hcju/isabelleps huggingface dataset:
pdevs_of_list_sum: fixes xs :: "'a list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "distinct xs"   and "e \<in> UNIV \<rightarrow> I" obtains f :: "'a \<Rightarrow> real" where "f \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list xs) = (\<Sum>P\<in>set xs. f P *\<^sub>R P)"
rhs: shows "maintained (translation e\<^sub>L, translation (A_Int e\<^sub>L e\<^sub>R)) G"
invertible_right_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "u \<cdot> local.inverse u = \<one>"
simps_1: fixes xs :: "'h list"   and opt :: "('f, 'g, 'h) env option" shows "??.Nested_Environment.update_option xs opt None = (if xs = [] then opt else None)"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "leq1 a b"   and "leq1 b c" shows "leq1 a c"
exp_minus: fixes x :: "'b" shows "exp (- x) = inverse (exp x)"
pi_iteratei: fixes S :: "('b, 'c) assoc_list" shows "proper_it (lm.iteratei S) (lm.iteratei S)"
spanning_subset_independent: fixes B :: "'d set"   and A :: "'d set" assumes "B \<subseteq> A"   and "cindependent A"   and "A \<subseteq> cspan B" shows "A = B"
simps_2: fixes k :: "nat"   and s :: "'b"   and t :: "'b" shows "alternating_list (Suc k) s t = alternating_list k s t @ [if gcd_nat.greater_eq k 2 then s else t]"
sdrop_stl: fixes n :: "nat"   and s :: "'d stream" shows "sdrop n (stl s) = stl (sdrop n s)"
fps_neg_nth: fixes f :: "'a fps"   and n :: "nat" shows "- f $ n = - (f $ n)"
right_neutral: fixes a :: "'a" shows "inf a top_class.top = a"
lookup_update_some: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and e :: "('a, 'b, 'c) env"   and env' :: "('a, 'b, 'c) env" assumes "lookup env xs = Some e" shows "lookup (update xs (Some env') env) xs = Some env'"
subst_type_add_degenerate_instance: fixes idx :: "variable"   and s :: "String.literal set"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" assumes "(idx, s) \<notin> set (map fst insts)" shows "subst_typ insts T = subst_typ (((idx, s), Tv idx s) # insts) T"
pp_init_f_def: shows "pp_init_f \<equiv> \<lambda>(u, v). if u = s then c (u, v) else (0::'capacity)"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" shows "((a1, a2) \<in> encoding.indRelR Enc) = (\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S))"
eq'_1: fixes p :: "nat" assumes "prime p" shows "multiplicity p s1 = multiplicity p s2"
borderD_neq: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x \<noteq> w"
Act_Basic_6: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<diamond>\<phi>) in v]"
word_less_alt: fixes a :: "'a word"   and b :: "'a word" shows "(a < b) = (uint a < uint b)"
mult_ac_2: fixes a :: "'h"   and b :: "'h" shows "a * b = b * a"
eq_i_o_def: shows "eq_i_o \<equiv> \<lambda>x. pred.Pred (Predicate.eq x)"
fourier_unity_root: fixes k :: "nat"   and f :: "nat \<Rightarrow> complex"   and m :: "nat" assumes "0 < k" shows "poly (finite_fourier_poly' f k) (unity_root k (int m)) = (\<Sum>n<k. complex_of_real (1 / real k) * (\<Sum>m<k. f m * unity_root k (- int n * int m)) * unity_root k (int (m * n)))"
eqfelem_imp_iff: fixes x :: "'a"   and y :: "'a"   and A :: "'a fset" assumes "x = y" shows "(x |\<in>| A) = (y |\<in>| A)"
Liouville_weak: fixes f :: "complex \<Rightarrow> complex"   and l :: "complex"   and z :: "complex" assumes "f holomorphic_on UNIV"   and "(f \<longlongrightarrow> l) at_infinity" shows "f z = l"
lossless_exec_gpv: fixes gpv :: "('c, 'a, bool list) gpv"   and s :: "'a set \<times> ('a \<Rightarrow> bool list option)" assumes "lossless_gpv \<I>_full gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>_full; case s of (s, s_h) \<Rightarrow> s = dom s_h\<rbrakk> \<Longrightarrow> lossless_spmf (hash_oracle'' s out)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "case s of (s, s_h) \<Rightarrow> s = dom s_h" shows "lossless_spmf (exec_gpv hash_oracle'' gpv s)"
monomials_add: shows "monomials (p1 + p2) = insert m (monomials p1)"
inf_ident: fixes x :: "'a" shows "x \<sqinter> \<top> = x"
