hcju/isabelleps huggingface dataset:
nonzero_imp_inverse_nonzero: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a \<noteq> zero"
heap_is_wellformed_disc_nodes_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node \<in> set disc_nodes" shows "node |\<in>| node_ptr_kinds h"
esinvP: shows "ts_inv P I (thr s) (shr s)"
evaluate_clock_mono_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r3 :: "(v, v) result" assumes "evaluate ck env s e (s', r3)" shows "clock s' \<le> clock s"
ctr: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "siterate f x = x ## siterate f (f x)"
wf_def: fixes r :: "('e \<times> 'e) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
hausdorff: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t2_space open"   and "x \<noteq> y" shows "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
snd_sum: fixes f :: "'f \<Rightarrow> 'e \<times> 'd"   and A :: "'f set" shows "snd (sum f A) = (\<Sum>x\<in>A. snd (f x))"
assms: fixes t :: "term" assumes "t \<in> fmran' env" shows "wellformed t"
AE_E: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "almost_everywhere M P" obtains N :: "'c set" where "{x \<in> space M. \<not> P x} \<subseteq> N" and "emeasure M N = 0" and "N \<in> sets M"
CompleteLattice_def: shows "CompleteLattice = {cl \<in> PartialOrder. (\<forall>S\<subseteq>pset cl. \<exists>L. isLub S cl L) \<and> (\<forall>S\<subseteq>pset cl. \<exists>G. isGlb S cl G)}"
simps_1: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and s :: "'a" shows "On_Off.steps step s [] [] = s"
P3: shows "Col P Q C' \<and> P Q Perp C C'"
dims_matricize_2: fixes rmodes :: "nat set"   and T :: "'b tensor" shows "dim_col (matricize rmodes T) = prod_list (nths (dims T) (- rmodes))"
left_minus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) a = zero"
sup_top_right: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and top :: "'e"   and x :: "'e" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup x top = top"
gallery_map: fixes Cs :: "'a set list" assumes "folding_g.gallery Cs" shows "folding_g.gallery (g \<Turnstile> Cs)"
uncurry0_apply: fixes c :: "'a"   and x :: "unit" shows "uncurry0 c x = c"
labeled_sat_eqs_subseqs: fixes Di :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and i :: "'a"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and ac :: "poscheckvariant"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "Di \<in> set (subseqs D)"   and "\<forall>(j, p)\<in>{(i, t, s)} \<union> set D. \<forall>(k, q)\<in>{(i, t, s)} \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"   and "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) Di\<rbrakk>\<^sub>d \<I>" shows "Di \<in> set (subseqs (dbproj i D))"
distinct_4: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x11 :: "'a"   and x12 :: "'b" shows "x31@[x32] \<noteq> x11 \<leftarrow> x12"
L: shows "equivalence_of_categories (\<cdot>) (\<cdot>) F ((\<otimes>) \<I>) \<eta> \<epsilon>"
D_sync: fixes P :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "D (P \<lbrakk>A\<rbrakk> Q) = {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
main_lemma_reachability_s: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> card (reachable_s PROB s) - 1"
map_sel: fixes f1 :: "'k \<Rightarrow> 'h"   and f2 :: "'l \<Rightarrow> 'i"   and a :: "('k, 'l, 'j) gpv" shows "the_gpv (map_gpv f1 f2 a) = map_spmf (map_generat f1 f2 ((\<circ>) (map_gpv f1 f2))) (the_gpv a)"
MORE: shows "igb_graph_rec.more G' = ti_Lcnv ecnv G"
hyperpow_one: fixes r :: "'b star" shows "r pow 1 = r"
cring_lucas_lehmer_ring_mod: fixes m :: "nat" assumes "1 < m" shows "cring (lucas_lehmer_ring_mod m)"
