hcju/isabelleps huggingface dataset:
succ_ne_self: fixes i :: "hf" shows "i \<noteq> succ i"
hyps: fixes u :: "'a" assumes "p \<noteq> []"   and "u = tail G (hd p)"   and "v = head G (last p)"   and "\<And>i. Suc i < length p \<Longrightarrow> head G (p ! i) = tail G (p ! Suc i)" shows "cas u p v"
pair_sym_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs H"   and "w \<notin> pverts H" shows "pair_sym_digraph (subdivide H e w)"
nnegD: fixes P :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "nneg P" shows "(0::'d) \<le> P x"
simps_1: fixes t :: "trm"   and k :: "nat" shows "t[k \<mapsto>\<^sub>s []] = t"
rel_list_all2_Cons2: fixes S :: "('b \<times> 'c) set"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "list_all2 (fun_of S) xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> (z, y) \<in> S \<and> list_all2 (fun_of S) zs ys)"
principal_g3mod1: fixes r :: "(C, BD) sequent list \<times> (C, BD) sequent"   and R :: "((C, BD) sequent list \<times> (C, BD) sequent) set"   and A :: "(C, BD) form" assumes "r \<in> g3mod1"   and "R = Ax \<union> g3up \<union> g3mod1 \<union> g3mod2"   and "rightPrincipal r (\<box> A) R" shows "( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
last_tl: fixes xs :: "'a list" assumes "xs = [] \<or> tl xs \<noteq> []" shows "last (tl xs) = last xs"
not_asym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(\<not> x \<sqsubset> y) = (y \<sqsubseteq> x)"
surj_const_mult: fixes a :: "'a vec" assumes "a \<in> carrier_vec n" shows "(+) a ` carrier_vec n = carrier_vec n"
closure_of_eq_topspace: fixes X :: "'c topology"   and S :: "'c set" shows "(X closure_of S = topspace X) = (X interior_of (topspace X - S) = {})"
pseudo_simplex_like_above_pseudominimal_is_top: fixes P :: "'a set"   and f :: "'a \<Rightarrow> 'c set"   and A :: "'c set"   and t :: "'a" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A"   and "t \<in> P"   and "\<And>x. pseudominimal_in P x \<Longrightarrow> t \<succeq> x" shows "f t = A"
L3_5v_lemma5: fixes b :: "'a set"   and c :: "'a com"   and k :: "nat"   and s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a"   and i :: "'a set" assumes "(fwhile b c k, s) -P*\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "(While b i c, s) -P*\<rightarrow> (Parallel Ts, t)"
butlast_rev: fixes xs :: "'b list" shows "butlast (rev xs) = rev (tl xs)"
simps_1: fixes allowed :: "char list list"   and x :: "char list"   and xs :: "char list list"   and ts :: "char list" shows "??.Parser_Monad.oneof_aux allowed (x # xs) ts = (if map snd (zip x ts) = x then Inr (x, trim (drop (length x) ts)) else ??.Parser_Monad.oneof_aux allowed xs ts)"
coeff_map_poly_hom: fixes p :: "real poly"   and i :: "nat" shows "coeff (map_poly of_real p) i = of_real (coeff p i)"
isStaticLoc_simps_2: fixes f :: "AttId" shows "isStaticLoc (staticLoc f) = True"
less_numeral_extra_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
depth_decr: fixes lev :: "nat"   and t :: "Core.term" shows "depth (decr lev t) = depth t"
h': shows "h3 \<turnstile> set_disconnected_nodes document_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes_h3) \<rightarrow>\<^sub>h h'"
continuous_Im: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> complex" assumes "continuous F g" shows "continuous F (\<lambda>x. Im (g x))"
iso_in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"   and "C.iso f" shows "\<guillemotleft>f : a \<cong>\<^sub>C b\<guillemotright>"
simps_5: fixes I :: "'a"   and b :: "bexp"   and P :: "'a"   and C :: "'a acom"   and Q :: "'a" shows "strip ({I} WHILE b DO {P} C {Q}) = WHILE b DO strip C"
zp_sg: shows "subgroup_generated (relative_homology_group (int n) (lsphere n) (equator n)) {zp} = relative_homology_group (int n) (lsphere n) (equator n)"
