hcju/isabelleps huggingface dataset:
upd_rev_cs_Cons_Call_Cons: 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 a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and cs :: "'a list"   and as :: "'a list"   and c' :: "'a"   and cs' :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "CFG.upd_rev_cs kind cs as = c' # cs'" shows "CFG.upd_rev_cs kind cs (a # as) = cs'"
f_def: shows "f = (\<lambda>u. u powr (- 1 / 2) / (1 + u) powr - a)"
sin_of_real: fixes x :: "real" shows "sin (of_real x) = of_real (sin x)"
Red: shows "s -\<triangleright>ttas\<rightarrow>* s'"
inject_7: fixes x2 :: "pi"   and x1 :: "pi"   and y2 :: "pi"   and y1 :: "pi" shows "(x2 \<parallel> x1 = y2 \<parallel> y1) = (x2 = y2 \<and> x1 = y1)"
set_homE_1: fixes f :: "'b set_arrow"   and U :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "f \<in> Hom\<^bsub>set_cat U\<^esub> A B" shows "A \<subseteq> U"
set_restrictionD: fixes f :: "'a set \<Rightarrow> 'a set" assumes "set_restriction f" shows "\<exists>P. \<forall>A. f A = {x \<in> A. P x}"
norm_2': fixes \<xi> :: "'a" assumes "\<parallel>\<xi>\<parallel> < r" shows "\<parallel>f *\<^sub>v (x + \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` ball (0::'a) r"
Ints_0: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "zero \<in> ring_1.Ints one plus zero minus"
lists_gen_to_hull: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists G\<^sub>+" shows "us \<in> lists \<langle>G\<rangle>\<^sub>+"
cyclic_on_orbit': fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "permutation f" shows "cyclic_on f (orbit f x)"
progress_ge: fixes \<phi> :: "Formula.formula"   and i :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace" assumes "Formula.future_bounded \<phi>" shows "\<exists>j. i \<le> Monitor.progress \<sigma> Map.empty \<phi> j"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = insert x21 (set x22)"
elims_2: fixes x :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "('a \<times> 'd) list \<times> 'o"   and xb :: "('a \<times> 'e) list \<times> 'o" assumes "prod_ord_raw x xa xb" obtains f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" and xs :: "('a \<times> 'd) list" and ox :: "'o" and ys :: "('a \<times> 'e) list \<times> 'o" where "x = f" and "xa = (xs, ox)" and "xb = ys" and "prod_ord_pair ox f xs (sort_oalist_aux ox ys)"
singleton_simplex: fixes X :: "'b set set"   and v :: "'b" assumes "ChamberComplex X"   and "v \<in> \<Union> X" shows "{v} \<in> X"
m'_inst_2: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup m' xs = Some tstp" shows "tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I)"
monad_set: shows "monad (\<lambda>x. {x}) (\<lambda>A f. \<Union> (f ` A))"
prv_scnj2_imp_cnj: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (scnj {\<phi>, \<psi>}) (cnj \<phi> \<psi>))"
bij_mono_map_set_to_nat_1: fixes A :: "'a set" assumes "finite A" shows "bij_betw (bij_mono_map_set_to_nat A) A {..<card A}"
simps_4: fixes a :: "Cooper.num" shows "bound0 (Le a) = numbound0 a"
AXH_def: shows "AXH \<equiv> {x. AXHp x}"
Cauchy_next_derivative_circlepath_1: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and k :: "nat"   and g :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "continuous_on (path_image (circlepath z r)) f"   and "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ k) has_contour_integral g w) (circlepath z r)"   and "k \<noteq> 0"   and "w \<in> ball z r" shows "(\<lambda>u. f u / (u - w) ^ Suc k) contour_integrable_on circlepath z r"
ev_blocks_part_leD: fixes m :: "nat"   and A :: "'a mat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "local.ev_blocks_part m A"   and "i \<le> j"   and "j \<le> k"   and "k < m"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"
parallel_converter2_eq_\<I>_cong: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>1' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and conv1' :: "('i, 'j, 'k, 'l) converter"   and \<I>2 :: "('m, 'n) \<I>"   and \<I>2' :: "('o, 'p) \<I>"   and conv2 :: "('m, 'n, 'o, 'p) converter"   and conv2' :: "('m, 'n, 'o, 'p) converter" assumes "\<I>1,\<I>1' \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>2' \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1' \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<sim> conv1' |\<^sub>= conv2'"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel"