remove_vertex_E: fixes x :: "'a" shows "E\<^bsub>remove_vertex x\<^esub> = Restr E (V - {x})"
starset_def: fixes A :: "'b set" shows "*s* A = Iset (star_of A)"
AE_PiM_component: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and i :: "'d"   and P :: "'e \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "i \<in> I"   and "almost_everywhere (M i) P" shows "AE x in Pi\<^sub>M I M. P (x i)"
div_div_div_same: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and d :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times d b"   and "dvd.dvd times b a" shows "divide (divide a d) (divide b d) = divide a b"
elims_2: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list" assumes "abc_notfinal x xa" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s, lm)" and "xa = p" and "s < length p"
r_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier V"   and "a \<in> carrier V" shows "(x = a \<oplus>\<^bsub>V\<^esub> x) = (a = \<zero>\<^bsub>V\<^esub>)"
nat_term_compD1': fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and u :: "'b"   and v :: "'b" assumes "comparator cmp"   and "nat_term_comp cmp"   and "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"   and "fst (rep_nat_term u) = 0" shows "cmp u v \<noteq> Gt"
vector_space_def: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "vector_space scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'b) x = x)"
AE_iff_nn_integral: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M" shows "almost_everywhere M P = (integral\<^sup>N M (indicator {x. \<not> P x}) = 0)"
fold_const: fixes xs :: "'b list"   and a :: "'b" shows "fold (\<lambda>x _. x) xs a = last (a # xs)"
simps_1: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_i [] A i bezout = A"
emeasure_Un_null_set: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> null_sets M" shows "emeasure M (A \<union> B) = emeasure M A"
dinfl: shows "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"
tau_s_prod: fixes x :: "'a"   and y :: "'a" shows "\<tau> (x \<cdot> y) = x \<cdot> \<tau> y"
rel_fset_alt: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a fset"   and b :: "'b fset" shows "rel_fset R a b = ((\<forall>t\<in>fset a. \<exists>u\<in>fset b. R t u) \<and> (\<forall>t\<in>fset b. \<exists>u\<in>fset a. R u t))"
sum_restrict': fixes g :: "'c \<Rightarrow> 'b"   and I :: "'c set" shows "sum' (restrict g I) I = sum' g I"
has_integral_integral_real: fixes f :: "'c \<Rightarrow> real" assumes "integrable lborel f" shows "(f has_integral integral\<^sup>L lborel f) UNIV"
natural_transformation_axioms: shows "natural_transformation A C (K \<circ> G) (K \<circ> H) (K \<circ> \<tau>')"
one_sqmatrix_def: shows "1 \<equiv> SqMatrix (\<lambda>i j. if i = j then 1::'c else (0::'c))"
convex_hull_Times: fixes s :: "'b set"   and t :: "'c set" shows "convex hull s \<times> t = (convex hull s) \<times> (convex hull t)"
frechet_linear: fixes I :: "('d, 'e, 'f) interp"   and \<theta> :: "('d, 'f) trm"   and v :: "(real, 'f) vec" assumes "is_interp I"   and "dfree \<theta>" shows "bounded_linear (frechet I \<theta> v)"
simps_9: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and l :: "formula list" shows "sat \<sigma> V v i (Ands l) = Ball (set l) (sat \<sigma> V v i)"
hv_him: fixes q :: "octo" shows "Hv (HIm q) = Octo 0 (Im1 q) (Im2 q) (Im3 q) (Im4 q) (Im5 q) (Im6 q) (Im7 q)"
iffb_surj: shows "top_pres (\<P> fb\<^sup>-\<^sub>\<F>)"
natural_isomorphism_axioms: shows "natural_isomorphism CCC.comp (\<cdot>) T.ToTC T.ToCT \<alpha>.map"
smc_Comp_vrange: shows "\<R>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
