hcju/isabelleps huggingface dataset:
NF_eq: fixes t :: "lam" shows "NF t = (\<forall>t'. \<not> t \<rightarrow>\<^sub>\<beta> t')"
WT_gpv_outs_gpv: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>"
zor_poly_pole_eq: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "is_pole f z" shows "\<forall>\<^sub>F w in at z. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
factorize_polys_square_free: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and f :: "rat poly" assumes "factorize_polys ps = (fs, data)"   and "f \<in> set fs" shows "square_free f"
disc_15: fixes x1 :: "('d, 'e) Term.term" shows "\<not> stateful_strand_step.is_Equality (send\<langle>x1\<rangle>)"
Iff_E1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A"   and "insert B H \<turnstile> C" shows "insert (A IFF B) H \<turnstile> C"
bdd_above_insert: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and A :: "'c set" assumes "class.lattice inf less_eq less sup" shows "preorder.bdd_above less_eq (insert a A) = preorder.bdd_above less_eq A"
isomorphic_implies_hpar_1: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>B f'" shows "B.ide f"
evalRR_simps_5: fixes x :: "'a" assumes "x \<in> carrier R" shows "eval R R id x \<one>\<^bsub>P\<^esub> = \<one>"
psubsetD: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subset> B"   and "c \<in> A" shows "c \<in> B"
bernoulli_gen_filtration: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "Filtration.filtration N G"
cases: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and a1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)"   and a2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)"   and dma :: "'b \<Rightarrow> Sec" assumes "sifum_security stop eval some_val some_val'"   and "(a1, a2) \<in> sifum_security.mm_equiv dma eval" obtains \<R> :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set" and lc\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" and lc\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" where "a1 = lc\<^sub>1" and "a2 = lc\<^sub>2" and "sifum_security.strong_low_bisim_mm dma eval \<R>" and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>"
elims: fixes x :: "(int \<times> int) \<times> int \<times> int"   and xa :: "(int \<times> int) \<times> int \<times> int"   and xb :: "int"   and xc :: "(int \<times> int) list"   and y :: "(int \<times> int) \<times> int \<times> int" assumes "combine x xa xb xc = y" obtains p\<^sub>0\<^sub>L :: "int \<times> int" and p\<^sub>1\<^sub>L :: "int \<times> int" and p\<^sub>0\<^sub>R :: "int \<times> int" and p\<^sub>1\<^sub>R :: "int \<times> int" and l :: "int" and ps :: "(int \<times> int) list" where "x = (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L)" and "xa = (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R)" and "xb = l" and "xc = ps" and "y = (let (c\<^sub>0, c\<^sub>1) = if dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in Let (filter (\<lambda>p. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps) (find_closest_pair (c\<^sub>0, c\<^sub>1)))"
id_dom_cod_2: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Cod\<^bsub>CC\<^esub> (Id\<^bsub>CC\<^esub> A) = A"
abs_not_less_zero: fixes abs :: "'a \<Rightarrow> 'a"   and 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 a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "\<not> less (abs a) zero"
assms: shows "x ^ 3 + y ^ 3 = z ^ 3"
nd_def: fixes F :: "'a formula" shows "nd F \<equiv> {} \<turnstile> F"
N2_proj_subset: fixes l :: "'lbl"   and A :: "('lbl strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list" shows "N2 (proj_unl l A) \<subseteq> N2 (unlabel A)"
simps_4: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list"   and x1 :: "'b" shows "Fun x21 x22 \<noteq> Var x1"
