hcju/isabelleps huggingface dataset:
ranking_def: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and f :: "nat \<times> 'b \<Rightarrow> nat" shows "ranking A w f \<equiv> (\<forall>v\<in>gunodes A w. f v \<le> 2 * card (nodes A)) \<and> (\<forall>v\<in>gunodes A w. \<forall>u\<in>gusuccessors A w v. f u \<le> f v) \<and> (\<forall>v\<in>gunodes A w. gaccepting A v \<longrightarrow> even (f v)) \<and> (\<forall>v\<in>gunodes A w. \<forall>r k. gurun A w r v \<longrightarrow> smap f (gtrace r v) = sconst k \<longrightarrow> odd k)"
visible_method_exists: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>(b, Ts, T, m)\<rfloor>"
normalize_poly_def: fixes p :: "'b poly" shows "normalize p = p div [:unit_factor (lead_coeff p):]"
True: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
finite_field_factorization_int_def: fixes p :: "int" shows "finite_field_factorization_int p = (if p \<le> 65535 then finite_field_factorization_main p (finite_field_ops32 (uint32_of_int p)) else if p \<le> 4294967295 then finite_field_factorization_main p (finite_field_ops64 (uint64_of_int p)) else finite_field_factorization_main p (finite_field_ops_integer (integer_of_int p)))"
has_domain_iff_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "(domains f \<noteq> {}) = arr f"
non_Limit_ord_of_nat: fixes m :: "nat" shows "\<not> Limit (m\<^sub>\<nat>)"
D_base_finite: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> distinct xs}"
assms_4: shows "length (predecessors g n) \<noteq> 1"
intros_14: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com" shows "\<Gamma>\<turnstile>c \<down> Stuck"
hom_mult: fixes x :: "real poly"   and y :: "real poly" shows "map_poly of_real (x * y) = map_poly of_real x * map_poly of_real y"
arctan_less_zero_iff: fixes x :: "real" shows "(arctan x < 0) = (x < 0)"
OO_def: fixes R :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'h \<Rightarrow> 'g \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
presCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presCons h hA MOD \<equiv> presVar h MOD \<and> presAbs h hA MOD \<and> presOp h hA MOD"
cf_h_coeff: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "f \<in> rHom S B"   and "pol_coeff S c" shows "pol_coeff B (cf_h f c)"
igSubstBinpIPresIGWlsBinpSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstBinpIPresIGWlsBinpSTR MOD \<equiv> \<forall>ys y Y delta binp. igWls MOD (asSort ys) Y \<longrightarrow> igWlsBinp MOD delta (igSubstBinp MOD ys Y y binp) = igWlsBinp MOD delta binp"
c_union_is_pr: shows "c_union \<in> PrimRec2"
hyps: assumes "singleCombinators xs"   and "distinct xs"   and "allNetsDistinct xs" shows "wellformed_policy2Pr (removeShadowRules2 xs)"
2: shows "V.arr \<lparr>Chn = \<mu>.dtrg, Dom = \<lparr>Leg0 = \<mu>.dtrg, Leg1 = \<mu>.dtrg\<rparr>, Cod = \<lparr>Leg0 = \<mu>.dtrg, Leg1 = \<mu>.dtrg\<rparr>\<rparr>"
module_inc_zero: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "R module M" shows "\<zero>\<^bsub>M\<^esub> \<in> carrier M"
dilating_fun_image_right: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and m :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating_fun f r" shows "{k. f m < k \<and> k \<le> f n \<and> hamlet (Rep_run r k c)} = f ` {k. m < k \<and> k \<le> n \<and> hamlet (Rep_run r (f k) c)}"
Abs_reg_lan_cases: fixes x :: "'a reg_lan" obtains y :: "'a list set" where "x = Abs_reg_lan y" and "y \<in> range lang"
map_key_PP_plus: fixes p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c"   and q :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.map_key PP (p + q) = Poly_Mapping.map_key PP p + Poly_Mapping.map_key PP q"
extensional_restrict: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" assumes "f \<in> extensional A" shows "restrict f A = f"
quasi_isometry_on_empty: fixes C :: "real"   and lambda :: "real"   and f :: "'b \<Rightarrow> 'c" assumes "0 \<le> C"   and "1 \<le> lambda" shows "lambda C -quasi_isometry_on {} f"
