hcju/isabelleps huggingface dataset:
assms_2: shows "Enc_keys_clean (G \<union> H)"
leadsTo_Basis': fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A co A \<union> B"   and "F \<in> transient A" shows "F \<in> A \<longmapsto> B"
join_self_below_1: fixes x :: "'a"   and y :: "'a" shows "(x = x \<squnion> y) = (y \<sqsubseteq> x)"
extremum_uniqueI: fixes a :: "'d upred" assumes "false \<sqsubseteq> a" shows "a = false"
filtrationE2: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and s :: "'d"   and t :: "'d" assumes "Filtration.filtration M F"   and "s \<le> t" shows "subalgebra (F t) (F s)"
big_sound: shows "sound ([(p # \<Gamma>, q # closeI \<Delta> j), (q # \<Gamma>, p # closeI \<Delta> j)], \<Gamma>, \<Delta>)"
fmrel_upd: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and x :: "'c"   and y :: "'d"   and k :: "'e" assumes "fmrel P m n"   and "P x y" shows "fmrel P (fmupd k x m) (fmupd k y n)"
s_is_bounded_by_g1_and_g2: shows "s = {(y, x). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g1 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g2 x \<bullet> i)}"
Mp_to_int_poly: fixes m :: "int"   and f :: "'a mod_ring poly" assumes "poly_mod_type TYPE('a) m" shows "poly_mod.Mp m (to_int_poly f) = to_int_poly f"
Gr_empty: fixes f :: "'b \<Rightarrow> 'c" shows "BNF_Def.Gr {} f = {}"
Smith_nx2_def: fixes A :: "'a mat" shows "Smith_nx2 A = (let (P, S, Q) = Smith_2xn A\<^sup>T in (Q\<^sup>T, S\<^sup>T, P\<^sup>T))"
unit_vectors_def: fixes n :: "nat" shows "unit_vectors n \<equiv> {unit_vec n i |i. 0 \<le> i \<and> i < n}"
id_galcon_def: fixes A :: "'d gorder" shows "I\<^sub>g A = \<lparr>orderA = A, orderB = A, lower = id, upper = id\<rparr>"
ParTR: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>t s'" shows "(c1 | c2, s) \<rightarrow>c (c1, s')"
mapWithIndex_def: fixes f :: "nat \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "'d list" shows "mapWithIndex f xs = map (\<lambda>(i, t). f i t) (enumerate 0 xs)"
lexordp_linear: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp xs ys \<or> xs = ys \<or> signed.lexordp ys xs"
wfp_on_mono: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "wfp_on Q B" shows "wfp_on P A"
sat\<^sub>2: shows "eval\<^sub>c F G C\<^sub>2"
r_less: fixes x :: "nat"   and y :: "nat" shows "eval r_less [x, y] \<down>= (if x < y then 0 else 1)"
black_def: fixes r :: "'d" shows "black r \<equiv> \<lambda>s. marked r s \<and> \<not> grey r s"
compact_convex_combinations: fixes S :: "'b set"   and T :: "'b set" assumes "compact S"   and "compact T" shows "compact {(1 - u) *\<^sub>R x + u *\<^sub>R y |x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T}"
continuous_on: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x ** g x)"
sel_3: fixes x21 :: "'i"   and x22 :: "'g" shows "continuation (IO x21 x22) = x22"
get_disconnected_nodes_pure: fixes document_ptr :: "(_) document_ptr"   and h :: "(_) heap" shows "pure (get_disconnected_nodes document_ptr) h"
eval_Comp: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Dom t = Cod u" shows "\<lbrace>t \<^bold>\<cdot> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot>\<^sub>D \<lbrace>u\<rbrace>"
index_not_p_between: fixes p :: "nat \<Rightarrow> bool"   and n :: "nat"   and i :: "nat" assumes "infinitely_many p"   and "infinitely_many.index p n < i"   and "i < infinitely_many.index p (Suc n)" shows "\<not> p i"
f5: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a = (0::'a) \<or> (b = c / a) = (b * a = c)"
rel_spmf_try_spmf: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and p' :: "'b spmf"   and q :: "'a spmf"   and q' :: "'b spmf" assumes "rel_spmf R p p'"   and "\<not> lossless_spmf p' \<Longrightarrow> rel_spmf R q q'" shows "rel_spmf R (TRY p ELSE q) (TRY p' ELSE q')"
subdesign_iso_implies: fixes V :: "'a set"   and B :: "'a set multiset" assumes "sub_set_system V B \<V> \<B>" shows "sub_set_system (\<pi> ` V) (blocks_image B \<pi>) \<V>' \<B>'"
GSMP_proj_is_homogeneous: fixes A :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and l :: "'b" assumes "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' A)) Sec"   and "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l A))"   and "t \<notin> Sec" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t A Sec"