StepCondFalse: fixes b :: "expr"   and s :: "char list \<Rightarrow> val option"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" assumes "interpret b s = Some false" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> \<langle>if (b) c\<^sub>1 else c\<^sub>2,s,0\<rangle> \<leadsto> \<langle>c\<^sub>2,s,#:c\<^sub>1 + 1\<rangle>"
mod_by_0: fixes a :: "'a" shows "a mod (0::'a) = a"
linord_of_listP_rev: fixes z :: "'a"   and zs :: "'a list"   and xs :: "'a list"   and y :: "'a" assumes "z # zs \<in> set (subseqs xs)"   and "y \<in> set zs" shows "linord_of_listP z y (rev xs)"
nrename_Cut: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and M :: "trm"   and z :: "name"   and N :: "trm" assumes "R[x\<turnstile>n>y] = Cut <c>.M z.N"   and "c \<sharp> (N, R)"   and "z \<sharp> (x, y, M, R)" shows "\<exists>M' N'. R = Cut <c>.M' z.N' \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N \<and> c \<sharp> N' \<and> z \<sharp> M'"
T: shows "(\<exists>T1. \<Gamma> \<turnstile> e1 : T1) \<and> (\<exists>T2. \<Gamma> \<turnstile> e2 : T2)"
f3: fixes n :: "nat"   and na :: "nat" shows "u' = u \<or> indexlt n na k u \<or> \<not> indexlt n na k' u'"
neqE: fixes x :: "complex"   and y :: "complex" assumes "x \<noteq> y" obtains "x <\<^sub>\<complex> y"   | "y <\<^sub>\<complex> x"
distinct_11: fixes x7 :: "fm" shows "T \<noteq> NOT x7"
simps_7: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
power_float_interval_real_interval: fixes x :: "real"   and X :: "float interval"   and n :: "nat"   and prec :: "nat" assumes "x \<in>\<^sub>r X" shows "x ^ n \<in>\<^sub>r power_float_interval prec n X"
find_csas_lemma_onezero_gen: fixes qs :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list"   and csa :: "rat list" assumes "factorize_polys qs = (fs, data)"   and "fss = cast_rat_list fs" shows "(csa \<in> consistent_sign_vectors_r fss UNIV \<and> \<not> has_no_zeros csa) = (csa \<in> set (find_sgas_aux fss))"
ks_def: shows "ks \<equiv> SOME ks. alw (\<lambda>ks. (shd (stl ks), shd ks) \<in> Ord) ks \<and> alw (ev (\<lambda>ks. shd (stl ks) \<noteq> shd ks)) ks"
vanishesI: fixes X :: "nat \<Rightarrow> rat" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r" shows "Real.vanishes X"
DERIV_power_series': fixes R :: "real"   and f :: "nat \<Rightarrow> real"   and x0 :: "real" assumes "\<And>x. x \<in> {- R<..<R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x ^ n)"   and "x0 \<in> {- R<..<R}"   and "0 < R" shows "((\<lambda>x. \<Sum>n. f n * x ^ Suc n) has_real_derivative (\<Sum>n. f n * real (Suc n) * x0 ^ n)) (at x0)"
l8_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C" shows "Per C B A"
simps_3: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and b :: "bool" shows "typeof\<^bsub>h\<^esub> (Bool b) = \<lfloor>Boolean\<rfloor>"
top_xor: fixes x :: "'g" shows "\<top> \<oplus> x = - x"
foldSetD_imp_finite: fixes A :: "'e set"   and x :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "(A, x) \<in> foldSetD D f e" shows "finite A"
filter_M_pure_I: fixes xs :: "'l list"   and P :: "'l \<Rightarrow> ('m, 'n, bool) prog"   and h :: "'m" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "pure (filter_M P xs) h"
redT_updWs_def: fixes t :: "'e" shows "redT_updWs t = rtrancl3p (redT_updW t)"
simps_4: fixes p :: "perm"   and \<alpha> :: "'d"   and t :: "('b, 'c, 'd) Tree" shows "p \<bullet> tAct \<alpha> t = tAct (p \<bullet> \<alpha>) (p \<bullet> t)"
wt_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "0 < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s))"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_loop1 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "inv_loop1_loop n (l, r) \<or> inv_loop1_exit n (l, r)"
fcompD: fixes a :: "V"   and c :: "V"   and r :: "V"   and s :: "V" assumes "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s" shows "\<exists>b. [b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<and> [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> s"