Max_less_iff: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(linordered_monoid.Max A \<prec> x) = (\<forall>a\<in>A. a \<prec> x)"
l_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "o[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
strict_trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<sqsubset> a"   and "c \<sqsubset> b" shows "c \<sqsubset> a"
Inl_in_sum_iff: fixes a :: "hf"   and A :: "hf"   and B :: "hf" shows "(HF.Inl a \<^bold>\<in> A + B) = (a \<^bold>\<in> A)"
k'_def: shows "threshold \<phi> (suffix i w) \<G>' = Some k'"
abrupt_if_True_None: fixes x :: "abopt" shows "abrupt_if True x None = x"
S_transform_observable_transition_iff: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" shows "P \<Rightarrow>{S_action.Act \<alpha>}\<^sub>S P' = P \<Rightarrow>{\<alpha>} P'"
step_preserves_inst: fixes g :: "'g"   and u :: "'node \<Rightarrow> 'val set"   and p :: "'node \<times> 'val \<Rightarrow> 'val list option" assumes "i.inst' g (u, p)"   and "CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs (uses(g := u__, g := u)) (phis(g := p__, g := p)) g" shows "i.inst' g (i.step g (u, p))"
rreq_rrep_sn_simps_1: fixes hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "rreq_rrep_sn (Rreq hops rreqid dip dsn dsk oip osn sip) = (1 \<le> osn)"
R2: fixes p :: "nat"   and xs :: "pratt list" shows "valid_cert (Prime p # xs) = (1 < p \<and> valid_cert xs \<and> (\<exists>a. [a ^ (p - 1) = 1] (mod p) \<and> Triple p a (p - 1) \<in> set xs))"
Pz_nn: fixes x :: "'e" assumes "x \<in> space P" shows "0 \<le> Pz x"
countable_type_imp_count_space: fixes t :: "pdf_type" assumes "countable_type t" shows "stock_measure t = count_space (type_universe t)"
c1_on_open_euclidean_congI: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x" shows "c1_on_open_euclidean g f' Y"
assms_2: shows "x \<notin> r \<down>s S"
meet_col_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and P :: "Points" assumes "meet_in (line A B) (line C D) P" shows "col A B P"
qp1: shows "is_quantum_predicate (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1))"
simps: fixes s :: "16 word"   and e :: "16 word"   and p_p :: "16 word" shows "simple_match_port (s, e) p_p = (p_p \<in> {s..e})"
S_edge_reachable: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set"   and u :: "'a"   and v :: "'a" assumes "Prim_Invar1_loc w g r A"   and "u \<in> Prim.S r A"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges (Prim.rg g r)"
\<delta>_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
PR1: fixes p :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! p"   and "\<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! q"
fold_true_fold_false_not_invalid: fixes l :: "trilean list" assumes "fold (\<or>?) l true = true" shows "fold (\<or>?) (rev l) false \<noteq> invalid"
Disj_CI: fixes B :: "fm"   and H :: "fm set"   and A :: "fm" assumes "insert (Neg B) H \<turnstile> A" shows "H \<turnstile> A OR B"
assms: fixes a :: "'a state" shows "pre_post (mk a) = a"
inter_aform_plane_def: fixes prec :: "nat"   and Xs :: "'a \<times> 'a pdevs"   and sctn :: "'a sctn" shows "inter_aform_plane prec Xs sctn = inter_aform_plane_ortho_nres prec Xs (normal sctn) (pstn sctn) \<bind> (\<lambda>cxs. case cxs of None \<Rightarrow> SUCCEED | Some cxs \<Rightarrow> if normal sctn \<in> set Basis_list then ASSERT (index Basis_list (normal sctn) < DIM('a)) \<bind> (\<lambda>_. RETURN (project_ncoord_aform cxs (index Basis_list (normal sctn)) (pstn sctn))) else if - normal sctn \<in> set Basis_list then ASSERT (index Basis_list (- normal sctn) < DIM('a)) \<bind> (\<lambda>_. RETURN (project_ncoord_aform cxs (index Basis_list (- normal sctn)) (- pstn sctn))) else SUCCEED)"
spec_imp_SafeStickyNrecVacantUpto_aux: fixes s :: "('c, 'd) configuration stream" assumes "alw next s" shows "alw SafeStickyNrecVacantUpto s"
psubstInp_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and inp :: "'h \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodEnv rho"   and "goodInp inp" shows "goodInp inp %[rho]"
