hcju/isabelleps huggingface dataset:
abs_inner: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c"   and x :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (abs x) i = \<bar>inner x i\<bar>"
matches_MatchNot_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchNot m) a], Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[], Undecided\<rangle> \<Rightarrow> t"
one_closed: shows "\<zero>\<^bsub>UP R\<^esub> \<in> carrier (UP R)"
flush_append_Prog\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and m :: "nat \<Rightarrow> nat" shows "flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])) m = flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) m"
totient_aux2_def: fixes xs :: "(nat \<times> nat) list" shows "totient_aux2 xs = (\<Prod>(p, k)\<leftarrow>xs. p ^ k * (p - 1))"
alw_alw: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "alw (alw \<phi>) = alw \<phi>"
Rep_Abs_Sync: fixes P :: "'c process"   and Q :: "'c process"   and A :: "'c set" shows "Rep_process (Abs_process ({(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}, {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)})) = ({(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}, {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)})"
sup_inf_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<cdot> z + x = (y + x) \<cdot> (z + x)"
less_irrefl: fixes ko :: "'a key_order"   and x :: "'a" shows "\<not> lt_of_key_order ko x x"
index_map_mat_1: fixes i :: "nat"   and A :: "'d mat"   and j :: "nat"   and f :: "'d \<Rightarrow> 'e" assumes "i < dim_row A"   and "j < dim_col A" shows "map_mat f A $$ (i, j) = f (A $$ (i, j))"
schnorr_base_def: fixes \<G> :: "'a cyclic_group" shows "schnorr_base \<G> \<equiv> prime (order \<G>)"
intros: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> S"   and "\<forall>x'\<in>S. f x \<le> f x'" shows "x \<in> least_map f S"
abs_eq: fixes xa :: "Vars.var set"   and x :: "CoCalls" shows "ccTTree xa x = Abs_ttree (valid_lists xa x)"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"   and "injective_map \<eta> R R'" shows "ring_monomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"
distinct_adj_altdef: fixes xs :: "'a list" shows "distinct_adj xs = (remdups_adj xs = xs)"
finite_measure_count_space: fixes A :: "'d set" assumes "finite A" shows "finite_measure (count_space A)"
TunerMode_CTRL_Delta_def: shows "TunerMode_CTRL_Delta = {(''1'', (En ''Next'', defaultguard, defaultaction), ''2''), (''2'', (En ''Next'', defaultguard, defaultaction), ''3''), (''3'', (En ''Next'', defaultguard, defaultaction), ''4''), (''4'', (En ''Next'', defaultguard, defaultaction), ''1''), (''1'', (En ''Back'', defaultguard, defaultaction), ''4''), (''4'', (En ''Back'', defaultguard, defaultaction), ''3''), (''3'', (En ''Back'', defaultguard, defaultaction), ''2''), (''2'', (En ''Back'', defaultguard, defaultaction), ''1'')}"
lunit_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<mu> \<cdot> \<l>[local.dom \<mu>] = \<l>[cod \<mu>] \<cdot> (trg \<mu> \<star> \<mu>)"
simps_1: fixes uu :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and uv :: "action"   and uw :: "'c" shows "ternary_to_bool_unknown_match_tac uu uv uw TernaryTrue = True"
t_Node_1: fixes p :: "ref" assumes "Dag p low high lt"   and "st \<le> lt" shows "\<exists>stp. Dag stp low high st"
lm124: fixes list :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "set (takeAll P list) = (!) list ` set (filterpositions2 P list)"
confluentp_imp_semiconfluentp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "confluentp r" shows "semiconfluentp r"
substitution_lemma: fixes \<A> :: "'b \<Rightarrow> bool"   and n :: "'b"   and G :: "'b formula"   and F :: "'b formula" shows "\<A> \<Turnstile> F[G/n] = \<A>(n := \<A> \<Turnstile> G) \<Turnstile> F"
par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subset: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and B :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" assumes "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"   and "set B \<subseteq> set A" shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t B Sec"
ordLess_ordLeq_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r <o r'"   and "r' \<le>o r''" shows "r <o r''"
multiset_inter_single: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "(a, {#}) \<inter># (b, {#}) = {#}"
simps_9: fixes f :: "'m \<Rightarrow> 'l"   and x2 :: "'m" shows "map_option f (Some x2) = Some (f x2)"
