hcju/isabelleps huggingface dataset:
Pfunc_def: fixes A :: "'a set"   and B :: "'b set" shows "Pfunc A B \<equiv> {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and> (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
clock_trans_aux: fixes t' :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a' :: "'a"   and t :: "'s Trace"   and eact :: "'eAct"   and aact :: "'a \<Rightarrow> 'aAct" assumes "t' \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a' t'"   and "t \<in> Clock.jkbpCn (tLength t')"   and "eact \<in> set (envAction (tLast t))" shows "(aact \<in> set (listToFuns (\<lambda>a. local.clock_simAction a (fst ec, clock_knowledge envObs (fst ec) a (tLast t))) (toList agents))) = (\<forall>a. aact a \<in> set (jAction (Clock.MCn (tLength t')) t a))"
le_degree: fixes p :: "'a poly"   and n :: "nat" assumes "coeff p n \<noteq> (0::'a)" shows "n \<le> degree p"
reducecoeff_def: fixes t :: "Ferrack.num" shows "reducecoeff t = (let g = numgcd t in if g = 0 then C 0 else if g = 1 then t else reducecoeffh t g)"
simps_15: fixes x2 :: "'a" shows "set_option \<lfloor>x2\<rfloor> = {x2}"
cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes document :: "('m, 'n, 'o, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'p option\<rparr> + 'q) option) RDocument_scheme" shows "cast document = (case RDocument.more document of None \<Rightarrow> None | Some (Inl shadow_root) \<Rightarrow> Some (RDocument.extend (RDocument.truncate document) shadow_root) | Some (Inr x) \<Rightarrow> Map.empty x)"
dvd_gcd_mult: fixes k :: "'b"   and p :: "'b"   and q :: "'b"   and r :: "'b" assumes "k dvd p * q"   and "k dvd p * r" shows "k dvd p * gcd q r"
RSLE_eq: fixes X :: "'a set"   and Y :: "'a set"   and x :: "'a"   and y :: "'a" assumes "X \<in> Q"   and "Y \<in> Q"   and "x \<in> X"   and "y \<in> Y"   and "x \<sqsubseteq> y" shows "X \<sqsubseteq>\<^sup>s Y"
right_neutral: fixes a :: "'b \<times> 'c" shows "inf_prod a top_prod = a"
nset_Tr52: fixes j :: "nat" assumes "j \<noteq> Suc (Suc 0)"   and "Suc 0 \<le> j - Suc 0" shows "\<not> j - Suc 0 \<le> Suc 0"
sup_compl_top: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and 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 bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (uminus x) = top"
fps_integral_deriv: fixes a :: "'a fps" shows "fps_integral (fps_deriv a) (fps_nth a 0) = a"
leq_refl: fixes leq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.quasi_order leq" shows "leq x x"
groupWIs1_groupWIs2_equi: fixes c :: "parts_connection"   and rs :: "'b simple_rule list" shows "groupWIs2 c rs = groupWIs1 c rs"
Partitions_def: fixes x :: "'a list" shows "Partitions x \<equiv> {(x\<^sub>p, x\<^sub>s). x\<^sub>p @ x\<^sub>s = x}"
prod_fun_imageE: fixes c :: "'c \<times> 'd"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and R :: "('e \<times> 'f) set" assumes "c \<in> map_prod f g ` R" obtains x :: "'e" and y :: "'f" where "c = (f x, g y)" and "(x, y) \<in> R"
left_diff_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ring plus zero minus uminus times" shows "times (minus a b) c = minus (times a c) (times b c)"
t_heapify_induct: fixes i :: "nat"   and xs :: "'b list" assumes "i \<le> length xs" shows "t_heapify i xs + height (fst (heapify i xs)) \<le> 5 * i + 1"
61163: fixes x :: "'a" shows "- (x \<squnion> - - x) = - x"
wf_filterQuery_1: fixes I :: "nat set"   and V :: "nat set"   and n :: "nat"   and Qp :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQp :: "(nat set \<times> 'a option list set) set"   and QQn :: "(nat set \<times> 'a option list set) set" assumes "I \<subseteq> V"   and "1 \<le> card I"   and "rwf_query n V Qp Qn"   and "QQp = filterQuery I Qp"   and "QQn = filterQueryNeg I Qn" shows "wf_query n I QQp QQn"