gcd_neg1: 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 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 uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (uminus a) b = gcd a b"
wf_ci_stk_offerD: fixes ci :: "('a, 'b) check_instr"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and stk' :: "'a val list" assumes "ci_app ci ins P h stk loc C M pc frs" shows "ci_app ci ins P h (stk @ stk') loc C M pc frs"
mkAutoSim_simps_1: fixes aOps :: "('e, 'c, 'd list) MapOps"   and tOps :: "('f, 'c \<times> 'b, 'c) MapOps"   and simObs :: "'g \<Rightarrow> 'c \<Rightarrow> 'b"   and simInit :: "'g \<Rightarrow> 'b \<Rightarrow> 'c"   and simTrans :: "'g \<Rightarrow> 'c \<Rightarrow> 'c list"   and simAction :: "'g \<Rightarrow> 'c \<Rightarrow> 'd list"   and frontier :: "'g \<Rightarrow> 'c list"   and a :: "'g" shows "pInit (mkAlgAuto aOps tOps simObs simInit simTrans simAction frontier a) = simInit a"
assms_1: shows "sig_red_zero (\<preceq>\<^sub>t) (set bs) r"
Gromov_converging_at_boundary_converges: fixes u :: "nat \<Rightarrow> 'b" assumes "Gromov_converging_at_boundary u" shows "\<exists>x\<in>Gromov_boundary. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"
P: fixes n :: "nat" obtains "n = 0"   | "n = 1"   | "1 < n"
real_transfer_operator_cong: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "AE x in M. f x = g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. real_transfer_operator f x = real_transfer_operator g x"
rep_eq: fixes x :: "'b poly" shows "Rep_ratfps (ratfps_of_poly x) = to_fract x"
Send: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
ar_c_top: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "c_top (b_ag R) I (a \<uplus>\<^bsub>R\<^esub> I) (b \<uplus>\<^bsub>R\<^esub> I) = (a \<plusminus> b) \<uplus>\<^bsub>R\<^esub> I"
disc_rfr_proc_borel_adapted: fixes r :: "real" shows "borel_adapt_stoch_proc G (disc_rfr_proc r)"
real_circle_sgn_r: fixes H :: "circline"   and a :: "complex"   and r :: "real" assumes "is_circle H"   and "(a, r) = euclidean_circle H" shows "sgn r = - circline_type H"
aform_val_uminus_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" shows "aform_val e (uminus_aform X) = - aform_val e X"
rev_it_def: fixes xs :: "'a list" shows "rev_it xs = rev_it' [] xs"
elims: fixes x :: "nat set"   and xa :: "nat"   and xb :: "nat"   and y :: "nat" assumes "fresh2 x xa xb = y" obtains xs :: "nat set" and x :: "nat" and y :: "nat" where "xa__ = xs" and "xa = x" and "xb = y" and "ya__ = (if x \<notin> xs \<or> infinite xs then x else if y \<notin> xs then y else fresh2 xs (x - 1) (y + 1))"
pcr_cr_eq: shows "pcr_fmap (=) (=) = cr_fmap"
one_right_assertion: fixes x :: "'a" assumes "x \<in> assertion" shows "x * (1::'a) = x"
update_convs_10: fixes undef' :: "bool \<Rightarrow> bool"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "undef_update undef' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef' undef, \<dots> = more\<rparr>"
hnorm_triangle_ineq2: fixes a :: "'a star"   and b :: "'a star" shows "hnorm a - hnorm b \<le> hnorm (a - b)"
simps_2: fixes l :: "'a list"   and i :: "nat"   and M :: "mon" shows "Imon l (Minj i M) = Imon (drop i l) M"
simps_1: fixes bs :: "real list" shows "Ifm bs T = True"
card_grab: fixes N :: "nat set"   and n :: "nat" assumes "infinite N" shows "card (fst (grab N n)) = n"
ccpo_Sup_below_iff: fixes Y :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) Y" shows "(\<Squnion> Y \<le> x) = (\<forall>y\<in>Y. y \<le> x)"
le_F_T: fixes P :: "'d process"   and Q :: "'d process" assumes "P \<sqsubseteq>\<^sub>F Q" shows "P \<sqsubseteq>\<^sub>T Q"
continuous_map_pairwise: fixes Z :: "'b topology"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'b \<Rightarrow> 'c \<times> 'd" shows "continuous_map Z (prod_topology X Y) f = (continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f))"
WWa_def: fixes D :: "('c, 'd) Order_scheme"   and g :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "WWa D g a = {W. Wa D W g a}"
