hcju/isabelleps huggingface dataset:
sel_2: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.one (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x2"
assms: shows "\<A> \<Turnstile> \<Phi> \<Pi> t"
one_final_Seed: fixes p :: "nat" shows "one_final Seed\<^bsub>p\<^esub>"
kuratowski_card: fixes X :: "'b set" assumes "kuratowski X" shows "card {f. CK_nf_pos f} = 7"
simps_3: fixes k :: "nat" shows "\<J> (Suc k) 0 = \<pi> (Suc k) {} (\<I> k)"
card_word: shows "CARD('a word) = 2 ^ LENGTH('a)"
remdups_adj_singleton: fixes xs :: "'b list"   and x :: "'b" assumes "remdups_adj xs = [x]" shows "xs = replicate (length xs) x"
multiplicative_dirichlet_prod'_def: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and f_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and g_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f_prime :: "nat \<Rightarrow> 'a"   and g_prime :: "nat \<Rightarrow> 'a" shows "multiplicative_dirichlet_prod' f g f_prime_power g_prime_power f_prime g_prime \<equiv> multiplicative_function' f f_prime_power f_prime \<and> multiplicative_function' g g_prime_power g_prime"
semiring_norm_169: fixes v :: "num"   and w :: "num"   and z :: "'a" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
countable_set_pmf: fixes p :: "'d pmf" shows "countable (set_pmf p)"
distinct_29: fixes x3 :: "nat"   and x5 :: "nat" shows "Nonce x3 \<noteq> Key x5"
not_less_zero: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
ldropn_eq_LConsD: fixes n :: "nat"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "ldropn n xs = y % ys" shows "enat n < llength xs"
ang_const_o: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "\<not> Col A B P"   and "QCongA a"   and "QCongAnNull a"   and "QCongAnFlat a" shows "\<exists>C. a A B C \<and> A B OS C P"
sees_methods_idemp: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and M :: "char list"   and m :: "staticb \<times> ty list \<times> ty \<times> 'a"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>Mm'. P \<turnstile> D sees_methods Mm' \<and> Mm' M = \<lfloor>(m, D)\<rfloor>"
ideal_prod_mono1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "A \<subseteq> B" shows "A \<diamondsuit>\<^sub>r C \<subseteq> B \<diamondsuit>\<^sub>r C"
bisim_bad: fixes x :: "bool list"   and cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a"   and s :: "bool list \<Rightarrow> bool list option"   and r :: "bool list" shows "rel_spmf (\<lambda>(x, s, bad) (y, s', bad'). bad = bad' \<and> (\<not> bad' \<longrightarrow> x = y)) (exec_gpv (rf_encrypt_bad x) (\<A>2 cipher \<sigma>) (s(x \<mapsto> r), False)) (exec_gpv (rf_encrypt_bad x) (\<A>2 cipher \<sigma>) (s, False))"
path_tail: shows "path AB (ws || r1s || r2s) (r1, r2)"
normalize_dvd_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (normalize a) b = dvd.dvd times a b"
list_gcd_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "gcd_list (x # xs) = gcd x (gcd_list xs)"
cod_simps_3: shows "f.cod.leg1 = f.leg1"
typeII_cube_line_integral_exists_boundary: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
pairwise_cp: fixes q :: "real poly" assumes "q \<in> set (cast_rat_list fs)" shows "algebraic_semidom_class.coprime (coprime_r (cast_rat_list fs)) q"
cr_strm_def: shows "cr_strm \<equiv> \<lambda>x y. x = raw_term y"
RHS4: shows "Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r = a \<odot>\<^bsub>P\<^esub> Cring_Poly.compose R (q [^]\<^bsub>P\<^esub> n) r"
mono: fixes x :: "'a"   and y :: "'a" assumes "Q x"   and "Q y"   and "x \<le> y" shows "f x \<le> f y"
is_Read\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_Read\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = True"
