hcju/isabelleps huggingface dataset:
finite_null_set: fixes M :: "'a measure"   and A :: "'a set" assumes "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> null_sets M"   and "finite A"   and "A \<subseteq> space M" shows "A \<in> null_sets M"
Red_F_of_subset: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set"   and N' :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "N \<subseteq> N'" shows "Red_F N \<subseteq> Red_F N'"
assms: shows "outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>"
induct: fixes x1 :: "BExpr"   and x2 :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and x3 :: "(Location \<times> Location) set"   and x4 :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x5 :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and P :: "BExpr \<Rightarrow> (((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> (Location \<times> Location) set \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "(x1, x2, x3, x4, x5) \<in> var"   and "\<And>b t s \<beta> \<Phi>. \<lbrakk>\<not> evalB b (fst t); s \<equiv>\<^sub>\<beta> t\<rbrakk> \<Longrightarrow> P b \<Phi> \<beta> s t"   and "\<And>b t \<Phi> r \<beta> s. \<lbrakk>evalB b (fst t); \<Phi> (r, t, \<beta>); \<forall>w \<gamma>. r \<equiv>\<^sub>\<gamma> w \<longrightarrow> (b, \<Phi>, \<gamma>, s, w) \<in> var \<and> P b \<Phi> \<gamma> s w\<rbrakk> \<Longrightarrow> P b \<Phi> \<beta> s t" shows "P x1 x2 x3 x4 x5"
le_less: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x < y \<or> x = y)"
annot_def: fixes t :: "('b, 'a) FingerTree" shows "FingerTree.annot t \<equiv> FingerTreeStruc.annot (Rep_FingerTree t)"
transfer: shows "rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV g)) (rel_gpv (BNF_Def.Grp UNIV f) (rel_sum (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV g))) right_gpv right_gpv"
Image_def: fixes r :: "('e \<times> 'd) set"   and s :: "'e set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
distrib: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
sep_conj_true: shows "sep_true ** sep_true = sep_true"
gcd_mod_left: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm"   and "b \<noteq> zero" shows "gcd (modulo a b) b = gcd a b"
emeasure_eq_measure: fixes s :: "'a cfg"   and A :: "'a cfg stream set" shows "emeasure (MC.T s) A = ennreal (MC.T.prob s A)"
beta_equiv_eq_1_1: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and v :: "i"   and x :: "\<nu>" assumes "IsProperInX \<phi>"   and "IsProperInX \<psi>"   and "\<And>x. [\<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) in v]" shows "[\<lparr>\<^bold>\<lambda>y. \<phi> (y\<^sup>P),x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>y. \<psi> (y\<^sup>P),x\<^sup>P\<rparr> in v]"
trg_dom: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (B.dom \<mu>) = src \<mu>"
mpoly_Const_uminus: fixes a :: "'b" shows "Const (- a) = - Const a"
fps: shows "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts\<^sub>2"
black_heap_reachable: fixes m :: "'e"   and y :: "'f"   and s :: "'e process_name \<Rightarrow> ('g, 'e, 'h, 'f) Model.local_state" assumes "mut_m.reachable m y s"   and "black_heap s"   and "valid_refs_inv s" shows "black y s"
baEA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and \<phi> :: "'KS \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "sb.eval c t t' n [\<phi>]\<^sub>b" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)"