starset_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "*s* f ` A = (*f* f) ` (*s* A)"
polyfun_Prod: fixes I :: "'a set"   and N :: "nat set"   and f :: "'a \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> polyfun N (f i)" shows "polyfun N (\<lambda>x. \<Prod>i\<in>I. f i x)"
While_P_def: fixes vars :: "nat set"   and M0 :: "complex mat"   and M1 :: "complex mat"   and C :: "com" shows "ps_P.While_P vars M0 M1 C = While (\<lambda>n. if n = 0 then mat_extension ps2_P.dims0 vars M0 else if n = 1 then mat_extension ps2_P.dims0 vars M1 else undefined) C"
square_inequality: fixes n :: "nat" assumes "1 < n" shows "n + n \<le> n * n"
abs_eq: fixes xb :: "'b"   and xa :: "PreSimplyTyped.type"   and x :: "'b ptrm" shows "Fn xb xa (abs_trm x) = abs_trm (PFn xb xa x)"
tree_sigma_def: fixes M :: "'b measure" shows "tree_sigma M = sigma (trees (space M)) (trees_cyl ` trees (sets M))"
to_nat_from_nat: fixes j :: "'a"   and k :: "nat" assumes "mod_type_class.to_nat j = k" shows "mod_type_class.from_nat k = j"
primeD1: fixes p :: "nat" assumes "prime p" shows "1 < p"
RedTry: fixes v :: "val"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>try Val v catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>Val v,s,b\<rangle>"
asubg_nsubg: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "aGroup A"   and "A +> H" shows "b_ag A \<triangleright> H"
map_of_map: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "('e \<times> 'g) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = (map_option f \<circ>\<circ> map_of) xs"
simps: fixes ko :: "'o"   and xs :: "('a \<times> 'c) list"   and ox :: "'o" shows "sort_oalist_aux ko (xs, ox) = (if ko = ox then xs else sort_oalist ko xs)"
abs_eq: fixes x :: "ereal" assumes "eq_onp ((\<le>) 0) x x" shows "inverse (e2ennreal x) = e2ennreal (inverse x)"
r_maxlt_def: shows "r_maxlt \<equiv> r_shrink r_maxletr"
pre_Frame: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<forall>s. P s \<longrightarrow> pre C Q s"   and "vc C Q" shows "\<exists>C'. strip C = strip C' \<and> vc C' (Q \<and>* F) \<and> (\<forall>s. (P \<and>* F) s \<longrightarrow> pre C' (Q \<and>* F) s)"
bot_eq_sup_iff: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(bot = sup x y) = (x = bot \<and> y = bot)"
angle_construction_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 A' :: "'a"   and B' :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "\<not> Tarski_neutral_dimensionless.Col Bet A' B' P" shows "\<exists>C'. Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C' \<and> (Tarski_neutral_dimensionless.OS Bet A' B' C' P \<or> Tarski_neutral_dimensionless.Col Bet A' B' C')"
bound_of_nat_times: fixes n :: "nat"   and v :: "'a" shows "bound (of_nat n * v) \<le> n * bound v"
set_rotate: fixes n :: "nat"   and xs :: "'g list" shows "set (rotate n xs) = set xs"
induct: fixes x1 :: "fterm literal set set"   and x2 :: "fterm literal set set"   and P :: "fterm literal set set \<Rightarrow> fterm literal set set \<Rightarrow> bool" assumes "resolution_step x1 x2"   and "\<And>C\<^sub>1 Cs C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>. \<lbrakk>C\<^sub>1 \<in> Cs; C\<^sub>2 \<in> Cs; applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>\<rbrakk> \<Longrightarrow> P Cs (Cs \<union> {resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>})"   and "\<And>C Cs C'. \<lbrakk>C \<in> Cs; var_renaming_of C C'\<rbrakk> \<Longrightarrow> P Cs (Cs \<union> {C'})" shows "P x1 x2"
has_field_derivative_cong_ev: fixes x :: "'a"   and y :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a"   and t :: "'a set" assumes "x = y"   and "\<forall>\<^sub>F x in nhds x. x \<in> S \<longrightarrow> f x = g x"   and "u = v"   and "S = t"   and "x \<in> S" shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative v) (at y within t)"