lrev_def: shows "lrev \<equiv> \<mu> lrev. \<Lambda> v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>lnil) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(lrev\<cdot>xs :++ x :@ lnil)))"
multiset_le_def: fixes a :: "'b multiset"   and b :: "'b multiset"   and r :: "('b \<times> 'b) set" shows "multiset_le a b r \<equiv> a = b \<or> (a, b) \<in> mult r"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "sum g {m..n} = g m + sum g {Suc m..n}"
I_def: shows "I = Restr init_seg_of {r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
algebraicI: fixes p :: "'b list"   and K :: "'b set"   and x :: "'b" assumes "p \<in> carrier (K [X]\<^bsub>S\<^esub>)"   and "p \<noteq> []"   and "Hom.S.eval p x = \<zero>\<^bsub>S\<^esub>" shows "(Hom.S.algebraic over K) x"
Collect_subset: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
sets: fixes i :: "'a"   and A :: "'b set" assumes "i \<in> I"   and "A \<in> sets (M i)" shows "{x \<in> space N. x i \<in> A} \<in> sets N"
fi_f_eq: fixes x :: "'u"   and y :: "'v" assumes "f x = y"   and "x \<in> V" shows "fi y = x"
ascii_of_idem: fixes c :: "char" assumes "\<not> digit7 c" shows "String.ascii_of c = c"
lemma_2_4_5: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<le> (c r\<rightarrow> a) r\<rightarrow> c r\<rightarrow> b"
pivotandupdate_valuation_other_nolhs: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"   and "x \<notin> lvars (\<T> s)"   and "x \<noteq> x\<^sub>j" shows "Mapping.lookup (\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)) x = Mapping.lookup (\<V> s) x"
prev: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S"   and "\<forall>z\<in>S. f z \<noteq> 0" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = exp (g z))"
1: fixes u :: "'t" assumes "d (pp_of_term u) \<le> dgrad_max d"   and "component_of_term u < length fs" shows "is_sig_GB_in d G u"
elims: fixes x :: "'a literal"   and y :: "char list \<times> 'a list" assumes "get_atom x = y" obtains p :: "char list" and ts :: "'a list" where "x = Pos p__ ts__" and "y = (p__, ts__)"   | p :: "char list" and ts :: "'a list" where "x = Neg p ts" and "y = (p, ts)"
prems: shows "(v, x) \<notin> E\<^sup>+"
dist_fs_infinite1: fixes z1 :: "complex" shows "dist_fs (of_complex z1) \<infinity>\<^sub>h = 2 / sqrt (1 + (cmod z1)\<^sup>2)"
FtorCompComp': fixes F :: "('a, 'd, 'b, 'e, 'c, 'f, 'g) Functor_scheme"   and f :: "'b"   and g :: "'b"   and G :: "('d, 'h, 'e, 'i, 'f, 'j, 'k) Functor_scheme" assumes "f \<approx>>\<^bsub>CatDom F\<^esub> g"   and "F \<approx>>;;; G" shows "G ## F ## f ;;\<^bsub>CatDom F\<^esub> g = G ## F ## f ;;\<^bsub>CatCod G\<^esub> (G ## F ## g)"
IH: assumes "\<And>t'. Join t' \<in> set cas \<Longrightarrow> \<not> not_final_thread s t' \<and> t \<noteq> t'" shows "cond_action_oks s t cas"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b bot"   and b :: "'c bot" assumes "rel_bot R a b" obtains x :: "'b" and y :: "'c" where "a = bot.Value x" and "b = bot.Value y" and "R x y"   | "a = bot.Bot" and "b = bot.Bot"
CR_imp_conversionIff_join: fixes r :: "('b \<times> 'b) set" assumes "CR r" shows "r\<^sup>\<leftrightarrow>\<^sup>* = r\<^sup>\<down>"
select_convs_1: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and sa_L :: "'a \<Rightarrow> 'b"   and more :: "'c" shows "sa_L \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, sa_L = sa_L, \<dots> = more\<rparr> = sa_L"
dgrad_set_le_dgrad_set: fixes d :: "'b \<Rightarrow> nat"   and F :: "'b set"   and G :: "'b set"   and m :: "nat" assumes "dgrad_set_le d F G"   and "G \<subseteq> dgrad_set d m" shows "F \<subseteq> dgrad_set d m"
not_in_dom': fixes dom :: "'c set"   and carrier :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'c" assumes "preorder_family dom carrier R"   and "i \<notin> dom" shows "R i = (\<lambda>_ _. False)"
