hcju/isabelleps huggingface dataset:
gm_expr: shows "LINT x|M. f x * enn2real (nn_transfer_operator gm x) = LINT x|M. f (T x) * enn2real (gm x)"
Suc_4: shows "\<mu>_small_row i fs (Suc j)"
two_less_deg2: shows "verts3 (with_proj G) = pverts G"
assms: shows "x \<in> SINK (g \<upharpoonleft> \<Gamma> / f)"
1: shows "r =o |Field r| \<and> |Field r| =o r"
coeffs_eq_Nil: fixes p :: "'c poly" shows "(Polynomial.coeffs p = []) = (p = 0)"
distinct_41: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Await x71 x72"
get_M_Mshadow_root_preserved3a: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('bb \<Rightarrow> 'bb) \<Rightarrow> ('ar, 'ap, 'aq, 'ay option) RShadowRoot_scheme \<Rightarrow> ('ar, 'ap, 'aq, 'ay option) RShadowRoot_scheme"   and v :: "'bb"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'bc" assumes "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
l: shows "(\<lambda>n. f (xs n)) \<longlonglongrightarrow> l"
hyps: fixes \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" assumes "weak_sharing_consistent \<O> sb"   and "read_only \<S> \<inter> \<O> = {}"   and "\<S> a \<noteq> Some False"   and "share sb \<S> a = Some False" shows "a \<in> \<O> \<union> all_acquired sb"
AL_update2: fixes a :: "'a"   and b :: "'a"   and L :: "('a \<times> 'b) list"   and v :: "'b" assumes "a \<noteq> b" shows "L[a\<mapsto>v]\<down>b = L\<down>b"
poly_divides_trans: fixes p :: "'a list"   and q :: "'a list"   and r :: "'a list" assumes "p divides q"   and "q divides r" shows "p divides r"
assms: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
list_comb: fixes f :: "'a"   and xs :: "'a list" shows "shadows_consts (list_comb f xs) = (shadows_consts f \<or> list_ex shadows_consts xs)"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list" assumes "\<And>acc. P acc []"   and "\<And>acc x xs. P (acc + 1) xs \<Longrightarrow> P acc (x # xs)" shows "P a0 a1"
simps: fixes A :: "('a, 'b, 'c) FSM"   and B :: "('a, 'b, 'd) FSM"   and FAIL :: "'c \<times> 'd"   and AB :: "('a, 'b, 'c \<times> 'd) FSM" shows "productF A B FAIL AB = (inputs A = inputs B \<and> fst FAIL \<notin> nodes A \<and> snd FAIL \<notin> nodes B \<and> AB = \<lparr>succ = \<lambda>a (p1, p2). if p1 \<in> nodes A \<and> p2 \<in> nodes B \<and> fst a \<in> inputs A \<and> snd a \<in> outputs A \<union> outputs B then if succ A a p1 = {} \<and> succ B a p2 \<noteq> {} then {FAIL} else succ A a p1 \<times> succ B a p2 else {}, inputs = inputs A, outputs = outputs A \<union> outputs B, initial = (initial A, initial B)\<rparr>)"
disj_assoc: fixes P :: "'a upred"   and Q :: "'a upred"   and S :: "'a upred" shows "((P \<or> Q) \<or> S) = (P \<or> Q \<or> S)"
inj: fixes A :: "('d \<times> 'e) set" shows "inj_on (\<lambda>(x, y). (x, x, y)) A"
nth_Cons_Suc: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
KBasic2_11: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<box>\<^bold>\<box>\<^bold>\<not>\<phi> in v]"
left_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "(- a) \<cdot> b = - a \<cdot> b"
sum: fixes s :: "'s" shows "sum (P s) (supp (P s)) \<le> 1"
LAss_lem: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'b list"   and m\<^sub>1 :: "'a \<Rightarrow> 'b option"   and m\<^sub>2 :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "x \<in> set xs"   and "length xs \<le> length ys"   and "m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2(xs [\<mapsto>] ys)" shows "m\<^sub>1(x \<mapsto> y) \<subseteq>\<^sub>m m\<^sub>2(xs [\<mapsto>] ys[last_index xs x := y])"
encode_goal_state_is_cnf: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf ((\<Phi>\<^sub>G \<Pi>) t)"
CD_on_XD_smallest: shows "CD_on ds XD_smallest = X"