get_odd_double: fixes n :: "nat" shows "\<exists>i. get_odd n = 2 * i + 1"
BasicRule: fixes p :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and q :: "'a set" assumes "p \<subseteq> {s. f s \<in> q}" shows "\<parallel>- p Basic f q"
allDefs_def: fixes g :: "'g"   and n :: "'node" shows "step.allDefs g n \<equiv> defs g n \<union> step.phiDefs g n"
set_atoms0_iff: fixes \<phi> :: "atom fm"   and a :: "atom" assumes "qfree \<phi>" shows "(a \<in> set (atoms\<^sub>0 \<phi>)) = (a \<in> atoms \<phi> \<and> hd_coeff a \<noteq> 0)"
DSpace_def: shows "DSpace = Abs_dataspace [range V0, range V1]"
mult_strict_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less zero b"   and "less_eq zero c" shows "less (times a c) (times b d)"
swap_def: fixes p :: "'l \<times> 'k" shows "prod.swap p = (snd p, fst p)"
simps_1: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_aux_PQ [] bezout (P, A, Q) = (P, A, Q)"
weakenBisimE_4: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx>\<^sub>w Q" shows "\<Psi> \<rhd> Q \<approx>\<^sub>w P"
scaleR_diff_right: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
irrefl: fixes a :: "'a Word.word" shows "\<not> a <s a"
set_iterator_genord_filter_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" assumes "set_iterator_genord it S R" shows "set_iterator_genord (set_iterator_filter P it) {x \<in> S. P x} R"
float_distinct_finite_1: fixes a :: "('c, 'd) IEEE.float" shows "\<not> (is_nan a \<and> is_finite a)"
omega_unfold: fixes x :: "'a" shows "x\<^sup>\<omega> \<le> x \<cdot> x\<^sup>\<omega>"
I_def: shows "I \<equiv> Abs_pmf (distr M (count_space UNIV) (X 0))"
cinner_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and r :: "complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (scaleC r x) y = cnj r * cinner x y"
mset_swap: fixes i :: "nat"   and ls :: "'a list"   and j :: "nat" assumes "i < length ls"   and "j < length ls" shows "mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
fun_upd_upd: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'h"   and z :: "'h" shows "f(x := y, x := z) = f(x := z)"
B: fixes x :: "'a" assumes "x \<in> S" shows "norm x \<le> B"
scaleRe_ivl_spec_def: fixes l :: "ereal"   and u :: "ereal"   and X :: "('a \<times> 'b) set" shows "scaleRe_ivl_spec l u X = SPEC (\<lambda>Y. Y = scaleR2 l u X)"
left_impl_one: fixes a :: "'b" shows "a l\<rightarrow> a = (1::'b)"
wf_graph_remove_edges_union: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set" assumes "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
limitin_const: fixes l :: "'b"   and F :: "'c filter" shows "limitin euclidean (\<lambda>a. l) l F"
perp_in_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X B A C D"
local_and_fork_commute: fixes s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and x :: "('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r'' :: "'r"   and e :: "('r, 'l, 'v) expr" assumes "s\<^sub>2 = s\<^sub>1(r \<mapsto> x)"   and "s\<^sub>2' = s\<^sub>1(r' \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r'')]), r'' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"   and "s\<^sub>2 r' = Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r'' \<notin> RID\<^sub>G s\<^sub>2"   and "revision_step r s\<^sub>2' (s\<^sub>1(r' \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r'')]), r'' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e), r \<mapsto> x))"   and "r \<noteq> r'"   and "r \<noteq> r''" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
circ_back_loop_prefixpoint: fixes z :: "'a"   and y :: "'a" shows "z \<sqinter> top \<sqinter> y \<squnion> z \<le> z \<sqinter> top"
posaux: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "position a l < position b l" shows "a \<noteq> b"
g: fixes i :: "nat" shows "g i \<in> M \<rightarrow>\<^sub>M N"
intros_6: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and fs :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and n :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a)" shows "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n None (Some (p, False)) False n'"
filter_for_biflows_un_filter_for_uniflows: fixes E :: "('a \<times> 'a) list" shows "set (filter_for_biflows E) \<union> set (filter_for_uniflows E) = set E"
collapse: fixes interval :: "'a interval" shows "Interval (interval.lower interval) (interval.upper interval) = interval"
