hcju/isabelleps huggingface dataset:
fls_divide_base_factor': fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor f / fls_base_factor g = fls_shift (fls_subdegree f - fls_subdegree g) (f / g)"
delete_edges_simp2: fixes G :: "'a graph"   and E :: "('a \<times> 'a) set" shows "delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
v\<^sub>0_2: shows "v\<^sub>0 \<in> allUses g m"
Nil_in_lists: fixes A :: "'a set" shows "[] \<in> lists A"
Units_l_inv_ex: fixes x :: "'a" assumes "x \<in> Units \<G>" shows "\<exists>y\<in>carrier \<G>. y \<otimes> x = \<one>"
Under_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Under r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. (b, a) \<in> r}"
opt_MatchAny_match_expr_correct: fixes \<gamma> :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'d match_expr" shows "matches \<gamma> (opt_MatchAny_match_expr m) = matches \<gamma> m"
Legendre_zmult: fixes p :: "int"   and a :: "int"   and b :: "int" assumes "2 < p"   and "prime p" shows "Legendre (a * b) p = Legendre a p * Legendre b p"
intro: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set" assumes "ll_on_open T f X" shows "ll_on_open_it T f X"
power_diff_inverse: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<noteq> (0::'a)"   and "m \<le> n" shows "inverse a ^ (n - m) = a ^ m / a ^ n"
arbitrary_union_of_Union: fixes \<U> :: "'c set set"   and P :: "'c set \<Rightarrow> bool" assumes "\<And>S. S \<in> \<U> \<Longrightarrow> (arbitrary union_of P) S" shows "(arbitrary union_of P) (\<Union> \<U>)"
typeof_NoneD: fixes v :: "'a val"   and x :: "ty" assumes "typeof v = \<lfloor>x\<rfloor>" shows "\<not> is_Addr v"
add_order: fixes a :: "'a" shows "a +^ add_order a = (0::'a)"
power_le_dvd: fixes a :: "'a"   and n :: "nat"   and b :: "'a"   and m :: "nat" assumes "a ^ n dvd b"   and "m \<le> n" shows "a ^ m dvd b"
mod_prop1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat"   and c :: "nat" assumes "Matrix_Tensor.mult id f" shows "a mod (b * c) mod c = a mod c"
ideD_1: fixes a :: "'a \<times> 'a" assumes "H.VV.ide a" shows "H.VV.arr a"
simps_6: fixes P :: "'a prog"   and uu :: "char list"   and v :: "val" shows "icheck P uu (Val v) = False"
thread_ok_new_thread: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f"   and m' :: "'i"   and x :: "'j" assumes "thread_ok ts (NewThread t m' x)" shows "ts t = None"
fmrel_restrict_set: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and A :: "'e set" assumes "fmrel P m n" shows "fmrel P (fmrestrict_set A m) (fmrestrict_set A n)"
cB: fixes i :: "'n"   and j :: "'n" shows "coeff (to_fun (\<^bold>X - \<^bold>C A) i j) (Suc 0) = (if i = j then 1::'a else (0::'a))"
no: fixes z :: "complex" assumes "cmod z < 1" shows "cmod (f z) < 1"
has_field_derivative_eval_fps: fixes z :: "'a"   and f :: "'a fps"   and A :: "'a set" assumes "ereal (norm z) < fps_conv_radius f" shows "(eval_fps f has_field_derivative eval_fps (fps_deriv f) z) (at z within A)"
edge_at_def: fixes c :: "'form"   and "is" :: "nat list" shows "edge_at c is = (edge_from c is, edge_to c is)"
lift_eligible_literal: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.eligible_literal trm_ord sel L C \<sigma>"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "basic_superposition.eligible_literal trm_ord sel L C \<theta>"
simps_1: fixes l :: "lambda"   and \<beta> :: "nat \<Rightarrow> contour option" shows "contours_in_d (DC (l, \<beta>)) = ran \<beta>"
red_indE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "(\<exists>f\<in>F. f \<noteq> 0 \<and> lt f adds\<^sub>t lt p \<and> q = p - monom_mult (lc p / lc f) (lp p - lp f) f) \<or> red F (tail p) (q - monomial (lc p) (lt p))"
rel_def: fixes Rv :: "('V \<times> 'c) set"   and Rw :: "('W \<times> 'd) set" shows "\<langle>Rv, Rw\<rangle>rel \<equiv> br \<alpha> invar O \<langle>Rv, Rw\<rangle>graph_rel"