onllD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'f, 'd, 'g) seqp"   and P :: "('h \<times> 'g) \<times> 'i \<times> 'h \<times> 'g \<Rightarrow> bool"   and \<xi> :: "'h"   and p :: "('e, 'f, 'd, 'g) seqp"   and a :: "'i"   and \<xi>' :: "'h"   and p' :: "('e, 'f, 'd, 'g) seqp" assumes "onll \<Gamma> P ((\<xi>, p), a, \<xi>', p')" shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
neval_SucE: fixes x :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and n :: "nat"   and y :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" assumes "conc.neval x (Suc n) y" obtains ya :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" where "eval_abv\<^sub>C x ya" and "conc.neval ya n y"
tailrec_transform': fixes m :: "'c nres"   and a :: "'c \<Rightarrow> 'c nres"   and b :: "'c \<Rightarrow> 'b nres" shows "m \<bind> REC (tailrec_body a b) = lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b"
Spy_analz_private_Key: fixes evs :: "event list"   and b :: "bool"   and A :: "agent" assumes "evs \<in> set_cr" shows "(Key (invKey (publicKey b A)) \<in> analz (knows Spy evs)) = (A \<in> bad)"
ord_spmf_lossless: fixes p :: "'b spmf"   and q :: "'b spmf" assumes "ord_spmf (=) p q"   and "lossless_spmf p" shows "p = q"
measure_frontier: fixes S :: "'b set" assumes "bounded S" shows "Sigma_Algebra.measure lebesgue (frontier S) = Sigma_Algebra.measure lebesgue (closure S) - Sigma_Algebra.measure lebesgue (interior S)"
sum_strict_mono: fixes B :: "nat set"   and A :: "nat set" assumes "finite B"   and "A \<subset> B"   and "0 \<notin> B" shows "\<Sum> A < \<Sum> B"
coprime_iff_invertible'_int: fixes m :: "int"   and a :: "int" assumes "0 < m" shows "coprime a m = (\<exists>x\<ge>0. x < m \<and> [a * x = 1] (mod m))"
t1: shows "bdd_above (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite) = bdd_above ((\<lambda>A. sqrt (\<Sum>i\<in>A. ((cmod \<circ> x) i)\<^sup>2)) ` Collect finite)"
transfer_bforall_def: shows "transfer_bforall \<equiv> \<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x"
sub_block_fresh: fixes i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and block :: "('c, 'b) fm list \<times> 'b"   and f :: "'b \<Rightarrow> 'd"   and j :: "'d" assumes "i \<notin> branch_nominals branch"   and "block \<in>. branch" shows "sub_block (f(i := j)) block = sub_block f block"
sel_rews_3: shows "head\<cdot>[] = \<bottom>"
padic_mult_assoc: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)"   and "z \<in> carrier (padic_int p)" shows "x \<otimes>\<^bsub>padic_int p\<^esub> y \<otimes>\<^bsub>padic_int p\<^esub> z = x \<otimes>\<^bsub>padic_int p\<^esub> (y \<otimes>\<^bsub>padic_int p\<^esub> z)"
finfun_upd_apply_other: fixes a :: "'a"   and a' :: "'a"   and f :: "'a \<Rightarrow>f 'b"   and b :: "'b" assumes "a \<noteq> a'" shows "f(a $:= b) $ a' = f $ a'"
set_map: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "set (map f xs) = f ` set xs"
sets_Sup_sigma: fixes M :: "'b set set set"   and \<Omega> :: "'b set" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>" shows "sets (Sup (sigma \<Omega> ` M)) = sets (sigma \<Omega> (\<Union> M))"
PiE_iff: fixes f :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and X :: "'e \<Rightarrow> 'f set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
MSF_def: fixes B :: "nat uprod set" shows "s.MSF B \<equiv> s.SpanningForest B \<and> (\<forall>B'. s.SpanningForest B' \<longrightarrow> sum w B \<le> sum w B')"
wf_sorted_list: shows "wf ((\<lambda>(x, y). (sorted_list_of_set x, sorted_list_of_set y)) ` finite_psubset)"
assms_4: fixes act :: "'c"   and c :: "'d"   and formula :: "('a, 'b, 'c) formula" assumes "bn act \<sharp>* c"   and "\<And>c. P c formula" shows "P c (Act act formula)"
simps_1: shows "len \<langle>\<rangle> = 0"