l7_9_bis: fixes A :: "'p"   and P :: "'p"   and X :: "'p"   and Q :: "'p" assumes "A Midpoint P X"   and "A Midpoint X Q" shows "P = Q"
l_get_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
knows_Cons: fixes A :: "agent"   and ev :: "event"   and evs :: "event list" shows "knows A (ev # evs) = (case ev of Says A' B X \<Rightarrow> insert X (knows A evs) | Gets A' X \<Rightarrow> knows A evs | Notes A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs)"
inout_arcs_arc_simps_3: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d" assumes "e \<in> arcs G" shows "out_arcs G u \<inter> {} = {}"
conjugate_vec_def: fixes v :: "'a vec" shows "conjugate v = vec (dim_vec v) (\<lambda>i. conjugate (v $ i))"
goal1_2: shows "lowest_tops [i, t, e] = Some a_"
M1: shows "u \<in> {u. Le (0::'b) \<preceq> M 0 0 \<and> (\<forall>c. v c \<le> n \<longrightarrow> dbm_entry_val u None (Some c) (M 0 (v c)) \<and> dbm_entry_val u (Some c) None (M (v c) 0)) \<and> (\<forall>c1 c2. v c1 \<le> n \<and> v c2 \<le> n \<longrightarrow> dbm_entry_val u (Some c1) (Some c2) (M (v c1) (v c2)))}"
cur: shows "cur = (if ne + length auxlist = 0 then 0 else \<tau> \<sigma> (ne + length auxlist - 1))"
self_append_conv2: fixes ys :: "'g list"   and xs :: "'g list" shows "(ys = xs @ ys) = (xs = [])"
degen: shows "nr \<noteq> 0 \<or> 0 \<noteq> 0 \<or> (0::'a) \<le> Bnd"
div2_Suc_Suc: fixes m :: "nat" shows "Suc (Suc m) div 2 = Suc (m div 2)"
assert_bound_nolhs_atoms_imply_bounds_index: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and ats :: "('a \<times> 'b atom) set"   and a :: "'a \<times> 'b atom" assumes "AssertBoundNoLhs assert_bound"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "ats \<Turnstile>\<^sub>i \<B>\<I> s" shows "insert a ats \<Turnstile>\<^sub>i \<B>\<I> (assert_bound a s)"
to_bl_word_rev: fixes w :: "'a word" shows "to_bl (word_reverse w) = rev (to_bl w)"
rev_star2: fixes s :: "'b list"   and A :: "'b list set" assumes "s \<in> A\<star>" shows "rev s \<in> (rev ` A)\<star>"
rel_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_option R\<inverse>\<inverse> = (rel_option R)\<inverse>\<inverse>"
snga_assn_def: fixes r :: "'b array"   and x :: "'b list" shows "r \<mapsto>\<^sub>a x = Abs_assn (Assn (\<lambda>h. Array.get (heapOf h) r = x \<and> addrOf h = {addr_of_array r} \<and> addr_of_array r < lim (heapOf h)))"
prefixTauCases: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a prefix \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<alpha>\<cdot>P \<longmapsto> \<tau> \<prec> P'"   and "\<Psi> \<rhd> P \<sim> P' \<Longrightarrow> Prop pTau P'" shows "Prop \<alpha> P'"
iwlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "iwlsFSw MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSwapCls MOD \<and> igAbsCongS MOD"
prems_5: shows "ref \<notin> set (interp_ins ops)"
distinct_9: fixes x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool"   and x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" shows "(x2)\<^sub>\<surd> \<noteq> x41\<hookleftarrow>\<^bsub>x42\<^esub>x43"
MTF_def: shows "MTF = (\<lambda>_. (), \<lambda>s r. ((length (fst s) - 1, []), ()))"
assms_2: shows "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
left_commute: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "abel_semigroup f" shows "f b (f a c) = f a (f b c)"
flatten_pi_2_intersection_eq: fixes N :: "'b ltln set"   and \<phi> :: "'b ltln"   and N' :: "'b ltln set" assumes "N \<inter> subformulas\<^sub>\<nu> \<phi> = N'" shows "\<phi>[N']\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2"
hr_comp_Id2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "hr_comp R Id = R"
card_gt_0_iff: fixes A :: "'g set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> vs1.span S"
MWLsSteps_det_def: shows "MWLsSteps_det \<equiv> {((xa, x), xd, xc, xb). MWLsSteps_detp xa x xd xc xb}"
eq_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "(A ===> A ===> (=)) (=) (=)"
assms_2: shows "lpfp\<^sub>c f c = Some c'"
skip_doesnt_read: fixes annos :: "'Var ModeUpd list"   and x :: "'Var" shows "doesnt_read (Skip \<otimes> annos) x"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) Cod_f.map Cod_g.map Fun_g.map"
assms_2: shows "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"
spec\<^sub>6_def: fixes p :: "prog" shows "spec\<^sub>6 p \<equiv> p = p\<^sub>0"
eq_vecI: fixes w :: "'a Matrix.vec"   and v :: "'a Matrix.vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $ i = w $ i"   and "dim_vec v = dim_vec w" shows "v = w"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "H.ide a"   and "H.seq f a" shows "f \<star> a = f"
