hcju/isabelleps huggingface dataset:
"2_1": fixes thesis :: "bool" assumes "\<And>p a q b. ((p, i, a), q, j, b) \<in>\<^sub>p snd ` set ps \<Longrightarrow> thesis"   and "pair_in_list ps i j" shows "thesis"
countable_nat_UN: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "range A \<subseteq> sets M" shows "\<Union> (range A) \<in> sets M"
simps_7: fixes \<phi> :: "atom fm" shows "clearQuantifiers (ExQ \<phi>) = (let \<phi>' = clearQuantifiers \<phi> in if freeIn 0 \<phi>' then lowerFm 0 1 \<phi>' else ExQ \<phi>')"
lock_actions_ok'_must_acquire_lock_lock_actions_ok: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list" assumes "lock_actions_ok' l t Ls"   and "must_acquire_lock Ls \<longrightarrow> may_lock l t" shows "lock_actions_ok l t Ls"
eexp_le_eexp_iff_le: fixes x :: "ereal"   and y :: "ereal" shows "(eexp x \<le> eexp y) = (x \<le> y)"
graph_homomorphism_def2: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> univalent f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union (map_graph f G\<^sub>1) G\<^sub>2 = G\<^sub>2)"
trancl_unfold_left: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>+ = r ; r\<^sup>*"
simps_2: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'd option" shows "Option.bind (Some x) f = f x"
assms_5: shows "adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
square_free_def: fixes p :: "'a poly" shows "square_free p = (p \<noteq> 0 \<and> (\<forall>q. 0 < degree q \<longrightarrow> \<not> q * q dvd p))"
exists_base: fixes K :: "'c set"   and n :: "nat"   and E :: "'c set" assumes "subfield K S"   and "S.dimension n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier S \<and> S.independent K Vs \<and> length Vs = n \<and> S.Span K Vs = E"
in_closure_of: fixes x :: "'c"   and X :: "'c topology"   and S :: "'c set" shows "(x \<in> X closure_of S) = (x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<in> S \<and> y \<in> T)))"
in_rtrancl_on_in_F_2: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> rtrancl_on F r" shows "b \<in> F"
simps_2: fixes t :: "Core.term"   and u :: "Core.term" shows "eta_norm (t $ u) = eta_norm t $ eta_norm u"
nnfis_unique: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real" assumes "measure_space M"   and "a \<in> nnfis f M"   and "b \<in> nnfis f M" shows "a = b"
imageI: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> A" shows "f x \<in> f ` A"
asum_multTr_mm: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "m < 0"   and "n < 0" shows "aSum A (nat (- m)) (-\<^sub>a aSum A (nat (- n)) (-\<^sub>a a)) = aSum A (nat (- m * - n)) a"
valid_reusables_under_some_conditions: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "local.well_valued0 t"   and "well_shaped t"   and "l t < fst w"   and "r t \<le> snd w" shows "Ball (set (reusables t w)) local.valid"
pf_e: fixes t :: "real" assumes "t \<in> {0..1}" shows "norm (pf t - p t) < e"
times_bool_def: shows "(\<cdot>) \<equiv> (\<and>)"
assoc_simps_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.arr \<a>\<^sub>B[f, g, h]"
extend_distinct_path: fixes v' :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v" assumes "finite E"   and "is_trail v' ps v"   and "even (degree v' G) \<and> v' \<noteq> v \<or> odd (degree v' G) \<and> v' = v" shows "\<exists>e v1. is_trail v1 (e # ps) v"
rank_def: fixes tree :: "'b tree" shows "rank tree = (case tree of Node x1 x2 x3 \<Rightarrow> x1)"
