hcju/isabelleps huggingface dataset:
simps_6: fixes U :: "Unary_PCF.type"   and e :: "expr" shows "fv (\<Lambda>\<langle>U\<rangle> e) = fv e"
simps_2: fixes A :: "'a" shows "complement (Neg A) = Pos A"
distinct_5: fixes x11 :: "String.literal"   and x12 :: "typ"   and x41 :: "typ"   and x42 :: "Core.term" shows "Ct x11 x12 \<noteq> Abs x41 x42"
assms: fixes x :: "'a" assumes "x \<in> U" shows "\<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U"
assms_5: shows "Domainp B = (\<lambda>x. x \<in> \<UU>B)"
sub_distrib_pconj_def: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "sub_distrib_pconj prog \<equiv> \<forall>P Q. unitary P \<longrightarrow> unitary Q \<longrightarrow> wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)"
force_enabled: fixes x :: "'a"   and \<omega> :: "'a stream" assumes "K1.enabled x \<omega>" shows "K1.force_enabled x \<omega> = \<omega>"
oalist_inv_update_by_fun_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.oalist_inv rep_key_order (oalist_raw.update_by_fun_raw rep_key_order k f xs)"
basis_in_max_indep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> X"   and "x \<in> \<E> - X" shows "\<not> indep_in \<E> (insert x X)"
fds_nth_one_Suc_0: shows "fds_nth 1 (Suc 0) = (1::'b)"
continuous_on_closed_Collect_le: fixes s :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "continuous_on s f"   and "continuous_on s g"   and "closed s" shows "closed {x \<in> s. f x \<le> g x}"
degree_mod_less: fixes y :: "'b poly"   and x :: "'b poly" assumes "y \<noteq> 0" shows "x mod y = 0 \<or> degree (x mod y) < degree y"
index_row_2: fixes A :: "'a Matrix.mat"   and i :: "nat" shows "dim_vec (Matrix.row A i) = dim_col A"
zero_ffun_def: shows "{}\<^sub>f \<equiv> ffun.Abs_pfun {}\<^sub>p"
tm_pi_def: fixes prec :: "nat" shows "tm_pi prec = (let pi_ivl = the (compute_bound_fa prec Pi []) in TaylorModel (mid pi_ivl)\<^sub>p (centered pi_ivl))"
reach: fixes x :: "C" shows "(\<Squnion>n. C_take n\<cdot>x) = x"
d_7: fixes x :: "'a"   and y :: "'a" shows "ad x + d x \<cdot> ad y = ad x + ad y"
\<rho>'_def: shows "\<rho>' \<equiv> run\<^sub>t (\<delta>\<^sub>m x) q\<^sub>0 w"
assms_1: shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"
simps_6: fixes r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "satisfies_gen r \<AA> (FEx k \<phi>) = (\<exists>P. r k P (Length \<AA>) \<and> satisfies_gen r (Extend k 0 \<AA> P) \<phi>)"
isShortestPath_length_less_V: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "s \<in> V"   and "isShortestPath s p t" shows "length p < card V"
wt_\<delta>_imp_\<delta>_eq_\<epsilon>: fixes s :: "('s, 'v) tm" assumes "wt s = \<delta>" shows "\<delta> = \<epsilon>"
distinct_twice_contradiction: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b"   and j :: "nat" assumes "xs ! i = x"   and "xs ! j = x"   and "i < j"   and "j < length xs" shows "\<not> distinct xs"
sub1: shows "local.sub1 q k l (r, A) = (r'', A'')"
BrnFT_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "BrnFT cl dl \<in> UNpart01 cl dl P"
lsemimodule1: fixes s :: "'b"   and a :: "'a"   and b :: "'a" shows "s \<circ> (a + b) = s \<circ> a + s \<circ> b"
inv_inv: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "category.inv C (category.inv C f) = f"
rel_T_pos_distr_imp: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "(('z, 'n, 'q, 't, 'w, 'aa) T \<times> ('ab, 'o, 'r, 'u, 'x, 'aa) T \<times> ('ac, 'p, 's, 'v, 'y, 'aa) T \<times> 'z \<times> 'ab \<times> 'ac \<times> 'aa) itself"   and tytok_T :: "('z \<times> 'ab \<times> 'ac \<times> 'aa) itself" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_T_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_T"