sel_2: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "ltl (llist_of_tllist xs) = (case xs of TCons x xs' \<Rightarrow> llist_of_tllist xs')"
rel_cong: fixes x :: "'a iarray"   and ya :: "'a iarray"   and y :: "'b iarray"   and xa :: "'b iarray"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_iarray ya; yb \<in> set_iarray xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_iarray R x y = rel_iarray Ra ya xa"
star_of_le_HNatInfinite: fixes N :: "nat star"   and n :: "nat" assumes "N \<in> HNatInfinite" shows "hypnat_of_nat n \<le> N"
word_sless_alt: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a <s b) = (sint a < sint b)"
bounded_linear_axis: fixes i :: "'d" shows "bounded_linear (axis i)"
lists_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "lists A \<subseteq> lists B"
mapRule_def: shows "mapRule = (\<lambda>f (conc, prems). (map f conc, map f ` prems))"
divide_inverse_commute: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a b = times (inverse b) a"
sorted_add_spmat_helper: fixes arr :: "(nat \<times> (nat \<times> 'a) list) list"   and brr :: "(nat \<times> (nat \<times> 'a) list) list"   and ab :: "nat"   and bb :: "(nat \<times> 'a) list"   and list :: "(nat \<times> (nat \<times> 'a) list) list" assumes "add_spmat arr brr = (ab, bb) # list" shows "arr \<noteq> [] \<and> ab = fst (hd arr) \<or> brr \<noteq> [] \<and> ab = fst (hd brr)"
rdv: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<sqinter> x \<cdot> \<top>) \<cdot> y = z \<cdot> y \<sqinter> x \<cdot> \<top>"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) K (N \<prec>' P)"
WTrtBlock: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e :: "char list exp"   and T' :: "ty" assumes "P,E(V \<mapsto> T),h \<turnstile> e : T'" shows "P,E,h \<turnstile> {V:T; e} : T'"
that: fixes x :: "real" assumes "{..<ereal x} \<subseteq> A" shows "thesis"
star_unfold_left: fixes A :: "'a list set" shows "star A = A @@ star A \<union> {[]}"
lotteries_on_nonempty: fixes outcomes :: "'b set" assumes "outcomes \<noteq> {}" shows "lotteries_on outcomes \<noteq> {}"
gdpr_att: shows "gdpr_Kripke \<turnstile> EF {x. \<not> global_policy' x ''Eve''}"
assms_4: shows "X = (\<Union>n\<le>length xs. \<Union>W\<in>R n. W) \<and> (\<forall>W\<in>R 0. xs \<notin> sentences P \<and> None \<notin> set xs \<and> (xs, W) \<in> failures P \<or> xs \<in> sentences P \<and> (\<exists>U V. (xs, U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length xs}. \<forall>W\<in>R n. take (length xs - n) xs \<in> sentences P \<and> (drop (length xs - n) xs, W) \<in> failures Q)"
mergeFrameRes1: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>G :: "'d"   and A\<^sub>G :: "name list"   and x :: "name"   and \<Psi>\<^sub>F :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"   and "A\<^sub>F \<sharp>* A\<^sub>G"   and "x \<sharp> A\<^sub>F"   and "x \<sharp> \<Psi>\<^sub>F"   and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F" shows "assertionAux.mergeFrame SCompose (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) (\<lparr>\<nu>x\<rparr>\<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle>) = \<langle>(A\<^sub>F @ x # A\<^sub>G), SCompose \<Psi>\<^sub>F \<Psi>\<^sub>G\<rangle>"
sup_aforms_def: fixes prec :: "nat"   and xs :: "('a \<times> 'a pdevs) list" shows "sup_aforms prec xs = map (Sup_aform' prec) xs"
\<T>_subset_TokensAt: fixes k :: "nat"   and u :: "nat" shows "\<T> k u \<subseteq> TokensAt k (\<J> k u)"
join_NF_imp_eq: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> r\<^sup>\<down>"   and "x \<in> NF r"   and "y \<in> NF r" shows "x = y"