winning_moves_remain_winning_Odd: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a" assumes "odd (length p)"   and "strategy_winning_by_Odd \<sigma> p" shows "strategy_winning_by_Odd \<sigma> (augment_list \<sigma> p)"
succ_rvar_valuation_id: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and x :: "nat" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x \<in> rvars (\<T> s)"   and "x \<in> rvars (\<T> s')" shows "\<langle>\<V> s\<rangle> x = \<langle>\<V> s'\<rangle> x"
null_uni_uniq_1: fixes v :: "i" shows "[\<^bold>\<exists>!x. Null (x\<^sup>P) in v]"
zero_min: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq zero x"
poly_subst_poly_subst: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and g :: "'f \<Rightarrow> ('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('f \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_subst f (poly_subst g p) = poly_subst (\<lambda>x. poly_subst f (g x)) p"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "sum g A \<noteq> (0::'b)" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> (0::'b)"
rep_eq: fixes x :: "'b interval"   and xa :: "'b interval" shows "bounds_of_interval (min_interval x xa) = (case bounds_of_interval x of (l1, u1) \<Rightarrow> \<lambda>(l2, u2). (min l1 l2, min u1 u2)) (bounds_of_interval xa)"
sc_minus_am1: fixes a :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M" shows "-\<^sub>a a \<cdot>\<^sub>s m = (-\<^sub>a\<^bsub>R\<^esub> a) \<cdot>\<^sub>s m"
inline_parametric': fixes S :: "'a \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'h \<Rightarrow> bool"   and C' :: "'d \<Rightarrow> 'i \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'j \<Rightarrow> bool"   and A :: "'k \<Rightarrow> 'l \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun C (rel_gpv'' (rel_prod R S) C' R'))) (rel_fun (rel_gpv'' A C R) (rel_fun S (rel_gpv'' (rel_prod A S) C' R'))) inline inline"
Image_def: fixes r :: "('f \<times> 'e) set"   and s :: "'f set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
dom_map_upd: fixes m :: "'d \<Rightarrow> 'e option"   and a :: "'d"   and b :: "'e" shows "dom (m(a \<mapsto> b)) = insert a (dom m)"
mfinalD_2: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and x :: "'a"   and ln :: "'b \<Rightarrow>f nat" assumes "final_thread.mfinal final s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" shows "ln = no_wait_locks"
hcard_hinsert_if: fixes x :: "hf"   and y :: "hf" shows "hcard (y \<triangleleft> x) = (if x \<^bold>\<in> y then hcard y else Suc (hcard y))"
indRelLTPO_impl_TRel_is_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_bisimulation (indRelLTPO TRel) (STCal Source Target)" shows "weak_reduction_bisimulation (TRel\<^sup>+) Target"
rep_eq: fixes x :: "'b" shows "(*\<^sub>v) (blinfun_mult_right x) = (*) x"
simps_2: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" shows "tree_invar (Node e a (Suc r) ts) = (\<exists>e1 a1 ts1 e2 a2 ts2 e' a'. tree_invar (Node e1 a1 r ts1) \<and> tree_invar (Node e2 a2 r ts2) \<and> (Node e a (Suc r) ts = link (Node e1 a1 r ts1) (Node e2 a2 r ts2) \<or> Node e a (Suc r) ts = skewlink e' a' (Node e1 a1 r ts1) (Node e2 a2 r ts2)))"
CNOT_def: shows "CNOT \<equiv> mat 4 4 (\<lambda>(i, j). if i = 0 \<and> j = 0 then 1 else if i = 1 \<and> j = 1 then 1 else if i = 2 \<and> j = 3 then 1 else if i = 3 \<and> j = 2 then 1 else 0)"
Abs_freeword_conv_freeletter_sum_list: fixes xs :: "('a \<times> bool) list" assumes "proper_signed_list xs" shows "Abs_freeword xs = sum_list (map (apply_sign Abs_freeletter) xs)"
1: fixes x :: "'s" assumes "x \<in> S.set a" shows "\<Phi> x = \<phi> x"
cases: fixes a :: "C"   and b :: "nat" assumes "a \<in> hbase_ext b" obtains n :: "C" where "a = na__" and "na__ \<in> hbase b"   | n :: "C" and m :: "C list" where "a = C (n # m)" and "n \<in> hbase_ext b" and "C m \<in> hbase b" and "\<And>m'. m' \<in> set m \<Longrightarrow> evalC b n \<le> evalC b m'"
times_ow_def: fixes U :: "'a set"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "times_ow U times \<equiv> \<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> times a b \<in> U"