wx_equiv: fixes X :: "'b program set" shows "wx X = {F. \<forall>G. F ok G \<longrightarrow> F \<squnion> G \<in> X}"
ssubst_Call: fixes z :: "char list"   and u :: "sterm"   and t :: "sterm"   and l :: "Label"   and s :: "sterm" shows "[z \<rightarrow> u] Call t l s = Call ([z \<rightarrow> u] t) l ([z \<rightarrow> u] s)"
less_add_same_cancel1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
74: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<squnion> y"
wellformed_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and wellformed :: "'b \<Rightarrow> bool"   and all_frees :: "'b \<Rightarrow> name fset"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b" assumes "class.pre_strong_term size frees subst consts app unapp const unconst free unfree abs_pred wellformed all_frees" shows "wellformed (app u\<^sub>1 u\<^sub>2) = (wellformed u\<^sub>1 \<and> wellformed u\<^sub>2)"
not_outside_2: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "vNM_utility outcomes relation u"   and "p \<succeq>[relation] q" shows "q \<in> \<P> outcomes"
Q_Suc: fixes i :: "nat"   and j :: "nat" assumes "P i j"   and "Q i j" shows "Q i (Suc j)"
polyfun_det: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a mat"   and n :: "nat"   and N :: "nat set" assumes "\<And>x. A x \<in> carrier_mat n n"   and "\<And>x i j. \<lbrakk>i < n; j < n\<rbrakk> \<Longrightarrow> polyfun N (\<lambda>x. A x $$ (i, j))" shows "polyfun N (\<lambda>x. det (A x))"
lifting: fixes real_resource :: "nat \<Rightarrow> ('h + 'i, 'j + 'k) resource"   and ideal_resource :: "nat \<Rightarrow> ('l + 'i, 'm + 'k) resource"   and sim :: "nat \<Rightarrow> ('h, 'j, 'l, 'm) converter"   and \<I>_real :: "nat \<Rightarrow> ('h, 'j) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('l, 'm) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('i, 'k) \<I>"   and bound :: "nat \<Rightarrow> enat"   and lossless :: "bool"   and w :: "nat \<Rightarrow> ('l \<Rightarrow> 'h) \<times> ('j \<Rightarrow> 'm)"   and \<I>_common' :: "nat \<Rightarrow> ('n, 'o) \<I>"   and conv :: "nat \<Rightarrow> ('n, 'o, 'i, 'k) converter"   and bound_conv :: "nat \<Rightarrow> enat"   and bound' :: "nat \<Rightarrow> enat" assumes "constructive_security real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w"   and "\<And>\<eta>. \<I>_common' \<eta>, \<I>_common \<eta> \<turnstile>\<^sub>C conv \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_converter (conv \<eta>) (bound_conv \<eta>)"   and "\<And>\<eta>. bound' \<eta> * max (bound_conv \<eta>) 1 \<le> bound \<eta>"   and "\<And>\<eta>. lossless \<Longrightarrow> plossless_converter (\<I>_common' \<eta>) (\<I>_common \<eta>) (conv \<eta>)" shows "constructive_security (\<lambda>\<eta>. 1\<^sub>C |\<^sub>= conv \<eta> \<rhd> real_resource \<eta>) (\<lambda>\<eta>. 1\<^sub>C |\<^sub>= conv \<eta> \<rhd> ideal_resource \<eta>) sim \<I>_real \<I>_ideal \<I>_common' bound' lossless w"
is_hashmap_def: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable" shows "is_hashmap m ht = (\<exists>\<^sub>Al. is_hashmap' m l ht)"
natPredCases: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>n. P n) \<or> \<not> P 0 \<or> (\<exists>n. P n \<and> \<not> P (Suc n))"
proves_eq_symmetric_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and s :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "term_ok \<Theta> s"   and "typ_of s = typ_of t" shows "\<Theta>,{} \<turnstile> mk_eq s t \<longmapsto> mk_eq t s"
transpose_mat_of_rows: fixes n :: "nat"   and vs :: "'a vec list" shows "mat_of_rows n vs\<^sup>t = mat_of_cols n vs"
a_loc: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> d y) = ad (x \<cdot> y)"