ltl_prop_entailment_monotonI: fixes S :: "'a ltln set"   and \<phi> :: "'a ltln"   and S' :: "'a ltln set" assumes "S \<Turnstile>\<^sub>P \<phi>"   and "S \<subseteq> S'" shows "S' \<Turnstile>\<^sub>P \<phi>"
ccw_convex: fixes u :: "real"   and v :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real"   and d :: "real \<times> real" assumes "0 \<le> u"   and "0 \<le> v"   and "u + v = 1"   and "ccw a b c"   and "ccw a b d"   and "\<lbrakk>coll a b c; coll a b d\<rbrakk> \<Longrightarrow> Counterclockwise_2D_Arbitrary.lex b a" shows "ccw a b (u *\<^sub>R c + v *\<^sub>R d)"
insert3_def: fixes a :: "'a"   and t :: "'a tree \<times> nat" shows "insert3 a t = Time_Monad.val (insert3_tm a t)"
invar_tree_def: fixes t :: "'b tree" shows "invar_tree t = (invar_btree t \<and> invar_otree t)"
simps_2: fixes i :: "'d"   and "is" :: "'d list"   and cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and B :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and pc :: "nat"   and s :: "'c" shows "wtl_inst_list (i # is) cert f r T B step pc s = (let s' = wtl_cert cert f r T B step pc s in if s' = T \<or> s = T then T else wtl_inst_list is cert f r T B step (pc + 1) s')"
simplex_in_fold_A_im_image: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and x :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "x \<in> (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> A" shows "(ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) ` x = x"
search_nonempty_eq0_2: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat"   and y :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "\<phi> i y \<down>"   and "e_length b \<le> y" shows "\<phi> i (the (eval (total_cons.r_search s) [b])) \<down>"
perfect_choose_dist: fixes r :: "real"   and x :: "'b" assumes "0 < r" shows "\<exists>a. a \<noteq> x \<and> dist a x < r"
all_acquired_takeWhile: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_acquired (takeWhile P sb) \<subseteq> all_acquired sb"
vsv_vdomain_vempty_vrange_vempty: assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>) \<noteq> []\<^sub>\<circ>" shows "\<R>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>) \<noteq> []\<^sub>\<circ>"
transient_def: fixes A :: "'d set" shows "transient A \<equiv> {F. \<exists>act\<in>Acts F. act `` A \<subseteq> - A}"
elims: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> 'a + 'c"   and xa :: "'d"   and xb :: "'b \<times> 'e"   and y :: "'a + 'b \<times> 'c" assumes "assert_cs x xa xb = y" obtains ass :: "'d \<Rightarrow> 'e \<Rightarrow> 'a + 'c" and i :: "'d" and cs :: "'b" and s :: "'e" where "x = ass" and "xa = i" and "xb = (cs, s)" and "y = (case ass i s of Inl x \<Rightarrow> Inl x | Inr s' \<Rightarrow> Inr (cs, s'))"
erf_0: shows "erf (0::'a) = (0::'a)"
offs_nth: fixes ns :: "nat list"   and n :: "nat"   and k :: "nat" assumes "find ((<) (Suc 0)) ns = None"   and "foldl (+) 0 ns = n"   and "k < n" shows "\<exists>i<length ns. offs ns 0 ! i = k"
intros_1: fixes e :: "'b"   and D :: "'b set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "e \<in> D" shows "({}, e) \<in> foldSetD D f e"
n1_ch: shows "|~ \<langle>n1\<rangle>_vars = n1"
rel_option_unfold: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" shows "rel_option R x y = (Option.is_none x = Option.is_none y \<and> (\<not> Option.is_none x \<longrightarrow> \<not> Option.is_none y \<longrightarrow> R (the x) (the y)))"
vars_polynomial_of_mset: fixes za :: "(char list multiset \<times> int) multiset" shows "vars (polynomial_of_mset za) \<subseteq> \<Union> ((`) \<phi> ` (set_mset \<circ> fst) ` set_mset za)"