simps_2: fixes ds :: "bool list"   and T\<^sub>1 :: "tree"   and T\<^sub>2 :: "tree" shows "delete (True # ds) (Branching T\<^sub>1 T\<^sub>2) = Branching (delete ds T\<^sub>1) T\<^sub>2"
Units_inv_closed: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "inv x \<in> carrier Zp"
Units_mult_eq_Units: shows "Units (Ring_Divisibility.mult_of R) = Units R"
exists_gcd: fixes 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 a :: "'b"   and b :: "'b" assumes "class.GCD_ring times one plus zero minus uminus" shows "\<exists>d. dvd.dvd times d a \<and> dvd.dvd times d b \<and> (\<forall>d'. dvd.dvd times d' a \<and> dvd.dvd times d' b \<longrightarrow> dvd.dvd times d' d)"
measurable_distr: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(\<lambda>M'. distr M' N f) \<in> subprob_algebra M \<rightarrow>\<^sub>M subprob_algebra N"
assms: shows "constant_term (snd x) \<noteq> (0::'a)"
2: shows "{xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> xs ! e} \<union> {xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> \<not> xs ! e} = {xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m}"
antisym: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b"   and "b \<le> a" shows "a = b"
indexed_nominal_ts_axioms: shows "indexed_nominal_ts TYPE('idx) (\<turnstile>) (\<rightarrow>)"
l11_4_2: 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"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E"   and "\<forall>A' C' D' F'. Tarski_neutral_dimensionless.Out Bet B A' A \<and> Tarski_neutral_dimensionless.Out Bet B C' C \<and> Tarski_neutral_dimensionless.Out Bet E D' D \<and> Tarski_neutral_dimensionless.Out Bet E F' F \<and> Cong B A' E D' \<and> Cong B C' E F' \<longrightarrow> Cong A' C' D' F'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"
simps_5: fixes b :: "bexp"   and c :: "com"   and t :: "char list \<Rightarrow> int option" shows "defs (WHILE b DO c) t = t |` (- lvars c)"
nn_integral_uniform_measure: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and S :: "'d set" assumes "f \<in> borel_measurable M"   and "S \<in> sets M" shows "integral\<^sup>N (uniform_measure M S) f = set_nn_integral M S f / emeasure M S"
foldD_insert: fixes B :: "'c set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and e :: "'d" assumes "LCD B D f"   and "finite A"   and "x \<notin> A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e (insert x A) = f x (foldD D f e A)"
sconst_streams: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "sconst x \<in> streams A"
comp_assoc: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VoV.comp (VoV.comp h g) f = VoV.comp h (VoV.comp g f)"
dvd_mult: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (times b c)"
lookup_tl': fixes xs :: "'x"   and k :: "'a" shows "lookup (local.tl xs) k = (if k = fst (local.hd xs) then 0::'b else lookup xs k)"
f1: shows "\<not> ((fst instr = load_store_type LDD \<or> fst instr = load_store_type LDDA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) \<noteq> 0 \<or> (fst instr = load_store_type LD \<or> fst instr = load_store_type LDA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) \<noteq> 0 \<or> (fst instr = load_store_type LDUH \<or> fst instr = load_store_type LDUHA \<or> fst instr = load_store_type LDSH \<or> fst instr = load_store_type LDSHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) \<noteq> 0) \<and> \<not> ((fst instr = load_store_type LDD \<or> fst instr = load_store_type LDDA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<noteq> 0 \<or> (fst instr = load_store_type LD \<or> fst instr = load_store_type LDA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<noteq> 0 \<or> (fst instr = load_store_type LDUH \<or> fst instr = load_store_type LDUHA \<or> fst instr = load_store_type LDSH \<or> fst instr = load_store_type LDSHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<noteq> 0)"
X: fixes i :: "'i" assumes "i \<in> J" shows "X i \<in> events i"
