseteq> A" and "B = f ` C"
assms_2: shows "h \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
wlog_non_Rf: fixes N :: "'a literal multiset set"   and CC :: "'a literal multiset multiset"   and E :: "'a literal multiset"   and D :: "'a literal multiset" assumes "\<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD + CC \<longrightarrow> I \<Turnstile> E) \<and> (\<forall>D'. D' \<in># DD \<longrightarrow> D' < D)" shows "\<exists>DD. set_mset DD \<subseteq> N - Rf N \<and> (\<forall>I. I \<Turnstile>m DD + CC \<longrightarrow> I \<Turnstile> E) \<and> (\<forall>D'. D' \<in># DD \<longrightarrow> D' < D)"
assms_3: shows "proj_list_5 xs' (ys, w, ws, z, zs) = (ys', w', ws', z', zs')"
extremum_uniqueI: fixes a :: "'b upred" assumes "a \<sqsubseteq> true" shows "a = true"
mcont_mono: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "mcont luba orda lubb ordb f" shows "monotone orda ordb f"
in_set_simps_3: fixes x :: "'a" shows "(x \<in> set []) = False"
xivp1: shows "X \<in> Sols (\<lambda>t. f) (ll_on_open.existence_ivl UNIV (\<lambda>t. f) UNIV 0) UNIV 0 s"
chine_assoc_def: shows "doms.chine_assoc \<equiv> \<langle>doms.Prj\<^sub>1\<^sub>1 \<lbrakk>doms.\<mu>.leg0, doms.\<nu>.leg1 \<cdot> doms.\<nu>\<pi>.prj\<^sub>1\<rbrakk> \<langle>doms.Prj\<^sub>0\<^sub>1 \<lbrakk>doms.\<nu>.leg0, doms.\<pi>.leg1\<rbrakk> doms.Prj\<^sub>0\<rangle>\<rangle>"
domintros: fixes B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "\<And>aa ba xa ya. \<lbrakk>Suc 0 < card B; (aa, ba) = (SOME b. b \<in> B \<and> card (snd b) = Suc m); (xa, ya) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)\<rbrakk> \<Longrightarrow> shift1_dom (shift_list (SOME b. b \<in> B \<and> card (snd b) = Suc m) (SOME y. y \<in> ya) qs, B - {SOME b. b \<in> B \<and> card (snd b) = Suc m})" shows "shift1_dom (qs, B)"
minus_r_inv_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A - A = 0\<^sub>m nr nc"
quot_of_ratfps_eq_iff: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "(quot_of_ratfps x = quot_of_ratfps y) = (x = y)"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t]"
inject: fixes x21 :: "'c"   and x22 :: "'c list"   and y21 :: "'c"   and y22 :: "'c list" shows "(Cons x21 x22 = Cons y21 y22) = (x21 = y21 \<and> x22 = y22)"
simps_5: fixes I :: "'b"   and b :: "bexp"   and P :: "'b"   and C :: "'b acom"   and Q :: "'b" shows "strip ({I} WHILE b DO {P} C {Q}) = WHILE b DO strip C"
map_transform_cong: fixes m1 :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and m2 :: "(Vars.var \<times> exp) list"   and t :: "'a \<Rightarrow> exp \<Rightarrow> exp" assumes "\<And>x. x \<in> domA m1 \<Longrightarrow> ae x = ae' x"   and "m1 = m2" shows "map_transform t ae m1 = map_transform t ae' m2"
homeomorphic_space_sym: fixes X :: "'c topology"   and Y :: "'d topology" shows "(X homeomorphic_space Y) = (Y homeomorphic_space X)"
ennreal_spmf_bind: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf"   and x :: "'a" shows "ennreal (spmf (p \<bind> f) x) = \<integral>\<^sup>+ xa. ennreal (spmf (f xa) x) \<partial>measure_spmf p"
div_by_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a zero = zero"
Nats_altdef1: shows "\<nat> = {of_int n |n. 0 \<le> n}"
prod_zero_iff': fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(prod f A = (0::'b)) = (\<exists>x\<in>A. f x = (0::'b))"
composite_simps_2: shows "trg (g \<star> h) = trg g"
cos0_scalprod0: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "(cos (\<angle> z1 z2) = 0) = (scalprod z1 z2 = 0)"
lipschitz_on_def: fixes C :: "real"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C \<cdot> dist x y))"
porder_on_consistent_linorder_on_trancl_antisym: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set" assumes "partial_order_on A r"   and "linear_order_on B s"   and "order_consistent r s"   and "B \<subseteq> A" shows "antisym ((r \<union> s)\<^sup>+)"
proj_topology_def: shows "proj_topology = map_topology Proj euclidean"
List_hd_not_in_tl: fixes h :: "nat \<Rightarrow> nat option"   and b :: "nat"   and as :: "nat list"   and a :: "nat" assumes "List h b as"   and "h a = Some b" shows "a \<notin> set as"