bij_betw_finite: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "finite A = finite B"
g_ty: shows "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"
sum: fixes a :: "real"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "a *\<^sub>R sum g A = (\<Sum>x\<in>A. a *\<^sub>R g x)"
rel_conversep: fixes R1 :: "'s \<Rightarrow> 'q \<Rightarrow> bool"   and R2 :: "'t \<Rightarrow> 'r \<Rightarrow> bool" shows "rel_sum R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_sum R1 R2)\<inverse>\<inverse>"
max_min_same_4: fixes y :: "'a"   and x :: "'a" shows "max y (min x y) = y"
distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<cdot> z = x \<cdot> z \<squnion> y \<cdot> z"
pair_sigma_finite_def: fixes M1 :: "'c measure"   and M2 :: "'d measure" shows "pair_sigma_finite M1 M2 \<equiv> sigma_finite_measure M1 \<and> sigma_finite_measure M2"
span_unique: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "module scale"   and "S \<subseteq> T"   and "module.subspace scale T"   and "\<And>T'. \<lbrakk>S \<subseteq> T'; module.subspace scale T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'" shows "module.span scale S = T"
induct: fixes x1 :: "string_ternop + collection_ternop"   and x2 :: "call_kind"   and x3 :: "'a OCL_Types.type"   and x4 :: "'b OCL_Types.type"   and x5 :: "'a OCL_Types.type"   and x6 :: "'a OCL_Types.type"   and P :: "string_ternop + collection_ternop \<Rightarrow> call_kind \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "ternop_type x1 x2 x3 x4 x5 x6"   and "\<And>op \<tau> \<sigma> \<rho> \<upsilon>. string_ternop_type op \<tau> \<sigma> \<rho> \<upsilon> \<Longrightarrow> P (Inl op) DotCall \<tau> \<sigma> \<rho> \<upsilon>"   and "\<And>op \<tau> \<sigma> \<rho> \<upsilon>. collection_ternop_type op \<tau> \<sigma> \<rho> \<upsilon> \<Longrightarrow> P (Inr op) ArrowCall \<tau> \<sigma> \<rho> \<upsilon>" shows "P x1 x2 x3 x4 x5 x6"
triangular_to_jnf_steps_dims_2: fixes A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.step_1 A) = dim_col A"
qp: shows "is_quantum_predicate (adjoint U2 * P * U2)"
one_reducible_lm_1: fixes A :: "nat set" assumes "one_reducible_to univ_ce A" shows "\<not> computable A"
Arctan_in_Reals: fixes z :: "complex" assumes "z \<in> \<real>" shows "Arctan z \<in> \<real>"
nhop_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "ip \<noteq> dip" shows "nhop (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = nhop rt ip"
tyvrs_of_subst: fixes B :: "binding"   and X :: "tyvrs"   and T :: "ty" shows "tyvrs_of (B[X \<mapsto> T]\<^sub>b) = tyvrs_of B"
real_interval_abs: fixes A :: "float interval" shows "real_interval (abs_interval A) = abs_interval (real_interval A)"
simps: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and ta :: "('b, 'c, 'f, 'd, 'e, 'g) thread_action" shows "final_thread.actions_ok final s t ta = (lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> \<and> thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> \<and> final_thread.cond_action_oks final s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> \<and> wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
equiv_subst_var_eq: fixes v :: "nat \<Rightarrow> 'a"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and eq :: "nat \<times> linear_poly" assumes "v \<Turnstile>\<^sub>e (x\<^sub>j, lp')" shows "v \<Turnstile>\<^sub>e eq = v \<Turnstile>\<^sub>e subst_var_eq x\<^sub>j lp' eq"
le_filter_def: fixes F :: "'c filter"   and F' :: "'c filter" shows "(F \<le> F') = (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
abs_eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(abs_fset x = abs_fset y) = list_eq x y"
d: fixes k :: "nat"   and la :: "complex" assumes "(k, la) \<in> set n_as"   and "cmod la = 1" shows "la ^ d la = 1 \<and> d la \<noteq> 0"
unique_connecting_path_properties_3: fixes v :: "'c"   and w :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>"   and "w \<in> V\<^bsub>T\<^esub>" shows "hd (v \<leadsto>\<^bsub>T\<^esub> w) = v"
mult_smult_distrib: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and k :: "'a" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * (k \<cdot>\<^sub>m B) = k \<cdot>\<^sub>m (A * B)"
that: fixes y :: "'a" assumes "y \<in> S"   and "y \<in> T"   and "y \<noteq> x" shows "thesis"
IH2: shows "dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j) \<le> B / L * (exp (L * (t j - t 0) + 1) - 1) * max_stepsize j ^ p"