le_left_mono: fixes x :: "'b"   and y :: "'b"   and a :: "'b" assumes "x \<le> y" shows "y \<le> a \<longrightarrow> x \<le> a"
eventually_inf: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and F' :: "'b filter" shows "eventually P (inf F F') = (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
mod_in_carrier: fixes a :: "int"   and n :: "nat" shows "a mod p ^ n \<in> carrier (residue_ring (p ^ n))"
H2_UNIV: shows "UNIV = set [H1, H2]"
longest_Lyndon_suf_shorter: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "\<^bold>|w\<^sup><\<inverse>LynSuf w \<^bold>| < \<^bold>|w\<^bold>|"
outgoing_def: fixes v :: "nat" shows "cf.outgoing v \<equiv> {(v, u) |u. (v, u) \<in> cf.E}"
d_reaches_unique: fixes M :: "('d, 'e, 'f) FSM"   and p :: "'f"   and xs :: "'d list"   and q1 :: "'f"   and q2 :: "'f" assumes "d_reaches M p xs q1"   and "d_reaches M p xs q2" shows "q1 = q2"
pareto_efficient_scf_axioms: shows "pareto_efficient_scf agents alts scf"
msum_pdevs_zero_ident: fixes x :: "'b pdevs" shows "msum_pdevs 0 zero_pdevs x = x"
fprod_fimage: fixes a :: "('c \<times> 'd \<times> 'a) fset"   and b :: "('e \<times> 'f \<times> 'b) fset" shows "(\<lambda>(uu_, uu_, id). [id]) |`| a |\<times>| (\<lambda>(uu_, uu_, id). [id]) |`| b = (\<lambda>((uu_, uu_, id1), uu_, uu_, id2). ([id1], [id2])) |`| (a |\<times>| b)"
a_mul_d': fixes x :: "'a" shows "ad x \<cdot> d x = (0::'a)"
induct: fixes P :: "('a, 'rule) graph_node \<Rightarrow> ('form, 'var) out_port \<Rightarrow> bool"   and a0 :: "('a, 'rule) graph_node"   and a1 :: "('form, 'var) out_port" assumes "\<And>r h a. P (Rule r) (Hyp h a)"   and "\<And>v uv_. P (Assumption v) uv_"   and "\<And>v uv_. P (Conclusion v) uv_"   and "\<And>uv_. P Helper uv_"   and "\<And>uu_ v. P uu_ (Reg v)" shows "P a0 a1"
nonempty: fixes G :: "'a set" assumes "AbGroup G" shows "G \<noteq> {}"
const_res_subseq_prop_0: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "\<exists>l. const_res_subseq k s = filtered_seq s (kth_res_equals k l) \<and> is_subseq_of s (const_res_subseq k s) \<and> (\<forall>m. const_res_subseq k s m k = l)"
assms_1: shows "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"
subcls1_def2: fixes G :: "prog" shows "subcls1 G = (SIGMA C:{C. is_class G C}. {D. C \<noteq> Object \<and> super (the (class G C)) = D})"
middle_1: fixes x :: "'a"   and y :: "'a" shows "dist x (middle x y) = dist x y / 2"
ratfps_nth_bigo_square_free_factorization: fixes q :: "complex poly"   and b :: "complex"   and cs :: "(complex poly \<times> nat) list"   and R :: "real"   and k :: "nat"   and p :: "complex poly" assumes "square_free_factorization q (b, cs)"   and "q \<noteq> 0"   and "0 < R"   and "\<And>c l. (c, l) \<in> set cs \<Longrightarrow> \<forall>x\<in>ball 0 (1 / R). poly c x \<noteq> 0"   and "\<And>c l. \<lbrakk>(c, l) \<in> set cs; k < l\<rbrakk> \<Longrightarrow> \<forall>x\<in>sphere 0 (1 / R). poly c x \<noteq> 0" shows "($) (fps_of_poly p / fps_of_poly q) \<in> O(\<lambda>n. of_nat n ^ k * complex_of_real R ^ n)"
newrght_lemma: fixes p :: "nat"   and r :: "nat"   and a :: "nat" shows "rec_exec rec_newrght [p, r, a] = newrght p r a"
in_singleton: fixes m :: "nat"   and n :: "nat" shows "m \<^bold>\<in> Abs_nat_int {n} \<longrightarrow> m = n"
distinct_255: fixes x11 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Prj1 x11 \<noteq> Unroll x14"
prod_in_terms_of_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "arr g" shows "HF'.prod f g = hfsetcat.tuple (local.comp f (hfsetcat.pr1 (local.dom f) (local.dom g))) (local.comp g (hfsetcat.pr0 (local.dom f) (local.dom g)))"