create_name_gba_aimpl_refine: fixes \<phi> :: "'b ltlr" shows "create_name_gba_aimpl \<phi> \<le> \<Down> Id (create_name_gba \<phi>)"
pathVertices_edgeset: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "u \<in> V"   and "isPath u p v" shows "set (pathVertices u p) \<subseteq> V"
succ_neq_zero: fixes m :: "'a" shows "succ m \<noteq> zero"
Ik_intT_int: 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 pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and T :: "'a trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "Signature.wt wtFsym arOf resOf T"   and "Struct.wtE (ModelIkPolMcalc2C.I_intT infTp intT) \<xi>"   and "infTp (Signature.tpOf resOf T) \<or> (\<forall>x\<in>ProblemIkPol.nv2T T. Signature.tpOfV x \<noteq> Signature.tpOf resOf T)" shows "intT (Signature.tpOf resOf T) (Struct.int (ModelIkPolMcalc2C.I_intF arOf intT intF) \<xi> T)"
coeff_lift2_lift: shows "(\<lambda>a. [:[:a:]:]) = (\<lambda>a. [:a:]) \<circ> (\<lambda>a. [:a:])"
assms_4: shows "vs \<in> lists (no_head_gen \<C> b)"
inf_less_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> z \<or> y \<le> z)"
Int_Inter_eq_2: fixes \<B> :: "'b set set"   and A :: "'b set" shows "\<Inter> \<B> \<inter> A = (if \<B> = {} then A else \<Inter>B\<in>\<B>. B \<inter> A)"
g_def: shows "g = (\<lambda>xa. \<lfloor>frac (real xa * x) * real s\<rfloor>)"
WHILEI_unfold: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm"   and x :: "'a" shows "WHILEI I b f x = (if I x then if b x then bind (f x) (WHILEI I b f) else return x else top)"
l: shows "\<forall>U. l \<in> U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
induct: fixes P :: "'b mat \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) list \<Rightarrow> bool"   and a0 :: "'b mat"   and a1 :: "nat"   and a2 :: "(nat \<times> nat) list" assumes "\<And>A list. P A 0 list"   and "\<And>A i_end list. (\<And>x. x = ??.Jordan_Normal_Form_Existence.identify_block A i_end \<Longrightarrow> P A x ((x, i_end) # list)) \<Longrightarrow> P A (Suc i_end) list" shows "P a0 a1 a2"
2: fixes n :: "nat"   and m :: "int" assumes "0 < n" shows "\<exists>k. plusi (exp (\<i> * (of_int m * complex_of_real pi) + complex_of_real (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
linorder_of_list_def: fixes xs :: "'b list" shows "linorder_of_list xs = {(a, b). a \<in> set xs \<and> b \<in> set xs \<and> index xs a \<le> index xs b}"
AbruptProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> =n\<Rightarrow> Abrupt s"
ExFalso: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<not> \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
OK_le_conv: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" shows "(OK x \<sqsubseteq>\<^bsub>le r\<^esub> e) = (e = Err \<or> (\<exists>y. e = OK y \<and> x \<sqsubseteq>\<^bsub>r\<^esub> y))"
n_max_def: fixes A :: "nat set" shows "n_max A = (THE m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m))"
strict_trans1: fixes b :: "'b word"   and a :: "'b word"   and c :: "'b word" assumes "b \<le>s a"   and "c <s b" shows "c <s a"
memberdecl_mdecl_acc_modi_simp: fixes m :: "mdecl" shows "accmodi (mdecl m) = accmodi m"
mutual_information_distr: fixes S :: "'e measure"   and T :: "'f measure"   and X :: "'a \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and Pxy :: "'e \<times> 'f \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>y. y \<in> space T \<Longrightarrow> 0 \<le> Py y"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)" shows "mutual_information b S T X Y = LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