simps_10: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Cod \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Cod t \<^bold>\<star> Cod u) \<^bold>\<star> Cod v"
minus: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real"   and x :: "'b" assumes "seminorm V norm"   and "vectorspace V"   and "x \<in> V" shows "norm (- x) = norm x"
arev_arev: fixes a :: "'b" shows "edge_rev M (edge_rev M a) = a"
add_mal: fixes x :: "char list"   and a :: "'b" shows "Malformed\<lparr>x:a\<rparr> = Malformed"
hyp2_cltn2_def: fixes p :: "hyp2"   and A :: "cltn2" shows "hyp2_cltn2 p A \<equiv> Abs_hyp2 (apply_cltn2 (Rep_hyp2 p) A)"
lookup_add: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and g :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "lookup (f + g) k = lookup f k + lookup g k"
regular_comp: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" assumes "regular f"   and "regular g" shows "regular (g \<circ> f)"
P'B'Q': shows "(P', B', Q') = Diagonal_to_Smith_row_i_PQ i bezout (P, B, Q)"
card_insert_le: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card A \<le> card (insert x A)"
assms_4: shows "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t"
content_free_unit: fixes p :: "'a poly" assumes "content p = (1::'a)" shows "(p dvd 1) = (degree p = 0)"
on_graph_id: fixes B :: "('f, 'e) labeled_graph" shows "on_graph B id = Id_on (vertices B)"
infinite_nat_iff_unbounded: fixes S :: "nat set" shows "infinite S = (\<forall>m. \<exists>n>m. n \<in> S)"
simps_1: fixes t :: "'b" shows "has_locks None t = 0"
Posix_injval: fixes s :: "'b list"   and c :: "'b"   and r :: "'b rexp"   and v :: "'b val" assumes "s \<in> deriv c r \<rightarrow> v" shows "(c # s) \<in> r \<rightarrow> injval r c v"
ldistinct_lprefix: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "ldistinct xs"   and "ys \<sqsubseteq> xs" shows "ldistinct ys"
time_find_closest_bf_tm: fixes p :: "int \<times> int"   and ps :: "(int \<times> int) list" shows "time (find_closest_bf_tm p ps) \<le> length ps + 1"
simps_6: fixes uv :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uv: t \<in> t'\<rangle> = fv t \<union> fv t'"
degree_add_le_max: fixes p :: "'b poly"   and q :: "'b poly" shows "degree (p + q) \<le> max (degree p) (degree q)"
simps: shows "res.wf_residual_network = ((\<forall>x xa. (x, xa) \<in> \<^bold>E\<^bsub>residual_network f\<^esub> \<longrightarrow> (xa, x) \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>) \<and> (source (residual_network f), sink (residual_network f)) \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>)"
dagger_plus_one: fixes x :: "'a list set" shows "p_one \<union> p_star x = p_star x"
has_derivative_linear: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "(f has_derivative f') F" shows "linear f'"
power_minus': fixes one :: "'a"   and times :: "'a \<Rightarrow> '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 x :: "'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "NO_MATCH one x" shows "power.power one times (uminus x) n = times (power.power one times (uminus one) n) (power.power one times x n)"
Ri_def: fixes N :: "'a literal multiset set" shows "Ri N = {\<gamma> \<in> \<Gamma>. redundant_infer N \<gamma>}"
parts_overlap_Fs: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "\<forall>y. P y (\<sigma>x. F x) \<longrightarrow> (\<exists>z. F z \<and> O y z)"
s_Prj2: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Prj2 e\<ggreater> m\<rightarrow> \<lless>\<pi>', Prj2 e'\<ggreater>"
ex_br_conj_iff: fixes y :: "'f"   and a :: "'f \<Rightarrow> 'e"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" shows "(\<exists>x. (y, x) \<in> br a I \<and> P x) = (I y \<and> P (a y))"
leg1_is_map: shows "is_left_adjoint u\<^sub>1"
dom_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.dom \<a>\<^sub>D[a, b, c] = (a \<otimes>\<^sub>D b) \<otimes>\<^sub>D c"