cod_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "cod (mkarr t) = mkarr (Cod t)"
keys_idx_pm_of_pm_subset: fixes xs :: "'d list"   and f :: "'d \<Rightarrow>\<^sub>0 'c" shows "keys (idx_pm_of_pm xs f) \<subseteq> {0..<length xs}"
in_arcs_imp_in_arcs_ends: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs sG" shows "(fst e, snd e) \<in> parcs sG"
env_restr_fun_upd: fixes x :: "'a"   and S :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and v :: "'b" assumes "x \<in> S" shows "m1(x := v) f|` S = (m1 f|` S)(x := v)"
mcont_ltake: shows "mcont (prod_lub Sup lSup) (rel_prod (\<le>) (\<sqsubseteq>)) lSup (\<sqsubseteq>) (\<lambda>(x, y). ltake x y)"
assms_2: shows "x\<^sup>2 = 1 + D * y\<^sup>2"
simps_1: fixes w :: "int" shows "bl_to_bin_aux [] w = w"
replicate_eqI: fixes xs :: "'c list"   and n :: "nat"   and x :: "'c" assumes "length xs = n"   and "\<And>y. y \<in> set xs \<Longrightarrow> y = x" shows "xs = replicate n x"
elims_2: fixes x :: "atom set \<times> 'a"   and xa :: "atom set \<times> 'a" assumes "x \<approx>abs_res xa" obtains bs :: "atom set" and xb :: "'a" and cs :: "atom set" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<exists>p. (bs, xb) \<approx>res (=) supp p (cs, y)"
zero_left: fixes b :: "'c" shows "blinfun_apply 0 b = (0::'b)"
parts_cut: fixes Y :: "msg"   and X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "Y \<in> parts (insert X G)"   and "X \<in> parts H" shows "Y \<in> parts (G \<union> H)"
GRep_def: shows "GRep = FinGroupRepresentation G fgsmult"
ExecMsgLen: shows "length execMsg - 1 = length traceMsg"
mL: shows "dfa_is_node (binop_dfa f A B) m"
from_nat_to_nat: fixes x :: "'b" shows "from_nat (to_nat x) = x"
exp_of_derived_is_subgroup: fixes H :: "'a set"   and n :: "nat" assumes "subgroup H G" shows "subgroup ((derived G ^^ n) H) G"
equiv_msgrel_iff: fixes x :: "freemsg"   and y :: "freemsg" shows "(msgrel `` {x} = msgrel `` {y}) = (x \<sim> y)"
iso_sym: fixes H :: "('a, 'b) monoid_scheme" assumes "DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n) \<cong> H" shows "H \<cong> DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)"
inject_7: fixes x71 :: "int"   and x72 :: "MIR.num"   and y71 :: "int"   and y72 :: "MIR.num" shows "(Mul x71 x72 = Mul y71 y72) = (x71 = y71 \<and> x72 = y72)"
wellformed_policy1_charn: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1 p"   and "DenyAll \<in> set p" shows "\<exists>p'. p = DenyAll # p' \<and> DenyAll \<notin> set p'"
optimize_no_cond_sol_sat: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat"   and x :: "rat vec" assumes "rat_LP A b c m n"   and "optimize_no_cond A b c = Inr x" shows "x \<in> sat_primal A b"
evaluate_iff_sym_1: fixes env :: "v sem_env"   and st :: "'a state"   and e :: "exp0"   and r :: "'a state \<times> (v, v) result" shows "evaluate True env st e r = (eval env e st = r)"
higher_deriv_linear: fixes n :: "nat"   and c :: "'a" shows "(deriv ^^ n) ((*) c) = (\<lambda>z. if n = 0 then c * z else if n = 1 then c else (0::'a))"
annir: fixes x :: "'a" shows "\<bottom> \<cdot> x = \<bottom>"
generate_incl: fixes H :: "(nat \<Rightarrow> nat) set" assumes "H \<subseteq> carrier (alt_group n)" shows "generate (alt_group n) H \<subseteq> carrier (alt_group n)"
cnt_Y_pres: fixes xa :: "'a" assumes "count (mset (x # xs)) xa < count (mset (y # ys)) xa"   and "xa \<noteq> y" shows "count (mset xs) xa < count (mset ys) xa"
closure_image_closed_sum: fixes U :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "bounded_linear U" shows "closure (U ` (A +\<^sub>M B)) = closure (U ` A) +\<^sub>M closure (U ` B)"
P_f: shows "{x \<in> space M. P (f x)} \<in> sets M"
zero_lin_dep: fixes S :: "'c set" assumes "\<zero>\<^bsub>M\<^esub> \<in> S"   and "carrier R \<noteq> {\<zero>}" shows "lin_dep S"
strongBisimSubstWeakBisimSubst: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim>\<^sub>s Q" shows "\<Psi> \<rhd> P \<approx>\<^sub>s Q"
set_2: fixes x21 :: "'g"   and x22 :: "'g list" shows "set (x21 # x22) = insert x21 (set x22)"