no_cs_change_if_no_channel: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "channel i = None" shows "cs c i = cs c' i"
diff_subset_eq_self: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M - N \<subseteq># M"
light_edge_def: fixes C :: "'v set"   and u :: "'v"   and v :: "'v" shows "light_edge C u v \<equiv> u \<in> C \<and> v \<notin> C \<and> (u, v) \<in> edges rg \<and> (\<forall>(u', v')\<in>edges rg \<inter> C \<times> - C. w {u, v} \<le> w {u', v'})"
rsr5: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad (rel_ad (x \<union> y)) = rel_ad (rel_ad x) \<union> rel_ad (rel_ad y)"
holomorphic_on_balls_imp_entire: fixes A :: "real set"   and f :: "complex \<Rightarrow> complex"   and c :: "complex"   and B :: "complex set" assumes "\<not> bdd_above A"   and "\<And>r. r \<in> A \<Longrightarrow> f holomorphic_on ball c r" shows "f holomorphic_on B"
Diagonalize_in_Hom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> \<in> Hom \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
assms_1: shows "J \<subseteq>\<^sub>\<circ> I"
remdups_adj_length_ge1: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "Suc 0 \<le> length (remdups_adj xs)"
1: shows "\<forall>i. Cring_Poly.compose R f (monom (UP R) d 1) i = d [^] i \<otimes> f i"
S_K2_empty: shows "S \<inter> hyp2 = {}"
ts''_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : ts'' _> ts'"
vardict_inv_updateI: fixes ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list"   and vs :: "(String.literal, variable) assoc_list"   and x :: "String.literal"   and v :: "variable" assumes "vardict_inv ss proc vs"   and "x \<in> process_names ss proc"   and "variable_inv v" shows "vardict_inv ss proc (lm.update x v vs)"
a2: shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
init_conv: fixes gco :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option"   and prog :: "program" shows "(gco \<in> li.init prog) = (\<exists>gc. gco = Some gc \<and> gc \<in> {init_gc prog})"
matrix_to_iarray_snd_Gauss_Jordan_PA: fixes A :: "(('d, 'e) vec, 'f) vec" shows "matrix_to_iarray (snd (Gauss_Jordan_PA A)) = snd (Gauss_Jordan_iarrays_PA (matrix_to_iarray A))"
iT_Plus_neg_cut_le: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus>- k) \<down>\<le> t = I \<down>\<le> (t + k) \<oplus>- k"
Diff_iff: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
WS_silent_moves: fixes S :: "'node SDG_node set"   and ms\<^sub>1 :: "'node list"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms\<^sub>1' :: "'node list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms\<^sub>2 :: "'node list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) =as\<Rightarrow>\<^sub>\<tau> (ms\<^sub>1',s\<^sub>1')"   and "((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> WS S" shows "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>2, s\<^sub>2) \<in> WS S"
simps_2: fixes f :: "char list"   and ts :: "fterm list" shows "vars\<^sub>t (Fun f ts) = vars\<^sub>t\<^sub>s ts"
tail_face_cycle_succ: fixes a :: "'b" assumes "a \<in> arcs H" shows "tail H (H.face_cycle_succ a) = head H a"
prems: fixes x :: "nat" assumes "x \<in> free_vars (LET e1 IN e2)" shows "\<Gamma> x = \<Gamma>' x"
stone_phi_base_filter: fixes x :: "'c regular" shows "ord_class.filter (stone_phi_base x)"
toplevel_inters_lang: fixes r :: "'b rexp"   and s :: "'b rexp"   and n :: "nat" assumes "r \<in> toplevel_inters s" shows "lang n s \<subseteq> lang n r"
in_homE: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and a :: "'d"   and b :: "'d" assumes "category C"   and "partial_magma.in_hom C f a b" obtains "partial_magma.arr C f" and "partial_magma.dom C f = a" and "partial_magma.cod C f = b"
is_generalisation_of_apply_guards: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat"   and j :: "vname \<Rightarrow> value option" assumes "is_generalisation_of t' t i r"   and "apply_guards (Guards t) j" shows "apply_guards (Guards t') j"