assms_3: shows "F \<in> hom unity a \<rightarrow> hom unity b"
padl_image: fixes U :: "'b vec set"   and n :: "nat"   and m :: "nat" assumes "U \<subseteq> carrier_vec n" shows "(@\<^sub>v) (0\<^sub>v m) ` U \<subseteq> carrier_vec (m + n)"
ltl_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "ltl (ldropn n xs) = ldropn n (ltl xs)"
mset_single_iff: fixes xs :: "'b list"   and x :: "'b" shows "(mset xs = {#x#}) = (xs = [x])"
open_exclude_set_def: fixes x :: "'c"   and A :: "'c set" shows "open_exclude_set x A = (if \<exists>U. open U \<and> U \<inter> A = {x} then SOME U. open U \<and> U \<inter> A = {x} else {})"
left_inverse_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and g :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "g \<circ> f = id" shows "Vector_Spaces.linear (*s) (*s) g"
right_distrib': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<cdot> z = x \<cdot> z \<squnion> y \<cdot> z"
simps_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v1 :: "'a"   and w :: "'b"   and v2 :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_trail G v ((v1, w, v2) # ps) v' = (v = v1 \<and> (v1, w, v2) \<in> edges G \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> valid_unMultigraph.is_trail G v2 ps v')"
w_def: fixes i :: "'n" shows "w i = (\<Sum>x\<in>UNIV. g x * t x i)"
assms_1: fixes A :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "i \<notin> A" shows "\<exists>j. j \<notin> f ` A"
valid_program_history: shows "valid_program_history ts\<^sub>s\<^sub>b"
assms: fixes n :: "nat" assumes "0 < n" shows "fds_nth f n = fds_nth g n"
Diff_eq_empty_iff_mset: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A - B = {#}) = (A \<subseteq># B)"
simps_4: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "median_le_5 [a, b, c, d] = median_3 a b c"
root_int_def: fixes p :: "nat"   and x :: "int" shows "root_int p x \<equiv> if p = 0 then [] else if x = 0 then [0] else let e = even p; s = sgn x; x' = \<bar>x\<bar> in if x < 0 \<and> e then [] else case root_int_main p x' of (y, True) \<Rightarrow> if e then [y, - y] else [s * y] | (y, False) \<Rightarrow> []"
\<Sigma>_protocol: fixes \<eta> :: "nat" shows "local.\<Sigma>_protocols_base.\<Sigma>_protocol \<eta>"
basis_enum_def: shows "basis_enum \<equiv> basis_enum_of_vec"
i_Exec_Stream_Pre_Post2_Suc: fixes c_n :: "'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat"   and x_n1 :: "'b"   and P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "c_n = i_Exec_Comp_Stream trans_fun input c n"   and "x_n1 = input (Suc n)" shows "(P c_n \<longrightarrow> Q (input (Suc n)) (i_Exec_Comp_Stream trans_fun input c (Suc n))) = (P c_n \<longrightarrow> Q x_n1 (trans_fun x_n1 c_n))"
simps: fixes m :: "Body"   and v :: "'a" shows "cTest m v = ((0::'a) < v \<and> (\<forall>x y. (\<exists>p. Ph p \<and> m sees p at x \<and> m sees p at y) = (space2 x y = v * v * time2 x y)))"
prj1_def: shows "prj1 \<equiv> \<lambda>x. x \<bullet> (SOME i. i \<in> Basis)"
update_convs_1: fixes break_status' :: "bool \<Rightarrow> bool"   and break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "break_status_update break_status' \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status' break_status, return_status = return_status, \<dots> = more\<rparr>"
discI_2: fixes llist :: "'c llist"   and x21 :: "'c"   and x22 :: "'c llist" assumes "llist = LCons x21 x22" shows "\<not> lnull llist"
check_On_CTRL: shows "(On_CTRL_States, On_CTRL_Init, On_CTRL_Labels, On_CTRL_Delta) \<in> seqauto"
canonical_subs_empty: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "canonical_subs n {} m"
absorb1: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" shows "sup a b = a"
empty_is: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "is = []"
