hcju/isabelleps huggingface dataset:
cring_simprules_15: fixes x :: "int"   and y :: "int" shows "x \<ominus> y = x \<oplus> \<ominus> y"
inext_predicate_change_exists2_all: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P a"   and "\<forall>k\<in>I \<down>\<ge> b. P k" shows "\<exists>n\<in>I. a \<le> n \<and> n < b \<and> \<not> P n \<and> (\<forall>k\<in>I \<down>> n. P k)"
BEGIN_def: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and i :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "arelations.BEGIN e m ov d s f i = {j |j. (j, i) \<in> ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>}"
rtrancl_path_simps_2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "'a"   and bs :: "'a list"   and y :: "'a" shows "rtrancl_path R x (a # bs) y = (R x a \<and> rtrancl_path R a bs y)"
abc_bcd_abd: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[b c d]]" shows "[[a b d]]"
conforms_deallocL: fixes s :: "state"   and G :: "prog"   and L :: "lenv"   and vn :: "lname"   and T :: "ty" assumes "s\<Colon>\<preceq>(G, L(vn \<mapsto> T))"   and "L vn = None" shows "s\<Colon>\<preceq>(G, L)"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion>1 y \<squnion>1 z = x \<squnion>1 (y \<squnion>1 z)"
11069: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- - x \<sqinter> y) \<squnion> - x \<sqinter> - z = - (- - x \<sqinter> y)"
subst_cls_list_length: fixes Cs :: "'a literal multiset list"   and \<sigma> :: "'s" shows "length (Cs \<cdot>cl \<sigma>) = length Cs"
zero_in_circline_point_0: shows "0\<^sub>h \<in> circline_set circline_point_0"
flip_conv: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and b :: "'i"   and a :: "'h" shows "flip f b a = f a b"
pr_key_unused: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Key K \<notin> U" shows "analz (insert (Key K) (A \<union> spies evs)) = insert (Key K) (analz (A \<union> spies evs))"
content_ff_1_coeffs_to_fract: fixes p :: "'b fract poly" assumes "content_ff p =dff 1" shows "set (coeffs p) \<subseteq> range to_fract"
fps_divide_add: fixes f :: "'a fps"   and g :: "'a fps"   and h :: "'a fps" shows "(f + g) / h = f / h + g / h"
nn_pp_np_1: fixes f :: "'b \<Rightarrow> 'c" assumes "nonnegative f" shows "pp f = f"
valid_and_I: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<upsilon> x"   and "\<tau> \<Turnstile> \<upsilon> y" shows "\<tau> \<Turnstile> \<upsilon> (x and y)"
help_lemma: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'a, 'e) Combined.configuration_scheme"   and p :: "'d"   and loc0 :: "'a"   and t0 :: "'c"   and loc1 :: "'a"   and t1 :: "'c"   and s2 :: "'b"   and loc2 :: "'a" assumes "dataflow_topology summary results_in"   and "0 < zcount (c_pts (prop_config c p) loc0) t0"   and "dataflow_topology.cri_less_eq summary results_in (loc0, t0) (loc1, t1)"   and "s2 \<in>\<^sub>A graph.path_weight summary loc1 loc2"   and "dataflow_topology.safe summary results_in (prop_config c p)" shows "\<exists>t2\<le>results_in t1 s2. t2 \<in>\<^sub>A frontier (c_imp (prop_config c p) loc2)"
e_tl_elementary: fixes e :: "nat" shows "e_tl e = (if e = 0 then 0 else if pdec1 (e - 1) = 0 then 0 else Suc (prod_encode (pdec1 (e - 1) - 1, pdec22 (e - 1))))"
smap_applicative: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c stream" shows "smap f x = Applicative.ap (pure f) x"
fls_shift_fls_shift_reorder: fixes m :: "int"   and k :: "int"   and f :: "'a fls" shows "fls_shift m (fls_shift k f) = fls_shift k (fls_shift m f)"
msg_sender_def: fixes m :: "msg" shows "msg_sender m \<equiv> case m of Pkt x ipc \<Rightarrow> ipc"
r_leap_prim: shows "prim_recfn 3 r_leap"
path_decomp': fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "Q.path (xs @ x # ys)" shows "Q.path (xs @ [x])"
combine_append_zero: fixes Us :: "'a list"   and Ks :: "'a list" assumes "set Us \<subseteq> carrier R" shows "combine (Ks @ [\<zero>]) Us = combine Ks Us"