axioms_1: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
split_face_edges_f12_subset: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []" shows "{(hd vs, ram1), (ram2, last vs)} \<union> Edges (rev vs) \<subseteq> \<E> f12"
Contingent_def: shows "Contingent \<equiv> \<lambda>F. \<^bold>\<not>(Necessary F \<^bold>\<or> Impossible F)"
bad_runs_def: fixes s :: "'a l2_state_scheme" shows "bad_runs s = {R. owner (guessed_runs R) \<in> bad s \<or> partner (guessed_runs R) \<in> bad s}"
simps_4: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form" shows "subc c s (Neg p) = Neg (subc c s p)"
intros: fixes F :: "nat \<Rightarrow> 'b set" assumes "\<And>n. closed (F n)" shows "fsigma (\<Union> (range F))"
x_stays_before_y_if_y_not_moved_to_front: fixes q :: "'a"   and xs :: "'a list"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "y \<noteq> q"   and "x < y in xs" shows "x < y in mtf2 n q xs"
is_lubI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a" assumes "class.po below"   and "po.is_ub below S x"   and "\<And>u. po.is_ub below S u \<Longrightarrow> below x u" shows "po.is_lub below S x"
simps_9: fixes e\<^sub>1 :: "expr"   and F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr" shows "fv (e\<^sub>1\<bullet>F{Cs} := e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
proj2_rep_cart2_append1: fixes p :: "proj2" assumes "z_non_zero p" shows "proj2_rep p = proj2_rep p $ 3 *\<^sub>R cart2_append1 p"
ldrop_mono: fixes xs :: "'c llist" shows "mono_llist_lift (\<lambda>ldrop. case xs of LNil \<Rightarrow> \<lambda>_. LNil | LCons x xs \<Rightarrow> \<lambda>n. if n = 0 then LCons x xs else ldrop xs (n - 1))"
apps_preserves_betas: fixes rs :: "dB list"   and ss :: "dB list"   and r :: "dB" assumes "rs => ss" shows "r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> r \<degree>\<degree> ss"
upper_uminus: fixes A :: "'b interval" shows "upper (- A) = - lower A"
assms: shows "\<forall>w\<in>space M. X w = Z w"
disc_iff_2: fixes p :: "'c \<Rightarrow> bool"   and g1 :: "'c \<Rightarrow> 'b"   and g21 :: "'c \<Rightarrow> 'a"   and g22 :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "(\<not> is_TNil (unfold_tllist p g1 g21 g22 a)) = (\<not> p a)"
cinfinite_def: fixes r :: "('a \<times> 'a) set" shows "cinfinite r = infinite (Field r)"
Red_F_def: fixes N :: "'f set" shows "Red_F N = {C. \<exists>DD\<subseteq>N. DD \<Turnstile> {C} \<and> (\<forall>D\<in>DD. D < C)}"
select_convs_3: fixes Obj :: "'h set"   and Mor :: "'g set"   and Dom :: "'g \<Rightarrow> 'h"   and Cod :: "'g \<Rightarrow> 'h"   and Id :: "'h \<Rightarrow> 'g"   and Comp :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and more :: "'i" shows "Dom \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Dom"
evaluate_net_Conv_copy_first: fixes m :: "real mat convnet"   and input :: "real vec list"   and j :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "input_sizes m = map dim_vec input"   and "j < nr"   and "0 < output_size' m" shows "evaluate_net (Conv (copy_first_matrix nr (output_size' m)) m) input $ j = evaluate_net m input $ 0"
simps_1: fixes lev :: "nat"   and i :: "nat" shows "decr lev (Bv i) = (if lev \<le> i then Bv (i - 1) else Bv i)"
simps_51: fixes x31 :: "poscheckvariant"   and x32 :: "('d, 'c) Term.term"   and x33 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>t\<^sub>p \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t = fv x32 \<union> fv x33"