hyps_2: fixes x :: "nat" assumes "j + 2 < card X"   and "x \<in> {j + 3..card X - 1}"   and "0 < x"   and "x + 1 \<le> card X" shows "aa x + bb x \<le> Dube_aux (card X) (poly_deg f) x"
is_pos_neg_not_is_pos: fixes L :: "'a literal" shows "is_pos (- L) = is_neg L"
closed_mult_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "(r, s) \<in> carrier rel"   and "(t, u) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (t |\<^bsub>rel\<^esub> u) \<in> set_class_of\<^bsub>rel\<^esub>"
prFreshAll_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'd"   and SEM :: "('e, 'f, 'g, 'sort, 'h, 'b, 'i) semDom_scheme" shows "prFreshAll g gA SEM \<equiv> prFresh g SEM \<and> prFreshAbs gA SEM"
op_subset_ivl: fixes le :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "('d \<times> 'e) set" assumes "GEN_OP le (\<le>) (A \<rightarrow> A \<rightarrow> bool_rel)" shows "(\<lambda>(a, b) (c, d). le a b \<longrightarrow> le c a \<and> le b d, op_subset_ivl) \<in> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel \<rightarrow> bool_rel"
ereal_real: fixes x :: "ereal" shows "ereal (real_of_ereal x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
p: shows "p = (\<Sum>g\<in>G0. q g * g)"
fcompE: fixes ac :: "V"   and r :: "V"   and s :: "V" assumes "ac \<in>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s" obtains a :: "V" and b :: "V" and c :: "V" where "ac = [a, c]\<^sub>\<circ>" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> s" and "[b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
edge_from_valid_out_port: fixes p :: "nat list"   and c :: "'form" assumes "p \<in> it_paths (it' c)"   and "c \<in> set conclusions" shows "valid_out_port (edge_from c p)"
V_def: shows "V \<equiv> BD_Security.filtermap \<phi> f"
trans_eq_elapsed: fixes u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a" assumes "local.trans u u'"   and "u \<in> V" shows "eq_elapsed u u'"
solvable_condition: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and K :: "('f, 'g) monoid_scheme"   and g :: "'d \<Rightarrow> 'f" assumes "group_hom G H f"   and "group_hom H K g"   and "g ` carrier H = carrier K"   and "kernel H K g \<subseteq> f ` carrier G"   and "solvable G"   and "solvable K" shows "solvable H"
less_asym: fixes x :: "'a word"   and y :: "'a word"   and P :: "bool" assumes "x <s y"   and "\<not> P \<Longrightarrow> y <s x" shows "P"
seqE: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.seq g f" obtains "VV.arr f" and "VV.arr g" and "VV.dom g = VV.cod f"
label_wrt_adjacent: fixes B :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and C :: "'a set"   and D :: "'a set"   and v :: "'a"   and w :: "'a" assumes "label_wrt B f"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "v \<in> C - D"   and "w \<in> D - C" shows "f v = f w"
hyps_3: shows " s1 \<longrightarrow>\<^sub>1 s2"
pochhammer_of_nat: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "nat"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero (semiring_1.of_nat one plus zero x) n = semiring_1.of_nat one plus zero (pochhammer x n)"
iso_\<psi>: fixes f :: "'c" assumes "C.ide f" shows "D.iso (\<tau>.\<psi> f)"
finite_measure_Diff: fixes A :: "'d set"   and M :: "'d pmf"   and B :: "'d set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "B \<subseteq> A" shows "measure_pmf.prob M (A - B) = measure_pmf.prob M A - measure_pmf.prob M B"
init: shows "\<forall>x v'. v = x > v' \<longrightarrow> (\<forall>xa. xa \<noteq> x \<longrightarrow> (\<nexists>v u. v'=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts xa = ext v \<and> restrict v (res ts) xa = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)))"
Carmichael_number_imp_dvd: fixes n :: "nat"   and p :: "nat" assumes "Carmichael_number n"   and "prime p"   and "p dvd n" shows "p - 1 dvd n - 1"
isos_compose: fixes f :: "'a"   and f' :: "'a" assumes "local.iso f"   and "local.iso f'"   and "seq f' f" shows "local.iso (f' \<cdot> f)"