normal: fixes i :: "nat" assumes "i + 1 < length \<HH>" shows "\<HH> ! i \<lhd> G\<lparr>carrier := H, carrier := \<HH> ! (i + 1)\<rparr>"
brc_det_algo_def: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" shows "brc_det_algo rqrm \<delta> \<equiv> \<lparr>dwa_cond = brc_cond, dwa_step = brc_step rqrm, dwa_initial = brc_initial \<delta>, dwa_invar = brc_invar (ls.\<alpha> \<delta>)\<rparr>"
induct: fixes x1 :: "('e \<times> 'f) list"   and x2 :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and P :: "('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) set \<Rightarrow> bool" assumes "(x1, x2) \<in> c_failures step out s\<^sub>0"   and "P [] {(x, p). p \<noteq> out s\<^sub>0 x}"   and "\<And>xps uu_ s x. \<lbrakk>(xps, uu_) \<in> c_failures step out s\<^sub>0; P xps uu_; s = foldl step s\<^sub>0 (map fst xps)\<rbrakk> \<Longrightarrow> P (xps @ [(x, out s x)]) {(y, p). p \<noteq> out (step s x) y}"   and "\<And>xps Y X. \<lbrakk>(xps, Y) \<in> c_failures step out s\<^sub>0; P xps Y; X \<subseteq> Y\<rbrakk> \<Longrightarrow> P xps X" shows "P x1 x2"
path_from_to_ends: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and ys :: "'a list" assumes "v \<leadsto>(xs @ w # ys)\<leadsto>\<^bsub>H\<^esub> w" shows "ys = []"
Aenc_synth_eq2: fixes X :: "msg"   and H :: "msg set"   and K :: "msg" assumes "X \<notin> synth H" shows "(Aenc X K \<in> synth H) = (Aenc X K \<in> H)"
Holder_inequality_1: fixes p :: "real"   and q :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "0 < p"   and "0 < q"   and "1 / p + 1 / q = 1"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"   and "integrable M (\<lambda>x. \<bar>g x\<bar> powr q)" shows "integrable M (\<lambda>x. f x * g x)"
is_acc_run_def: fixes G :: "('c, 'd) igb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'c" assumes "igb_graph G" shows "igb_graph.is_acc_run G r \<equiv> graph_defs.is_run G r \<and> igb_graph.is_acc G r"
cong: fixes s1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" shows "vector_space_pair.construct s1 s2 \<equiv> vector_space_pair.construct s1 s2"
mult2_s_locally_cancel1: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and z :: "'a"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "s O ns \<subseteq> s"   and "ns O s \<subseteq> s"   and "refl ns"   and "trans ns"   and "locally_irrefl s (add_mset z X)"   and "locally_irrefl s (add_mset z Y)"   and "trans s"   and "(add_mset z X, add_mset z Y) \<in> mult2_s ns s" shows "(X, Y) \<in> mult2_s ns s"
acyclicI_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'a" assumes "class.preorder less_eq less"   and "\<And>a b. (a, b) \<in> r \<Longrightarrow> less (f b) (f a)" shows "acyclic r"
divisors_int_pos_def: fixes x :: "int" shows "divisors_int_pos x \<equiv> map int (divisors_nat (nat \<bar>x\<bar>))"
assms_2: fixes x :: "'a" assumes "x \<in># B" shows "prime x"
conf_NT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> NT) = (v = Null)"
Un_iff: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
assms_4: shows "x \<sharp> \<Psi>\<^sub>F"
uomega_def: fixes P :: "'b hrel" shows "P\<^sup>\<omega> = \<mu> ((;;) P)"
sum_mset_replicate_mset: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum_mset plus zero (replicate_mset n a) = times (semiring_1.of_nat one plus zero n) a"
observable_move_notempty: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')"   and "as = []" shows "False"
sq: fixes p :: "real poly" assumes "p \<in> set fss" shows "square_free p"
assms_1: shows "sys \<Turnstile>\<^bsub>pre\<^esub> P"
arc_subpath_arc: fixes g :: "real \<Rightarrow> 'b"   and u :: "real"   and v :: "real" assumes "arc g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<noteq> v" shows "arc (subpath u v g)"
cut_ge_0_all: fixes I :: "nat set" shows "I \<down>\<ge> 0 = I"
contour_integral_subpath_combine_less: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real"   and w :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}"   and "u < v"   and "v < w" shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f"
CR_iff_conversion_imp_join: fixes r :: "('b \<times> 'b) set" shows "CR r = (r\<^sup>\<leftrightarrow>\<^sup>* \<subseteq> r\<^sup>\<down>)"
