hcju/isabelleps huggingface dataset:
that: fixes us :: "'a list"   and vs :: "'a list" assumes "xs = us @ vs"   and "filter P us = ys"   and "filter P vs = zs" shows "thesis"
rbt_map_entry_rbt_less: fixes a :: "'a"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "(rbt_map_entry k f t |\<guillemotleft> a) = (t |\<guillemotleft> a)"
nonempty_some: fixes A :: "'c set" assumes "A \<noteq> {}" shows "(SOME x. x \<in> A) \<in> A"
that: assumes "nop_bij A (1\<^sub>\<nat>) f" shows "thesis"
assms_5: shows "\<Delta> s' vl' s1' vl1'"
lfp_fusion_pointed: fixes C :: "'d \<rightarrow> 'e"   and F :: "'d \<rightarrow> 'd"   and G :: "'e \<rightarrow> 'e" assumes "\<And>f. C\<cdot>(F\<cdot>f) = G\<cdot>(C\<cdot>f)"   and "C\<cdot>\<bottom> = \<bottom>" shows "C\<cdot>(fix\<cdot>F) = fix\<cdot>G"
intros_1: fixes P :: "'a prog"   and T :: "ty" shows "P \<turnstile> T \<le> T"
plus_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "plus_ivl i1 i2 = (if is_empty i1 \<or> is_empty i2 then Abs_Int2_ivl.empty else case (i1, i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1 + l2) (h1 + h2))"
partial_magma_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "partial_magma C \<equiv> \<exists>!n. \<forall>f. C n f = n \<and> C f n = n"
assms_1: shows "netgmap sr s1 = netmask (net_tree_ips n1) (\<sigma>, snd (netgmap sr s1))"
imp2: fixes x :: "'a" assumes "x \<in># mset_set A" shows "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "'c list" shows "(State x1 x2 x3 = State y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
stutter_extend_edgesE: fixes u :: "'a"   and v :: "'a"   and V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "(u, v) \<in> stutter_extend_edges V E" obtains "(u, v) \<in> E"   | "u \<in> V" and "u \<notin> Domain E" and "u = v"
exe_impl_exists: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> 'a \<Rightarrow> \<o>"   and x :: "'a" shows "[\<lparr>\<^bold>\<lambda>x. \<^bold>\<forall>p. p \<^bold>\<rightarrow> p,\<^bold>\<iota>y. \<phi> y x\<rparr> \<^bold>\<equiv> (\<^bold>\<exists>!y. \<^bold>\<A>\<phi> y x) in v]"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "decseq A" shows "(\<lambda>n. Sigma_Algebra.measure M (A n)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Inter> (range A))"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "??.Sublist.prefix a b"   and "??.Sublist.prefix b c" shows "??.Sublist.prefix a c"
assms_3: shows "y \<in> set (local.filter_min_aux xs ys)"
rotate_root_self: fixes r :: "'b list"   and k :: "nat" shows "rotate \<^bold>|r\<^bold>| (r \<^sup>@ k) = r \<^sup>@ k"
zip_append2: fixes xs :: "'k list"   and ys :: "'l list"   and zs :: "'l list" shows "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
abs_prob_def: shows "abs_prob = \<lparr>sas_plus_problem.variables_of = abs_ast_variable_section, operators_of = abs_ast_operator_section, initial_of = abs_ast_initial_state, goal_of = abs_ast_goal, range_of = abs_range_map\<rparr>"
E2: shows "(\<Sum>i<length qs. sum (ALG' A qs init i) (set init)) = (\<Sum>x\<in>set init. \<Sum>i<length qs. ALG' A qs init i x)"
conv: fixes xy :: "nat \<times> nat" shows "convergent (\<lambda>n. h' (k n) xy)"
Ord_VsetI: fixes i :: "V"   and b :: "V" assumes "Ord i"   and "rank b < i" shows "b \<in> elts (Vset i)"
CD_on_range': fixes c :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "c \<in> PdXXX.CD_on ds A" shows "c \<in> A \<inter> (\<Union>d\<in>ds. Field (Pd' d))"
diffs_equiv: fixes c :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "summable (\<lambda>n. diffs c n * x ^ n)" shows "(\<lambda>n. of_nat n * c n * x ^ (n - Suc 0)) sums (\<Sum>n. diffs c n * x ^ n)"
add_left_mono: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f"   and b :: "'f"   and c :: "'f" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus c a) (plus c b)"
assms_2: shows "nat_of_fatom (get_atom l) < length ds"
ordinal_of_nat_less: fixes x :: "nat"   and y :: "nat" shows "(ordinal_of_nat x < ordinal_of_nat y) = (x < y)"
gcds_def: fixes xs :: "'a list" shows "gcds xs = gcd_list xs"
assms_3: shows "A \<subseteq> outs_\<I> \<I>"
distrib_3: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
simply_connected_imp_path_connected: fixes S :: "'b set" assumes "simply_connected S" shows "path_connected S"