1: fixes n :: "nat" assumes "a n \<noteq> \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>" shows "(\<ominus>\<^bsub>padic_int p\<^esub> a) n \<noteq> \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>"
typing_alt_simps_40: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<pi> :: "'a OCL_Types.type list" shows "(\<Gamma> \<turnstile>\<^sub>L [] : \<pi>) = (\<pi> = [])"
c_def: shows "c \<equiv> \<lambda>i. if i = 0 then x else y"
pwd_spy_sep_agr_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_sep_agr n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_sep_agr_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
goedel_first_theEasyHalf: assumes "consistent" shows "\<not> prv \<phi>G"
surj_of_rat_nat_to_rat_surj: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "class.field_char_0 divide inverse times one plus zero minus uminus"   and "r \<in> field_char_0.Rats divide one plus zero minus" shows "\<exists>n. r = field_char_0.of_rat divide one plus zero minus (nat_to_rat_surj n)"
smallest_closed_cdi2: shows "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
NONCE: fixes N :: "nat" shows "msgrelp (NONCE N) (NONCE N)"
im_sheaf_axioms: shows "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open (identity S)"
fpairwise_alt_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b fset" shows "fpairwise P M = fBall M (\<lambda>m. fBall M (P m))"
group_axioms_def: fixes G :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d" shows "Group_Theory.group_axioms G composition unit \<equiv> \<forall>u. u \<in> G \<longrightarrow> monoid.invertible G composition unit u"
not_disjoint_intervals_def: fixes A :: "'b word \<times> 'b word"   and B :: "'b word \<times> 'b word" shows "not_disjoint_intervals A B \<equiv> \<not> disjoint (interval_of A) (interval_of B)"
diff_0_right: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"
word_sub_mono2: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word"   and d :: "'b word" assumes "a + b \<le> c + d"   and "c \<le> a"   and "b \<le> a + b"   and "d \<le> c + d" shows "b \<le> d"
exception_\<I>_def: fixes \<I> :: "('c, 'd) \<I>" shows "exception_\<I> \<I> = map_\<I> id exception_of_option (stop_\<I> \<I>)"
mono_transD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "mono_trans t"   and "sound P"   and "sound Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
echelon_form_of_upt_k_iarrays_def: fixes A :: "'a iarray iarray"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "echelon_form_of_upt_k_iarrays A k bezout = fst (foldl echelon_form_of_column_k_iarrays (A, 0, bezout) [0..<Suc k])"
infinite_chain_function_iff_infinite_chain_llist: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<exists>f. \<forall>i. r (f (Suc i)) (f i)) = (\<exists>c. \<not> lfinite c \<and> chain r\<inverse>\<inverse> c)"
simps_2: fixes \<phi> :: "'fmla"   and \<phi>s :: "'fmla list" shows "ldsj (\<phi> # \<phi>s) = dsj \<phi> (ldsj \<phi>s)"
rbt_comp_lookup: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_lookup c = ord.rbt_lookup (lt_of_comp c)"
ennreal_add_left_cancel_le: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a + b \<le> a + c) = (a = \<infinity> \<or> b \<le> c)"
Sublists_conv_Suffixes: fixes A :: "'a list set" shows "Sublists A = Suffixes (Prefixes A)"
graph_minus_set_aimpl: fixes E1 :: "('a \<times> 'a) set"   and E2 :: "('a \<times> 'a) set" shows "E1 - E2 = E_of_succ (\<lambda>u. {v \<in> E1 `` {u}. (u, v) \<notin> E2})"
put_in_tape_small_enough0: fixes rsx :: "nat"   and xs :: "nat list"   and x :: "nat"   and anything :: "nat list" assumes "0 < rsx" shows "\<exists>y. (xs @ x # rsx # anything)[Suc (length xs) := rsx - Suc 0] = xs @ x # y # anything \<and> y \<le> rsx"