is_prefix_chars: fixes a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" assumes "is_prefix a b" shows "is_prefix (chars a) (chars b)"
IH: fixes ss :: "nat list list" assumes "ss \<in> list n A"   and "p < length ss" shows "merges f ps' ss ! p = map snd (filter (\<lambda>a. case a of (p', t') \<Rightarrow> p' = p) ps') \<Squnion>\<^bsub>f\<^esub> ss ! p"
valid_path_ITP2015: fixes Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap"   and s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap"   and rest :: "('c, 'd) fmap list" shows "valid_path Pi [] = True \<and> valid_path Pi [s] = (s \<in> valid_states Pi) \<and> valid_path Pi (s1 # s2 # rest) = (s1 \<in> valid_states Pi \<and> (\<exists>a. a \<in> Pi \<and> exec_plan s1 [a] = s2) \<and> valid_path Pi (s2 # rest))"
scaleR_eq_iff: fixes b :: "'a"   and u :: "real"   and a :: "'a" shows "(b + u *\<^sub>R a = a + u *\<^sub>R b) = (a = b \<or> u = 1)"
gcd_greatest: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c dvd a"   and "c dvd b" shows "c dvd normalization_euclidean_semiring_class.gcd a b"
swap_nested_emb_step_at: fixes p :: "dir list"   and q :: "dir list"   and d2 :: "dir"   and d1 :: "dir"   and t :: "('c, 'd) tm" shows "emb_step_at (p @ q) d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t)"
simps_2: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
injectivity_subexpr1_1: fixes a :: "'b"   and A :: "'b set"   and X :: "'b set"   and P :: "'b set set"   and X' :: "'b set"   and P' :: "'b set set"   and k' :: "nat" assumes "a \<notin> A"   and "X \<in> P \<and> X' \<in> P'"   and "insert (insert a X) (P - {X}) = insert (insert a X') (P' - {X'})"   and "(partition_on A P \<and> card P = Suc k') \<and> partition_on A P' \<and> card P' = Suc k'" shows "P = P'"
scale: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and r :: "'b"   and b :: "'c" assumes "module_hom s1 s2 f" shows "f (s1 r b) = s2 r (f b)"
funcset_to_empty_iff: fixes A :: "'a set" shows "A \<rightarrow> {} = (if A = {} then UNIV else {})"
decisionD: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>rs, Decision X\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "t = Decision X"
inclusion: fixes f :: "'a \<times> 'a" assumes "fst f \<star> snd f \<noteq> null" shows "VxV.arr f"
finite_enumerate_initial_segment: fixes S :: "'b set"   and n :: "nat"   and s :: "'b" assumes "finite S"   and "n < card (S \<inter> {..<s})" shows "wellorder_class.enumerate (S \<inter> {..<s}) n = wellorder_class.enumerate S n"
upd_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt"   and upd :: "'a ModeUpd"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c@[upd], mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((c, sifum_lang_no_dma.update_modes upd mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
Suc_diff_le: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "Suc m - n = Suc (m - n)"
fresh_4: fixes a :: "atom"   and var :: "Syntax.var"   and "term" :: "Syntax.term" shows "a \<sharp> Rec var term = (a \<sharp> term \<or> a \<in> set [atom var])"
norm_diff_ineq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (plus a b)"
n_in: shows "N \<in> Red_F (lnth Ns i0)"
finite_imageI: fixes F :: "'d set"   and h :: "'d \<Rightarrow> 'e" assumes "finite F" shows "finite (h ` F)"
map_add_eval_left': fixes x :: "'c"   and h' :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom h'"   and "h \<bottom> h'" shows "(h ++ h') x = h x"
simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and v :: "'c" shows "ord.rbt_ins less f k v rbt.Empty = Branch color.R rbt.Empty k v rbt.Empty"
cdf_lim_at_bot: shows "(cdf M1 \<longlongrightarrow> 0) at_bot"
analytic_pre_zeta: fixes a :: "real"   and A :: "complex set" assumes "0 < a" shows "pre_zeta a analytic_on A"