finite_Psegments_less_eq1: fixes P :: "real \<Rightarrow> bool"   and a :: "real"   and c :: "real"   and b :: "real" assumes "finite_Psegments P a c"   and "b \<le> c" shows "finite_Psegments P a b"
widen_ObjectD: fixes G :: "prog"   and T :: "ty" assumes "G\<turnstile>Class Object\<preceq>T" shows "T = Class Object"
tensor_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>f \<otimes> g : a \<otimes> c \<rightarrow> b \<otimes> d\<guillemotright>"
next_step_step: fixes i :: "nat"   and l :: "'b" shows "i \<le> next_step_of l i \<and> (\<not> can_step l (s (next_step_of l i)) \<or> rstep l s (next_step_of l i))"
backwards_wdc_res_stab: fixes ts :: "traffic"   and d :: "cars"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "(ts \<^bold>\<midarrow>wdc( d ) \<^bold>\<rightarrow> ts') \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1 \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
partitionsE: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p partitions n" obtains "\<And>i. p i \<noteq> 0 \<Longrightarrow> 1 \<le> i \<and> i \<le> n" and "(\<Sum>i\<le>n. p i * i) = n"
in_degree_def: fixes G :: "('b, 'c) pre_digraph"   and v :: "'b" shows "in_degree G v \<equiv> card (in_arcs G v)"
big_mono: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> norm (g x)" shows "f \<in> O[F](g)"
add_right_mono: fixes a :: "'b zmultiset"   and b :: "'b zmultiset"   and c :: "'b zmultiset" assumes "a \<subseteq>#\<^sub>z b" shows "a + c \<subseteq>#\<^sub>z b + c"
mono_D2: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P [+] S) \<longrightarrow> Ra (P [+] S) s = Ra (Q [+] S) s"
\<C>_vars_finite: fixes x :: "'Var" shows "finite (\<C>_vars x)"
string_to_nat_aux_inv: fixes s :: "char list"   and n :: "nat" assumes "is_nat_string s"   and "0 < n \<or> is_strong_nat_string s" shows "nat_to_string (string_to_nat_aux n s) = (if n = 0 then [] else nat_to_string n) @ s"
div0_uint8_def: fixes x :: "uint8" shows "div0_uint8 x = undefined (div) x 0"
intro: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "blinding_respects_hashes h bo"
rt_Some_sqn: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "rt ip = Some (dsn, dsk, flag, hops, nhip, pre)" shows "sqn rt ip = dsn"
invariant_imp_sorted: fixes l :: "nat" assumes "Suc l < length (ext_ids s)"   and "a (ext_ids s ! l) < a \<lbrakk>I m\<rbrakk> \<and> a \<lbrakk>I m\<rbrakk> < a (ext_ids s ! (l + 1))" shows "sorted_wrt (<) (map a (ext_ids (take l s @ to_woot_char m # drop l s)))"
takeWhile_cong: fixes l :: "'g list"   and k :: "'g list"   and P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "takeWhile P l = takeWhile Q k"
simps_1: fixes x :: "'a"   and y :: "'a" shows "Fin x + Fin y = Fin (x + y)"
oreceived_msg_inv: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> msg \<Rightarrow> bool"   and Q :: "state \<Rightarrow> state \<Rightarrow> bool"   and i :: "nat" assumes "\<And>\<sigma> \<sigma>' m. \<lbrakk>P \<sigma> m; other Q {i} \<sigma> \<sigma>'\<rbrakk> \<Longrightarrow> P \<sigma>' m"   and "\<And>\<sigma> m. P \<sigma> m \<Longrightarrow> P (\<sigma>(i := \<sigma> i\<lparr>msg := m\<rparr>)) m" shows "opaodv i \<Turnstile> (otherwith Q {i} (orecvmsg P), other Q {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<sigma>, l). l \<in> {PAodv-:1} \<longrightarrow> P \<sigma> (msg (\<sigma> i)))"
conv_radius_add_ge: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "min (conv_radius f) (conv_radius g) \<le> conv_radius (\<lambda>x. f x + g x)"
True: fixes F :: "('g \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
respectfulness_2: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and \<nu> :: "'b set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "af_letter_opt \<phi> \<nu> \<equiv>\<^sub>P af_letter_opt \<psi> \<nu>"