xvecFreshTvec: shows "(p \<bullet> xvec) \<sharp>* Tvec"
erase_insrt: fixes t :: "'a tree"   and x :: "'a" assumes "hinv t" shows "erase (insrt x t) = insrt\<^sub>0 x (erase t)"
cmh: shows "continuous_map (subtopology (Euclidean_space n) S) (Euclidean_space n) h"
cases: fixes a :: "intv" assumes "isIntv a" obtains uu_ :: "nat" where "a = Intv uu_"
unintersect_def: fixes X :: "'d set" shows "unintersect X = SPEC ((\<subseteq>) X)"
mult_assoc: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> E" shows "(a * b) \<cdot> x = a \<cdot> b \<cdot> x"
preserves_arr: fixes f :: "'c local.term set" assumes "arr f" shows "\<F>\<^sub>SC.arr (D f)"
F1set1_incl_JF1set: fixes a :: "'d JF1" shows "F1set1 (dtor1 a) \<subseteq> JF1set a"
simps: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" shows "inverse_1 (p, l, r) = real_alg_2 (abs_int_poly (reflect_poly p), inverse r, inverse l)"
vpra_empty_valid_call_list_rev: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_call_list sourcenode kind valid_edge get_proc cs n" shows "CFG.valid_path_rev_aux kind get_return_edges [] (rev cs)"
simps_50: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x1 :: "('d, 'e) Term.term" shows "(case send\<langle>x1\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f1 x1"
rep_eq: fixes x :: "'a ACI_rexp" shows "ACI_lang x = lang (rep_ACI_rexp x)"
fmdrop_set_single: fixes a :: "'c"   and m :: "('c, 'd) fmap" shows "fmdrop_set {a} m = fmdrop a m"
small_set_comprehension: fixes A :: "V \<Rightarrow> 'a"   and I :: "V" shows "small {A i |i. i \<in>\<^sub>\<circ> I}"
simps_1: fixes a :: "('a, 'b) atm" shows "M.wtL (Pos a) = M.wtA a"
conc_trans_additional_4: fixes A :: "'b nres"   and B :: "'b nres"   and C :: "'b nres" assumes "A \<le> \<Down> Id B"   and "B \<le> \<Down> Id C" shows "A \<le> C"
assms: fixes k :: "nat" assumes "k \<in> {1..n}" shows "p k \<in> \<rat>"
Sup_llist_LCons: fixes X :: "'b set"   and Xs :: "'b set llist" shows "Sup_llist (LCons X Xs) = X \<union> Sup_llist Xs"
adjacent_int_facet2: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y"   and "x \<noteq> y" shows "x \<inter> y \<lhd> y"
analz_analz_Un: fixes G :: "msg set"   and H :: "msg set" shows "analz (analz G \<union> H) = analz (G \<union> H)"
bij_pointE: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "bij f" obtains x :: "'b" where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
assms_1: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
Chaos_is_CSP: shows "is_CSP_process (R (false \<turnstile> true))"
1: shows "C = {y \<in> C. ord x y} \<union> {y \<in> C. ord y x}"
closure_of_subtopology_mono: fixes T :: "'b set"   and U :: "'b set"   and X :: "'b topology"   and S :: "'b set" assumes "T \<subseteq> U" shows "subtopology X T closure_of S \<subseteq> subtopology X U closure_of S"
set_zip: fixes xs :: "'d list"   and ys :: "'e list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
some_neq_not_eq: fixes i :: "nat" shows "(SOME x. x \<noteq> i) \<noteq> i"
Sup_aform': fixes X :: "'a \<times> 'a pdevs"   and p :: "nat" shows "Sup_aform X \<le> Sup_aform' p X"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> C.CCC.arr f" shows "C.\<alpha>' f = C.null"
sumr_sum: fixes R :: "nat set"   and f :: "nat \<Rightarrow> real"   and k :: "nat" shows "(\<Sum>i = 0..<k. if i \<in> R then f i else 0) = sum f (R \<inter> {..<k})"
rep_eq: fixes x :: "'b mat_impl" shows "dim_col_impl x = fst (snd (Rep_mat_impl x))"
word_pred_m1: fixes a :: "'a word" shows "word_pred a = a - 1"
all_security_requirements_fulfilled_imp_no_offending_flows: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" assumes "valid_reqs M"   and "all_security_requirements_fulfilled M G" shows "(\<Union>m\<in>set M. \<Union> (c_offending_flows m G)) = {}"
not_outside_1: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq>[relation] q" shows "p \<in> \<P> outcomes"
