hcju/isabelleps huggingface dataset:
postdominate_variant: fixes n' :: "'node"   and n :: "'node" assumes "n' postdominates n" shows "\<forall>as. n -as\<rightarrow>\<^sub>\<surd>* (_Exit_) \<longrightarrow> n' \<in> set (sourcenodes as)"
pos: fixes i :: "nat"   and x :: "'a"   and C :: "'a set" shows "0 \<le> f ((T ^^ i) x) * indicator C x"
1: shows "Domain a = Domain {(x, Y) |Y x. Y \<in> finestpart (a ,, x) \<and> x \<in> Domain a}"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Var x1 = Var y1) = (x1 = y1)"
finite_measure_subadditive_finite: fixes I :: "'a set"   and A :: "'a \<Rightarrow> nat set set set" assumes "finite I"   and "A ` I \<subseteq> events" shows "prob (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. prob (A i))"
unique_idxI: fixes bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "\<And>f g. \<lbrakk>f \<in> set bs; g \<in> set bs; fst (snd f) = fst (snd g)\<rbrakk> \<Longrightarrow> f = g"   and "\<And>f. f \<in> set bs \<Longrightarrow> fst (snd f) < fst data" shows "unique_idx bs data"
open_Int: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "open T" shows "open (S \<inter> T)"
comp_cod_arr: fixes f :: "'s"   and b :: "'s" assumes "arr f"   and "cod f = b" shows "b \<cdot> f = f"
some1_equality: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "\<exists>!x. P x"   and "P a" shows "(SOME x. P x) = a"
less_numeral_extra_1: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
elSuct3: shows "z \<in> set (s_TS init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i + 1))"
relcompp_mono: fixes r' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "r' \<le> r"   and "s' \<le> s" shows "r' OO s' \<le> r OO s"
d_OUT_alt_def: fixes x :: "'k"   and G :: "('k, 'l) graph_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "\<And>y. (x, y) \<notin> \<^bold>E\<^bsub>G\<^esub> \<Longrightarrow> g (x, y) = 0" shows "d_OUT g x = (\<Sum>\<^sup>+ y\<in>\<^bold>O\<^bold>U\<^bold>T\<^bsub>G\<^esub> x. g (x, y))"
funpow_simps_right_2: fixes n :: "nat"   and f :: "'k \<Rightarrow> 'k" shows "f ^^ Suc n = f ^^ n \<circ> f"
SN_I: fixes A :: "('a \<times> 'a) set" assumes "\<And>a. SN_on A {a}" shows "SN A"
one_stalk_def: fixes V :: "'c set" shows "stfx.one_stalk V \<equiv> stfx.class_of V (one_str\<^sub>Y V)"
in_lists_conv_set: fixes xs :: "'c list"   and A :: "'c set" shows "(xs \<in> lists A) = (\<forall>x\<in>set xs. x \<in> A)"
fds_abs_converges_of_real: fixes f :: "real fds"   and s :: "real" shows "fds_abs_converges (fds_of_real f) (of_real s) = fds_abs_converges f s"
a_set_child_nodes_def: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr children"
Nmlize_Src_1: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Src t\<^bold>\<rfloor> = Src \<^bold>\<lfloor>t\<^bold>\<rfloor>"
full_abstraction_impl_SRel_iff_TRel_is_refl: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)" shows "refl SRel = (\<forall>S. (\<lbrakk>S\<rbrakk>, \<lbrakk>S\<rbrakk>) \<in> TRel)"
f_def: fixes i :: "nat" shows "f i = insertion (nth_default 0 (L[var := B / C])) (isolate_variable_sparse p var i) * B ^ i"
succsr_isNode: fixes x :: "'a \<times> 'b \<times> 'a"   and y :: "'a \<times> 'b \<times> 'a" assumes "(x, y) \<in> succsr\<^sup>*"   and "x \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0" shows "y \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"
isomorphic_reflexive: fixes f :: "'a \<times> 'a" assumes "VoV.ide f" shows "VoV.isomorphic f f"
inject_2: fixes x2 :: "('a, 'b) atm"   and y2 :: "('a, 'b) atm" shows "(Neg x2 = Neg y2) = (x2 = y2)"
l2_\<phi>I_DERIV: fixes i' :: "int"   and l' :: "nat"   and \<Phi>_n :: "real \<Rightarrow> real"   and l :: "nat"   and i :: "int"   and \<Phi>_p :: "real \<Rightarrow> real" assumes "\<And>x. x \<in> {(real_of_int i' - 1) / 2 ^ (l' + 1)..real_of_int i' / 2 ^ (l' + 1)} \<Longrightarrow> (\<Phi>_n has_real_derivative \<phi> (l', i') x * \<phi> (l, i) x) (at x)"   and "\<And>x. x \<in> {real_of_int i' / 2 ^ (l' + 1)..(real_of_int i' + 1) / 2 ^ (l' + 1)} \<Longrightarrow> (\<Phi>_p has_real_derivative \<phi> (l', i') x * \<phi> (l, i) x) (at x)" shows "l2_\<phi> (l', i') (l, i) = \<Phi>_n (real_of_int i' / 2 ^ (l' + 1)) - \<Phi>_n ((real_of_int i' - 1) / 2 ^ (l' + 1)) + (\<Phi>_p ((real_of_int i' + 1) / 2 ^ (l' + 1)) - \<Phi>_p (real_of_int i' / 2 ^ (l' + 1)))"