NatTransExt0: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NTDom \<eta>1 = NTDom \<eta>2"   and "NTCod \<eta>1 = NTCod \<eta>2"   and "\<And>X. X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub> \<Longrightarrow> \<eta>1 $$ X = \<eta>2 $$ X"   and "($$) \<eta>1 \<in> extensional (obj\<^bsub>NTCatDom \<eta>1\<^esub>)"   and "($$) \<eta>2 \<in> extensional (obj\<^bsub>NTCatDom \<eta>2\<^esub>)" shows "\<eta>1 = \<eta>2"
monoid_list_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "monoid_list f z \<equiv> monoid f z"
nn_integral_cong_AE: fixes u :: "'d \<Rightarrow> ennreal"   and v :: "'d \<Rightarrow> ennreal"   and M :: "'d measure" assumes "AE x in M. u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
tendsto_at_left_continuous_on_avoidI: fixes a :: "'c"   and i :: "'c"   and I :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on ({a..i} - I) g"   and "\<And>x. \<lbrakk>a < x; x < i\<rbrakk> \<Longrightarrow> g x = f x"   and "i \<notin> I"   and "finite I"   and "a < i" shows "(f \<longlongrightarrow> g i) (at_left i)"
unique_expression1: fixes H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "free_generator R M H"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "m \<in> {j. j \<le> n} \<rightarrow> H"   and "inj_on m {j. j \<le> n}"   and "l_comb R M n s m = \<zero>" shows "\<forall>j\<in>{j. j \<le> n}. s j = \<zero>\<^bsub>R\<^esub>"
ex_idirgen: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier G" shows "\<exists>gs. set gs \<subseteq> generate G A \<and> cycle gs \<and> is_idirprod (generate G A) (\<lambda>g. generate G {g}) (set gs) \<and> successively (dvd) (map ord gs) \<and> card (set gs) \<le> card A"
nAct_mono: fixes n :: "enat"   and n' :: "enat"   and c :: "'id"   and t :: "cnf llist" assumes "n \<le> n'" shows "\<langle>c #\<^bsub>n\<^esub>t\<rangle> \<le> \<langle>c #\<^bsub>n'\<^esub>t\<rangle>"
circuit_subset_eq: fixes C\<^sub>1 :: "'a set"   and C\<^sub>2 :: "'a set" assumes "circuit C\<^sub>1"   and "circuit C\<^sub>2"   and "C\<^sub>1 \<subseteq> C\<^sub>2" shows "C\<^sub>1 = C\<^sub>2"
Says_imp_analz_Spy: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> analz (knows Spy evs)"
simps_1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.Derivation \<NN> \<TT> \<RR> a [] b = (a = b)"
dvd_multiplicity_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "dvd.dvd times x y = (\<forall>p. factorial_semiring.multiplicity times one p x \<le> factorial_semiring.multiplicity times one p y)"
star_of_scaleR: fixes r :: "real"   and x :: "'b" shows "star_of (r *\<^sub>R x) = r *\<^sub>R star_of x"
mrbst_diff_altdef: shows "mrbst_diff = mrbst_inter_diff False"
assms_6: assumes "\<exists>b. s\<lparr>b\<rparr> = c \<and> r\<lparr>a\<rparr> = b" shows "P"
elims_2: fixes x :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and xa :: "char list \<times> char list list" assumes "is_subobj x xa" obtains P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" and C :: "char list" and D :: "char list" where "x = Pa__" and "xa = (Ca__, [D__])" and "is_class Pa__ Ca__ \<and> Ca__ = D__ \<or> (\<exists>X. Pa__ \<turnstile> Ca__ \<preceq>\<^sup>* X \<and> Pa__ \<turnstile> X \<prec>\<^sub>S D__)"   | P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" and C :: "char list" and D :: "char list" and E :: "char list" and Xs :: "char list list" where "x = P" and "xa = (C, D # E # Xs)" and "let Ys = butlast (D # E # Xs); Y = last (D # E # Xs); X = last Ys in is_subobj P (C, Ys) \<and> P \<turnstile> X \<prec>\<^sub>R Y"