47: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> (y \<squnion> - - x) = y \<squnion> top"
sep_conj_empty: fixes P :: "'a \<Rightarrow> bool" shows "(P \<and>* \<box>) = P"
ge_One: fixes r :: "'a language" shows "(One \<le> r) = \<oo> r"
minus_def: fixes x :: "'a"   and y :: "'a" shows "f.source.additive.minus x y \<equiv> f.source.subtraction x y"
assms_1: shows "PAC_Format (\<V>, A) (\<V>', A')"
decomps\<^sub>e\<^sub>s\<^sub>t_subterms: fixes A' :: "('fun, 'var) extstrand_step list"   and M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "A' \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>" shows "subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A') \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<union> N)"
Dg: shows "(g has_field_derivative g') (at (f x) within f ` S)"
intersect'_language: fixes A :: "('c, 'd) dba"   and B :: "('c, 'e) dba" shows "DGBA.language (intersect' A B) = DBA.language A \<inter> DBA.language B"
thms_falseE: fixes H :: "'b pl set"   and q :: "'b pl" assumes "H |- false" shows "H |- q"
is_singleton_def: fixes A :: "'c set" shows "is_singleton A = (\<exists>x. A = {x})"
seq_char: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.seq g f = (J_C.arr f \<and> J_C.arr g \<and> J_C.Dom g = J_C.Cod f)"
prems_1: shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..Suc k} \<longrightarrow> (0::'a) \<le> len m i i xs"
distinct_299: fixes x151 :: "floatarith"   and x152 :: "nat"   and x18 :: "float" shows "Power x151 x152 \<noteq> Num x18"
must_have_one_false: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd G (Bc True)) s = false" shows "\<exists>g\<in>set G. gval g s = false"
maxsimpchain_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "folding.gallery (xs @ ys)" shows "folding.gallery xs"
func_space_def: fixes S :: "'d set" shows "func_space S = \<lparr>carrier = S \<rightarrow>\<^sub>E carrier R, mult = \<lambda>f g. \<lambda>v\<in>S. \<zero>, one = \<lambda>v\<in>S. \<zero>, zero = \<lambda>v\<in>S. \<zero>, add = \<lambda>f g. \<lambda>v\<in>S. f v \<oplus> g v, smult = \<lambda>c f. \<lambda>v\<in>S. c \<otimes> f v\<rparr>"
assms: shows "f holomorphic_on convex hull {a, b, c}"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "String.literal \<Rightarrow> 'a"   and f2 :: "ty \<Rightarrow> nat \<Rightarrow> 'a"   and htype :: "htype" shows "P (case htype of Class_type x \<Rightarrow> f1 x | Array_type x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. htype = Class_type x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. htype = Array_type x21 x22 \<and> \<not> P (f2 x21 x22))))"
zdiv_zmult2_eq: fixes c :: "int"   and a :: "int"   and b :: "int" assumes "0 \<le> c" shows "a div (b * c) = a div b div c"
If_cases: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(\<Sum>x\<in>A. if P x then h x else g x) = sum h (A \<inter> {x. P x}) + sum g (A \<inter> - {x. P x})"
\<phi>_in_hom_2: shows "\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0\<guillemotright>"
odsn_rrep: fixes i :: "nat" shows "opaodv i \<Turnstile> (otherwith quality_increases {i} (orecvmsg msg_fresh), other quality_increases {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (seql i (\<lambda>(\<xi>, l). l \<in> {PAodv-:6, PAodv-:7} \<union> {PRrep-:n |n. True} \<longrightarrow> 1 \<le> dsn \<xi>))"
lang_biderivs: fixes n :: "nat"   and r :: "'b rexp"   and w1 :: "'a list"   and w2 :: "'a list" assumes "local.wf n r"   and "wf_word n w1"   and "wf_word n w2" shows "lang n (biderivs w1 w2 r) = biQuots w1 w2 (lang n r)"
distinct_41: shows "string_unop.ToIntegerOp \<noteq> ToRealOp"
uminus_sheaf_spec_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "uminus_sheaf_spec U s \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.uminus_rel (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> (s \<pp>)"