inline_sel: fixes callee :: "'e \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'e, 'f, 'g) gpv"   and c :: "('d, 'h, 'i) gpv"   and s :: "'e" shows "the_gpv (inline callee c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. oracle input \<bind> (\<lambda>(x, s'). inline callee (rpv x) s'))) (inline1 callee c s)"
fps_add_nth: fixes f :: "('d, 'c) fps"   and g :: "('d, 'c) fps"   and n :: "'d list" shows "(f + g) $ n = f $ n + g $ n"
queue_invar_def: fixes q :: "('c, 'd) SkewBinomialTree list" shows "queue_invar q \<equiv> \<forall>t\<in>set q. tree_invar t"
mat_times_vec_leqD_3: fixes A :: "'a mat"   and x :: "'a vec"   and b :: "'a vec"   and i :: "nat" assumes "[A*\<^sub>vx]\<le>b"   and "i < dim_vec b" shows "(A *\<^sub>v x) $ i \<le> b $ i"
r'_def: shows "r' \<equiv> map_poly inv_embed cr"
left_trpl_fst: fixes l :: "nat"   and st :: "nat"   and r :: "nat" shows "left (trpl l st r) = l"
rimg_def: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('c, 'f) Ring_scheme"   and f :: "'d \<Rightarrow> 'c" shows "rimg A R f = \<lparr>carrier = f ` carrier A, pop = (\<plusminus>\<^bsub>R\<^esub>), mop = mop R, zero = \<zero>\<^bsub>R\<^esub>, tp = (\<cdot>\<^sub>r\<^bsub>R\<^esub>), un = 1\<^sub>r\<^bsub>R\<^esub>\<rparr>"
hom_from_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and A :: "'a set" assumes "Group.group G"   and "h \<in> hom G H" shows "h \<in> hom (subgroup_generated G A) H"
assms_6: shows "pathfinish f $ 2 = a $ 2"
word_bw_comms_1: fixes x :: "'b word"   and y :: "'b word" shows "x && y = y && x"
ge_mono: fixes x :: "real"   and bs :: "real list" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (ge n s) = Ifm (x # bs) (Ge a)"
assms_1: shows "saturated_binary_rule (ordered_sel_resolvent Sel) S"
neg_numeral_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus m) = uminus (numeral.numeral one plus n)) = (m = n)"
fold_insert_idem: fixes A :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option"   and x :: "'a\<^sub>m" assumes "finite A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (insert x A) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A \<bind> m x"
aform_val_project_coord_aform: fixes e :: "nat \<Rightarrow> real"   and sctn :: "'a sctn"   and X :: "'a \<times> 'a pdevs" shows "aform_val e (project_coord_aform sctn X) = project_coord (aform_val e X) (normal sctn) (pstn sctn)"
nn_integral_count_space_reindex: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'b \<Rightarrow> ennreal" assumes "inj_on f A" shows "integral\<^sup>N (count_space (f ` A)) g = (\<Sum>\<^sup>+ x\<in>A. g (f x))"
block_size_lt_order: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "card bl \<le> card \<V>"
semiring_of_sets_axioms: fixes M :: "'c measure" shows "semiring_of_sets (space M) (sets M)"
lset_mono: fixes x :: "'c llist" shows "lfp.mono_body (\<lambda>f. case x of LNil \<Rightarrow> {} | LCons x xs \<Rightarrow> insert x (f xs))"
example_det_state_cover: shows "is_det_state_cover M\<^sub>S V\<^sub>I"
hyps: assumes "finite A"   and "k \<le> card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1..card A}"
fps_const_lr_inverse_1: fixes a :: "'a"   and x :: "'a" shows "fps_left_inverse (fps_const a) x = fps_const x"
row_zero_A_imp_row_zero_H: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "echelon_form A"   and "\<not> is_zero_row i A"   and "is_zero_row j A" shows "is_zero_row j (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)"