rstep_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'d"   and s :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "df_system s\<^sub>0 lstep" shows "df_system.rstep lstep l s i \<equiv> lstep l (s i) (s (Suc i))"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "bisimp a1 a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>[{(xa, x). bisimp xa x}] Q" and "bisimp Q P"
52421: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - (- x \<sqinter> y)) \<sqinter> y = y"
RES_rule: fixes S :: "'c set"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "\<And>x. x \<in> S \<Longrightarrow> \<Phi> x" shows "RES S \<le> SPEC \<Phi>"
h1: shows "\<turnstile> F \<longrightarrow> G"
join_vmsaux_def: shows "join_vmsaux = (\<lambda>args rel1 (t, auxlist). (t, map (\<lambda>(t, rel). (t, join rel (args_pos args) rel1)) auxlist))"
filter_sup_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'c set"   and G :: "'c set" assumes "class.semilattice_inf inf less_eq less" shows "semilattice_inf.filter_sup inf less_eq F G \<equiv> {z. \<exists>x\<in>F. \<exists>y\<in>G. less_eq (inf x y) z}"
inject: fixes x1 :: "'a simple_match"   and x2 :: "simple_action"   and y1 :: "'a simple_match"   and y2 :: "simple_action" shows "(SimpleRule x1 x2 = SimpleRule y1 y2) = (x1 = y1 \<and> x2 = y2)"
factorization_preserves_finiteness: fixes P1 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl P1)"   and "factorization P1 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
simps_11: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm" shows "Ifm bbs bs (NOT p) = (\<not> Ifm bbs bs p)"
transD: fixes r :: "('e \<times> 'e) set"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
out_reg_mod_privilege: fixes s' :: "'b sparc_state"   and a :: "word32"   and w :: "'b word"   and r :: "word5"   and s :: "'b sparc_state" assumes "s' = out_reg_mod a w r s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
path_in_\<alpha>n: fixes g :: "'g"   and ns :: "'node list"   and n :: "'node" assumes "local.path g ns"   and "n \<in> set ns" shows "n \<in> set (\<alpha>n g)"
local: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "\<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a fset"   and g :: "'c \<Rightarrow> 'b"   and y :: "'c fset" shows "rel_fset Sa x (g |`| y) = rel_fset (\<lambda>x y. Sa x (g y)) x y"
less_1: fixes y :: "('a, 'b) tm" assumes "hsize y < hsize t"   and "y \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "\<exists>us. us \<noteq> [] \<and> hd us = y \<and> last us = s \<and> (\<forall>i. Suc i < length us \<longrightarrow> prio_emb_step prio (us ! i) s = us ! Suc i \<and> us ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b us ! Suc i)"
adjunction_preserved_by_iso_right: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<phi> :: "'a"   and g' :: "'a" assumes "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>"   and "local.iso \<phi>" shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> local.inv \<phi>))"
responses_plus_\<I>_Inl: fixes \<I>l :: "('c, 'a) \<I>"   and \<I>r :: "('d, 'b) \<I>"   and x :: "'c" shows "responses_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) (Inl x) = Inl ` responses_\<I> \<I>l x"
remove_shadow_root_def: shows "remove_shadow_root \<equiv> l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root Shadow_DOM.get_child_nodes get_shadow_root set_shadow_root get_disconnected_nodes"
pcr_cr_eq: shows "pcr_freeword (=) = cr_freeword"
homeomorphic_maps_sym: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "homeomorphic_maps X Y f g = homeomorphic_maps Y X g f"
connected_iff_connected_component_eq: fixes S :: "'a set" shows "connected S = (\<forall>x\<in>S. \<forall>y\<in>S. connected_component_set S x = connected_component_set S y)"
matrix_vector_mul_linear_gen: fixes A :: "(('b, 'c) vec, 'd) vec" shows "linear (*s) (*s) ((*v) A)"
assoc_coherence: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "C'.trg f = C'.src g"   and "C'.trg g = C'.src h" shows "F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (g \<star>\<^sub>C\<^sup>o\<^sup>p f, h) \<cdot>\<^sub>D (F h \<star>\<^sub>D\<^sup>o\<^sup>p \<Phi> (f, g)) = \<Phi> (f, h \<star>\<^sub>C\<^sup>o\<^sup>p g) \<cdot>\<^sub>D (\<Phi> (g, h) \<star>\<^sub>D\<^sup>o\<^sup>p F f) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
hom_prod_list: fixes xs :: "int poly list" shows "of_int_poly (prod_list xs) = prod_list (map of_int_poly xs)"