lm078: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and Y :: "'a set" assumes "finite X" shows "sum f X = sum f (X - Y) + sum f (X \<inter> Y)"
preserves_lunit_1: fixes f :: "'c" assumes "C.ide f" shows "H \<l>\<^sub>C[f] = \<l>\<^sub>D[H f] \<cdot>\<^sub>D (D.inv (H.unit (trg\<^sub>C f)) \<star>\<^sub>D H f) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (trg\<^sub>C f, f))"
healthy_nnegD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "nneg (t P)"
1: shows "continuous_map (subtopology (powertop_real UNIV) (standard_simplex (p - Suc 0))) (subtopology (powertop_real UNIV) (standard_simplex p)) (simplical_face k)"
eq_\<I>_comp_cong: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and conv1' :: "('a, 'b, 'c, 'd) converter"   and \<I>3 :: "('e, 'f) \<I>"   and conv2 :: "('c, 'd, 'e, 'f) converter"   and conv2' :: "('c, 'd, 'e, 'f) converter" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>3 \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1,\<I>3 \<turnstile>\<^sub>C comp_converter conv1 conv2 \<sim> comp_converter conv1' conv2'"
valid_reads: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "reads_consistent False \<O> m sb"
u_simps_3: shows "trg (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = src (f \<star> g \<star> h)"
galleryD_adj: fixes xs :: "'b set list" assumes "codomain.gallery xs" shows "adjacentchain xs"
deg_sectD_2: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> deg_sect X d" shows "deg_pm t = d"
list_of_set_set_is_empty: fixes s :: "'a set" shows "(list_of_set_set s = {}) = infinite s"
rel_intros_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "'c"   and y2 :: "'d" assumes "R x2 y2" shows "rel_option R (Some x2) (Some y2)"
integral_eucl_le: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f integrable_on s"   and "g integrable_on s"   and "\<And>x. x \<in> s \<Longrightarrow> f x \<le> g x" shows "integral s f \<le> integral s g"
cnt_0_I: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "cnt x xs = 0"
differing_lists_finite: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat" assumes "sifum_security stop eval some_val some_val'" shows "finite (sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)"
RedVar: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and V :: "char list"   and v :: "val"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "lcl s V = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Var V,s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
simps_11: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Ln f) env = compute_tm_by_comp prec ord I a (Ln (Var 0)) (approx_tm prec ord I a f env) (\<lambda>x. 0 < lower x)"
term_pp_rel_trans_eq_right: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a"   and b :: "'t \<times> 'a"   and c :: "'t \<times> 'a" assumes "ord_term_lin.is_le_rel r"   and "term_pp_rel r a b"   and "term_pp_rel (=) b c" shows "term_pp_rel r a c"
residuated_pair_multl: fixes y :: "'a" shows "residuated_pair (\<lambda>x. x \<cdot> y) (\<lambda>x. x \<leftarrow> y)"
ell2_norm_triangle_1: fixes x :: "'c \<Rightarrow> complex"   and y :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x"   and "has_ell2_norm y" shows "has_ell2_norm (\<lambda>i. x i + y i)"
Do: fixes f :: "state \<Rightarrow> state set"   and P :: "state \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {\<lambda>s. (\<forall>t\<in>f s. P t) \<and> f s \<noteq> {}} Do f {P}"
has_integral_spike_finite_eq: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f has_integral y) T = (g has_integral y) T"
chain_def2: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph" shows "chain S = (\<forall>i j. i \<le> j \<longrightarrow> subgraph (S i) (S j))"