vpeq_s_nt: fixes t :: "'state_t"   and execs2 :: "'dom_t \<Rightarrow> 'action_t list list"   and u :: "'dom_t"   and s :: "'state_t" assumes "precondition (next_state t execs2) (next_action t execs2)"   and "unrelated (current t) u"   and "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s t" shows "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s (step (next_state t execs2) (next_action t execs2))"
Closed_Kernel_axioms: shows "Closed_Kernel \<O>' (set \<O>\<^sub>s)"
eqvtI: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "P \<leadsto>[Rel] Q"   and "Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "perm \<bullet> P \<leadsto>[Rel'] perm \<bullet> Q"
subtermeqI': fixes x :: "('c, 'd) Term.term"   and y :: "('c, 'd) Term.term" assumes "x = y" shows "x \<sqsubseteq> y"
inject: fixes x :: "'a \<Rightarrow> 'b + 'c \<times> 'a"   and ya :: "'a \<Rightarrow> 'b + 'c \<times> 'a" shows "(Prog x = Prog ya) = (x = ya)"
extend_2: shows "Ind (set w\<^sub>2) (sset u)"
p_nonneg: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "0 \<le> p x y n"
set_var_coeff_def: fixes v :: "nat"   and c :: "rat"   and lp :: "(nat, rat) fmap" shows "set_var_coeff v c lp \<equiv> if c = 0 then fmdrop v lp else fmupd v c lp"
EndPhase1_HInv3_nL: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' t"   and "\<not> HInv3_L s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "\<not> HInv3_L s' p q d"
swaprows_mat_def: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "swaprows_mat n k l = mat n n (\<lambda>(i, j). if k = i \<and> l = j \<or> k = j \<and> l = i \<or> i = j \<and> i \<noteq> k \<and> i \<noteq> l then 1::'c else (0::'c))"
inducts: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and prod :: "'a \<times> 'b" assumes "\<And>a b. P (a, b)" shows "P prod"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "g n = (\<Prod>p\<in>prime_factors n. g (p ^ multiplicity p n))"
suppI: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "f x \<noteq> (0::'a)" shows "x \<in> supp f"
Red_I_of_Red_F_subset_F: fixes N' :: "'f set"   and N :: "'f set" assumes "N' \<subseteq> local.Red_F_\<G> N" shows "Red_I_\<G> N \<subseteq> Red_I_\<G> (N - N')"
Pa: fixes ivl :: "float interval" assumes "approx prec a vs = Some ivl" shows "interpret_floatarith a xs \<in>\<^sub>r ivl"
card_of_empty4: fixes A :: "'b set" shows "(|{}| <o |A|) = (A \<noteq> {})"
prod_1: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N" shows "(a l\<rightarrow> b) l\<rightarrow> c \<le> ((b l\<rightarrow> a) l\<rightarrow> c) l\<rightarrow> c"
swap_image_eq: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "a \<in> A"   and "b \<in> A" shows "Fun.swap a b f ` A = f ` A"
run_put_state: fixes s :: "'e"   and m :: "('e, 'd) stateT"   and s' :: "'e" shows "run_state (put_state s m) s' = run_state m s"
nempty_list_append_length_add: fixes l2 :: "'a list"   and l1 :: "'a list"   and l3 :: "'a list" assumes "l2 \<noteq> []" shows "length (l1 @ l3) < length (l1 @ l2 @ l3)"
chambers_share_facet: fixes C :: "'a set"   and v :: "'a"   and z :: "'a set" assumes "chamber C"   and "chamber (insert v z)"   and "z \<lhd> C" shows "z \<lhd> insert v z"
segments_subset_convex_hull_3: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "closed_segment b c \<subseteq> convex hull {a, b, c}"
assms_5: shows "a \<in> hyp2 \<union> S"
gpush_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "gpush_code node_eq_impl node_hash_impl G_impl \<equiv> \<lambda>x xa. let y' = xa in case y' of (xb, xc) \<Rightarrow> (as_push xb (igbgi_acc G_impl x), push_code node_eq_impl node_hash_impl G_impl x xc)"
strict_boundedE: fixes b :: "'t"   and c :: "'t"   and a :: "'t" assumes "ord_term_lin.max b c \<prec>\<^sub>t a" obtains "b \<prec>\<^sub>t a" and "c \<prec>\<^sub>t a"
not_iff_asym: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(\<not> r x y) = asympartp r y x"
id_iso: fixes G :: "('c, 'd) monoid_scheme" shows "id \<in> Group.iso G G"