zcount_inject: fixes M :: "'a zmultiset"   and N :: "'a zmultiset" shows "(zcount M = zcount N) = (M = N)"
Domainp_eq_Collect: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<UU> :: "'a set" shows "(Domainp A = (\<lambda>x. x \<in> \<UU>)) = (\<UU> = Collect (Domainp A))"
ins2_iff_ins_3: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and n' :: "nat"   and h' :: "nat" shows "(ins2 n d x t = Unbal2 t' n' h') = (ins n d x t = Unbal t' \<and> n' = size t' \<and> h' = height t')"
border_example: shows "{b. border b ''aabaabaa''} = {[], ''a'', ''aa'', ''aabaa'', ''aabaabaa''}"
\<rho>_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<rho> a = runit a"
matrix_mult_sum: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s column i A)"
mk_lossless_eq_return_pmf_None: fixes p :: "'c spmf" shows "(mk_lossless p = return_pmf None) = (p = return_pmf None)"
Compl_eq_Compl_iff: fixes A :: "'d set"   and B :: "'d set" shows "(- A = - B) = (A = B)"
less_appendD: fixes xs :: "nat list"   and ys :: "nat list"   and vs :: "nat list"   and ws :: "nat list" assumes "xs @ ys <\<^sub>v vs @ ws"   and "length xs = length vs" shows "xs <\<^sub>v vs \<or> ys <\<^sub>v ws"
of_complex_inverse: fixes x :: "complex" shows "of_complex (inverse x) = inverse (of_complex x)"
singleActionChain_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'" shows "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"
no_self_channel: shows "\<forall>i. \<nexists>p. channel i = Some (p, p)"
inj_double_dual_iso: shows "inj_on double_dual_iso (carrier G)"
components_are_equivalences: fixes a :: "'a" assumes "C.obj a" shows "D.equivalence_map (\<rho>\<^sub>0 a)"
res_i1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "f (residual f x) \<le> x"
append_Cons_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 x :: "'g"   and xs :: "'g list"   and ys :: "'g list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "x # xs @ ys = (x # xs) @ ys"
hyp2: shows "fst (snd (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<Suc k])) = snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
\<Phi>_comp: fixes g :: "'t"   and f :: "'t" assumes "S'.seq g f" shows "\<Psi> (g \<cdot>\<acute> f) = \<Psi> g \<cdot> \<Psi> f"
nn_integral_eq_integrable: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "0 \<le> x" shows "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x) = (integrable M f \<and> integral\<^sup>L M f = x)"
relative_to_subset: fixes S :: "'c set"   and T :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "S \<subseteq> T \<and> P S" shows "(P relative_to T) S"
pick_eq_iff_inf: fixes S :: "nat set"   and x :: "nat"   and y :: "nat" assumes "infinite S" shows "(x = y) = (pick S x = pick S y)"
swapEnvIm_preserves_good: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "goodEnv (swapEnvIm xs x y rho)"
contg: shows "continuous_map (top_of_set {0..1}) X g"
wf_induct_rule: fixes r :: "('e \<times> 'e) set"   and P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "wf r"   and "\<And>x. (\<And>y. (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
assms_3: shows "\<forall>u\<in>set U. \<exists>y. u = Var y"
1: shows "[\<lbrace>a\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>\<rbrace> in v]"
weak_supplementation: fixes y :: "i"   and x :: "i" assumes "MM3"   and "PP y x" shows "\<exists>z. P z x \<and> \<not> O z y"
no_collisionI: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and S :: "real set" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "\<And>t. t \<in> S \<Longrightarrow> movement.s a\<^sub>e v\<^sub>e s\<^sub>e t \<noteq> movement.s a\<^sub>o v\<^sub>o s\<^sub>o t" shows "\<not> safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o S"
PseudoHA_HAInitStates: fixes A :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "HAInitStates (PseudoHA A D) = {InitState A}"
denot_empty_diverge: fixes e :: "exp" assumes "E e [] = {}"   and "FV e = {}" shows "diverge e \<or> goes_wrong e"
tauInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "\<And>P C. F C (\<tau>.P) P"   and "\<And>P P' c C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C ([c\<frown>c]P) P'"   and "\<And>P P' c d C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'; c \<noteq> d\<rbrakk> \<Longrightarrow> F C ([c\<noteq>d]P) P'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) P'"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto> \<tau> \<prec> Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) Q'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q)"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto> \<tau> \<prec> Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P \<parallel> Q')"   and "\<And>P a x P' Q b Q' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a[b] \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P'[x::=b] \<parallel> Q')"   and "\<And>P a b P' Q x Q' C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q'[x::=b])"   and "\<And>P a x P' Q y Q' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a<\<nu>y> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C; y \<sharp> P; y \<sharp> Q; y \<sharp> C; x \<noteq> y\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>y>(P'[x::=y] \<parallel> Q'))"   and "\<And>P a y P' Q x Q' C. \<lbrakk>P \<longmapsto> a<\<nu>y> \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C; y \<sharp> P; y \<sharp> Q; y \<sharp> C; x \<noteq> y\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>y>(P' \<parallel> Q'[x::=y]))"   and "\<And>P P' x C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; x \<sharp> C; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) (<\<nu>x>P')"   and "\<And>P P' C. \<lbrakk>P \<parallel> !P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C (P \<parallel> !P) P'\<rbrakk> \<Longrightarrow> F C (!P) P'" shows "F C P P'"
2: shows "finite ((\<lambda>(v, a). v $ <a>) ` ({v. llength v = enat n} \<times> UNIV))"
len_Suc_mult2: fixes x :: "nat" shows "len (Suc (2 * x)) = Suc (len x)"
assms_4: shows "mix_pmf \<beta> local.some_best local.some_worst \<approx>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst"
hyps_2: shows "foldl\<cdot>f\<cdot>z\<cdot>xs = foldr\<cdot>f\<cdot>z\<cdot>xs"
append_rows_mult_left_id: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and m :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat 1 n"   and "B \<in> carrier_mat (m - 1) n"   and "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"   and "D \<in> carrier_mat (m - 1) (m - 1)" shows "C * (A @\<^sub>r B) = A @\<^sub>r D * B"
memberid_Ext_foo_simp: shows "memberid (mdecl Ext_foo) = mid foo_sig"
phils_update_take: fixes n :: "nat"   and ps :: "nat list"   and e :: "dining_event" shows "take n (Up\<^sub>P ps e) = Up\<^sub>P (take n ps) e"
unfold1: fixes s1 :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and z :: "'e" shows "dtor1 (unfold1 s1 s2 z) = F1map id (unfold1 s1 s2) (unfold2 s1 s2) (s1 z)"
power2_norm_eq_cinner: fixes x :: "'b" shows "(complex_of_real (norm x))\<^sup>2 = cinner x x"
stopping_time_epred_SCons: fixes M :: "'b measure"   and T :: "'b stream \<Rightarrow> enat"   and x :: "'b"   and \<omega> :: "'b stream" assumes "stopping_time (stream_filtration M) T"   and "x \<in> space M"   and "\<omega> \<in> streams (space M)"   and "0 < T (x ## \<omega>)" shows "stopping_time (stream_filtration M) (\<lambda>\<omega>. epred (T (x ## \<omega>)))"
L274: fixes E :: "'d \<Rightarrow> 'e upred"   and X :: "'e upred"   and Y :: "'e upred" assumes "\<forall>n. (E n \<and> X) = (E n \<and> Y)" shows "(\<Or>range E \<and> X) = (\<Or>range E \<and> Y)"
elims_1: fixes x :: "atom"   and y :: "bool" assumes "is_Less x = y" obtains r :: "real" and rs :: "real list" where "x = Less r rs" and "y"   | v :: "real" and va :: "real list" where "x = Eq v va" and "\<not> y"
assms_3: fixes x :: "'a" assumes "x \<in> X" shows "indep (X - {x})"