suffix_appendI: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "suffix xs ys" shows "suffix xs (zs @ ys)"
action_proj_in_prob_proj: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" assumes "a \<in> PROB" shows "action_proj a vs \<in> prob_proj PROB vs"
dims0_def: shows "b.dims0 = nths dims b.vars0"
assms_1: shows "net_ips s1 \<inter> net_ips s2 = {}"
led: shows "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
xor_zero_right: fixes x :: "'a" shows "x XOR (0::'a) = x"
isomorphic_def: fixes a :: "('a S.E.term, 'a) S.arr"   and a' :: "('a S.E.term, 'a) S.arr" shows "S.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>S a'\<guillemotright> \<and> S.iso f)"
one_minus_one_minus_q_x_n_nn_integral: fixes q :: "real"   and n :: "nat" assumes "q \<in> {0<..<1}" shows "\<integral>\<^sup>+x\<in>{0..}. ennreal (1 - (1 - q powr x) ^ n)\<partial>lborel = ennreal (LBINT x=0..\<infinity>. 1 - (1 - q powr x) ^ n)"
filter_not_empty: fixes x :: "'d"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> set xs"   and "P x" shows "filter P xs \<noteq> []"
rightInvertible: fixes R' :: "('a sequent list \<times> 'a sequent) set"   and R :: "('a sequent list \<times> 'a sequent) set"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form multiset"   and F :: "'a"   and Fs :: "'a form list"   and n :: "nat"   and \<Gamma>' :: "'a form multiset"   and \<Delta>' :: "'a form multiset" assumes "R' \<subseteq> upRules \<and> R = Ax \<union> R'"   and "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Compound F Fs, n) \<in> derivable R*"   and "\<forall>r'\<in>R. rightPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable R*"
AbstrLevels_A2_A23: fixes i :: "AbstrLevelsID" assumes "sA2 \<in> AbstrLevel i" shows "sA23 \<notin> AbstrLevel i"
symmetric_bibdII: assumes "\<Lambda> * (\<v> - 1) = \<k> * (\<k> - 1)" shows "symmetric_bibd \<V> \<B> \<k> \<Lambda>"
Ball_def: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
rebase_dvd_def: shows "rebase_dvd TYPE('a) TYPE('b) \<equiv> CARD('b) dvd CARD('a)"
Mp_square_free_m: fixes f :: "int poly" shows "square_free_m (Mp f) = square_free_m f"
sin_squared_eq: fixes x :: "'a" shows "(sin x)\<^sup>2 = (1::'a) - (cos x)\<^sup>2"
when_less_one: fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "arctan x = (\<Sum>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
rev_to_bl_eq: shows "rev (to_bl w) = map (bit w) [0..<LENGTH('a)]"
wf_inv_image: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a" assumes "wf r" shows "wf (inv_image r f)"
nonpos_int_cases: fixes k :: "int" assumes "k \<le> 0" obtains n :: "nat" where "k = - int n"
while_option_finite_increasing: shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f a = Some P"
Least_equality: fixes P :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "P x"   and "\<And>y. P y \<Longrightarrow> x \<preceq> y" shows "ordered_powerprod_lin.Least P = x"
assms_4: fixes s2 :: "'s2" assumes "bad2 s2"   and "I2 s2" shows "callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"
possible_assignments_for_set_is: fixes v :: "'d"   and \<Psi> :: "('d, 'e) sas_plus_problem" assumes "v \<in> dom (range_of \<Psi>)" shows "set (possible_assignments_for \<Psi> v) = {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v}"
reds_into_equiv: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t" shows "s \<leftrightarrow> t"
possit_sit_1: fixes v :: "i"   and x :: "\<nu>" shows "[Situation (x\<^sup>P) \<^bold>\<equiv> \<^bold>\<box>Situation (x\<^sup>P) in v]"