tr_lookup_def: shows "tr_lookup = (\<lambda>T i j. if i = j then False else if j < i then T ! (i - 1) ! j else T ! (j - 1) ! i)"
linorder_cases: fixes y :: "'d"   and x :: "'d" obtains "y < x"   | "x = y"   | "x < y"
list_update_last: fixes ls :: "'b list"   and i :: "nat"   and x :: "'b" assumes "length ls = Suc i" shows "ls[i := x] = take i ls @ [x]"
Ge_def: shows "Ge \<equiv> {(x, y). y \<le> x}"
ell2_norm_ket: fixes a :: "'c" shows "ell2_norm (\<lambda>i. if a = i then 1 else 0) = 1"
monad_altc_axioms: shows "monad_altc NI_return NI_bind NI_altc"
push_bit_or: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "push_bit n (or a b) = or (push_bit n a) (push_bit n b)"
intros_1: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and F :: "'d fs_set"   and P :: "'a"   and Q :: "'a" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q" shows "effect_nominal_ts.L_bisimilar satisfies transition effect_apply L (EF (F, P)) (EF (F, Q))"
set_Cons_remove1: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "set (x # remove1 x xs) = set xs"
image_map_of': fixes xs :: "('b \<times> 'c) list" assumes "distinct (map fst xs)" shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)"
Arr_iff_cod_in_codomain: fixes f :: "'a" shows "right (src \<mu>) f = (cod f \<in> Right.codomains f)"
WT_gpv_simps: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, ('f, 'd, 'e) rpv) generat spmf" shows "\<Gamma> \<turnstile>g GPV gpv \<surd> = (\<forall>out c. IO out c \<in> set_spmf gpv \<longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>))"
iD_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> iD rt" shows "nsqn rt ip = sqn rt ip - 1"
S1ClerkUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [MClkNext memCh crCh cst p]_(c p) \<and> $S1 rmhist p \<longrightarrow> unchanged c p"
partition_on_mset_distinct_inter: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and p1 :: "'a multiset"   and p2 :: "'a multiset" assumes "partition_on_mset A P"   and "distinct_mset A"   and "p1 \<in># P"   and "p2 \<in># P"   and "p1 \<noteq> p2" shows "p1 \<inter># p2 = {#}"
Gramian_determinant_def: fixes n :: "nat"   and G :: "'a vec list"   and k :: "nat" shows "gram_schmidt.Gramian_determinant n G k = det (gram_schmidt.Gramian_matrix n G k)"
Oops_range_spies1: fixes A :: "agent"   and KeyA :: "nat"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt KeyA \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "authK \<notin> range shrK \<and> authK \<in> symKeys"
qintersect: fixes Q :: "'a set"   and Q' :: "'a set" assumes "Q \<in> Quorum"   and "Q' \<in> Quorum" shows "Q \<inter> Q' \<noteq> {}"
induct: fixes x :: "['b]"   and P :: "['b] \<Rightarrow> bool" assumes "distinct x"   and "P []"   and "\<And>x xs. \<lbrakk>distinct xs; P xs; x \<notin> set xs\<rbrakk> \<Longrightarrow> P (x : xs)" shows "P x"
plus_pres_lens_indep: fixes X :: "'b \<Longrightarrow> 'c"   and Z :: "'d \<Longrightarrow> 'c"   and Y :: "'e \<Longrightarrow> 'c" assumes "X \<bowtie> Z"   and "Y \<bowtie> Z" shows "X +\<^sub>L Y \<bowtie> Z"
nsum_add_nm: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier M"   and "\<forall>j\<le>m. g j \<in> carrier M" shows "\<Sigma>\<^sub>e M jointfun n f m g Suc (n + m) = \<Sigma>\<^sub>e M f n \<plusminus> \<Sigma>\<^sub>e M g m"
distinct_Ex1: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < |xs| \<and> xs ! i = x"
set_iterator_product_def: fixes it_a :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it_b :: "'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" shows "set_iterator_product it_a it_b \<equiv> \<lambda>c f. it_a c (\<lambda>a. it_b a c (\<lambda>b. f (a, b)))"
convex_hull_in_cone: fixes C :: "'a vec set" shows "convex_hull C \<subseteq> cone C"