simps_14: fixes v :: "atom fm" shows "push_forall (AllQ (Neg v)) = (if freeIn 0 (Neg v) then lowerFm 0 1 (Neg v) else AllQ (Neg v))"
circuit_elim: fixes C\<^sub>1 :: "'a set"   and C\<^sub>2 :: "'a set"   and x :: "'a" assumes "\<E>.circuit C\<^sub>1"   and "\<E>.circuit C\<^sub>2"   and "C\<^sub>1 \<noteq> C\<^sub>2"   and "x \<in> C\<^sub>1 \<inter> C\<^sub>2" shows "\<exists>C\<^sub>3\<subseteq>C\<^sub>1 \<union> C\<^sub>2 - {x}. \<E>.circuit C\<^sub>3"
assms_6: shows "\<forall>q\<in>ps\<^sub>L. fst q \<le> l"
induct: fixes P :: "atom fm \<Rightarrow> bool"   and a0 :: "atom fm" assumes "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (And \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (Or \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>r c cs. P (Atom (Less r (c # cs)))"   and "\<And>r c cs. P (Atom (Eq r (c # cs)))"   and "P TrueF"   and "P FalseF"   and "\<And>va. P (Atom (Less va []))"   and "\<And>va. P (Atom (Eq va []))"   and "\<And>v. P (Neg v)"   and "\<And>v. P (ExQ v)" shows "P a0"
Rep_nd_fun_inject: fixes x :: "'b nd_fun"   and y :: "'b nd_fun" shows "(x\<^sub>\<bullet> = y\<^sub>\<bullet>) = (x = y)"
hdeg_p_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and d :: "nat"   and p :: "'e" shows "hdeg_p R S X d p = snd (scf_d R S X p d) d \<cdot>\<^sub>r\<^bsub>R\<^esub> X^\<^bsup>R d\<^esup>"
comm_scalar_prod: fixes v\<^sub>1 :: "'b Matrix.vec"   and n :: "nat"   and v\<^sub>2 :: "'b Matrix.vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> v\<^sub>2 = v\<^sub>2 \<bullet> v\<^sub>1"
REC_def: fixes B :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" shows "REC B x \<equiv> if trimono B then lfp B x else top"
RED_props_1: fixes \<tau> :: "ty" shows "CR1 \<tau>"
counit_in_hom_2: shows "\<guillemotleft>\<epsilon> : f \<star> f\<^sup>* \<Rightarrow> trg f\<guillemotright>"
rel_envTI: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "(R ===> M) f g" shows "local.rel_envT R M (EnvT f) (EnvT g)"
weakCongSimCasePres: fixes CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and Eq :: "'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool"   and \<Psi> :: "'b"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> Eq \<Psi> P Q"   and "\<And>\<Psi>' P Q. Eq \<Psi>' P Q \<Longrightarrow> \<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q" shows "\<Psi> \<rhd> Cases CsP \<leadsto>\<guillemotleft>Rel\<guillemotright> Cases CsQ"
lookup_Abs_poly_mapping: fixes f :: "'b \<Rightarrow> 'c" assumes "finite {x. f x \<noteq> (0::'c)}" shows "lookup (Abs_poly_mapping f) = f"
real_sum_nat_ivl_bounded2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and K :: "'a"   and k :: "nat" assumes "\<And>p. p < n \<Longrightarrow> f p \<le> K"   and "(0::'a) \<le> K" shows "sum f {..<n - k} \<le> of_nat n * K"
init_fin_\<tau>diverge_bisim_inv: fixes t :: "'t"   and sxm1 :: "(status \<times> 'x1) \<times> 'm1"   and sxm2 :: "(status \<times> 'x2) \<times> 'm2" assumes "t \<turnstile> sxm1 \<approx>i sxm2" shows "\<tau>trsys.\<tau>diverge (r1.init_fin t) r1.init_fin_\<tau>move sxm1 = \<tau>trsys.\<tau>diverge (r2.init_fin t) r2.init_fin_\<tau>move sxm2"
set_tag_name_defs: fixes ptr :: "(_) element_ptr"   and tag :: "char list" shows "a_set_tag_name ptr tag = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr tag_name_update tag)"
nths_vars2': shows "nths c.dims0 (- c.vars1') = c.dims2"
unique_LID_M: fixes i :: "nat"   and LID_M :: "nat"   and P :: "agent"   and Trans :: "msg"   and evs :: "event list"   and C :: "agent"   and M :: "agent"   and OD :: "nat"   and PA :: "nat" assumes "Notes (Merchant i) \<lbrace>Number LID_M, Agent P, Trans\<rbrace> \<in> set evs"   and "Notes C \<lbrace>Number LID_M, Agent M, Agent C, Number OD, Number PA\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "M = Merchant i \<and> Trans = \<lbrace>Agent M, Agent C, Number OD, Number PA\<rbrace>"