cases: fixes R :: "('b \<times> 'b) set"   and a :: "'b list" assumes "Linorder_Relations.sorted_wrt R a" obtains "a = []"   | xs :: "'b list" and x :: "'b" where "a = x # xs" and "Linorder_Relations.sorted_wrt R xs" and "\<And>y. y \<in> set xs \<Longrightarrow> (x, y) \<in> R"
compare: fixes a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and L :: "(real mpoly \<times> nat) list"   and L' :: "(real mpoly \<times> nat) list" assumes "(a, b) = chophelper L L'" shows "chop L = b"
H: shows "Max {i. i < length ns \<and> offs ns 0 ! i < k} \<in> {i. i < length ns \<and> offs ns 0 ! i < k}"
brw_while_algo: fixes \<delta> :: "('a, 'b) ta_rule set"   and Qi :: "'a set" assumes "finite \<delta>" shows "while_algo (brw_algo Qi \<delta>)"
comp_fun_commute: fixes y :: "'b fset set"   and x :: "'b fset set" shows "(\<otimes>) y \<circ> (\<otimes>) x = (\<otimes>) x \<circ> (\<otimes>) y"
simps_2: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> Boolean) = (P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Boolean\<rfloor>)"
v11_vcomp_vconverse: shows "(T\<lparr>ArrVal\<rparr>)\<inverse>\<^sub>\<circ> \<circ>\<^sub>\<circ> T\<lparr>ArrVal\<rparr> = vid_on (\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>))"
cong: fixes q :: "real" shows "fbnorm.N q \<equiv> fbnorm.N q"
ind': assumes "\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"   and "\<forall>i\<le>x. 0 < rec_exec f (xs @ [i])" shows "\<exists>stp>x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
Iff_ltlc_def: fixes \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<phi> iff\<^sub>c \<psi> \<equiv> (\<phi> implies\<^sub>c \<psi>) and\<^sub>c (\<psi> implies\<^sub>c \<phi>)"
simps_2: fixes I :: "('a, 'b, 'c) interp"   and f :: "'a"   and args :: "'c \<Rightarrow> ('a, 'c) trm"   and v :: "(real, 'c) vec" shows "sterm_sem I ($f f args) v = Functions I f (\<chi>i. sterm_sem I (args i) v)"
Suc: shows "higher_differentiable_on {0<..} f k \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
Above_decr: fixes A :: "'a set"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> B" shows "Above r B \<subseteq> Above r A"
varnum_eq_zero_iff: fixes X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat" shows "(varnum X t = 0) = (t \<in> .[X])"
eq_equal: fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.equal equal" shows "(=) \<equiv> equal"
eq_Min_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(m = linorder.Min (\<sqsubseteq>) A) = (m \<in> A \<and> (\<forall>a\<in>A. m \<sqsubseteq> a))"
is_processT3_ST: fixes s :: "'a event list"   and t :: "'a event list"   and P :: "'a process" assumes "s @ t \<in> T P" shows "s \<in> T P"
assms_3: fixes n' :: "nat" assumes "\<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"   and "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "eval c t t' n' \<gamma>"
fe: shows "\<forall>v1 v2. (v1, v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)"
finite_partitions: fixes n :: "nat" shows "finite {p. p partitions n}"
iterate_Suc: fixes n :: "nat"   and F :: "'a \<rightarrow> 'a"   and x :: "'a" shows "iterate (Suc n)\<cdot>F\<cdot>x = F\<cdot>(iterate n\<cdot>F\<cdot>x)"
Cons_5: shows "lookup_pair (filter P xs) k = (let v = lookup_pair xs k in if P (k, v) then v else (0::'b))"
mod_add_right_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
item_rhs_def: fixes x :: "item" shows "item_rhs x = snd (item_rule x)"
inext_nth_closed: fixes I :: "nat set"   and n :: "nat" assumes "I \<noteq> {}" shows "I \<rightarrow> n \<in> I"
induce_subgraph_tail: fixes G :: "('d, 'c) pre_digraph"   and vs :: "'d set" shows "tail (G \<restriction> vs) = tail G"
that: fixes ns :: "nat list" assumes "DirProds (\<lambda>n. Z (ns ! n)) {..<length ns} \<cong> G"   and "\<forall>n\<in>set ns. n \<noteq> 0" shows "thesis"
trace_callee_conv_trace_callee_aux: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('d \<times> 'e) spmf"   and p :: "'e spmf"   and xs :: "('f \<times> 'd) list"   and a :: "'f" shows "trace_callee callee p xs a = trace_callee_aux callee p xs \<bind> (\<lambda>s. map_spmf fst (callee s a))"
ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "ring_hom hom \<equiv> semiring_hom hom"