hcju/isabelleps huggingface dataset:
scopep_scope_eq: fixes ps :: "'vertex \<times> ('form, 'var) antecedent" shows "scopep ps = (\<lambda>x. x \<in> scope ps)"
val_LI_noninf: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "j \<le> n" shows "LI K (\<nu>\<^bsub>K P j\<^esub>) I \<noteq> \<infinity>"
sub_sup_closed: fixes x :: "'b"   and y :: "'b" shows "- x \<squnion> - y = - - (- x \<squnion> - y)"
ESS_1: shows "(edges g\<^sub>1)\<^sup>* \<subseteq> (edges (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2)))\<^sup>*"
ipath_sim2: fixes r :: "nat \<Rightarrow> 'v" assumes "ipath G'.E r" shows "ipath E (fi \<circ> r)"
cdf_cts_step_2: fixes M :: "real measure"   and x :: "real"   and y :: "real" assumes "real_distribution M"   and "x < y" shows "integral\<^sup>L M (cts_step x y) \<le> cdf M y"
fst_snd_flip: fixes xy :: "'b \<times> 'c" shows "fst xy = (snd \<circ> (\<lambda>(x, y). (y, x))) xy"
divide_strict_right_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero c" shows "less (divide a c) (divide b c)"
recurrence_time_flip_sign_1: fixes T :: "real"   and x :: "'a" assumes "T \<in> existence_ivl0 x"   and "flow0 x T = x" shows "- T \<in> existence_ivl0 x"
lad_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "lad_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. card (f C) \<le> card (f B))"
size_add_mset: fixes a :: "'b"   and A :: "'b multiset" shows "size (add_mset a A) = Suc (size A)"
measure_Diff_null_set: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> sets M"   and "B \<in> null_sets M" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A"
simps_7: fixes v :: "Cooper.num" shows "not (Ge v) = NOT (Ge v)"
lnth_0: fixes x :: "'b"   and xs :: "'b llist" shows "lnth (LCons x xs) 0 = x"
distinct_count_atmost_1': fixes xs :: "'b list" shows "distinct xs = (\<forall>a. count (mset xs) a \<le> 1)"
apply_rsp': fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'b"   and y :: "'d" assumes "rel_fun R1 R2 f g"   and "R1 x y" shows "R2 (f x) (g y)"
preserves_reflects_arr: fixes f :: "'a" shows "Right_a.arr (Right_a.R f) = Right_a.arr f"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Cod \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = Cod t \<^bold>\<otimes> Cod u \<^bold>\<otimes> Cod v"
gso_oc_projection_span_2: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')" shows "gs.is_oc_projection (gs2.gso i) (gs.span (gs2.gso ` {0..<i})) (map of_int_hom.vec_hom fs' ! i)"
simps_6: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and x2 :: "'i" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
cases: 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 a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "char list" and D :: "char list" where "a1 = Class C__" and "a2 = Class D" and "P \<turnstile> Path C__ to D unique"   | C :: "char list" where "a1 = NT" and "a2 = Class C"
sel: fixes x :: "'a \<Rightarrow> 'b \<times> 'a" shows "runState (State x) = x"
isnonconstant_coefficients_length: fixes p :: "'b poly" assumes "isnonconstant p" shows "1 < length (coefficients p)"
prognosis_reorder: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityPrognosisShape prognosis"   and "map_of \<Gamma> = map_of \<Delta>" shows "prognosis ae as u (\<Gamma>, e, S) = prognosis ae as u (\<Delta>, e, S)"
eq_inf_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.eq_inf ind w\<^sub>1 w\<^sub>2 \<equiv> traces.le_inf ind w\<^sub>1 w\<^sub>2 \<and> traces.le_inf ind w\<^sub>2 w\<^sub>1"
sup_state_opt_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' s"
intros: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and lc\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "strong_low_bisim_mm \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "lc\<^sub>1 \<approx> lc\<^sub>2"