sylvester_carrier_mat: fixes p :: "'b poly"   and q :: "'b poly" shows "sylvester_mat p q \<in> carrier_mat (degree p + degree q) (degree p + degree q)"
smult_one: fixes R :: "('c, 'd) ring_scheme"   and M :: "('c, 'e, 'f) module_scheme"   and x :: "'e" assumes "module R M"   and "x \<in> carrier M" shows "\<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x"
degen_acc_run_complete: fixes r :: "nat \<Rightarrow> 'Q"   and m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" assumes "is_acc_run r" obtains r' :: "nat \<Rightarrow> 'Q \<times> nat" where "b_graph.is_acc_run (degeneralize_ext m) r'" and "r = fst \<circ> r'"
subst_apply_fv_empty: fixes t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "fv t = {}" shows "fv (t \<cdot> \<sigma>) = {}"
hoare_tripleD: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and h :: "heap"   and \<sigma> :: "heap option"   and r :: "'b" assumes "<P> c <Q>"   and "run c (Some h) \<sigma> r" shows "\<forall>as. pHeap h as \<Turnstile> P \<longrightarrow> \<sigma> \<noteq> None \<and> pHeap (the \<sigma>) (new_addrs h as (the \<sigma>)) \<Turnstile> Q r \<and> relH {a. a < lim h \<and> a \<notin> as} h (the \<sigma>) \<and> lim h \<le> lim (the \<sigma>)"
da_monotone: fixes Env :: "env"   and B :: "lname set"   and t :: "Term.term"   and A :: "assigned"   and B' :: "lname set"   and A' :: "assigned" assumes "Env\<turnstile> B \<guillemotright>t\<guillemotright> A"   and "B \<subseteq> B'"   and "Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'" shows "nrm A \<subseteq> nrm A' \<and> (\<forall>l. brk A l \<subseteq> brk A' l)"
iso_group_mul: fixes G :: "('a, 'b) monoid_scheme"   and A :: "'a set"   and B :: "'a set" assumes "group_disjoint_sum G A B"   and "comm_group G" shows "((\<lambda>(x, y). x \<otimes>\<^bsub>G\<^esub> y) \<in> Group.iso (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B \<subseteq> {\<one>\<^bsub>G\<^esub>} \<and> A <#>\<^bsub>G\<^esub> B = carrier G)"
linorder_cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
lm019: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set"   and a :: "('a \<times> 'b set) set" shows "summedBid bids ` (N \<times> (Pow G - {{}}) \<inter> a) = {summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}}) \<inter> a}"
val_minus_one: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v (-\<^sub>a 1\<^sub>r) = 0"
acceptD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Accept]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalAllow"
bij_betw_UN: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c set" assumes "bij_betw f A B" shows "(\<Union>n\<in>A. g (f n)) = \<Union> (g ` B)"
cols_upt_k_def: fixes A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" shows "cols_upt_k A k = {column i A |i. i \<le> mod_type_class.from_nat k}"
congs_map_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" assumes "inj_on f (set xs \<union> set ys)" shows "map f xs \<cong> map f ys = xs \<cong> ys"
zip_last: fixes xs :: "'d list"   and ys :: "'e list" assumes "xs || ys \<noteq> []"   and "length xs = length ys" shows "last (xs || ys) = (last xs, last ys)"
mu_rec_total_pure_rule: fixes R :: "('d \<times> 'd) set"   and B :: "('e, 'f) urel \<Rightarrow> ('e, 'f) urel"   and Pre :: "('e, 'f) urel"   and e :: "('d, 'e) uexpr"   and Post :: "('e, 'f) urel" assumes "wf R"   and "Monotonic B"   and "\<And>f st. \<lbrakk>(Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f; \<mu> B = f\<rbrakk> \<Longrightarrow> (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f" shows "(Pre \<Rightarrow> Post) \<sqsubseteq> \<mu> B"
TryThrow\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and h\<^sub>1 :: "heap"   and ls\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and i :: "nat"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile>\<^sub>1 \<langle>try e\<^sub>1 catch(C i) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>"