take_bit_not_take_bit: fixes n :: "nat"   and a :: "'a" shows "take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)"
m: shows "\<bar>m\<bar> \<le> max \<bar>l\<bar> (sum_list (map abs ks))"
return_cons_rule: fixes P :: "assn"   and Q :: "'a \<Rightarrow> assn"   and x :: "'a" assumes "P \<Longrightarrow>\<^sub>A Q x" shows "<P> return x <Q>"
da_elim_cases_45: fixes Env :: "env"   and B :: "lname set"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and e :: "expr"   and mn :: "mname"   and pTs :: "ty list"   and args :: "expr list"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>{accC,statT,mode}e\<cdot>mn( {pTs}args)\<rangle>\<guillemotright> A" obtains E :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> nrm E \<guillemotright>\<langle>args\<rangle>\<guillemotright> A"
TAbs_type': fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and s :: "trm"   and U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> (\<lambda><:S. s) : (\<forall><:U. U')" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
inf_compl_bot_left1: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<sqinter> y) = \<bottom>"
exhaust_sel: fixes prog :: "('l, 'm, 'n) prog" obtains "prog = Prog (prog.fun_env prog) (prog.heap prog) (main_fun prog)"
valuate_scaleRat: fixes c :: "rat"   and lp :: "linear_poly"   and v :: "nat \<Rightarrow> 'b" shows "c *R lp \<lbrace> v \<rbrace> = c *R lp \<lbrace> v \<rbrace>"
A_RCOSETS_def': fixes G :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "a_rcosets\<^bsub>G\<^esub> H \<equiv> \<Union>a\<in>carrier G. {H +>\<^bsub>G\<^esub> a}"
h3: shows "(\<exists>y. \<forall>x<y. (\<Sum>i\<le>n. k i x * x ^ i) = 0) = (\<forall>x y. (\<Sum>i\<le>n. k i x * y ^ i) = 0)"
prems_2: fixes x :: "'a" assumes "x \<in> insert x__ xs" shows "degree (f x) = 0"
intro: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "\<tau>inv trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2" shows "bisimulation_into_delay trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"
closed_interval_right: fixes a :: "'c" shows "closed {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}"
right_mult_one_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A * 1\<^sub>m nc = A"
degen_assoc: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "(p\<^sub>1 = Infinity \<or> p\<^sub>2 = Infinity \<or> p\<^sub>3 = Infinity) \<or> (p\<^sub>1 = opp p\<^sub>2 \<or> p\<^sub>2 = opp p\<^sub>3) \<or> opp p\<^sub>1 = local.add a p\<^sub>2 p\<^sub>3 \<or> opp p\<^sub>3 = local.add a p\<^sub>1 p\<^sub>2" shows "local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3) = local.add a (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
to_woot_character_keeps_P: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message" shows "P (to_woot_character M m) = P m"
wi_bintr: fixes n :: "nat"   and w :: "int" assumes "LENGTH('c) \<le> n" shows "word_of_int (take_bit n w) = word_of_int w"
basic_pseudo_hoop_algebra_axioms_def: fixes imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.basic_pseudo_hoop_algebra_axioms imp_l less_eq imp_r \<equiv> (\<forall>a b c. less_eq (imp_l (imp_l a b) c) (imp_l (imp_l (imp_l b a) c) c)) \<and> (\<forall>a b c. less_eq (imp_r (imp_r a b) c) (imp_r (imp_r (imp_r b a) c) c))"
intermediary_def: fixes d :: "'dom_t"   and u :: "'dom_t" shows "intermediary d u \<equiv> \<exists>v. ifp\<^sup>*\<^sup>* v d \<and> ifp d u \<and> \<not> ifp v u \<and> d \<noteq> u"
Par_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Par C D \<equiv> A B ParStrict C D \<or> A \<noteq> B \<and> C \<noteq> D \<and> Col A C D \<and> Col B C D"
nonneg_vel_other: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "0 \<le> v\<^sub>o"