ProblemIkTpartG_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool" shows "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl \<equiv> ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw \<and> ProblemIkTpartG_axioms arOf prot protCl"
r_null: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "x \<otimes>\<^bsub>poly_ring R\<^esub> \<zero>\<^bsub>poly_ring R\<^esub> = \<zero>\<^bsub>poly_ring R\<^esub>"
sup1I1: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and B :: "'a \<Rightarrow> bool" assumes "A x" shows "(A \<squnion> B) x"
\<theta>_upper_bound: fixes x :: "real" assumes "1 \<le> x" shows "\<theta> x < ln 4 * x"
exhaust: fixes y :: "f" obtains "y = val"   | "y = Aodv_Basic.inv"
cobounded1: fixes a :: "'h"   and b :: "'h" shows "a \<le> max a b"
nat_sing_prop: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "{i. i = k \<and> P i} = {i. i = k \<and> P k}"
dense_up_closed: fixes x :: "'a"   and y :: "'a" assumes "dense x"   and "x \<le> y" shows "dense y"
iT_Plus_cut_less2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>< (t + k) = I \<down>< t \<oplus> k"
add_cancel_left_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
seq_char: fixes g :: "('a, 'a) CC.arr"   and f :: "('a, 'a) CC.arr" shows "CC.seq g f = (CC.arr f \<and> CC.arr g \<and> CC.Dom g = CC.Cod f)"
finite_lookup_default: fixes d :: "'b"   and xs :: "('a, 'b) fmap" shows "finite {x. fmlookup_default d xs x \<noteq> d}"
upred_lattice_eq: shows "(.=\<^bsub>\<P>\<^esub>) = (=)"
l_put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas DocumentClass.type_wf"
linord_of_list_Field: fixes xs :: "'d list" shows "Field (linord_of_list xs) = set xs"
cauchy_minus: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "Real.cauchy (\<lambda>n. - X n)"
MWLfSteps_det_cases_6: fixes c :: "('exp, 'id) MWLfCom"   and V :: "('exp, 'id) MWLfCom list"   and m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (fork c V) m W m'" obtains "W = c # V" and "m' = m"
pre_oldest_configuration_range: fixes q :: "'b"   and n :: "nat" shows "pre_oldest_tokens q n \<subseteq> {0..Suc n}"
elims: fixes x :: "nat \<Rightarrow> nat"   and xa :: "tv"   and y :: "tv" assumes "change_tv x xa = y" obtains f :: "nat \<Rightarrow> nat" and b :: "bool" where "x = f__" and "xa = Det b" and "y = Det b"   | f :: "nat \<Rightarrow> nat" and n :: "nat" where "x = f" and "xa = Indet n" and "y = Indet (f n)"
left_null_space_orthogonal_complement_col_space: fixes A :: "((real, 'b) vec, 'a) vec" shows "left_null_space A = real_matrix.v.orthogonal_complement (col_space (\<chi>i j. cnj_class.cnj (A $ i $ j)))"
cut_leI: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I"   and "x \<le> t" shows "x \<in> I \<down>\<le> t"
has_densityD_1: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M N f" shows "f \<in> borel_measurable N"
sba1_bot: fixes z :: "'a" shows "sba1_bot = - (z \<squnion> - z)"
gauss_int_norm_pos_iff: fixes z :: "gauss_int" shows "(0 < gauss_int_norm z) = (z \<noteq> 0)"
ubounds_def: fixes as :: "atom list" shows "ubounds as = concat (map (\<lambda>x. case x of Less r [] \<Rightarrow> [] | Less r (c # cs) \<Rightarrow> if c < 0 then [(r / c, (- 1 / c) *\<^sub>s cs)] else [] | Eq real list \<Rightarrow> []) as)"
cases: fixes x :: "eint" obtains xa :: "int" where "x = eint xa"   | "x = \<infinity>"
ll_on_open_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set" shows "ll_on_open T f X \<equiv> interval T \<and> ll_on_open_axioms T f X"
rGamma_hurwitz_zeta_eq_contour_integral: fixes s :: "complex"   and r :: "real" assumes "s \<noteq> 0"   and "r \<in> {1..<2}"   and "0 < a" shows "2 * \<i> * complex_of_real pi * rGamma s * hurwitz_zeta a (1 - s) = cnj (contour_integral (part_circlepath 0 r pi 0) (f (cnj s))) - contour_integral (part_circlepath 0 r pi 0) (f s) + 2 * \<i> * sin (complex_of_real pi * s) * complex_set_lebesgue_integral lborel {r..} (g s)"
widen_array_array: fixes P :: "'a prog"   and A :: "ty"   and B :: "ty" assumes "P \<turnstile> A \<le> B" shows "P \<turnstile> A\<lfloor>\<rceil> \<le> B\<lfloor>\<rceil>"