assms: shows "orientable_4 (x \<sqinter> - (1::'a))"
lift_mult_TC_disjoint: fixes x :: "V"   and y :: "V"   and a :: "V" assumes "x \<noteq> y" shows "lift (a * x) (TC a) \<sqinter> lift (a * y) (TC a) = 0"
param_take: fixes Rb :: "('a \<times> 'b) set" shows "(take, take) \<in> nat_rel \<rightarrow> \<langle>Rb\<rangle>list_rel \<rightarrow> \<langle>Rb\<rangle>list_rel"
1: shows "bprv (PP \<langle>imp \<phi>G (neg (PP \<langle>\<phi>G\<rangle>))\<rangle>)"
lNact_active: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>the_singleton \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> = n"
mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> O[F](g1)"   and "f2 \<in> O[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)"
not_par_not_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A \<noteq> C"   and "\<not> A B Par A C" shows "\<not> Col A B C"
compact_continuous_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "inj_on f S" shows "continuous_on S f = (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact (f ` T))"
uminus_uminus_vec: fixes v :: "'a vec" shows "- (- v) = v"
inject_2: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and y31 :: "('a, 'b, 'c) Stmt"   and y32 :: "'a ModeUpd" shows "(x31@[x32] = y31@[y32]) = (x31 = y31 \<and> x32 = y32)"
mat_hom_one: fixes n :: "nat" shows "of_real_hom.mat_hom (1\<^sub>m n) = 1\<^sub>m n"
subseq_rev_drop_many: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "subseq xs ys" shows "subseq xs (ys @ zs)"
less_char_def: fixes c1 :: "char"   and c2 :: "char" shows "(c1 < c2) = (of_char c1 < of_char c2)"
var_pow_base_independent: fixes K :: "'a list set"   and n :: "nat" assumes "subfield K (K [X])" shows "ring.independent (K [X]\<^bsub>K [X]\<^esub>) (UP.poly_of_const ` K) (ring.exp_base (K [X]\<^bsub>K [X]\<^esub>) X\<^bsub>K [X]\<^esub> n)"
hd_in_set: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "list.hd xs \<in> set xs"
exec_n_end: fixes P :: "instr list"   and n :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and k :: "nat"   and n' :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "size P \<le> n" shows "P \<turnstile> (n, s, stk) \<rightarrow>^k (n', s', stk') = (n' = n \<and> stk' = stk \<and> s' = s \<and> k = 0)"
compare_sym: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "compare\<cdot>x\<cdot>y = (case compare\<cdot>y\<cdot>x of LT \<Rightarrow> GT | EQ \<Rightarrow> EQ | GT \<Rightarrow> LT)"
map_comp: fixes g1 :: "'g \<Rightarrow> 'e"   and g2 :: "'h \<Rightarrow> 'f"   and f1 :: "'i \<Rightarrow> 'g"   and f2 :: "'j \<Rightarrow> 'h"   and v :: "('i, 'j) F" shows "F g1 g2 (F f1 f2 v) = F (g1 \<circ> f1) (g2 \<circ> f2) v"
S\<^sub>r_nS2: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and s :: "'a" assumes "Reachability_Problem K S S1 S2"   and "s \<in> Reachability_Problem.S\<^sub>r K S S1 S2" shows "s \<notin> S2"
map_ms_altdef: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "('b \<times> real) msllist"   and g :: "real \<Rightarrow> real" shows "map_ms f (MS xs g) = MS (map_ms_aux f xs) g"
Append_to_free1: fixes m :: "node list"   and e :: "(nat \<times> nat) list"   and i :: "nat" assumes "Proper_Edges (m, e)" shows "Proper_Edges (m, Append_to_free (i, e))"
simps: fixes cs :: "QDelta ns_constraint list"   and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "check_simplex (Simplex_State (cs, asi_tv, s)) = (case check_s s of Inl x \<Rightarrow> Inl x | Inr s' \<Rightarrow> Inr (Simplex_State (cs, asi_tv, s')))"
pre: shows "\<forall>s. P s \<longrightarrow> (I \<and>* $ 1) s"