overlapping_def: fixes s :: "'a"   and t :: "'a" shows "overlapping s t = (\<exists>u. s \<lesssim> u \<and> t \<lesssim> u)"
I2: shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral (x2 - x1) * c / b ^ n) {x1..x2}"
trivial_limit_at_right_real: fixes x :: "'b" shows "at_right x \<noteq> bot"
add_mset_eq_singleton_iff: fixes x :: "'a"   and M :: "'a multiset"   and y :: "'a" shows "(x, M = y, {#}) = (M = {#} \<and> x = y)"
OstrowskiTr6: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "\<not> 0 \<le> v x" shows "1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x) \<in> carrier K - {\<zero>}"
sum_moebius_mu_divisors: fixes n :: "nat" assumes "1 < n" shows "(\<Sum>d | d dvd n. moebius_mu d) = (0::'b)"
P_new_decomp: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new" shows "P_new = DisjointPathsPlusOne.new_pre P_new @ [DisjointPathsPlusOne.new_last P_new]"
quotient_eq: shows "{f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // range_permutation A B = {F \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B. univ (\<lambda>f. f ` A = B) F}"
simps: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and b :: "'e"   and n :: "nat"   and s :: "'f" shows "list_update_raw g b (n, s) = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip (n, s') | Yield a s' \<Rightarrow> if n = 0 then Yield a (0, s') else if n = 1 then Yield b (0, s') else Yield a (n - 1, s'))"
qball: shows "\<forall>x\<in>proots q. Im x \<noteq> 0"
strongEqWeakCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s\<^sub>e Q" shows "P \<simeq>\<^sup>s Q"
spectrum_def: shows "cring0.spectrum \<equiv> {I. pr_ideal {0} I ring0.subtraction ring0.subtraction 0 0}"
no_leading_Nil: fixes P :: "'f \<Rightarrow> bool" shows "no_leading P []"
continuous_on_norm_id: fixes S :: "'b set" shows "continuous_on S norm"
transpose_zero: fixes A :: "(('d, 'f) vec, 'e) vec" shows "(Finite_Cartesian_Product.transpose A = 0) = (A = 0)"
unique_solution_closed_ivl: fixes X :: "real \<Rightarrow> 'a"   and t :: "real"   and s :: "'a"   and Y :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>t. f (X t)) on {0--t}"   and "X 0 = s"   and "X \<in> {0--t} \<rightarrow> S"   and "t \<in> T"   and " D Y = (\<lambda>t. f (Y t)) on {0--t}"   and "Y 0 = s"   and "Y \<in> {0--t} \<rightarrow> S"   and "s \<in> S" shows "X t = Y t"
cmp_def: shows "THfgh.cmp = THfgh_HHfgh.chine"
simps_3: fixes n :: "nat"   and e :: "String.literal"   and ts :: "Formula.trm list" shows "minit0 n (formula.Pred e ts) = MPred e ts"
subcring_axioms_def: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subcring_axioms H R \<equiv> \<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
is_symbol_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and s :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_symbol \<NN> \<TT> s = (CFG.is_terminal \<TT> s \<or> CFG.is_nonterminal \<NN> s)"
smult_zero_mat: fixes k :: "'c"   and nr :: "nat"   and nc :: "nat" shows "k \<cdot>\<^sub>m 0\<^sub>m nr nc = 0\<^sub>m nr nc"
notf: shows "\<forall>e\<in>{a, b, c}. r f \<noteq> e"
tag_finite_imageD: fixes tag :: "'b list \<Rightarrow> 'a"   and A :: "'b list set" assumes "finite (range tag)"   and "=tag= \<subseteq> \<approx>A" shows "finite (UNIV // \<approx>A)"
inf_idem: fixes x :: "'k multiset" shows "x \<inter># x = x"
idem: fixes a :: "'a" shows "a \<^bold>* a = a"
Says_S_message_form: fixes S :: "agent"   and A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says S A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "evs \<in> bankerberos" shows "K \<notin> range shrK \<and> X = Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<or> X \<in> analz (knows Spy evs)"
exec_gpv_invariant: fixes x :: "'c"   and s' :: "'s option \<times> 's"   and gpv :: "('c, 'a, 'b) gpv"   and s :: "'s option \<times> 's" assumes "(x, s') \<in> set_spmf (exec_gpv (\<lambda>(ss, s) x. map_spmf (\<lambda>(y, s'). (y, if I s' \<and> ss = None then Some s' else ss, s')) (callee s x)) gpv s)"   and "case s of (None, s) \<Rightarrow> True | (Some s', s) \<Rightarrow> f s = f s' \<and> I s' \<and> I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "case s' of (None, s) \<Rightarrow> True | (Some s', s) \<Rightarrow> f s = f s' \<and> I s' \<and> I s"