2: shows "cod\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> Y) = Y"
arctan_converges: fixes x :: "real" assumes "norm x < 1" shows "(\<lambda>n. arctan_coeffs n * x ^ n) sums arctan x"
assms_2: shows "f ` topspace X \<subseteq> S"
jkbpC_represents_uniquely: fixes T :: "'s Trace set" assumes "SPR.represents T" shows "T = SPR.jkbpC"
sum_equals_pmf_expectation: fixes p :: "'a pmf" assumes "p \<in> \<P> outcomes" shows "(\<Sum>z\<in>outcomes. pmf p z * u z) = measure_pmf.expectation p u"
compl_le_swap2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and y :: "'b"   and x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq (uminus y) x" shows "less_eq (uminus x) y"
pr_SN: shows "SN {(fn, gm). pr_strict fn gm}"
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'ma"   and M' :: "'ma" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> bind (ASSERT \<Phi>) (\<lambda>_. M')"
flush_update_other'': fixes a :: "nat"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and v :: "nat" assumes "a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb"   and "a \<notin> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb" shows "flush sb (m(a := v)) = (flush sb m)(a := v)"
degree_charpoly: shows "degree (charpoly A) = degree (\<Prod>i\<in>UNIV. to_fun (\<^bold>X - \<^bold>C A) i i)"
th: shows "row i (\<chi>k. if k = i then row j A else row k A) = row j (\<chi>k. if k = i then row j A else row k A)"
eval_list_conj_Uni: fixes A :: "atomUni fmUni list"   and x :: "real" shows "evalUni (list_conj_Uni A) x = (\<forall>l\<in>set A. evalUni l x)"
\<delta>_def: shows "\<delta> = real_of_ereal (min (ereal \<epsilon> / 2) (fps_conv_radius f / 2))"
LeftDerivationLadder_cut_prefix_propagate: fixes \<delta> :: "symbol list"   and \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and L' :: "(nat \<times> nat \<times> nat) list"   and D' :: "(nat \<times> symbol \<times> symbol list) list"   and index :: "nat" assumes "LeftDerivationLadder (\<delta> @ \<alpha>) D L \<gamma>"   and "is_word \<delta>"   and "derivation_ge D (length \<delta>)"   and "length \<delta> \<le> ladder_i L 0"   and "L' = ladder_cut_prefix (length \<delta>) L"   and "D' = derivation_shift D (length \<delta>) 0"   and "index < length L" shows "LeftDerivation \<alpha> (take (ladder_n L' index) D') (ladder_\<gamma> \<alpha> D' L' index) \<and> ladder_\<alpha> (\<delta> @ \<alpha>) D L index = \<delta> @ ladder_\<alpha> \<alpha> D' L' index \<and> ladder_\<gamma> (\<delta> @ \<alpha>) D L index = \<delta> @ ladder_\<gamma> \<alpha> D' L' index"
sets_measure_spmf: fixes p :: "'c spmf" shows "sets (measure_spmf p) = sets (count_space UNIV)"
l_sort_subset: fixes Y :: "'b list \<times> 'b list \<times> 'b list"   and X :: "'b list \<times> 'b list \<times> 'b list" assumes "Y \<in> l_sort_set X" shows "l_sort_set Y \<subseteq> l_sort_set X"
euler_approx_aux_def: fixes n :: "nat" shows "euler_approx_aux n = (\<Sum>k\<le>n. \<Prod>{k + 1..n})"
nth_eq_iff_index: fixes xs :: "'b list"   and n :: "nat"   and i :: "'b" assumes "distinct xs"   and "n < length xs" shows "(xs ! n = i) = (n = index xs i)"
lset_lnth: fixes x :: "'a"   and xs :: "'a llist" assumes "x \<in> lset xs" shows "\<exists>n. lnth xs n = x"
bit_sshiftr_word_iff: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "bit (w >>> m) n = bit w (if LENGTH('b) - m \<le> n \<and> n < LENGTH('b) then LENGTH('b) - 1 else m + n)"
negGorbit_list_def: fixes gs :: "'g list"   and T :: "'b \<Rightarrow> 'v"   and as :: "'b list" shows "negGorbit_list gs T as \<equiv> map (\<lambda>g. map ((*\<cdot>) (- g) \<circ> T) as) gs"