vimage_algebra_def: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'d measure" shows "vimage_algebra X f M = sigma X {f -` A \<inter> X |A. A \<in> sets M}"
FF_def: shows "UP.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (UP (fst \<mu>\<nu>), UP (snd \<mu>\<nu>)) else VV.null"
ceiling_log_nat_eq_if: fixes b :: "nat"   and n :: "nat"   and k :: "nat" assumes "b ^ n < k"   and "k \<le> b ^ (n + 1)"   and "2 \<le> b" shows "\<lceil>log (real b) (real k)\<rceil> = int n + 1"
evaln_eval: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and n :: "nat"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)" shows "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"
Collapse_dfa_def: shows "Collapse_dfa = \<lparr>states = HF ` states M // eq_right_lang, init = HF (eq_right_lang `` {init M}), final = {HF (eq_right_lang `` {q}) |q. q \<in> final M}, nxt = \<lambda>Q x. HF (\<Union>q\<in>hfset Q. eq_right_lang `` {nxt M q x})\<rparr>"
simps_86: fixes f1 :: "'a"   and f2 :: "'b basic_type \<Rightarrow> 'a"   and f3 :: "'b basic_type \<Rightarrow> 'a"   and f4 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f5 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f6 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f7 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f8 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f9 :: "String.literal \<rightharpoonup>\<^sub>f 'b OCL_Types.type \<Rightarrow> 'a"   and x6 :: "'b OCL_Types.type" shows "(case OrderedSet x6 of OclSuper \<Rightarrow> f1 | x[1] \<Rightarrow> f2 x | x[?] \<Rightarrow> f3 x | Collection x \<Rightarrow> f4 x | Set x \<Rightarrow> f5 x | OrderedSet x \<Rightarrow> f6 x | Bag x \<Rightarrow> f7 x | Sequence x \<Rightarrow> f8 x | Tuple x \<Rightarrow> f9 x) = f6 x6"
gfinprod_eq_finprod: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "finite A"   and "f \<in> A \<rightarrow> carrier G" shows "gfinprod G f A = finprod G f A"
eq_implies_lang: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "ltl_equivalence eq"   and "eq \<phi> \<psi>" shows "\<phi> \<sim>\<^sub>L \<psi>"
meldUniq_mset: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_to_multiset (meldUniq q q') = queue_to_multiset q + queue_to_multiset q'"
preserves_dom: fixes f :: "'j" assumes "J.arr f" shows "A_B.dom (cones_map f \<chi> f) = \<chi>'.A.map (J.dom f)"
sum_list_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero \<equiv> monoid_list.F plus zero"
card_image_le: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A" shows "card (f ` A) \<le> card A"
wls_swapAbs_ident: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and xs :: "'varSort"   and x :: "'var" assumes "wlsAbs (us, s) A" shows "A $[x \<and> x]_xs = A"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> w"
h3c: shows "MPoly_Type.degree p2 var = 0 \<or> MPoly_Type.degree p2 var = 1"
less_pp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'"   and "rep_nat_pp x < rep_nat_pp y" shows "less x y"
augment_saturate: fixes e :: "nat \<times> nat"   and edges :: "(nat \<times> nat) set" assumes "e \<in> edges" shows "e \<notin> Graph.E (cf.augment_cf edges (cf e))"
arr_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.arr \<l>[a]"
cases: fixes x :: "tm \<times> tm \<times> tm \<times> tm \<times> tm \<times> tm" obtains l :: "name" and s :: "tm" and k :: "tm" and v :: "tm" and i :: "tm" and sl :: "name" and sl' :: "name" and m :: "name" and n :: "name" and sm :: "name" and sm' :: "name" and sn :: "name" and sn' :: "name" and t :: "tm" and u :: "tm" where "atom l \<sharp> (s, k, v, i, sl, sl', m, n, sm, sm', sn, sn')" and "atom sl \<sharp> (s, v, i, sl', m, n, sm, sm', sn, sn')" and "atom sl' \<sharp> (s, v, i, m, n, sm, sm', sn, sn')" and "atom m \<sharp> (s, n, sm, sm', sn, sn')" and "atom n \<sharp> (s, sm, sm', sn, sn')" and "atom sm \<sharp> (s, sm', sn, sn')" and "atom sm' \<sharp> (s, sn, sn')" and "atom sn \<sharp> (s, sn')" and "atom sn' \<sharp> s" and "x = (v, i, t, u, s, k)"