span_def: fixes S :: "'a vec set" shows "local.span S = {lincomb a A |a A. finite A \<and> A \<subseteq> S}"
assms_2: shows "f ` topspace T1 \<subseteq> \<Union> S"
le_approx3: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q" shows "min_elems (D P) \<subseteq> T Q"
swap_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set" shows "X #[x \<and> y]_xs = asTerm (pick X #[[x \<and> y]]_xs)"
prefix_eq_fin_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a list" assumes "traces ind"   and "w\<^sub>1 \<le> w\<^sub>2"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>2 w\<^sub>3" shows "traces.le_fin ind w\<^sub>1 w\<^sub>3"
axioms_1: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "dynamically_complete_calculus Bot Inf entails Red_I Red_F" shows "calculus Bot Inf entails Red_I Red_F"
add_mono_thms_linordered_field_4: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i \<le> j \<and> k < l" shows "i + k < j + l"
selectlike_equal_cocond: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set"   and Y :: "(variable \<Rightarrow> real) set" shows "(selectlike X \<nu> (- V) = selectlike Y \<nu> (- V)) = (\<forall>\<mu>. Uvariation \<mu> \<nu> V \<longrightarrow> (\<mu> \<in> X) = (\<mu> \<in> Y))"
big_step_determ: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and u :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> t"   and "(c, s) \<Rightarrow> u" shows "u = t"
update_Gamma_remove_call_undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and m :: "'c match_expr"   and foo :: "char list"   and rs' :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list" assumes "\<Gamma>(chain \<mapsto> Rule m (Call foo) # rs'),\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"   and "matches \<gamma> m p" shows "\<Gamma>(chain \<mapsto> rs'),\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"
lowner_is_lub_matrix_sum: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "matrix_seq.lowner_is_lub (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>)) (matrix_seq.lowner_lub (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>)))"
hoare_triple_def: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'d"   and c :: "'d"   and q :: "'d" assumes "kat test not" shows "kat.hoare_triple p c q \<equiv> p \<cdot> c \<le> c \<cdot> q"
emeasure_pmf_single_eq_zero_iff: fixes M :: "'a pmf"   and y :: "'a" shows "(emeasure (measure_pmf M) {y} = 0) = (y \<notin> set_pmf M)"
oalist_inv_raw_update_by_fun_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.oalist_inv_raw comp (comparator.update_by_fun_pair comp k f xs)"
finite_arr: fixes A :: "'a set" assumes "finite A" shows "finite {ar. IArray.length ar \<le> max_array_size \<and> set (IArray.list_of ar) \<subseteq> A}"
p: shows "p = (\<Sum>b\<in>A. q0 b \<odot> b)"
strict_prefI: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "(y, x) \<notin> r" shows "x \<^bsub>r\<^esub>\<prec> y"
usubstr_union_undef2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and A :: "fml list"   and B :: "fml list"   and C :: "fml" assumes "usubstr \<sigma> (A @ B, C) \<noteq> None" shows "usubstr \<sigma> (A, C) \<noteq> None \<and> usubstr \<sigma> (B, C) \<noteq> None"
take_update_swap: fixes m :: "nat"   and xs :: "'d list"   and n :: "nat"   and x :: "'d" shows "take m (xs[n := x]) = (take m xs)[n := x]"
finite_rtrancl_delta_Image1: fixes r :: "'c" assumes "wellformed r" shows "finite ({(r, s). \<exists>a\<in>set alphabet. s = delta a r}\<^sup>* `` {r})"
perm_restrict_dom_subset: fixes f :: "'c \<Rightarrow> 'c"   and A :: "'c set" assumes "has_dom f A" shows "perm_restrict f A = f"
assms_2: shows "v' \<leadsto>(ys @ x # ys')\<leadsto> w'"