g1: fixes z :: "real" assumes "0 \<le> z"   and "z * 2 < 1"   and "z * 2 \<notin> s1" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2) within {0..1})"
ntEq: fixes \<Psi> :: "'b"   and x :: "name"   and M :: "'a" shows "\<Psi> \<turnstile> nameTerm x \<leftrightarrow> M = (M = nameTerm x)"
relabel_action_bound: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts" shows "length (filter is_RELABEL p) \<le> 2 * (card V)\<^sup>2"
distinct_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "distinct (linorder.sort_key less_eq f xs) = distinct xs"
joint_W_cmpser: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a set" assumes "W_cmpser G n f"   and "W_cmpser G m g"   and "\<natural>f n \<triangleright> g 0"   and "g 0 \<subset> f n" shows "W_cmpser G (Suc (n + m)) (jointfun n f m g)"
OUTfromVCorrect1_data6: shows "OUTfromVCorrect1 data6"
p_strict_antimono: fixes x :: "real"   and y :: "real" assumes "x < y"   and "delayed_safe_distance.other.t_stop \<le> x" shows "delayed_safe_distance.other.p y < delayed_safe_distance.other.p x"
surjective: fixes r :: "('a, 'b) graph_rec_scheme" shows "r = \<lparr>g_V = g_V r, g_E = g_E r, g_V0 = g_V0 r, \<dots> = graph_rec.more r\<rparr>"
IH: shows "fsafe \<phi> \<longrightarrow> (\<forall>i. dsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (FsubstFO \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (adjointFO I \<sigma> \<nu>) \<phi>))"
scale_eq_0_iff: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and x :: "'e" assumes "vector_space scale" shows "(scale a x = (0::'e)) = (a = (0::'d) \<or> x = (0::'e))"
function_factors_right: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c" shows "(\<forall>x. \<exists>y. g y = f x) = (\<exists>h. f = g \<circ> h)"
ccBinds_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "ccBinds \<Gamma>\<cdot>i = (\<Squnion>v\<mapsto>e\<in>map_of \<Gamma>. ccBind v e\<cdot>i)"
nth_mem: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "xs ! n \<in> ran\<^sub>l xs"
considers_def2: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" shows "considers f iA n = (\<exists>nAs. nforms (f n) = iA # nAs)"
zero_rel: shows "\<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor> \<in> carrier_direct_lim"
tr: shows "(s, a, \<xi>', p') \<in> automaton.trans A"
nonneg_Reals_one_I: shows "(1::'a) \<in> \<real>\<^sub>\<ge>\<^sub>0"
Lcm_Un: 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 Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<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 set"   and B :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)"
valid_5: shows "tmps_distinct ts\<^sub>s\<^sub>b'"
Limit_vempty_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>" shows "[]\<^sub>\<circ> \<in>\<^sub>\<circ> Vset \<alpha>"
prob_eq_0: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M"   and "A \<in> sets M" shows "(Sigma_Algebra.measure M A = 0) = (AE x in M. x \<notin> A)"
x_2: shows "\<forall>y\<in>s. infdist x t \<le> infdist y t"
assms_5: shows "s2 O ns2 \<subseteq> s2"
summable_gf_U': fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "norm z < 1" shows "summable (\<lambda>n. u x y n * real (Suc n) * z ^ n)"
\<phi>_simps_1: shows "arr r\<^sub>0s\<^sub>1.\<phi>"
b: shows "<a>:M \<in> \<parallel><B>\<parallel>"
add_less_le_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b"   and "less_eq c d" shows "less (plus a c) (plus b d)"
closure_eq_rel_interior_eq: fixes S1 :: "'a set"   and S2 :: "'a set" assumes "convex S1"   and "convex S2" shows "(closure S1 = closure S2) = (rel_interior S1 = rel_interior S2)"
hom_eq: fixes f :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and f' :: "'a \<Rightarrow> 'e" assumes "f \<in> hom G H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> f' x = f x" shows "f' \<in> hom G H"