map_conv_bind_option: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "map_option f x = Option.bind x (Some \<circ> f)"
HPhase1or2Write_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2a s"   and "HPhase1or2Write s s' p d" shows "Inv2a s'"
f22: fixes out' :: "'c"   and rpv :: "('a, 'c, 'b) rpv"   and gpv :: "('a, 'c, 'b) gpv"   and input :: "'b"   and out :: "'c" assumes "IO out' rpv \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> (\<I>_uniform UNIV {x. Domainp R x}) out'"   and "out \<in> outs_gpv (\<I>_uniform UNIV {x. Domainp R x}) (rpv input)"   and "Domainp (rel_gpv'' A C R) gpv" shows "Domainp (rel_gpv'' A C R) (rpv input)"
nnegI2: fixes P :: "'a \<Rightarrow> 'b" assumes "(\<lambda>s. 0::'b) \<le> P" shows "nneg P"
intros_5: fixes T :: "Javatype"   and U :: "Javatype"   and V :: "Javatype" assumes "treach T U"   and "treach U V" shows "treach T V"
refine: fixes ai :: "'b list list"   and a :: "'c set"   and sctnsi :: "real list sctn list"   and sctns :: "'c sctn set" assumes "(ai, a) \<in> clw_rel appr_rel"   and "(sctnsi, sctns) \<in> sctns_rel" shows "(sbelow_sctns_coll_impl ai sctnsi, sbelow_sctns_coll $ a $ sctns) \<in> \<langle>bool_rel\<rangle>nres_rel"
prems_1: shows "\<forall>x\<in>set (a # u). is_closed x"
Vset_non: fixes x :: "hf" assumes "\<not> Ord x" shows "Vset x = 0"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VV.comp g f = (if VV.arr f \<and> VV.arr g \<and> CC.seq g f then CC.comp g f else CC.null)"
Conseq: fixes p' :: "'a set"   and p :: "'a set"   and c :: "'a com"   and q :: "'a set"   and q' :: "'a set" assumes "p' \<subseteq> p"   and "\<parallel>- p c q"   and "q \<subseteq> q'" shows "\<parallel>- p' c q'"
intros_2: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Accept], Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
HeronConf_interp_stepwise_implies_not_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
inter_rel_internal: fixes AA :: "('d \<times> 'f set) set"   and BB :: "('e \<times> 'f set) set" shows "inter_rel AA BB = {((a, b), A \<inter> B) |a b A B. (a, A) \<in> AA \<and> (b, B) \<in> BB}"
univ_poly_is_principal: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "principal_domain (K [X]\<^bsub>K [X]\<^esub>)"
nth_least_restrict: fixes i :: "nat"   and s :: "nat set"   and k :: "enat" assumes "enat i < esize {i \<in> s. enat i < k}" shows "nth_least {i \<in> s. enat i < k} i = nth_least s i"
plus_tree_simps_1: fixes t :: "'b tree"   and t' :: "'b tree" shows "root (t + t') = root t + root t'"
ide_dom: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.ide (C.VV.dom f)"
Suc_less_le: fixes x :: "nat"   and n :: "nat" assumes "x < Suc n" shows "x \<le> n"
numeral_code_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
f: shows "finprod G f (insert x F) = f x \<otimes> finprod G f F"
intros_3: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and V :: "'c"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' (Some V) None False n'"   and "n'' \<noteq> n'" shows "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ [n'']) n'"
distinct_5: fixes x41 :: "('a, 'b, 'c) com"   and x42 :: "('a, 'b, 'c) com" shows "Skip \<noteq> Seq x41 x42"
next_state_precondition: shows "\<forall>s d a execs. AS_precondition s d a \<longrightarrow> AS_precondition (int.next_state s execs) d a"
normalized_gauss_int'_1: fixes z :: "gauss_int" assumes "normalize z = z"   and "z \<noteq> 0" shows "0 < ReZ z"
lens_plus_ub: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e" assumes "wb_lens Y" shows "X \<subseteq>\<^sub>L X +\<^sub>L Y"
add_0_r: fixes a :: "'a"   and p :: "'a point" shows "local.add a p Infinity = p"