zip_Cons1: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'b list" shows "zip (x # xs) ys = (case ys of [] \<Rightarrow> [] | y # ys \<Rightarrow> (x, y) # zip xs ys)"
right_closed: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E" shows "b \<in> S"
distinct_95: fixes x3 :: "val"   and x141 :: "'a exp"   and x142 :: "'a exp" shows "Val x3 \<noteq> x141;; x142"
dels_unsatisfied: fixes f :: "object atom formula"   and d :: "object atom formula list"   and a :: "object atom formula list"   and M :: "object atom formula set" assumes "f \<in> set d"   and "f \<notin> set a" shows "f \<notin> apply_effect (Effect a d) M"
ParComp: fixes R :: "pi"   and S :: "pi"   and T :: "pi"   and U :: "pi" assumes "(R, S) \<in> Rel"   and "(T, U) \<in> Rel'" shows "(R \<parallel> T, S \<parallel> U) \<in> Rel'"
neutral_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "neutral_sds_axioms agents alts sds \<equiv> \<forall>\<sigma> R. \<sigma> permutes alts \<longrightarrow> pref_profile_wf agents alts R \<longrightarrow> sds (permute_profile \<sigma> R) = map_pmf \<sigma> (sds R)"
snth_run: fixes r :: "('c \<times> 'd) stream"   and A :: "('c, 'd) nba"   and p :: "'d" assumes "\<And>k. fst (r !! k) \<in> alphabet A \<and> snd (r !! k) \<in> transition A (fst (r !! k)) (target (stake k r) p)" shows "run A r p"
bind_maybe_simps_3: fixes x :: "'d"   and f :: "'d \<rightarrow> 'c\<cdot>maybe" shows "Just\<cdot>x \<bind> f = f\<cdot>x"
Domain_nodes_connected: fixes G :: "('a, 'b) graph" assumes "valid_graph G" shows "Domain {(x, y) |x y. nodes_connected G x y} = nodes G"
bas: fixes x :: "'a set" assumes "x \<in> B" shows "P x"
SigmaE: fixes c :: "'b \<times> 'd"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'd set" assumes "c \<in> Sigma A B" obtains x :: "'b" and y :: "'d" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
map_lhs_pivot: shows "map lhs (\<T> (pivot' x\<^sub>i x\<^sub>j s)) = map (\<lambda>idx. lhs (\<T> s ! idx)) [0..<eq_idx_for_lvar (\<T> s) x\<^sub>i] @ [x\<^sub>j] @ map (\<lambda>idx. lhs (\<T> s ! idx)) [eq_idx_for_lvar (\<T> s) x\<^sub>i + 1..<length (\<T> s)]"
Inf_lower2: fixes u :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set"   and v :: "'c \<times> 'd" assumes "u \<in> A"   and "less_eq_prod u v" shows "less_eq_prod (Inf_prod A) v"
swap_def: fixes i :: "'c"   and j :: "'c"   and x :: "'c" shows "swap i j x = (if x = i then j else if x = j then i else x)"
variant_decreases: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "bval b s"   and "\<pi>: (c, s) \<Rightarrow> s'"   and "\<pi>: (WHILE b DO c, s') \<Rightarrow> t" shows "variant \<pi> b c s' < variant \<pi> b c s"
disc_iff: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c" assumes "ParityGame G" shows "\<not> lnull (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0)"
segments_subset_convex_hull_6: fixes c :: "'a"   and b :: "'a"   and a :: "'a" shows "closed_segment c b \<subseteq> convex hull {a, b, c}"
emb_prj: fixes x :: "udom" shows "EMB('e)\<cdot>(PRJ('e)\<cdot>x) = cast\<cdot>DEFL('e)\<cdot>x"
closed: fixes e :: "'a llist" assumes "e \<in> A" shows "\<exists>p. lprefix (llist_of p) e \<and> (\<forall>e'. lprefix (llist_of p) e' \<longrightarrow> llength e' = 2 * N \<longrightarrow> e' \<in> A)"
assms_1: shows "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> A)"
real_lebesgue_integral_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) - enn2real (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M)"
M'_def: shows "M' = Pi\<^sub>M UNIV (\<lambda>i. distr M borel (X i))"
Sup_subset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.greater_eq B A" shows "less_eq (Sup A) (Sup B)"
sup_closed_set_def: fixes X :: "'b set" shows "sup_closed_set X = (\<forall>x\<in>X. \<forall>y\<in>X. x \<squnion> y \<in> X)"
127: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
linorder_class_def: shows "OFCLASS('a, linorder_class) \<equiv> (OFCLASS('a, order_class) &&& class.linorder_axioms (\<le>))"