noNeg: shows "\<not> has_disc_negated disc False (rewrite_neg m)"
sig_merge: fixes q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "q \<in> set (add_spairs ps bs p')" shows "sig_of_pair p \<preceq>\<^sub>t sig_of_pair q"
less_I_def: fixes i :: "I"   and j :: "I" shows "i \<sqsubset> j \<equiv> j = init \<and> (i = loop \<or> i = final) \<or> j = loop \<and> i = final"
simps_2: fixes x :: "'b"   and ys :: "'b list" shows "sorted (x # ys) = (Ball (set ys) ((\<le>) x) \<and> sorted ys)"
lift_comp: fixes g :: "'g \<Rightarrow> 'f"   and f :: "'h \<Rightarrow> 'g"   and inp :: "'e \<Rightarrow> 'h option" shows "lift g (lift f inp) = lift (g \<circ> f) inp"
wf_ci_check_instr': shows "wf_ci check_instr'"
select_convs_1: fixes 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 "cpu_reg \<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> = cpu_reg"
conj_comm: fixes P :: "'d upred"   and Q :: "'d upred" shows "(P \<and> Q) = (Q \<and> P)"
fps_deriv_of_int: fixes n :: "int" shows "fps_deriv (of_int n) = 0"
simps_6: fixes v :: "ty" shows "is_Class (v\<lfloor>\<rceil>) = False"
bddmi_rel_def: fixes cs :: "bdd" shows "bddmi_rel cs \<equiv> {uu_. \<exists>a b c. uu_ = (a, c) \<and> (a, b) \<in> bf_ifex_rel \<and> (c, b) \<in> Rmi cs}"
ext: fixes g :: "'a"   and f :: "'a" assumes "V g f \<noteq> V.null" shows "V.seq g f"
stream_all2_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream"   and ys :: "'b stream" shows "stream_all2 (\<lambda>a b. R b a) xs ys = stream_all2 R ys xs"
minSeqInclu: fixes P :: "'c process"   and S :: "'c process" shows "min_elems (D (P `;` S)) \<subseteq> min_elems (D P) \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t1 \<notin> D P \<and> t2 \<in> min_elems (D S)}"
Pi_iff: fixes f :: "'f \<Rightarrow> 'g"   and I :: "'f set"   and X :: "'f \<Rightarrow> 'g set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
nvalid_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and n :: "nat"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A \<equiv> \<forall>s t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"
ik\<^sub>s\<^sub>t_subterm_exD: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "\<exists>x\<in>set S. t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>t\<^sub>p x"
preserves_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.CC.seq g f" shows "C.CC.map (C.CC.comp g f) = C.CC.comp (C.CC.map g) (C.CC.map f)"
float_power_bnds_def: fixes prec :: "nat"   and n :: "nat"   and l :: "float"   and u :: "float" shows "float_power_bnds prec n l u = (if 0 < l then (power_down_fl prec l n, power_up_fl prec u n) else if odd n then (- power_up_fl prec \<bar>l\<bar> n, if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n) else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n) else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))"
il_inf_left_unit: fixes x :: "'a" shows "top \<sqinter> x = x"
bin_snd_nemp: shows "u\<^sub>1 \<noteq> \<epsilon>"
not_acc_down: fixes x :: "'k"   and R :: "('k \<times> 'k) set" assumes "x \<notin> Wellfounded.acc R" obtains z :: "'k" where "(z, x) \<in> R" and "z \<notin> Wellfounded.acc R"
evals_final: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "finals es'"
right_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<cdot> z = x \<cdot> z \<squnion> y \<cdot> z"
f_def: fixes n :: "nat"   and xoyo :: "'v vertex \<times> 'v vertex" shows "f n xoyo = (case xoyo of (SOURCE, \<langle>y\<rangle>) \<Rightarrow> if y = b then d_IN (h n) b - d_IN (h 0) b else 0 | (SOURCE, _) \<Rightarrow> 0 | (vertex.SINK, \<langle>y\<rangle>) \<Rightarrow> if y \<in> A \<Gamma> then d_OUT (h 0) y - d_OUT (h n) y else 0 | (vertex.SINK, _) \<Rightarrow> 0 | (\<langle>x\<rangle>, SOURCE) \<Rightarrow> 0 | (\<langle>x\<rangle>, vertex.SINK) \<Rightarrow> if x \<in> A \<Gamma> then d_OUT (h n) x - d_OUT (h 0) x else 0 | (\<langle>x\<rangle>, \<langle>y\<rangle>) \<Rightarrow> if edge \<Gamma> x y then h 0 (x, y) - h n (x, y) else if edge \<Gamma> y x then h n (y, x) - h 0 (y, x) else 0)"