productF_path: fixes w :: "('d \<times> 'e) list"   and r1 :: "'f list"   and r2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and p1 :: "'f"   and p2 :: "'g" assumes "length w = length r1"   and "length r1 = length r2"   and "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path AB (w || r1 || r2) (p1, p2) = (path A (w || r1) p1 \<and> path B (w || r2) p2 \<or> target (w || r1 || r2) (p1, p2) = FAIL \<and> 0 < length w \<and> path A (butlast (w || r1)) p1 \<and> path B (butlast (w || r2)) p2 \<and> succ A (last w) (target (butlast (w || r1)) p1) = {} \<and> succ B (last w) (target (butlast (w || r2)) p2) \<noteq> {})"
bet__le2313: 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"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Le Bet Cong B C A C"
is_path_undir_split': fixes G :: "('a, 'b) graph"   and v :: "'a"   and p1 :: "('a \<times> 'b \<times> 'a) list"   and u :: "'a"   and w :: "'b"   and u' :: "'a"   and p2 :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G" shows "is_path_undir G v (p1 @ (u, w, u') # p2) v' = (is_path_undir G v p1 u \<and> ((u, w, u') \<in> edges G \<or> (u', w, u) \<in> edges G) \<and> is_path_undir G u' p2 v')"
lemma1: fixes P :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a"   and n :: "nat"   and m :: "nat" assumes "P A n"   and "\<forall>A n. P A n \<longrightarrow> P A (Suc n)" shows "P A (n + m)"
simps_1: fixes ts :: "ct list"   and ct1 :: "ct" shows "select_return_top ts ct1 TAny = TopType (take (length ts - 3) ts @ [ct1])"
queue_invar_simps_3: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.queue_invar (q @ q') = (SkewBinomialHeapStruc.queue_invar q \<and> SkewBinomialHeapStruc.queue_invar q')"
pred_dom_subset_succ_submap: fixes a :: "('a, bool) fmap \<times> ('a, bool) fmap"   and s1 :: "('a, bool) fmap"   and s2 :: "('a, bool) fmap" assumes "fmdom' (fst a) \<subseteq> fmdom' s1"   and "s1 \<subseteq>\<^sub>f s2" shows "state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a"
SUren: fixes I :: "('sf, 'b, 'sz) interp"   and x :: "'sz"   and y :: "'sz"   and \<theta> :: "('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec" shows "sterm_sem I (TUrename x y \<theta>) \<nu> = sterm_sem I \<theta> (RSadj x y \<nu>)"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'a \<Rightarrow> 'b"   and uw :: "'b" shows "ipurge_tr uu uv uw [] = []"
sum_of_powers_int: fixes x :: "int"   and n :: "nat" shows "(x - 1) * sum ((^) x) {0..n} = x ^ Suc n - 1"
assms_1: shows "id \<in> ring_iso (law_restrict A) (law_restrict B)"
int_of_vint_vint_of_int: fixes a :: "int" shows "int_of_vint (a\<^sub>\<int>) = a"
acquire_all_iff: fixes ls :: "('b, 'a) locks"   and t :: "'a"   and ln :: "'b \<Rightarrow>f nat"   and l :: "'b" shows "acquire_all ls t ln $ l = acquire_locks (ls $ l) t (ln $ l)"
l1: shows "left (Abs_real_int (left r4, right r2)) = left r4"
absorb2: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "inf a b = b"
converse_rtranclpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'c" where "r x y" and "r\<^sup>*\<^sup>* y z"
assms_3: shows "\<forall>q\<in>outs_\<I> \<I>. g ` responses_\<I> \<I>'' (f q) \<subseteq> responses_\<I> \<I> q"
geodesic_segment_param_in_geodesic_spaces_2: fixes x :: "'a"   and y :: "'a" shows "geodesic_segment_param {x--y} x (dist x y) = y"
mon_ww_uncons: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ee :: "('c, 'f) label list"   and ww :: "('c, 'f) label list list" shows "mon_ww fg (ee # ww) = mon_w fg ee \<union> mon_ww fg ww"
graph_def: fixes F :: "'b set"   and f :: "'b \<Rightarrow> real" shows "graph F f = {(x, f x) |x. x \<in> F}"
supset_h: shows "set (map (undo_factorize_polys data) sgas) \<subseteq> consistent_sign_vectors ps UNIV"
parent_def: fixes v :: "'a" shows "parent v \<equiv> hd (tl (v \<leadsto> root))"