correct_share_eval_def: fixes input1 :: "bool"   and input2 :: "bool" shows "share_correct.correct_share_eval input1 input2 \<equiv> \<forall>gate_eval\<in>F. \<exists>gate_protocol. share input1 \<bind> (\<lambda>(s1, s2). share input2 \<bind> (\<lambda>(s3, s4). gate_protocol (s1, s3) (s2, s4) \<bind> (\<lambda>(S1, S2). reconstruct (S1, S2)))) = gate_eval input1 input2"
s2'_in_S: shows "s2' \<in> S\<^bsub>SES\<^esub>"
isDERIV_FDERIV_floatariths: fixes N :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list"   and ds :: "floatarith list" assumes "isFDERIV N xs fas vs"   and "isFDERIV N xs ds vs"   and "length fas = length ds" shows "isFDERIV N xs (FDERIV_floatariths fas xs ds) vs"
substitute_not_carrier: fixes x :: "'b"   and t :: "'b ttree"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" assumes "x \<notin> carrier t"   and "\<And>x'. x \<notin> carrier (f x')" shows "x \<notin> carrier (substitute f T t)"
prv_eql_subst_trm_id_back_rev: fixes y :: "'var"   and \<phi> :: "'fmla"   and n :: "'trm" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "n \<in> num" shows "bprv (imp (eql n (Var y)) (imp (subst \<phi> n y) \<phi>))"
1: shows "\<l>\<^sub>C[a] \<in> C.hom (\<I>\<^sub>C \<otimes>\<^sub>C a) a \<and> \<I>\<^sub>C \<otimes>\<^sub>C \<l>\<^sub>C[a] = (\<iota>\<^sub>C \<otimes>\<^sub>C a) \<cdot>\<^sub>C C.assoc' \<I>\<^sub>C \<I>\<^sub>C a"
bot: shows "P (\<lambda>_. luba {}) (\<lambda>_. lubb {})"
subgraph_subset_1: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2" shows "vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2"
suffix_0: fixes x :: "nat \<Rightarrow> 'a" shows "suffix 0 x = x"
min_ground_head_in_ground_heads: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and \<zeta> :: "('b, 'a) hd" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.min_ground_head ground_heads_var \<delta> arity_sym wt_sym \<zeta> \<in> ground_heads.ground_heads ground_heads_var \<zeta>"
bind_split_asm: fixes P :: "'c option \<Rightarrow> bool"   and m :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "P (m \<bind> f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = Some x \<and> \<not> P (f x))))"
R1: fixes step :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and out :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and s\<^sub>0 :: "'e"   and xps :: "('f \<times> 'g) list"   and uu :: "('f \<times> 'g) set"   and s :: "'e"   and x :: "'f" assumes "c_failuresp step out s\<^sub>0 xps uu"   and "s = foldl step s\<^sub>0 (map fst xps)" shows "c_failuresp step out s\<^sub>0 (xps @ [(x, out s x)]) {(y, p). p \<noteq> out (step s x) y}"
ldropn_0: fixes xs :: "'c llist" shows "ldropn 0 xs = xs"
wf_converse_unif: shows "wf (unif\<inverse>)"
P: shows "f absolutely_integrable_on {y \<in> g ` S. 0 < f y}"
elims: fixes x :: "poly"   and y :: "nat \<Rightarrow> nat" assumes "degreen x = y" obtains c :: "poly" and n :: "nat" and p :: "poly" where "x = CN c n p" and "y = (\<lambda>m. if n = m then 1 + degreen p n else 0)"   | v :: "int \<times> int" where "x = C v__" and "y = (\<lambda>m. 0)"   | v :: "nat" where "x = Bound va__" and "y = (\<lambda>m. 0)"   | v :: "poly" and va :: "poly" where "x = Add vb__ vaa__" and "y = (\<lambda>m. 0)"   | v :: "poly" and va :: "poly" where "x = Sub vc__ vab__" and "y = (\<lambda>m. 0)"   | v :: "poly" and va :: "poly" where "x = Mul vd__ vac__" and "y = (\<lambda>m. 0)"   | v :: "poly" where "x = Neg ve__" and "y = (\<lambda>m. 0)"   | v :: "poly" and va :: "nat" where "x = Pw v va" and "y = (\<lambda>m. 0)"
INF_less_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and a :: "'b" shows "(Inf (f ` A) < a) = (\<exists>x\<in>A. f x < a)"
HAInitStates_HAStates: fixes A :: "('a, 'b, 'c) hierauto" shows "HAInitStates A \<subseteq> HAStates A"