proj_mcont2mcont: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> cnf llist"   and tCMP :: "'c \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'c \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'c" assumes "mcont lub ord lSup lprefix f" shows "mcont lub ord lSup lprefix (\<lambda>x. dynamic_component.proj tCMP active c (f x))"
mem_default_relI: fixes a :: "'a option"   and b :: "'b"   and d :: "'b"   and R :: "('a \<times> 'b) set" assumes "a = None \<Longrightarrow> b = d"   and "\<And>x. a = Some x \<Longrightarrow> (x, b) \<in> R" shows "(a, b) \<in> \<langle>R\<rangle>default_rel d"
assms_3: shows "eoutKnowsECorrect A (kE m)"
ext: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
monoid_add_hom_axioms: shows "monoid_add_hom (inv (inv hom))"
clm_vd: shows "1 \<le> |restrict vd (clm ts) c|"
cases: fixes g :: "'g"   and a :: "'val" assumes "liveVal g a" obtains n :: "'node" and val :: "'val" where "a = val" and "n \<in> set (\<alpha>n g)" and "val \<in> uses g n"   | v :: "'val" and v' :: "'val" where "a = v'" and "liveVal g v" and "phiArg g v v'"
leg1_simps_3: shows "src s\<^sub>1 = src s\<^sub>0"
starset_n_Un: fixes A :: "nat \<Rightarrow> 'a set"   and B :: "nat \<Rightarrow> 'a set" shows "*sn* (\<lambda>n. A n \<union> B n) = *sn* A \<union> *sn* B"
lns_normD: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" assumes "local_nonsatiation B P" shows "\<forall>x\<in>B. \<forall>e>0. \<exists>y\<in>B. norm (y - x) \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P"
pass1_Nil_iff: fixes hs :: "'b heap list" shows "(pass\<^sub>1 hs = []) = (hs = [])"
linear_eq_on_span: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and x :: "'c" assumes "clinear f"   and "clinear g"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x"   and "x \<in> cspan B" shows "f x = g x"
simps: fixes uu :: "'b extC" shows "uu\<^sup>\<star> = extC.Top"
normalize_coprime_degree: fixes f :: "rat poly"   and g :: "rat poly" assumes "normalize f = normalize g"   and "comm_monoid_mult_class.coprime f g" shows "degree f = 0"
node_deliver_messages_def: fixes cs :: "'msg event list" shows "node_deliver_messages cs \<equiv> List.map_filter (\<lambda>e. case e of Broadcast x \<Rightarrow> Map.empty x | Deliver m \<Rightarrow> Some m) cs"
orderI: fixes a :: "'b upred"   and b :: "'b upred" assumes "a = (a \<and> b)" shows "b \<sqsubseteq> a"
permutation_inverse_works_1: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "inv p \<circ> p = id"
adv: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and \<A> :: "('y + 'z, 'aa + 'ab) distinguisher_obsf"   and adv :: "real" assumes "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv"   and "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>" shows "advantage \<A> (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
distinct_1: fixes x1 :: "('e, 'f) Term.term"   and x2 :: "('e, 'f) Term.term" shows "send\<langle>x1\<rangle> \<noteq> receive\<langle>x2\<rangle>"
\<iota>_is_iso: shows "D.iso \<iota>\<^sub>D"
assms_2: shows "g piecewise_C1_differentiable_on i"
ord_less_eq_trans: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "less a b"   and "b = c" shows "less a c"
eq_iff: fixes a :: "'t"   and b :: "'t" shows "(a = b) = (a \<preceq>\<^sub>t b \<and> b \<preceq>\<^sub>t a)"
zimage_zexplode_eq: fixes f :: "ZF \<Rightarrow> ZF"   and z :: "ZF" shows "zimage f (zexplode z) = zexplode (Repl z f)"
concat_drop_suc_iff: fixes n :: "nat"   and tps :: "'a list list" assumes "Suc n < length tps" shows "concat (drop (Suc n) tps) = tps ! Suc n @ concat (drop (Suc (Suc n)) tps)"
simps_7: fixes C :: "'b"   and \<phi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (InContext C \<phi>) \<sigma> = (case SContexts \<sigma> C of None \<Rightarrow> InContext C (Fsubst \<phi> \<sigma>) | Some C' \<Rightarrow> PFsubst C' (\<lambda>_. Fsubst \<phi> \<sigma>))"
l3_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l3_step2 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xgnx = Some gnx \<and> guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> implInsec A B gnx \<in> ik s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgny, xgnx, xsk}), ik := {implAuth B A \<langle>Number 0, Exp Gen (NonceF (Rb $ ny)), gnx\<rangle>} \<union> ik s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit s\<rparr>}"
min_set_min_product_2: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<otimes>\<^sub>m min_set B = A \<otimes>\<^sub>m B"
subsequence_singleton: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "w [i \<rightarrow> Suc i] = [w i]"