pgen_lossless_gpv_mono: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('a, 'b) \<I>" assumes "pgen_lossless_gpv fail \<I> gpv"   and "\<I> \<le> \<I>'"   and "\<I> \<turnstile>g gpv \<surd>"   and "fail \<le> 1" shows "pgen_lossless_gpv fail \<I>' gpv"
iso_rep: fixes r :: "'a" assumes "ide r" shows "local.iso (rep r)"
Valid_def: fixes P :: "'b set"   and c :: "'b com"   and Q :: "'b set" shows "(\<turnstile> P c Q) = (\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> P \<longrightarrow> s' \<in> Q)"
ls1: shows "xs = fst (splitAt x xs) @ x # snd (splitAt x xs)"
real_of_nat_div3: fixes n :: "nat"   and x :: "nat" shows "real n / real x - real (n div x) \<le> 1"
Suc: fixes i :: "nat" shows "map_pmf (flip i) (bv n) = bv n"
augment_alt: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "cf_of (augment (augmentingFlow p)) = cf.augment_cf (set p) (resCap p)"
resume_def: fixes resumption :: "('g, 'h, 'f) Resumption.resumption" shows "Resumption.resumption.resume resumption = (case resumption of Resumption.resumption.Done a \<Rightarrow> \<lambda>inp. Resumption.resumption.Done None | Resumption.resumption.Pause x21 x22 \<Rightarrow> x22)"
order_trans: fixes x :: "('b, 'c) Combinators"   and l :: "'b set list"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators" assumes "in_list x l"   and "in_list y l"   and "in_list z l"   and "singleCombinators [x]"   and "singleCombinators [y]"   and "singleCombinators [z]"   and "smaller x y l"   and "smaller y z l" shows "smaller x z l"
quot_ModPon_imp_Pf: fixes e :: "(name, hf) finfun"   and \<alpha> :: "fm"   and \<beta> :: "fm" assumes "Pf \<lbrakk>\<guillemotleft>\<alpha> IMP \<beta>\<guillemotright>\<rbrakk>e"   and "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e" shows "Pf \<lbrakk>\<guillemotleft>\<beta>\<guillemotright>\<rbrakk>e"
design_def: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" shows "design point_set block_collection \<equiv> finite_incidence_system point_set block_collection \<and> design_axioms block_collection"
vwalk_joinI_vwalk': fixes p :: "'c list"   and G :: "('c, 'd) pre_digraph"   and q :: "'c list" assumes "vwalk p G"   and "vwalk q G"   and "last p = hd q" shows "vwalk (p \<oplus> q) G"
Inf_finite_empty: shows "\<Sqinter> {} = \<Squnion> UNIV"
linear_iff: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" shows "Vector_Spaces.linear s1 s2 f = (vector_space s1 \<and> vector_space s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
Mi_def: shows "Mi = snd (gauss_jordan M (1\<^sub>m n))"
integral_dominated_convergence: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>i. s i \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"   and "\<And>i. AE x in M. norm (s i x) \<le> w x" shows "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
intros_14: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<equiv>\<^sub>s P \<parallel> <\<nu>x>Q"
distinct_14: fixes x8 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Rjoin x8 \<noteq> VE x1"
simps_4: fixes las :: "lock_action list" shows "must_acquire_lock (UnlockFail # las) = must_acquire_lock las"
RG_mult_closed: fixes x :: "('a, 'g) aezfun"   and y :: "('a, 'g) aezfun" assumes "x \<in> Supgroup.group_ring"   and "y \<in> Supgroup.group_ring" shows "x * y \<in> Supgroup.group_ring"
map_add_upd: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option"   and x :: "'b"   and y :: "'c" shows "f ++ g(x \<mapsto> y) = (f ++ g)(x \<mapsto> y)"
carr_1: shows "y1 - y2 \<in> carrier_vec n"
z_facts_3: shows "z \<notin> Fvars \<sigma>"
bounded_linear_axioms_def: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
path_in_V: fixes xs :: "'a list" assumes "H.path xs" shows "set xs \<subseteq> V\<^bsub>H\<^esub>"