ideal_n_prodSn: fixes R :: "('c, 'd) Ring_scheme"   and n :: "nat"   and J :: "nat \<Rightarrow> 'c set" shows "i\<Pi>\<^bsub>R,Suc n\<^esub> J = i\<Pi>\<^bsub>R,n\<^esub> J \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> J (Suc n)"
eNorm_Norm: fixes x :: "'b"   and N :: "'b quasinorm" assumes "x \<in> space\<^sub>N N" shows "eNorm N x = ennreal (Norm N x)"
nonzero_fstI: fixes x :: "'a \<times> 'b" assumes "fst x \<noteq> (0::'a)" shows "x \<noteq> 0"
subst_fv_dom_img_subset: fixes t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<subseteq> subst_domain \<theta>" shows "fv (t \<cdot> \<theta>) \<subseteq> range_vars \<theta>"
prems_2: fixes s :: "'state"   and q :: "'request"   and a :: "'answer" assumes "P s" shows "P (step s q a)"
l2_inv7D: fixes x :: "l2_state" assumes "x \<in> l2_inv7" shows "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz generators)"
\<beta>_simps_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b"   and u :: "'b"   and w :: "'b"   and \<theta> :: "'b"   and w' :: "'b"   and \<theta>' :: "'b"   and \<beta> :: "'b" assumes "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "partial_magma.dom V \<beta> = H g w"
simps: fixes x :: "('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a + call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr" shows "abs_R\<cdot>x = (case undiscr x of Inl (PC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> b); ve' = ve \<union>. \<Union>.map2 (\<lambda>v. smap_singleton (v, b)) vs as in {Discr (Inr (c, \<beta>', ve', b))} else \<bottom> | Inl (PP (Plus c), [], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa, cnts], ve, b) \<Rightarrow> let b' = \<anb> b c; \<beta> = [c \<mapsto> b] in \<Union>cnt\<in>cnts. {Discr (Inl (cnt, [{}], ve, b'))} | Inl (PP (Plus c), x # xa # cnts # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts, cntfs], ve, b) \<Rightarrow> (let b' = \<anb> b ct; \<beta> = [ct \<mapsto> b] in \<Union>cnt\<in>cntts. {Discr (Inl (cnt, [], ve, b'))}) \<union> (let b' = \<anb> b cf; \<beta> = [cf \<mapsto> b] in \<Union>cnt\<in>cntfs. {Discr (Inl (cnt, [], ve, b'))}) | Inl (PP (prim.If ct cf), x # cntts # cntfs # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (AStop, [], bb) \<Rightarrow> \<bottom> | Inl (AStop, [x], xa, xb) \<Rightarrow> {} | Inl (AStop, x # a # lista, bb) \<Rightarrow> \<bottom> | Inr (App lab f vs, \<beta>, ve, b) \<Rightarrow> let fs = \<aA> f \<beta> ve; as = map (\<lambda>v. \<aA> v \<beta> ve) vs; b' = \<anb> b lab in \<Union>f'\<in>fs. {Discr (Inl (f', as, ve, b'))} | Inr (call.Let lab ls c', \<beta>, ve, b) \<Rightarrow> let b' = \<anb> b lab; \<beta>' = \<beta>(lab \<mapsto> b'); ve' = ve \<union>. \<Union>.map (\<lambda>(v, l). { (v, b') := \<aA> (L l) \<beta>' ve}.) ls in {Discr (Inr (c', \<beta>', ve', b'))})"
assms_6: shows "has_greatest_fixpoint (g \<circ> h)"
out_arcs_add_arc_iff: fixes a :: "'b"   and u :: "'a" shows "out_arcs (add_arc a) u = (if tail G a = u then insert a (out_arcs G u) else out_arcs G u)"
inv_inv: fixes x :: "(bool \<times> 'a) list" assumes "x \<in> carrier \<F>\<^bsub>gens\<^esub>" shows "inv\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> (inv\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> x) = x"
irrefl: fixes x :: "'a" assumes "x \<in> lset xs" shows "\<not> r x x"
skip_doesnt_read: fixes annos :: "'Var ModeUpd list"   and x :: "'Var" shows "doesnt_read_or_modify (Skip \<otimes> annos) x"
le1221: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Le Bet Cong A B B A"
isom_gch_transp: fixes n :: "nat"   and f :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Gchain n f"   and "i \<le> n"   and "j \<le> n"   and "i < j" shows "isom_Gchains n (transpos i j) f (cmp f (transpos i j))"
semiattractive_axioms: shows "semiattractive A (\<sqsubseteq>)"