inf_idem: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
append_self_conv: fixes xs :: "'e list"   and ys :: "'e list" shows "(xs @ ys = xs) = (ys = [])"
mkIde_set: fixes a :: "'t" assumes "S'.ide a" shows "S'.mkIde (S'.set a) = a"
step_z_dbm_sound: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))" shows "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto> \<langle>l', [D']\<^bsub>v,n\<^esub>\<rangle>"
simps_3: fixes A :: "'b"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" shows "(G \<^bold>\<or> H)[F/A] = G[F/A] \<^bold>\<or> H[F/A]"
left_impl_times: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
bounded_closed_imp_seq_compact: fixes S :: "'b set" assumes "bounded S"   and "closed S" shows "seq_compact S"
is_pd_def: fixes y :: "'n"   and x :: "'n" shows "(y pd\<rightarrow> x) = (x \<in> local.nodes \<and> (\<forall>\<pi> n. is_path \<pi> \<and> \<pi> 0 = x \<and> \<pi> n = local.return \<longrightarrow> (\<exists>k\<le>n. \<pi> k = y)))"
m1_secrecyE: fixes x :: "'a m1_state_scheme" assumes "x \<in> m1_secrecy" obtains "knC (runs x) \<subseteq> azC (runs x) \<union> Domain (m1_state.leak x) \<times> UNIV"
eval_wp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (a \<^bsub>Q\<^esub>\<oplus> b) P = (\<lambda>s. Q s * wp a P s + (1 - Q s) * wp b P s)"
mod_mult_self3: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times c b) a) b = modulo a b"
abs_dist_diff_le: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "\<bar>dist a b - dist b c\<bar> \<le> dist a c"
Confidentiality_S_temporal: fixes A :: "agent"   and K' :: "nat"   and T :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number T, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "\<not> expiredK T evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
ctermsl_dtermsl: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> dtermsl p" shows "q \<in> ctermsl p"
binomial_int_ineq_2: fixes y :: "int"   and x :: "int"   and k :: "nat" assumes "0 \<le> y"   and "y \<le> x" shows "2 * (x gchoose k) \<le> x - y gchoose k + (x + y gchoose k)"
Units_nonzero: fixes u :: "nat \<Rightarrow> int" assumes "u \<in> Units Zp"   and "\<one> \<noteq> \<zero>" shows "u \<in> nonzero Zp"
Bang_def: fixes x1 :: "pi" shows "!x1 \<equiv> Abs_pi (Bang_Rep (Rep_pi x1))"
simps: fixes a1 :: "val"   and a2 :: "val" shows "a1 \<sqsubseteq> a2 = ((\<exists>n. a1 = VNat n \<and> a2 = VNat n) \<or> (\<exists>t1 t2. a1 = VFun t1 \<and> a2 = VFun t2 \<and> fset t1 \<subseteq> fset t2))"
CONJ_def: fixes f :: "int \<Rightarrow> int" shows "CONJ f = inv ni_bij \<circ> f \<circ> ni_bij"
ideal_nsum_closedTr: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "Ring R"   and "ideal R I" shows "(\<forall>j\<le>n. f j \<in> I) \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> I"
A: fixes \<Gamma> :: "'a formula list" shows "A # \<Gamma> \<Rightarrow> [A]"
simps: fixes a1 :: "'c llist"   and a2 :: "'c llist" shows "a1 \<sqsubseteq> a2 = ((\<exists>xs. a1 = LNil \<and> a2 = xs) \<or> (\<exists>xs ys x. a1 = LCons x xs \<and> a2 = LCons x ys \<and> xs \<sqsubseteq> ys))"
N'_in_Red_F_N: shows "N' \<subseteq> local.Red_F_\<G> N"
aNDSetsEq: fixes a :: "('b set set, 'c) Combinators list"   and b :: "('b set set, 'c) Combinators list" assumes "singleCombinators a"   and "singleCombinators b"   and "set a = set b"   and "allNetsDistinct b" shows "allNetsDistinct a"
bijI': fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. (f x = f y) = (x = y)"   and "\<And>y. \<exists>x. y = f x" shows "bij f"
AllowedActs_eq: fixes init :: "'c set"   and acts :: "('c \<times> 'c) set set"   and allowed :: "('c \<times> 'c) set set" shows "AllowedActs (mk_program (init, acts, allowed)) = insert Id allowed"
length_partition_by_median_1: fixes l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a"   and ps :: "(real, 'a) vec list" assumes "(l, m, r) = partition_by_median k ps" shows "length ps = length l + length r"
perm_rel_lemma: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<pi> :: "perm"   and x :: "'c"   and y :: "'d" assumes "\<And>\<pi> x y. r (\<pi> \<bullet> x) (\<pi> \<bullet> y) \<Longrightarrow> r x y" shows "r (\<pi> \<bullet> x) (\<pi> \<bullet> y) = r x y"
max_min_distrib2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linorder less_eq less" shows "ord.max less_eq a (ord.min less_eq b c) = ord.min less_eq (ord.max less_eq a b) (ord.max less_eq a c)"