IH_7: assumes "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G - set X \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
valid_path_linepath: fixes a :: "'b"   and b :: "'b" shows "valid_path (linepath a b)"
suma_assoc_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and K :: "'p"   and L :: "'p"   and M :: "'p" assumes "SAMS A B C D E F"   and "SAMS D E F G H I"   and "A B C D E F SumA A' B' C'"   and "D E F G H I SumA D' E' F'"   and "A B C D' E' F' SumA K L M" shows "A' B' C' G H I SumA K L M"
add_ac_3: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b + (a + c) = a + (b + c)"
small_big_trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> O[F](h)" shows "f \<in> o[F](h)"
valid_Act: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and \<alpha> :: "'c"   and x :: "('d, 'b, 'c) Formula.formula" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid satisfies transition P (Formula.Act \<alpha> x) = (\<exists>\<alpha>' x' P'. Formula.Act \<alpha> x = Formula.Act \<alpha>' x' \<and> transition P \<langle>\<alpha>',P'\<rangle> \<and> nominal_ts.valid satisfies transition P' x')"
isomorphic_implies_hpar_1: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f f'" shows "Span.ide f"
g_eq: fixes a :: "real" shows "\<integral>\<^sup>+ x. ennreal (f x * indicat_real {..a} x) \<partial>lborel = ennreal (g a)"
eint_add_left_cancel_le: fixes a :: "eint"   and b :: "eint"   and c :: "eint" shows "(a + b \<le> a + c) = (a = \<infinity> \<or> b \<le> c)"
lens_plus_sub_assoc_1: fixes X :: "'d \<Longrightarrow> 'g"   and Y :: "'e \<Longrightarrow> 'g"   and Z :: "'f \<Longrightarrow> 'g" shows "X +\<^sub>L Y +\<^sub>L Z \<subseteq>\<^sub>L (X +\<^sub>L Y) +\<^sub>L Z"
arityOf_lt_var_def: fixes uu :: "'a"   and Delta :: "('b, 'c, 'd, 'e, 'f, 'g) signature_scheme" shows "arityOf_lt_var uu Delta \<equiv> \<forall>delta. wlsOpSym Delta delta \<longrightarrow> |{i. arityOf Delta delta i \<noteq> None}| <o |UNIV|"
T_vrestr_composed_2: fixes A :: "'a set"   and m :: "nat" assumes "A \<in> sets M" shows "T --` (T ^^ m) --` A = (T ^^ (m + 1)) --` A"
generate_idem'_right: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "generate G (A \<union> generate G B) = generate G (A \<union> B)"
step: fixes y :: "V" assumes "y \<in> elts x" shows "Transset (TC y)"
add_diff_cancel_left': fixes plus :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and minus :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "class.cancel_ab_semigroup_add (plus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (minus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(minus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) ((plus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type)) a = b"
distinct_10: fixes ty1' :: "ty"   and ty2' :: "ty" shows "ty1' \<rightarrow> ty2' \<noteq> tBOOL"
unit_simps_5: fixes a :: "'c" assumes "C.obj a" shows "D.cod (F.unit a) = F a"
st_red_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "st_red S t = (\<exists>t' p. subterm t p t' \<and> root_term S t' \<and> trm_rep t' S \<noteq> t')"
i_UN_set_def: shows "i_UN_set \<equiv> iFROM_UN_set \<union> iTILL_UN_set \<union> iIN_UN_set \<union> iMOD_UN_set \<union> iMODb_UN_set"
list_fset_rel_def: fixes xs :: "'b list"   and A :: "'b fset" shows "list_fset_rel xs A = (fset_of_list xs = A)"
\<Phi>_ide: fixes a :: "'t" assumes "S'.ide a" shows "\<Psi> a = \<Psi>o a"
nat_power_1_left: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.nat_power one times plus zero nat_power" shows "nat_power (Suc 0) x = one"
matchT_MT_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchT_MT c d \<equiv> \<forall>s t s'. s \<approx> t \<and> (c, s) \<rightarrow>t s' \<longrightarrow> (\<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t')"