h1: assumes "\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
gAbsRenS_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsRenS MOD \<equiv> \<forall>xs y x s X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> fresh xs y X' \<and> gFresh MOD xs y X' X \<longrightarrow> gAbs MOD xs y (X' #[y \<and> x]_xs) (gSwap MOD xs y x X' X) = gAbs MOD xs x X' X"
tOps: shows "MapOps (\<lambda>k. (simAbs (fst k), snd k)) ((\<Union>a. sim_equiv_class a ` jkbpC) \<times> UNIV) tOps"
Allowed_JN: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "Allowed (JOIN I F) = (\<Inter>i\<in>I. Allowed (F i))"
2: assumes "s\<^sub>1 = s\<^sub>2"   and "s\<^sub>1 \<leadsto> s\<^sub>2'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (s\<^sub>2, s\<^sub>3) \<in> [\<leadsto>]\<^sup>= \<and> (s\<^sub>2', s\<^sub>3') \<in> [\<leadsto>]\<^sup>="
deadlocked'_def: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "deadlocked' s \<equiv> \<forall>t. not_final_thread s t \<longrightarrow> t \<in> deadlocked s"
nrec_pts_delayed: fixes c :: "('b, 't, 'loc, 'c) Combined.configuration_scheme"   and x :: "'loc \<times> 't"   and p :: "'b" assumes "cri.InvGlobNonposImpRecordsNonpos (exchange_config c)"   and "0 < zcount (cri.records (exchange_config c)) x" shows "\<exists>x'. x'\<le>\<^sub>px \<and> 0 < zcount (c_glob (exchange_config c) p) x'"
oghoare_sound: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (pre P) c Q, A"
ub_def: shows "ub = 2 / ln 2 * sqrt x + 8 * ln 2 * x / (ln x)\<^sup>2"
D: fixes a :: "('a, 'b) alpha_rp_scheme"   and b :: "('a, 'b) alpha_rp_scheme" assumes "Q' (a, b)" shows "Q (a, b)"
bij_Abs': shows "bij_betw Abs' {0..<int CARD('a)} UNIV"
r_eq_orig: fixes c :: "char list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "all_chains (no_call_to c) \<Gamma> rs"   and "\<Gamma> c = Some rs" shows "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule MatchAny (Call c)], Undecided\<rangle> \<Rightarrow> t"
root_ptr': shows "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr'"
P_eq_list_filter: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x = Q x" shows "filter P xs = filter Q xs"
conflictFlagOrQEmptyAfterExhaustiveUnitPropagate: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state" shows "let state' = exhaustiveUnitPropagate state in getConflictFlag state' \<or> getQ state' = []"
assms: shows "cupcake_nsLookup static_cenv i = Some (len, b)"
set_partitions_itself: fixes A :: "'a set" assumes "A \<noteq> {}" shows "{A} partitions A"
Invariants_in_sets: fixes A :: "(nat \<Rightarrow> 'a) set" assumes "A \<in> sets Invariants" shows "A \<in> local.P.events"
nofail_SPEC_triv_refine: fixes m :: "'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "nofail m"   and "\<And>x. \<Phi> x" shows "m \<le> SPEC \<Phi>"
n: shows "N = old_concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) Q C)"
llast_lappend_LCons: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" assumes "lfinite xs" shows "llast (lappend xs (LCons y ys)) = llast (LCons y ys)"
mon_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "mon G H = {f \<in> hom G H. inj_on f (carrier G)}"
oprod_congR: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and t :: "('f \<times> 'f) set" assumes "r =o s"   and "Well_order t" shows "t *o r =o t *o s"
preserves_comp_1: fixes f' :: "'a"   and f :: "'a" assumes "A.seq f' f" shows "\<tau> (f' \<cdot>\<^sub>A f) = G f' \<cdot>\<^sub>B \<tau> f"
fwi_step: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and n :: "nat"   and j :: "nat" assumes "(0::'a) \<le> m k k"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "fwi m n k i j i j = min (m i j) (m i k + m k j)"
univ_rw: shows "UNIV = UNIV - {i} \<union> {i}"