prj_in_hom_1: shows "\<guillemotleft>\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<mu>\<nu>.apex\<guillemotright>"
nabla_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<nabla>x \<le> \<nabla>y"
newton_poly_sum: fixes n :: "nat" shows "local.newton_poly n = (\<Sum>i\<leftarrow>[0..<Suc n]. smult (local.c i) (local.N i))"
subset_imp_ensures: fixes A :: "'c set"   and B :: "'c set"   and F :: "'c program" assumes "A \<subseteq> B" shows "F \<in> A ensures B"
step: fixes r :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set"   and m :: "nat" assumes "r \<in> R"   and "fst r \<noteq> []"   and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R" shows "(snd r, m + 1) \<in> derivable R"
sort_oalist_id: fixes x :: "'o"   and xs :: "('a \<times> 'd) list" assumes "oalist_inv_raw x xs" shows "sort_oalist x xs = xs"
induct: fixes P :: "'g list \<Rightarrow> bool"   and a0 :: "'g list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. \<lbrakk>x = y \<Longrightarrow> P (x # xs); x \<noteq> y \<Longrightarrow> P (y # xs)\<rbrakk> \<Longrightarrow> P (x # y # xs)" shows "P a0"
lang_subset_lists: fixes n :: "nat"   and r :: "'b rexp" assumes "local.wf n r" shows "lang n r \<subseteq> lists (\<Sigma> n)"
kim: shows "k ` ({0..1} \<times> {0..1}) \<subseteq> C"
assms_2: assumes "B = add_mset s (B - {#s#})"   and "A = B - {#s#} + C" shows "P"
diff_eq_diff_less: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less a b = less c d"
case_2: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x2 :: "'d" shows "(case Nom x2 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f2 x2"
altc_sample_writer_def: fixes altc_sample :: "'d \<Rightarrow> ('e \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'d"   and f :: "'e \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
f: shows "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
space_bind_measurable: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and B :: "'b measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "space M \<noteq> {}" shows "space (M \<bind> f) = space B"
is_tautology_def: fixes C :: "'a literal list" shows "is_tautology C = (\<exists>A\<in>set (map atm_of C). Pos A \<in> set C \<and> Neg A \<in> set C)"
in_set_member: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = List.member xs x"
star_inductr': fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y ; x \<le> y" shows "z ; x\<^sup>\<star> \<le> y"
common_range_limit: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'b" assumes "finite (range x)"   and "finite (range y)" obtains i :: "nat" where "limit x = range (suffix i x)" and "limit y = range (suffix i y)"
p_sylow_mod_p: shows "card (subgroups_of_size (q ^ 1)) mod q = 1"
Vagree_refl: fixes \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "Vagree \<nu> \<nu> V"
Limsup_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" shows "Limsup F f = (\<Sqinter>P\<in>{P. eventually P F}. \<Squnion> (f ` {x. P x}))"
assms: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> vcard xs" shows "xs\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i"
assms: shows "bal_up\<^sub>i (node\<^sub>i k ts t)"
EM_remainder'_bounded_linear: fixes h :: "'a \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and n :: "nat" assumes "bounded_linear h"   and "continuous_on {a..b} f" shows "EM_remainder' n (\<lambda>x. h (f x)) a b = h (EM_remainder' n f a b)"
is_liftingD_2: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and C :: "'a" assumes "is_lifting f basis basis'"   and "is_expansion C basis" shows "is_expansion (f C) basis'"
assms: shows "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"
assms_13: shows "b2 * A2 \<noteq> B2 * A1 \<or> b1 * A2 \<noteq> B1 * A1"
prod_eqI: fixes p :: "'a \<times> 'b"   and q :: "'a \<times> 'b" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
arev_arev_raw: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev M (edge_rev M a) = a"
disj_zero_right: fixes x :: "'b" shows "x OR (0::'b) = x"
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and 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 wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
g_f_eq: fixes r :: "real"   and t :: "real" shows "Im (g r t) / Re (g r t) = (f \<circ> (\<lambda>x. 2 * r * x - r)) t"