b_e_type_testop_drop_cvt0: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and t :: "t"   and testop :: "testop"   and t1 :: "t"   and cvtop :: "cvtop"   and t2 :: "t"   and sx :: "sx option" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Testop t testop \<or> e = Drop \<or> e = Cvtop t1 cvtop t2 sx" shows "ts \<noteq> []"
bind_fv_subst_bv_cancel: fixes t :: "Core.term"   and v :: "variable \<times> typ" assumes "\<And>x. 0 < x \<Longrightarrow> \<not> loose_bvar1 t x"   and "\<not> occs (case v of (x, xa) \<Rightarrow> Fv x xa) t" shows "bind_fv v (subst_bv (case v of (x, xa) \<Rightarrow> Fv x xa) t) = t"
red_supset_def: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "A \<sqsupset>p B \<equiv> (\<exists>p. is_red A p \<and> \<not> is_red B p) \<and> (\<forall>p. is_red B p \<longrightarrow> is_red A p)"
stateT_map_def: shows "stateT_map \<equiv> \<Lambda> s a. \<mu> uu. stateT_abs oo cfun_map\<cdot>s\<cdot>(fmap\<cdot>(prod_map\<cdot>a\<cdot>s)) oo stateT_rep"
mat_mult_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and i :: "'c"   and j :: "'e" shows "(f \<otimes> g) i j \<equiv> \<Sum> {f i k * g k j |k. k \<in> UNIV}"
extend_map_to_freeword_hom1: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b" shows "(\<Sum>(s, b)\<leftarrow>freeword (Abs_freeletter s). if b then f s else - f s) = f s"
elims_3: fixes x :: "'d mat" assumes "\<not> square_mat x" obtains A :: "'d mat" where "x = A" and "dim_col A \<noteq> dim_row A"
hyps: fixes \<sigma> :: "'state" shows "csmall_nstep P \<sigma> (Suc n) = {a. case a of (\<sigma>2, coll) \<Rightarrow> \<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<sigma>2, coll2) \<in> csmall_nstep P \<sigma>1 n \<and> combine coll1 coll2 = coll}"
\<delta>_prod_h_def: fixes \<delta>1 :: "('a, 'c) ta_rule dlist"   and \<delta>2 :: "('b, 'c) ta_rule dlist" shows "\<delta>_prod_h \<delta>1 \<delta>2 \<equiv> lll_iflt_cp.inj_image_filter_cp (\<lambda>(r1, r2). r_prod r1 r2) (\<lambda>(r1, r2). rhsl r1 = rhsl r2 \<and> length (rhsq r1) = length (rhsq r2)) \<delta>1 \<delta>2"
AE_E3: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "almost_everywhere M P" obtains N :: "'b set" where "\<And>x. x \<in> space M - N \<Longrightarrow> P x" and "N \<in> null_sets M"
LSC_OrL_inv: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F \<^bold>\<or> G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F, \<Gamma> \<Rightarrow>\<^sub>n \<and> G, \<Gamma> \<Rightarrow>\<^sub>n"
assms_7: shows "g \<turnstile> n-ns\<rightarrow>defNode g r"
unchanged_def: fixes v :: "state \<Rightarrow> 'b"   and s :: "state"   and t :: "state" shows "(s, t) \<Turnstile> unchanged v \<equiv> v t = v s"
linear_add: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and b1 :: "('a, 'b) vec"   and b2 :: "('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (b1 + b2) = f b1 + f b2"
plus_QDelta_def: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" shows "qd1 + qd2 = QDelta (qdfst qd1 + qdfst qd2) (qdsnd qd1 + qdsnd qd2)"
lemma2_6_6_b': fixes r :: "('a \<times> 'a) set"   and Q :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "irrefl r"   and "(Q + M, Q + N) \<in> mul_eq r" shows "(M, N) \<in> mul_eq r"
baEANow: fixes c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and n :: "nat"   and \<phi> :: "'ND \<Rightarrow> bool" assumes "eval c t t' n [\<phi>]\<^sub>b"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"
wf_below_mono2: fixes A :: "'a set"   and A' :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> A'"   and "wf_below r A'" shows "wf_below r A"
ecard2: fixes e :: "nat uprod" assumes "e \<in> E" shows "proper_uprod e"
rfn_tool1: fixes r :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "0 < r"   and "k = i * r + j"   and "j < r" shows "k div r = i"
FORWEAK_mono_rule': fixes S :: "'c set"   and d :: "'d nres"   and P :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd nres" assumes "S = {} \<Longrightarrow> d \<le> SPEC P"   and "\<And>s. s \<in> S \<Longrightarrow> f s \<le> SPEC (I {s})"   and "\<And>ab bb xb. \<lbrakk>ab \<in> bb; bb \<subseteq> S; I (insert ab (S - bb)) xb\<rbrakk> \<Longrightarrow> I (S - bb) xb"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> S; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<le> SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I (insert x it)))"   and "\<And>\<sigma>. I S \<sigma> \<Longrightarrow> P \<sigma>" shows "FORWEAK S d f c \<le> SPEC P"