asTerm_pick: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
first_summand_in: fixes x :: "i"   and y :: "i" assumes "GEM1" shows "P x (x \<oplus> y)"
nth_equal_imp_dist_less: fixes i :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "\<And>j. j \<le> i \<Longrightarrow> fps_nth f j = fps_nth g j" shows "dist f g < inverse (2 ^ i)"
ordinality_of_utility_function: fixes f :: "real \<Rightarrow> real"   and y :: "'a"   and x :: "'a" assumes "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f" shows "((f \<circ> u) y < (f \<circ> u) x) = (u y < u x)"
cases: fixes x :: "tm \<times> tm \<times> tm \<times> tm \<times> tm" obtains g1 :: "name" and g2 :: "name" and f1 :: "tm" and k1 :: "tm" and f2 :: "tm" and k2 :: "tm" and g :: "tm" where "atom g1 \<sharp> (g2, f1, k1, f2, k2, g)" and "atom g2 \<sharp> (f1, k1, f2, k2, g)" and "x = (f1, k1, f2, k2, g)"
env_ext_db_def: shows "env_ext_db \<equiv> \<Lambda> x \<rho> v. case v of 0 \<Rightarrow> x | Suc v' \<Rightarrow> \<rho>\<cdot>v'"
nth_rewrite_eq: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < length xs" shows "rewrite xs i x ! i = x"
BNTr4: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W1 :: "'a set"   and W2 :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a"   and "\<exists>b\<in>carrier D. ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))" shows "W1 \<subseteq> W2"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b stream"   and b :: "'c stream" shows "stream_all2 R a b = (R (shd a) (shd b) \<and> stream_all2 R (stl a) (stl b))"
parts_mpair_key: fixes s :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and K :: "key" assumes "s\<^sub>0 \<Turnstile> s"   and "\<lbrace>X, Y\<rbrace> \<in> parts (used s)"   and "X = Key K \<or> Y = Key K \<and> K \<notin> range PubK" shows "{X, Y} \<subseteq> spied s"
inject_3: fixes x3 :: "name"   and x1 :: "name"   and x2 :: "pi"   and y3 :: "name"   and y1 :: "name"   and y2 :: "pi" shows "(x3<x1>.x2 = y3<y1>.y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
diff_gt_0_iff_gt: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (minus a b) = less b a"
cases: fixes x :: "'b equation \<times> ('b \<times> 'b trm) list" obtains u :: "'b trm" and v :: "'b trm" and s :: "('b \<times> 'b trm) list" where "x = (Eq u v, s)"
assms_9: shows "d (pp_of_term u) \<le> dgrad_max d"
big_iff_small: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = (P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle> \<and> final e')"
set_less_eq_aux'_infinite_complement_partitionD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "ord.set_less_eq_aux' less_eq A B" shows "A \<in> infinite_complement_partition \<and> B \<in> infinite_complement_partition"
pos: shows "\<forall>n. 0 < (\<Sum>k = n * 2..<n * 2 + 2. (- 1) ^ k / fact (2 * k + 1) * x ^ (2 * k + 1))"
assms: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and v :: "nat" assumes "p m \<noteq> (0::'a)"   and "lookup m v \<noteq> 0" shows "f v = g v"
abs_conv_abscissa_PInf_iff: fixes f :: "'a fds" shows "(abs_conv_abscissa f = \<infinity>) = (\<forall>s. \<not> fds_abs_converges f s)"
g_def: shows "g = (\<lambda>x. \<Sum>n. (1 / 2) ^ Suc n * indicat_real (A n) x / (1 + Sigma_Algebra.measure M (A n)))"
VSeq_implies_HPar: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "VSeq t u" shows "HPar t u"
init_fin_FWdelay_bisimulation_obs: shows "FWdelay_bisimulation_obs r1.init_fin_final r1.init_fin r2.init_fin_final r2.init_fin init_fin_bisim init_fin_bisim_wait r1.init_fin_\<tau>move r2.init_fin_\<tau>move"