s_def: shows "s = lcm b (lcm (fpxs_root_order f) (fpxs_root_order g))"
CFG_semantics_wf_axioms: shows "CFG_semantics_wf sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main sem identifies"
M_var_inst_cl: shows "is_TComp_var_instance_closed \<Gamma> M"
focus_Int: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and Y :: "'c set"   and X :: "'c set" assumes "p \<in> P[Y]" shows "focus (X \<inter> Y) p = focus X p"
ind: assumes "abc_notfinal (abc_steps_l (0, lm) A n) A" shows "abc_steps_l (0, lm) (A @ B) n = abc_steps_l (0, lm) A n"
neg: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and \<phi> :: "'c" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"   and "\<phi> \<in> fmla" shows "Syntax_with_Connectives_False.neg imp fls \<phi> \<in> fmla"
fixpt_False: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "\<not> fst (iter M T)" shows "fixpt M T = T"
exec_moves_append: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and es' :: "'addr expr1 list" assumes "exec_moves ci P t es h s ta h' s'" shows "exec_moves ci P t (es @ es') h s ta h' s'"
da_weakenE: fixes Env :: "env"   and B :: "lname set"   and t :: "Term.term"   and A :: "assigned"   and B' :: "lname set" assumes "Env\<turnstile> B \<guillemotright>t\<guillemotright> A"   and "B \<subseteq> B'" obtains A' :: "assigned" where "Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'" and "nrm A \<subseteq> nrm A'" and "\<And>l. brk A l \<subseteq> brk A' l"
P2_ideal_real_eq: fixes x1 :: "'a"   and x0 :: "'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" assumes "x1 \<in> carrier \<G>"   and "x0 \<in> carrier \<G>" shows "P2_real_model (x0, x1) \<sigma> z \<A> = P2_ideal_model (x0, x1) \<sigma> z \<A>"
cartesian_product: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and B :: "'c set"   and A :: "'b set" shows "(\<Sum>x\<in>A. sum (g x) B) = (\<Sum>(x, y)\<in>A \<times> B. g x y)"
Right_Coset_unit: shows "K |\<cdot> \<one> = K"
mcont_ltl: shows "mcont lSup (\<sqsubseteq>) lSup (\<sqsubseteq>) ltl"
linear_componentwise_iff: fixes f' :: "'b \<Rightarrow> 'c" shows "linear f' = (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"
extr_Chan: fixes c :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and H :: "chan set"   and bad :: "agent set"   and IK :: "msg set" assumes "Chan c A B M \<in> H"   and "c = insec \<or> c = auth \<or> A \<in> bad \<or> B \<in> bad" shows "M \<in> extr bad IK H"
ring_simprules_10: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
phi_2: shows "\<phi> piecewise_C1_differentiable_on {0..1}"
set_sum: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and K :: "'c set" assumes "aGroup A"   and "H \<subseteq> carrier A"   and "K \<subseteq> carrier A" shows "H \<minusplus>\<^bsub>A\<^esub> K = {x. \<exists>h\<in>H. \<exists>k\<in>K. x = h \<plusminus>\<^bsub>A\<^esub> k}"
Red_F_Bot: fixes B :: "'a"   and N :: "'a set" assumes "B \<in> Bot"   and "N \<Turnstile> {B}" shows "N - Red_F N \<Turnstile> {B}"
induct: fixes P :: "('d \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'd list \<Rightarrow> 'd list \<Rightarrow> bool"   and a0 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d list"   and a2 :: "'d list" assumes "\<And>uu_ xs. P uu_ xs []"   and "\<And>rel v va. P rel [] (v # va)"   and "\<And>rel x xs y ys. \<lbrakk>x = y \<Longrightarrow> P rel xs ys; \<lbrakk>x \<noteq> y; rel x y\<rbrakk> \<Longrightarrow> P rel xs (y # ys); \<lbrakk>x \<noteq> y; \<not> rel x y\<rbrakk> \<Longrightarrow> P rel (x # xs) ys\<rbrakk> \<Longrightarrow> P rel (x # xs) (y # ys)" shows "P a0 a1 a2"
P1: shows "Bet A Q A' \<and> Cong Q A' A Q"