assms_3: shows "f \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (f \<otimes>\<^sub>D f)"
InvGlobalInfoAtBetaUpright_def: fixes c :: "('c, 'd) configuration" shows "InvGlobalInfoAtBetaUpright c = (\<forall>k p q. beta_upright (InfoAt c k p q) (GlobalIncomingInfo c (k + 1) p q))"
bis_imp_1: fixes indis :: "('a \<times> 'a) set"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis"   and "(c1, c2) \<in> PL_Indis.Sbis tval aval indis" shows "(c1, c2) \<in> PL_Indis.ZObisT tval aval indis"
inteq: shows "to_stream -` S \<inter> space (Pi\<^sub>M UNIV PM) = imgSbs"
sign_simps_2: fixes x :: "'a" assumes "x = (0::'a)" shows "sign x = 0"
elims_3: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<not> sorted_wrt x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" and ys :: "'b list" where "x = P" and "xa = xb # ys" and "\<not> (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
inject: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "'e list"   and y1 :: "'c"   and y2 :: "'d"   and y3 :: "'e list" shows "(State x1 x2 x3 = State y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
WeakUntil_ltlp_def: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "\<phi> W\<^sub>p \<psi> \<equiv> \<psi> R\<^sub>p (\<phi> or\<^sub>p \<psi>)"
ide_to_hf_mapsto: shows "ide_to_hf \<in> Collect ide \<rightarrow> UNIV"
fps_to_fls_nth: fixes f :: "'a fps"   and n :: "int" shows "fls_nth (fps_to_fls f) n = (if n < 0 then 0::'a else f $ nat n)"
implmtD: fixes G :: "prog"   and C :: "qtname"   and J :: "qtname" assumes "G\<turnstile>C\<leadsto>J" shows "(\<exists>I. G\<turnstile>C\<leadsto>1I \<and> G\<turnstile>I\<preceq>I J) \<or> (\<exists>D. G\<turnstile>C\<prec>\<^sub>C1D \<and> G\<turnstile>D\<leadsto>J)"
has_contour_integral_part_circlepath_iff: fixes a :: "real"   and b :: "real"   and f :: "complex \<Rightarrow> complex"   and I :: "complex"   and c :: "complex"   and r :: "real" assumes "a < b" shows "(f has_contour_integral I) (part_circlepath c r a b) = ((\<lambda>t. f (c + complex_of_real r * cis t) * complex_of_real r * \<i> * cis t) has_integral I) {a..b}"
span_is_subset: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "V.module.span S \<subseteq> N"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Singleton_for_d.Cd d X" shows "x \<in> X \<inter> Field (Pd_singletons_for_ds X {Xd x} d)"
factorial_monoidI: assumes "\<And>a. \<lbrakk>a \<in> carrier (Ring_Divisibility.mult_of R); a \<notin> Units (Ring_Divisibility.mult_of R)\<rbrakk> \<Longrightarrow> \<exists>fs. set fs \<subseteq> carrier (Ring_Divisibility.mult_of R) \<and> wfactors (Ring_Divisibility.mult_of R) fs a"   and "\<And>a fs fs'. \<lbrakk>a \<in> carrier (Ring_Divisibility.mult_of R); set fs \<subseteq> carrier (Ring_Divisibility.mult_of R); set fs' \<subseteq> carrier (Ring_Divisibility.mult_of R); wfactors (Ring_Divisibility.mult_of R) fs a; wfactors (Ring_Divisibility.mult_of R) fs' a\<rbrakk> \<Longrightarrow> essentially_equal (Ring_Divisibility.mult_of R) fs fs'" shows "factorial_monoid (Ring_Divisibility.mult_of R)"
op_type_alt_simps_8: fixes op :: "boolean_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "boolean_unop_type op \<tau> \<sigma> = (op = NotOp \<and> \<sigma> = \<tau> \<and> \<tau> \<le> Boolean[?])"
smallest_accepting_rank_properties_1: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and i :: "nat" assumes "ltl_FG_to_rabin \<Sigma> \<G> w"   and "mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} = Some i" shows "mojmir_def.accept \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q}"
that_3: fixes x :: "'a" assumes "x \<in> src.carrier" shows "x \<in> U x"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B local.map cmp"