resCases': fixes x :: "name"   and P :: "pi"   and Rs :: "residual" assumes "<\<nu>x>P \<longmapsto> Rs" obtains P :: "pi" and a :: "name" and b :: "name" and P' :: "pi" where "<\<nu>x>Pa__ = <\<nu>b>Pb__" and "Rs = a__<\<nu>b> \<prec> P'__" and "Pb__ \<longmapsto> a__[b] \<prec> P'__" and "a__ \<noteq> b"   | P :: "pi" and a :: "subject" and xa :: "name" and P' :: "pi" and y :: "name" where "<\<nu>x>Pa__ = <\<nu>y__>Pc__" and "Rs = a\<guillemotleft>xa\<guillemotright> \<prec> <\<nu>y__>P'a__" and "Pc__ \<longmapsto> a\<guillemotleft>xa\<guillemotright> \<prec> P'a__" and "y__ \<sharp> a" and "y__ \<noteq> xa" and "xa \<sharp> Pc__" and "xa \<sharp> a"   | P :: "pi" and \<alpha> :: "freeRes" and P' :: "pi" and y :: "name" where "<\<nu>x>Pa__ = <\<nu>y>P" and "Rs = \<alpha> \<prec> <\<nu>y>P'" and "P \<longmapsto> \<alpha> \<prec> P'" and "y \<sharp> \<alpha>"
get_disconnected_nodes_delete_shadow_root: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) document_ptr" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
l_galois_connection: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice"   and y :: "'b::refinement_lattice" assumes "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "(((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)\<^sup>\<flat>) (x::'b::refinement_lattice) \<sqsubseteq> (y::'b::refinement_lattice)) = (x \<sqsubseteq> F y)"
red_Trg: fixes t :: "'a local.term" assumes "Ide t" shows "Trg t\<^bold>\<down> = Trg t"
HBall_def: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool" shows "HBall A P = (\<forall>x. x \<^bold>\<in> A \<longrightarrow> P x)"
satPB_gPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtE \<xi>"   and "GE.satPB \<xi> (gC ` \<Phi>)" shows "Ik.satPB \<xi> \<Phi>"
existence_ivl_maximal_segment: fixes x :: "real \<Rightarrow> real"   and t0 :: "real"   and t :: "real"   and x0 :: "real" assumes "(x solves_ode f) {t0--t} X"   and "x t0 = x0"   and "{t0--t} \<subseteq> T" shows "t \<in> existence_ivl t0 x0"
no_large_prime: fixes p :: "int" assumes "prime p"   and "n < p" shows "False"
size1_if_complete: fixes t :: "'c tree" assumes "complete t" shows "size1 t = 2 ^ height t"
zeta_partial_sum_pos_bigtheta: fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "sum_upto (\<lambda>n. real n powr - s) \<in> \<Theta>(\<lambda>x. x powr max 0 (1 - s))"
taylor_sum: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "p \<in> carrier P"   and "deg R p \<le> n"   and "a \<in> carrier R" shows "p = (\<Oplus>\<^bsub>P\<^esub>i\<in>{..n}. T\<^bsub>a\<^esub> p i \<odot>\<^bsub>P\<^esub> X_poly_minus R a [^]\<^bsub>P\<^esub> i)"
simps_1: fixes s :: "('d, 'e) Term.term" shows "\<box>\<langle>s\<rangle> = s"
cases: fixes a1 :: "diagram"   and a2 :: "interface"   and a3 :: "interface" assumes "prov_dia a1 a2 a3" obtains G :: "diagram" where "a1 = G" and "a2 = top_dia G" and "a3 = bot_dia G" and "wf_dia G" and "\<And>v. v \<in> fset G^V \<Longrightarrow> prov_ass (G^\<Lambda> v)" and "\<And>e. e \<in> set G^E \<Longrightarrow> prov_com (snd3 e) (\<Otimes>v|\<in>|fst3 e. bot_ass (G^\<Lambda> v)) (\<Otimes>v|\<in>|thd3 e. top_ass (G^\<Lambda> v))"
set_awalk_verts_not_Nil: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v"   and "p \<noteq> []" shows "set (awalk_verts u p) = set (map (tail G) p) \<union> set (map (head G) p)"
finite_Un: fixes F :: "'d set"   and G :: "'d set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
P2_total_imp_R2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<P>\<^sup>2"   and "total2 f" shows "f \<in> \<R>\<^sup>2"