card_set_mset_leq: fixes M :: "'b multiset" shows "card (set_mset M) \<le> size M"
graphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "graph G"   and "pre_digraph.digraph_isomorphism G hom" shows "graph (app_iso hom G)"
reads_insert_writes_set_right: fixes getter :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" assumes "reflp getter"   and "transp getter"   and "reads S f h h'" shows "reads (insert getter S) f h h'"
ProjInd_mem_eq1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set"   and S :: "('g, 'h) aGroup_scheme"   and h :: "'f \<Rightarrow> 'f" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "aGroup S"   and "h \<in> aHom (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f)"   and "\<forall>j\<in>I. compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (ProjInd I A f j) h = ProjInd I A f j" shows "h = aI\<^bsub>Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f\<^esub>"
og_mat_size: shows "M_mat_R signs subsets \<in> carrier_mat (length signs) (length signs)"
mult_poly_0_left: fixes q :: "'d poly" shows "0 * q = 0"
shift_bounds_nat_ivl: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..<n + k} = (\<Sum>i = m..<n. g (i + k))"
master_theorem_ltr: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi>" obtains X :: "'b ltln set" and Y :: "'b ltln set" where "X \<subseteq> subformulas\<^sub>\<mu> \<phi>" and "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>" and "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>" and "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)" and "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)"
vclosed_def: shows "vclosed \<equiv> value_pred.pred (\<lambda>\<Gamma>. list_all (\<lambda>(pat, t). closed_except t (fmdom \<Gamma> |\<union>| frees pat))) (\<lambda>_. True) (\<lambda>_. True)"
sshiftr1_eq_signed_drop_bit_Suc_0: shows "sshiftr1 = signed_drop_bit (Suc 0)"
strip_while_idem: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "no_trailing P xs" shows "strip_while P xs = xs"
assms: shows "reduced_row_echelon_form_upt_k A k"
FG_reducible: assumes "of_nat (card G) \<noteq> (0::'a)" shows "\<exists>Us. (\<forall>U\<in>set Us. IrrFinGroupRepresentation G (*) U) \<and> 0 \<notin> set Us \<and> group_ring = (\<Oplus>U\<leftarrow>Us. U)"
X: shows "B = u *\<^sub>R A + (1 - u) *\<^sub>R C"
rbt_height_bheight_if: fixes t :: "('b \<times> color) tree" assumes "invc t"   and "invh t" shows "height t \<le> 2 * bheight t + (if color t = Black then 0 else 1)"
case_optionE: fixes P :: "bool"   and Q :: "'f \<Rightarrow> bool"   and x :: "'f option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'f" where "x = Some y" and "Q y"
simps_1: fixes x :: "'b" shows "Weight_Balanced_Trees.insert x \<langle>\<rangle> = \<langle>\<langle>\<rangle>, (x, 1), \<langle>\<rangle>\<rangle>"
Lab: fixes Env :: "env"   and B :: "lname set"   and c :: "stmt"   and C :: "assigned"   and A :: "assigned"   and l :: "label" assumes "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C"   and "nrm A = nrm C \<inter> brk C l"   and "brk A = rmlab l (brk C)" shows "Env\<turnstile> B \<guillemotright>\<langle>Break l\<bullet> c\<rangle>\<guillemotright> A"
sqrt_nat_floor_def: fixes x :: "nat" shows "sqrt_nat_floor x = root_nat_floor 2 x"
HBL1: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "bprv (subst P \<langle>\<phi>\<rangle> xx)"
neutral_const: fixes A :: "'f set" shows "(\<Prod>_\<in>A. (1::'e)) = (1::'e)"
simps_1: fixes v :: "val" shows "val_of (Val v) = \<lfloor>v\<rfloor>"
cst_nn_integral: fixes f :: "bool stream \<Rightarrow> real"   and c :: "real" assumes "f \<in> borel_measurable (F 0)"   and "\<And>w. 0 \<le> f w"   and "f (sconst True) = c" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal c"