elims: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "nat"   and xb :: "'b fps"   and xc :: "nat"   and y :: "'b" assumes "radical x xa xb xc = y" obtains r :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" and a :: "'b fps" where "x = ra__" and "xa = 0" and "xb = aa__" and "xc = 0" and "y = (1::'b)"   | r :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" and a :: "'b fps" and n :: "nat" where "x = rb__" and "xa = 0" and "xb = ab__" and "xc = Suc n__" and "y = (0::'b)"   | r :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" and k :: "nat" and a :: "'b fps" where "x = rc__" and "xa = Suc k__" and "xb = ac__" and "xc = 0" and "y = rc__ (Suc k__) (ac__ $ 0)"   | r :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" and k :: "nat" and a :: "'b fps" and n :: "nat" where "x = r" and "xa = Suc k" and "xb = a" and "xc = Suc n" and "y = (a $ Suc n - (\<Sum>xs | xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs. \<Prod>j = 0..k. radical r (Suc k) a (xs ! j))) / (of_nat (Suc k) * radical r (Suc k) a 0 ^ k)"
empty_def: shows "local.empty \<equiv> \<lambda>i. Const 0"
sl_def: shows "sl = (\<lambda>(A, r, f). (err A, Err.le r, lift2 f))"
hid_Skip: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "hid P cs = (hid P cs ;; action.relation_of Skip)"
fn_monotone: shows "monotone (\<le>) (\<sqsubseteq>) (\<lambda>n. f^n \<bottom>)"
cr_cr_vfsequence_bi_unqie: shows "bi_unique cr_cr_vfsequence"
functor_axioms: shows "functor C (\<cdot>\<^sub>S) I.map"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NPreflow c s t f"   and "Flow c s t f" shows "NFlow c s t f"
echelon_form_of_column_k_det_def: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A' :: "'d \<times> (('d, 'e) vec, 'f) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k_det bezout A' k = (let (det_P, A, i) = A'; from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if i \<noteq> nrows A \<and> A $ from_nat_i $ from_nat_k = (0::'d) \<and> (\<exists>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)) then (- (1::'d) * det_P, echelon_form_of_column_k bezout (A, i) k) else (det_P, echelon_form_of_column_k bezout (A, i) k))"
HBL2_imp2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}"   and "bprv (PP \<langle>imp \<phi> \<chi>\<rangle>)" shows "bprv (imp (PP \<langle>\<phi>\<rangle>) (PP \<langle>\<chi>\<rangle>))"
f_shrink_nth_eq_f_last_message_hold_nth: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div> k ! n = xs \<longmapsto> k ! (n * k + k - Suc 0)"
dgrad_max_def: fixes d :: "'a \<Rightarrow> nat" shows "dgrad_max d = Max (d ` insert (0::'a) (Keys (set fs)))"
support_flow_Sup: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set" shows "support_flow (\<Squnion> Y) = \<Union> (support_flow ` Y)"
WHILEI_def: shows "WHILEI \<equiv> \<lambda>I b f. REC (WHILEI_body bind return I b f)"
foldl_Cons: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c"   and x :: "'d"   and xs :: "'d list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
nth_insert_nth_front: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i < j"   and "j \<le> length xs" shows "insert_nth j x xs ! i = xs ! i"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Nom x2 = Nom y2) = (x2 = y2)"
prefixI: fixes ys :: "'e list"   and xs :: "'e list"   and zs :: "'e list" assumes "ys = xs @ zs" shows "prefix xs ys"
aff_dim_geq: fixes V :: "'a set" shows "- 1 \<le> aff_dim V"
indexle_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "indexle k' u' k u = (indexlt k' u' k u \<or> k' = k \<and> u' = u)"
card_Diff_subset: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "B \<subseteq> A" shows "card (A - B) = card A - card B"
assms_1: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
ps1: fixes b :: "real"   and a :: "real" assumes "b * 2 \<le> 1"   and "0 \<le> b"   and "0 \<le> a"   and "a \<le> 1" shows "p ((1 - a) * (2 * b) + a) \<in> s"
cases: fixes a :: "('test, 'atom) com" assumes "discr a" obtains c :: "('test, 'atom) com" where "a = c" and "\<And>s c' s'. (c, s) \<rightarrow>c (c', s') \<Longrightarrow> s \<approx> s' \<and> discr c'" and "\<And>s s'. (c, s) \<rightarrow>t s' \<Longrightarrow> s \<approx> s'"
ce_rel_lm_24: fixes r :: "(nat \<times> nat) set"   and s :: "(nat \<times> nat) set" assumes "r \<in> ce_rels"   and "s \<in> ce_rels" shows "s O r \<in> ce_rels"
prod_F_zcong: shows "[prod id F = (- 1) ^ card E * prod id E] (mod int p)"