res_rc1: fixes x :: "'a"   and y :: "'a" shows "x \<le> inv_resr y (y \<cdot> x)"
select_convs_3: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "ichosen \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = ichosen"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_aft_erase_c (s, l, r) lm n ires = (\<exists>lnl lnr rn ml m. m = Suc (lm ! n) \<and> l = Bk \<up> lnr @ Oc \<up> m @ Bk \<up> lnl @ Bk # Bk # ires \<and> (r = <ml> @ Bk \<up> rn \<or> r = Bk # <ml> @ Bk \<up> rn))"
assms_2: shows "f \<in> carrier (function_ring (carrier R) R)"
i: shows "nat (1 + s ''x'') = 2 ^ k"
order_less_irrefl: fixes x :: "'b" shows "\<not> x < x"
intro: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_vfsequence \<alpha> \<SS>"   and "\<Z>_srng_axioms \<alpha> \<SS>" shows "\<Z>_srng \<alpha> \<SS>"
reduced_word_append_reduce_contra2: fixes A :: "'a set"   and bs :: "'a list"   and as :: "'a list" assumes "\<not> reduced_word A bs" shows "\<not> reduced_word A (as @ bs)"
rep_eq: fixes x :: "('e, 'f) generator" shows "generator (butlast_trans x) = butlast_raw (generator x)"
\<iota>_mapsto: shows "S.UP \<in> UNIV \<rightarrow> S.Univ"
r_minus: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<otimes>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> y = \<ominus>\<^bsub>K [X]\<^esub> (x \<otimes>\<^bsub>K [X]\<^esub> y)"
prefix_Nil: fixes xs :: "'a list" shows "prefix xs [] = (xs = [])"
abs_summable_on_cmult_right_iff: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. c * f x) abs_summable_on A) = (f abs_summable_on A)"
step_1: shows "LLL_bound_invariant False True i (fs[i := fs ! i - round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j) \<cdot>\<^sub>v fs ! j])"
continuous_map_in_subtopology: fixes X :: "'d topology"   and X' :: "'e topology"   and S :: "'e set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map X (subtopology X' S) f = (continuous_map X X' f \<and> f ` topspace X \<subseteq> S)"
run_one_step_basic_drop_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Drop) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
simps_2: fixes x :: "'b"   and \<iota> :: "nat"   and e :: "'a" shows "pp (x:=\<^bsub>\<iota>\<^esub> e) = \<iota>"
indep_card_eq_dim_span: fixes B :: "('c, 'd) vec set" assumes "vec.independent B" shows "finite B \<and> card B = vec.dim (vec.span B)"
lunit_commutes_with_L: fixes a :: "'c" assumes "C.ide a" shows "\<l>\<^sub>C[\<I>\<^sub>C \<otimes>\<^sub>C a] = \<I>\<^sub>C \<otimes>\<^sub>C \<l>\<^sub>C[a]"
LeadsTo_Un_Un: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "F \<in> B \<longmapsto>w B'" shows "F \<in> A \<union> B \<longmapsto>w A' \<union> B'"
pref_prod_cancel: fixes u :: "'b list"   and p :: "'b list"   and w :: "'b list"   and q :: "'b list" assumes "u \<le>p p \<cdot> w \<cdot> q"   and "\<^bold>|p\<^bold>| \<le> \<^bold>|u\<^bold>|"   and "\<^bold>|u\<^bold>| \<le> \<^bold>|p \<cdot> w\<^bold>|" obtains r :: "'b list" where "u = p \<cdot> r" and "r \<le>p w"
equivalence_map_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg" shows "bicategory.equivalence_map V H \<a> \<i> src trg f \<equiv> \<exists>g \<eta> \<epsilon>. equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
not_less_eq_eq: fixes m :: "nat"   and n :: "nat" shows "(\<not> m \<le> n) = (Suc n \<le> m)"
assms_2: fixes x :: "'c"   and h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> ok g x"