deriv_deriv_child: fixes n :: "nat"   and x :: "(nat \<times> form) list"   and y :: "(nat \<times> form) list" shows "((Suc n, x) \<in> deriv y) = (\<exists>z. z \<in> set (subs y) \<and> \<not> is_axiom (s_of_ns y) \<and> (n, x) \<in> deriv z)"
assms_3: shows "m \<cdot> e = m' \<cdot> e'"
has_derivative_within_singleton_iff: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative g) (at x within {x}) = bounded_linear g"
Quotient_to_Domainp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "Domainp T = (\<lambda>x. R x x)"
measurable_cong_sets: fixes M :: "'c measure"   and M' :: "'c measure"   and N :: "'d measure"   and N' :: "'d measure" assumes "sets M = sets M'"   and "sets N = sets N'" shows "M \<rightarrow>\<^sub>M N = M' \<rightarrow>\<^sub>M N'"
eucl_down_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "int"   and b :: "'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "eucl_down p b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (round_down p (inner b i)) i) Basis"
f_a: fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA" shows "f a b a = b"
TER_current_of_bipartite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "('a + 'a) \<times> ('a + 'a) \<Rightarrow> ennreal" assumes "web \<Gamma>"   and "current (bipartite_web_of \<Gamma>) f"   and "wave (bipartite_web_of \<Gamma>) f" shows "TER\<^bsub>\<Gamma>\<^esub> (web.current_of_bipartite \<Gamma> f) = web.separating_of_bipartite \<Gamma> (TER\<^bsub>bipartite_web_of \<Gamma>\<^esub> f)"
smallest_accepting_rank_def: shows "\<MM>.smallest_accepting_rank \<equiv> if \<MM>.accept then Some (LEAST i. finite \<MM>.fail \<and> finite (\<MM>.merge i) \<and> infinite (\<MM>.succeed i)) else None"
f12: shows "((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'', s \<otimes> s' \<otimes> s'') \<in> carrier rel"
negligible_K0: shows "Negligible.negligible (\<lambda>_. 0)"
locally_compact_compact: fixes S :: "'b set" shows "locally compact S = (\<forall>K. K \<subseteq> S \<and> compact K \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> V \<subseteq> S \<and> openin (top_of_set S) U \<and> compact V))"
mem_simps_2: fixes c :: "'e" shows "(c \<in> {}) = False"
undefined: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "h a \<noteq> h b"   and "a \<in> A" shows "m a b = None"
nonempty: fixes x :: "'e"   and A :: "'e set" assumes "x \<in> A" shows "A \<noteq> {}"
monofun_cfun: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "f \<sqsubseteq> g"   and "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> g\<cdot>y"
T3: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v)))"
prev_inI: fixes A :: "'c set"   and a :: "'c"   and b :: "'c"   and u :: "'c"   and v :: "'c" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a<..b}"   and "v \<in> A"   and "v < u"   and "{v<..<u} \<inter> A = {}" shows "prev_in A u = v"
AE_conj_iff: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and M :: "'c measure" shows "(AE x in M. P x \<and> Q x) = (almost_everywhere M P \<and> almost_everywhere M Q)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "rat" assumes "gs1.\<mu> x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if j < i then map of_int_hom.vec_hom fs ! i \<bullet> gs1.gso j / \<parallel>gs1.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
sim_valI: fixes M :: "('c, 'd, 'b) KripkeStructure"   and M' :: "('c, 'd, 'e) KripkeStructure"   and f :: "'b \<Rightarrow> 'e" assumes "\<And>u. u \<in> worlds M \<Longrightarrow> valuation M u = valuation M' (f u)" shows "sim_val M M' f"
X: shows "X \<in> sets (Pi\<^sub>M J M)"
subgraph_union: fixes H1 :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph"   and H2 :: "('c, 'd) pre_digraph" assumes "subgraph H1 G"   and "compatible H1 G"   and "subgraph H2 G"   and "compatible H2 G" shows "subgraph (Digraph_Component.union H1 H2) G"