stabiliser_subgroup_corollary: fixes g :: "'a"   and h :: "'a"   and x :: "'b" assumes "g \<in> carrier G"   and "h \<in> carrier G" shows "(g \<odot> x = h \<odot> x) = (inv g \<otimes> h \<in> stabiliser x)"
x'_def: shows "x' = ratfps_of_poly (ratfps_cutoff n x)"
sin_minus: fixes x :: "'b" shows "sin (- x) = - sin x"
axioms_2: fixes inner_cols :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a"   and inner_rows :: "('a, 'c) vec \<Rightarrow> ('a, 'c) vec \<Rightarrow> 'a" assumes "Generalizations2.matrix inner_cols inner_rows" shows "vec_real_inner (\<lambda>x y. x * cnj_class.cnj y) inner_rows"
simps_5: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "wt_cont_eff (Ch ch c1 c2) s i = (if i = 0 then cval ch s else 1 - cval ch s, if i = 0 then c1 else c2, s)"
AAs0'_def: shows "AAs0' = AAs0 \<cdot>\<cdot>aml \<rho>s"
lappend_eq_lappend_conv: fixes xs :: "'a llist"   and us :: "'a llist"   and ys :: "'a llist"   and vs :: "'a llist" assumes "llength xs = llength us" shows "(xs @\<^sub>l ys = us @\<^sub>l vs) = (xs = us \<and> (lfinite xs \<longrightarrow> ys = vs))"
cbig_def: fixes P :: "'prog"   and \<sigma> :: "'state" shows "cbig P \<sigma> \<equiv> {(\<sigma>', coll). \<exists>n. (\<sigma>', coll) \<in> csmall_nstep P \<sigma> n \<and> \<sigma>' \<in> endset}"
card_UNIV'_def: shows "card_UNIV' = Phantom('b) CARD('b)"
simps: fixes a :: "'c set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = aa \<triangleright> A \<and> finite A))"
GS_def: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt" shows "GS f \<equiv> \<not> manipulable f \<and> f ` Prof = UNIV"
map_add_restrict_nonmember_left: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd option"   and m' :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom m" shows "(m ++ m') |` {x} = m' |` {x}"
path_cons_simp: fixes v :: "'v"   and v' :: "'v"   and e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and pth :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" shows "path v v' (e # pth) = (fst (fst e) = v \<and> e \<in> edges \<and> path (fst (snd e)) v' pth)"
induct: fixes P :: "'d list \<Rightarrow> ('d, 'c) trie \<Rightarrow> bool"   and a0 :: "'d list"   and a1 :: "('d, 'c) trie" assumes "\<And>vo ts. P [] (Trie vo ts)"   and "\<And>k ks vo ts. (\<And>x2. map_of ts k = Some x2 \<Longrightarrow> P ks x2) \<Longrightarrow> P (k # ks) (Trie vo ts)" shows "P a0 a1"
Macaulay_red_def: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Macaulay_red vs fs = (let lts = map lt (filter (\<lambda>p. p \<noteq> 0) fs) in filter (\<lambda>p. p \<noteq> 0 \<and> lt p \<notin> set lts) (mat_to_polys vs (row_echelon (polys_to_mat vs fs))))"
conjugate: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and z :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc"   and "P (conjugate z) (conjugate u) (conjugate v) (conjugate w)" shows "P z u v w"
Close2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "Q \<longmapsto>\<^sub>e a<x> \<prec>\<^sub>e Q'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e <\<nu>x>(P' \<parallel> Q')"
split_i_j_x_def: fixes i :: "nat"   and j :: "nat"   and x :: "(nat, 'b) mapping" shows "split_i_j_x i j x = (vec i \<langle>x\<rangle>, vec (j - i) (\<lambda>y. \<langle>x\<rangle> (y + i)))"
ide_cod: fixes f :: "'j" assumes "J.arr f" shows "J.ide (J.cod f)"
abs_eq: fixes x :: "'b multiset \<times> 'b multiset" shows "mset_neg (abs_zmultiset x) = Abs_multiset (case x of (Mp, Mn) \<Rightarrow> count (Mn - Mp))"
asm: shows "NFlow c s t f \<and> (\<forall>f'. NFlow c s t f' \<longrightarrow> Flow.val c s f' \<le> val)"
foldr_Cons: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D"   and "y \<in> D" shows "x \<cdot> y = y \<cdot> x"