very_short_exact_sequence: fixes D :: "'a monoid"   and C :: "'a monoid"   and B :: "'a monoid"   and A :: "'a monoid"   and h :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "exact_seq ([D, C, B, A], [h, g, f])"   and "trivial_group A"   and "trivial_group D" shows "g \<in> Group.iso B C"
fset_eqI: fixes A :: "'c fset"   and B :: "'c fset" assumes "\<And>x. (x |\<in>| A) = (x |\<in>| B)" shows "A = B"
pw_bounded_WHILE_RES_rwof: fixes n :: "nat"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and m0 :: "'c nres"   and x :: "'c" assumes "nofail (bounded_WHILE n cond step m0)"   and "inres (bounded_WHILE n cond step m0) x" shows "rwof m0 cond step x"
MatchP: fixes r :: "formula Regex.regex"   and I :: "\<I>" assumes "safe_regex Past Strict r"   and "\<forall>\<phi>\<in>atms r. P \<phi>" shows "P (MatchP I r)"
cap_positive: fixes e :: "nat \<times> nat" assumes "e \<in> E" shows "(0::'capacity) < c e"
assms_2: fixes b :: "'a" assumes "b \<in> carrier G"   and "properfactor G b a" shows "b \<in> Units G"
LR: fixes m :: "real"   and a :: "real"   and b :: "real" assumes "m \<in> {a..b}"   and "{a..b} \<subseteq> {0..1}" shows "a \<le> leftcut a b m \<and> leftcut a b m \<le> m \<and> m \<le> rightcut a b m \<and> rightcut a b m \<le> b \<and> (\<forall>x\<in>{leftcut a b m..rightcut a b m}. f x = f m) \<and> (\<forall>x\<in>{a..<leftcut a b m}. f x \<noteq> f m) \<and> (\<forall>x\<in>{rightcut a b m<..b}. f x \<noteq> f m) \<and> (\<forall>x\<in>{a..<leftcut a b m}. \<forall>y\<in>{rightcut a b m<..b}. f x \<noteq> f y)"
elims: fixes x :: "nat"   and y :: "real" assumes "bernoulli x = y" obtains "x = 0" and "y = 1"   | n :: "nat" where "x = Suc n" and "y = - 1 / (real n + 2) * (\<Sum>k\<le>n. real (n + 2 choose k) * bernoulli k)"
var_same: fixes v :: "'a"   and t :: "('b, 'a) Term.term" shows "(Var(v := t) = Var) = (t = Var v)"
superclass_nameI: fixes dcl :: "char list"   and cl :: "cl"   and fds :: "fd list"   and meth_defs :: "meth_def list" shows "superclass_name (cld_def dcl cl fds meth_defs) cl"
exec_unit_SE: fixes \<sigma> :: "'d"   and P :: "bool" shows "(\<sigma> \<Turnstile> unit\<^sub>S\<^sub>E P) = P"
indep_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "X_insert.indep_in \<E> X = (X \<subseteq> \<E> \<and> indep_in (insert x X) X)"
inject_3: fixes x3 :: "real mpoly"   and y3 :: "real mpoly" shows "(Leq x3 = Leq y3) = (x3 = y3)"
norm_vec_def: fixes x :: "('a, 'b) vec" shows "norm x = L2_set (\<lambda>i. norm (x $ i)) UNIV"
IH: fixes i :: "nat"   and j :: "nat" assumes "local.sorted xs"   and "i < j"   and "j < length xs" shows "xs ! i \<prec> xs ! j"
assms_3: fixes s :: "'s"   and cfg :: "'s cfg" assumes "P s cfg" shows "action cfg \<in> K s"
indspaceD_range: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> BaseFGMod.indspace G FH V" shows "range f \<subseteq> V"
typeII_chain_line_integral_exists_boundary': fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
transition_eqvt': fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and \<alpha> :: "'d"   and Q :: "'b"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "transition P \<langle>\<alpha>,Q\<rangle>" shows "transition (p \<bullet> P) \<langle>p \<bullet> \<alpha>,p \<bullet> Q\<rangle>"
match_1: fixes h :: "'a"   and g :: "'a"   and f :: "'a" assumes "seq h g"   and "seq (h \<cdot> g) f" shows "seq g f"
natUnion_ex: fixes x :: "'b"   and f :: "nat \<Rightarrow> 'b set" assumes "x \<in> natUnion f" shows "\<exists>i. x \<in> f i"