pickEff_def: fixes eff :: "'b \<Rightarrow> 'a \<Rightarrow> 'a fset \<Rightarrow> bool"   and r :: "'b"   and s :: "'a" shows "RuleSystem_Defs.pickEff eff r s \<equiv> if RuleSystem_Defs.enabled eff r s then SOME sl. eff r s sl else the None"
connected_component_eq_eq: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "(connected_component_set S x = connected_component_set S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> connected_component S x y)"
lnth_lmap: fixes n :: "nat"   and xs :: "'b llist"   and f :: "'b \<Rightarrow> 'c" assumes "enat n < llength xs" shows "lnth (lmap f xs) n = f (lnth xs n)"
subset_Diff_insert: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d"   and C :: "'d set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
homotopic_with_compose_continuous_left: fixes p :: "('a \<Rightarrow> 'c) \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'c"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and Z :: "'c set" assumes "homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g"   and "continuous_on Y h"   and "h ` Y \<subseteq> Z" shows "homotopic_with_canon p X Z (h \<circ> f) (h \<circ> g)"
pt_swap_eq: fixes y :: "'a" assumes "pt TYPE('a) TYPE('b)" shows "(\<forall>a b. [(a, b)] \<bullet> y = y) = (\<forall>pi. pi \<bullet> y = y)"
rep_eq: fixes x :: "'a multiset"   and xa :: "'a multiset" shows "count (x + xa) = (\<lambda>a. count x a + count xa a)"
linorder_less_wlog: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.linorder less_eq less"   and "\<And>a b. less a b \<Longrightarrow> P a b"   and "\<And>a. P a a"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
PTrans: shows "P \<Longrightarrow>\<^sub>lb in P''\<rightarrow>a<x> \<prec> P'"
bounded_cbilinear_scaleC: shows "bounded_cbilinear (*\<^sub>C)"
order_lesseq_imp: fixes x :: "'b"   and y :: "'b" shows "(\<forall>z\<le>x. z \<le> y) = (x \<le> y)"
default_secure: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b"   and F :: "('a \<times> 'a) set"   and i :: "'a" assumes "wf_graph G"   and "\<not> sinvar G nP"   and "F \<in> set_offending_flows G nP" shows "(\<not> False \<longrightarrow> i \<in> fst ` F \<longrightarrow> \<not> sinvar G (nP(i := \<bottom>))) \<and> (False \<longrightarrow> i \<in> snd ` F \<longrightarrow> \<not> sinvar G (nP(i := \<bottom>)))"
logic_actual_nec_1: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<A>\<^bold>\<not>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<A>\<phi>]]"
EmptyNotInEquivSet: shows "{} \<notin> calM // RelM"
seqSubstOutput: fixes a :: "name"   and b :: "name"   and P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "a{b}.P[<\<sigma>>] = seq_subst_name a \<sigma>{seq_subst_name b \<sigma>}.(P[<\<sigma>>])"
right_neutral: fixes a :: "'r" shows "a + (0::'r) = a"
ring_irreducible_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" shows "ring_irreducible\<^bsub>R\<^esub> a = (a \<noteq> \<zero>\<^bsub>R\<^esub> \<and> irreducible R a)"
amSec_def: fixes c :: "('test, 'atom, 'choice) cmd" shows "amSec c = (\<forall>s1 s2 n t. s1 \<approx> s2 \<longrightarrow> T.prob (c, s1) {bT \<in> space (T (c, s1)). snd (((c, s1) ## bT) !! n) \<approx> t} = T.prob (c, s2) {bT \<in> space (T (c, s2)). snd (((c, s2) ## bT) !! n) \<approx> t})"
enorm_exI': fixes Z :: "'a \<Rightarrow> bool"   and P :: "assn"   and Q :: "'a \<Rightarrow> assn" assumes "\<And>x. Z x \<longrightarrow> (P \<Longrightarrow>\<^sub>A Q x)" shows "(\<exists>x. Z x) \<longrightarrow> (P \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x)"
select_convs_2: fixes role :: "role_t"   and owner :: "agent"   and partner :: "agent"   and more :: "'a" shows "owner \<lparr>role = role, owner = owner, partner = partner, \<dots> = more\<rparr> = owner"