headmarked_def: shows "headmarked = (\<lambda>(x, y). inmarked y)"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>Z {x1} x2 {x3}"   and "\<And>Pa. P (\<lambda>s. eSuc (Pa s)) SKIP Pa"   and "\<And>Pa a x. P (\<lambda>s. eSuc (Pa (s[a/x]))) (x ::= a) Pa"   and "\<And>Pa b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>Z {\<lambda>s. Pa s + \<up> (bval b s)} c\<^sub>1 {Q}; P (\<lambda>s. Pa s + \<up> (bval b s)) c\<^sub>1 Q; \<turnstile>\<^sub>Z {\<lambda>s. Pa s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}; P (\<lambda>s. Pa s + \<up> (\<not> bval b s)) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P (\<lambda>s. eSuc (Pa s)) (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>Z {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>Z {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>I b c. \<lbrakk>\<turnstile>\<^sub>Z {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}; P (\<lambda>s. I s + \<up> (bval b s)) c (\<lambda>t. I t + 1)\<rbrakk> \<Longrightarrow> P (\<lambda>s. I s + 1) (WHILE b DO c) (\<lambda>s. I s + \<up> (\<not> bval b s))"   and "\<And>Pa c Q P' Q'. \<lbrakk>\<turnstile>\<^sub>Z {Pa} c {Q}; P Pa c Q; \<And>s. Pa s \<le> P' s; \<And>s. Q' s \<le> Q s\<rbrakk> \<Longrightarrow> P P' c Q'"   and "\<And>k Pa c Q. \<lbrakk>\<turnstile>\<^sub>Z {\<lambda>s. enat k * Pa s} c {\<lambda>s. enat k * Q s}; P (\<lambda>s. enat k * Pa s) c (\<lambda>s. enat k * Q s); 0 < k\<rbrakk> \<Longrightarrow> P Pa c Q" shows "P x1 x2 x3"
a0n_anyP_observable: fixes P :: "a0n_state set" shows "observable (obs a0n) P"
basis_reduction_main_bound: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list"   and fs' :: "int vec list" assumes "LLL_bound_invariant True upw i fs"   and "basis_reduction_main (upw, i, fs) = fs'" shows "LLL_bound_invariant True True m fs'"
eventually_frequently: fixes F :: "'a filter"   and P :: "'a \<Rightarrow> bool" assumes "F \<noteq> bot"   and "eventually P F" shows "frequently P F"
rep_eq: fixes x :: "'a st" shows "fun x = fun_rep (rep_st x)"
l11_36_aux2: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and D :: "'p"   and E :: "'p"   and D' :: "'p"   and F :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A' \<noteq> B"   and "D \<noteq> E"   and "D' \<noteq> E"   and "Bet A B A'"   and "Bet D E D'"   and "D' E F LeA A' B C" shows "A B C LeA D E F"
swap_apply_2: fixes a :: "'e"   and b :: "'e"   and f :: "'e \<Rightarrow> 'd" shows "Fun.swap a b f b = f a"
alls_nth0: fixes A :: "nat"   and as :: "nat list" shows "alls A as ! 0 = (zeroes (length as), 0)"
assms_4: shows "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j"
psubstEnv_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodEnv rho"   and "goodEnv rho'" shows "goodEnv rho &[rho']"
simps_2: fixes s :: "('d, 'e) fmap"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" shows "exec_plan s (a # as) = exec_plan (state_succ s a) as"
4: shows "(c, s) \<rightarrow>*c (c'', s'')"
lm084: fixes N :: "'d set"   and G :: "'e list"   and bids :: "'d \<times> 'e set \<Rightarrow> 'f" assumes "N \<noteq> {}"   and "finite N"   and "distinct G"   and "set G \<noteq> {}" shows "winningAllocationRel N (set G) (\<inter>) bids (set (allAllocationsAlg N G)) \<noteq> {}"
add_0_left: fixes a :: "'a" shows "bot \<squnion> a = a"
assms_2: fixes P :: "ccs"   and Q :: "ccs" assumes "(P, Q) \<in> X" shows "P \<leadsto>[X] Q"
assms_6: shows "Q_J_neg = filterQuery J (Qn - Q_I_neg)"
simps_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x1 :: "'c" shows "(case Same x1 of Same x \<Rightarrow> f1 x | Diff x \<Rightarrow> f2 x) = f1 x1"
ffact_fact: fixes n :: "nat" shows "ffact n (of_nat n) = of_nat (fact n)"
comp_def: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "f \<circ> g = (\<lambda>x. f (g x))"
d1: shows "diagram ars (\<sigma>, \<tau>, \<tau>', \<sigma>')"
disjoint_family_on_bisimulation: fixes f :: "'b \<Rightarrow> 'c set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'd set" assumes "disjoint_family_on f S"   and "\<And>n m. \<lbrakk>n \<in> S; m \<in> S; n \<noteq> m; f n \<inter> f m = {}\<rbrakk> \<Longrightarrow> g n \<inter> g m = {}" shows "disjoint_family_on g S"
LLL_invariant_weak_def: fixes fs :: "int vec list" shows "LLL_invariant_weak fs = (gs.lin_indpt_list (map of_int_hom.vec_hom fs) \<and> lattice_of fs = L \<and> length fs = m)"