hcomp_arr_obj: fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr" assumes "S.arr f"   and "S.obj a"   and "S.src f = a" shows "f \<star>\<^sub>S a = f"
is_valid_operator_strips_def: fixes \<Pi> :: "'c strips_problem"   and op :: "'c strips_operator" shows "is_valid_operator_strips \<Pi> op \<equiv> let vs = \<Pi>\<^sub>\<V>; pre = strips_operator.precondition_of op; add = add_effects_of op; del = delete_effects_of op in list_all (\<lambda>v. ListMem v vs) pre \<and> list_all (\<lambda>v. ListMem v vs) add \<and> list_all (\<lambda>v. ListMem v vs) del \<and> list_all (\<lambda>v. \<not> ListMem v del) add \<and> list_all (\<lambda>v. \<not> ListMem v add) del"
select_convs_3: fixes c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c" shows "c_imp \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = c_imp"
le_iff_inf: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x y = (inf x y = x)"
rcis_in_complex_cone: fixes \<alpha> :: "real"   and a :: "real"   and b :: "real"   and r :: "real" assumes "\<alpha> \<in> {a..b}"   and "0 \<le> r" shows "rcis r \<alpha> \<in> complex_cone a b"
elims_2: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat"   and x :: "nat \<times> nat" assumes "TableInv f1 f2 f1' f2' e c l0"   and "TableInv.invar f1 f2 c l0 x" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "l0 \<le> l \<and> (int l0 \<le> \<lfloor>real l / c\<rfloor> \<longrightarrow> f1 * real l \<le> real n) \<and> real n \<le> f2 * real l"
slice_def: fixes Z :: "'c set"   and y :: "'b" shows "slice Z y \<equiv> {x. h (x, y) \<in> Z}"
bounded_linear_Blinfun_apply: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "(*\<^sub>v) (Blinfun f) = f"
effect_to_assignments_simp: fixes op :: "'c strips_operator" shows "effect_to_assignments op = map (\<lambda>v. (v, True)) (add_effects_of op) @ map (\<lambda>v. (v, False)) (delete_effects_of op)"
continuous_on_upperhalf_Arg2pi: shows "continuous_on ({z. 0 \<le> Im z} - {0}) Arg2pi"
span_is_subset2: fixes S :: "'c set" assumes "S \<subseteq> carrier M" shows "span S \<subseteq> carrier M"
ie6: shows "te q (i + 1) \<le> te p (i + 1)"
L2: fixes \<tau> :: "real" assumes "\<tau> \<in> cball t \<epsilon>\<^sub>2 \<inter> T" shows "L2-lipschitz_on (cball s \<epsilon>\<^sub>2 \<inter> S) (f2 \<tau>)"
real_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and real_of :: "'a \<Rightarrow> real"   and x :: "'a"   and z :: "real" assumes "class.real_embedding plus zero minus uminus less_eq less sgn abs times one real_of"   and "real_of x \<le> z" shows "less_eq x (ring_1.of_int one plus zero minus \<lceil>z\<rceil>)"
cname_of_def: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and a :: "'b" shows "heap_base.cname_of typeof_addr h a = the_Class (ty_of_htype (the (typeof_addr h a)))"
unfold: shows "op = (\<Lambda> pat v x. Fixrec.run\<cdot>(match_Pair\<cdot>v\<cdot>(\<Lambda> us vs. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>vs then (us :@ [:x:], stail\<cdot>vs) else If snull\<cdot>us then ([::], pat) else op\<cdot>pat\<cdot>(split\<cdot>pat\<cdot>(stail\<cdot>us))\<cdot>x))))"
leq_dual_def: fixes x' :: "'c dual"   and y' :: "'c dual" shows "x' \<sqsubseteq> y' \<equiv> undual y' \<sqsubseteq> undual x'"
tyenv_sec: shows "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
associativity: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and M1 :: "'a list list"   and M2 :: "'a list list"   and M3 :: "'a list list" assumes "Matrix_Tensor.mult id f"   and "Matrix_Legacy.mat (mult.row_length M1) (length M1) M1 \<and> Matrix_Legacy.mat (mult.row_length M2) (length M2) M2 \<and> Matrix_Legacy.mat (mult.row_length M3) (length M3) M3" shows "mult.Tensor f M1 (mult.Tensor f M2 M3) = mult.Tensor f (mult.Tensor f M1 M2) M3"
grounding_of_cls_empty: shows "grounding_of_cls {#} = {{#}}"