wlp_invD: fixes G :: "'b \<Rightarrow> bool"   and body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and I :: "'b \<Rightarrow> real"   and s :: "'b" assumes "wlp_inv G body I" shows "\<guillemotleft> G \<guillemotright> s * I s \<le> wlp body I s"
Red_Red_I_def: fixes N :: "'f set" shows "reduc_calc.Red_Red_I N = Red_Red_I N \<union> Inf_between UNIV (Red_F N)"
Cons_3: shows "check_no_overlap \<gamma> (a # ft)"
proj_image: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" shows "Equiv_Relations.proj r ` A = A // r"
cert_approx: fixes pc :: "nat" assumes "pc < length \<tau>s"   and "c ! pc \<noteq> \<bottom>" shows "c ! pc = \<tau>s ! pc"
t_ne_2: shows "x t_ne \<noteq> y t_ne"
swap_simps_1: fixes a :: "name"   and b :: "name" shows "[(a, b)] \<bullet> a = b"
in_components_maximal: fixes c :: "'a set"   and s :: "'a set" shows "(c \<in> components s) = (c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c))"
T_ge_half_k: shows "k div 2 \<le> T"
assms_3: shows "reduced_run q v\<^sub>1 ([a] @- v\<^sub>2) l w w\<^sub>1 w\<^sub>2 u"
less_eq_linord_helper_def: fixes a :: "('b, 'c) linord_helper"   and b :: "('b, 'c) linord_helper" shows "(a \<le> b) = linord_helper_less_eq1 a b"
Mf_factorization_m: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.factorization_m m f (poly_mod.Mf m cfs) = poly_mod.factorization_m m f cfs"
ftf_syn1: fixes a :: "'b event"   and u :: "'b event list"   and t :: "'b event list"   and s :: "'b event list"   and A :: "'b event set" shows "a \<notin> set u \<and> a \<notin> set t \<and> s setinterleaves ((t, u), A) \<longrightarrow> a \<notin> set s"
infinite: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "infinite A" shows "abel_semigroup_set.F f z g A = z"
ch_all_betw_f: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and yy :: "'a"   and z :: "'a"   and X :: "'a set"   and y :: "'a" assumes "[f[x .. yy ..  z]X]"   and "y \<in> X"   and "y \<noteq> x"   and "y \<noteq> z" shows "[[x y z]]"
sum_impl_correct: fixes S :: "'a set"   and gi :: "'a \<Rightarrow> 'b nres"   and g :: "'a \<Rightarrow> 'b" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> gi x \<le> (spec r. r = g x)" shows "sum_impl gi S \<le> (spec r. r = sum g S)"
lossless_exec_gpv: fixes \<I> :: "('b, 'c) \<I>"   and gpv :: "('d, 'b, 'c) gpv"   and callee :: "'e \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'e) spmf"   and s :: "'e" assumes "lossless_gpv \<I> gpv"   and "\<And>s out. out \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (callee s out)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s. \<I> \<turnstile>c callee s \<surd>" shows "lossless_spmf (exec_gpv callee gpv s)"
wtl_Suc: fixes pc :: "nat"   and "is" :: "'b list"   and c :: "'a list"   and s :: "'a" assumes "pc + 1 < length is"   and "wtl (take pc is) c 0 s \<noteq> \<top>" shows "wtl (take (pc + 1) is) c 0 s = wtc c pc (wtl (take pc is) c 0 s)"
atomize_conj: fixes A :: "bool"   and B :: "bool" shows "(A &&& B) \<equiv> A \<and> B"
project_aHom: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and j :: "'c" assumes "\<forall>k\<in>I. aGroup (A k)"   and "j \<in> I" shows "\<pi>\<^bsub>I,A,j\<^esub> \<in> aHom (a\<Pi>\<^bsub>I\<^esub> A) (A j)"
semiring_normalization_rules_28: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "'c"   and q :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times (power.power one times x q) x = power.power one times x (Suc q)"
fmdrop_empty: fixes a :: "'c" shows "fmdrop a {$$} = {$$}"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(b \<squnion> c \<le> a) = (b \<le> a \<and> c \<le> a)"
cong_add_rcancel_nat: fixes x :: "nat"   and a :: "nat"   and y :: "nat"   and n :: "nat" shows "[x + a = y + a] (mod n) = [x = y] (mod n)"