hcju/isabelleps huggingface dataset:
CastblocksEval: fixes ps :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and 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 E :: "char list \<Rightarrow> ty option"   and C' :: "char list"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option" assumes "length ps = length Ts"   and "length ps = length vs"   and "P,E \<turnstile> \<langle>\<lparr>C'\<rparr>blocks (ps, Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<exists>l'' vs'. P,E(ps [\<mapsto>] Ts) \<turnstile> \<langle>\<lparr>C'\<rparr>e,(h, l(ps [\<mapsto>] vs'))\<rangle> \<Rightarrow> \<langle>e',(h', l'')\<rangle> \<and> P \<turnstile> Ts Casts vs to vs'  \<and> length vs' = length vs"
prems_2: shows "path_conforms_with_strategy p P \<sigma>"
scaleR_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "'b"   and y :: "'b" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq x y"   and "0 \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b y)"
assms_1: shows "w \<in> set_tag_name_locs element_ptr"
uniformly_convergent_eq_Cauchy: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "uniformly_convergent_on X f = uniformly_Cauchy_on X f"
H_face_cycle_succ_a_neigh: shows "H.face_cycle_succ a_neigh = G.face_cycle_succ a'"
expectation_right_gpv: fixes fail :: "ennreal"   and \<I> :: "('a, 'c) \<I>"   and \<I>' :: "('b, 'd) \<I>"   and f :: "'e \<Rightarrow> ennreal"   and gpv :: "('e, 'b, 'd) gpv" shows "expectation_gpv fail (\<I> \<oplus>\<^sub>\<I> \<I>') f (right_gpv gpv) = expectation_gpv fail \<I>' f gpv"
not_is_unit_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "\<not> dvd.dvd times zero one"
steps'_distinct2: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list" assumes "length qs = length as"   and "n \<le> length as"   and "distinct s" shows "distinct (steps' s qs as n)"
gf_G_pos: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 < z"   and "z < 1"   and "(x, y) \<in> acc" shows "0 < gf_G x y z"
mono: fixes c :: "'a multiset"   and a :: "'a multiset"   and d :: "'a multiset"   and b :: "'a multiset" assumes "c \<subseteq># a"   and "d \<subseteq># b" shows "c \<union># d \<subseteq># a \<union># b"
b1: fixes x :: "'b"   and y :: "'a" shows "\<langle>(A\<^sup>\<dagger>) x, y\<rangle> = \<langle>x, A y\<rangle>"
strict_implies_order: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "a \<le> b"
not_ephemeral_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set" shows "MC_syntax.not_ephemeral K C = (C \<in> UNIV // MC_syntax.communicating K \<and> (\<nexists>x. C = {x} \<and> MC_syntax.p K x x 1 = 0))"
wagner_2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<omega> = (x \<cdot> y)\<^sup>\<omega>"
inc_ge: fixes u :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "length u = length y" shows "u \<le>\<^sub>v inc y i u"
accepting_pair\<^sub>G\<^sub>R_restrict: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e"   and F :: "('e \<times> 'd \<times> 'e) set"   and \<I> :: "('e \<times> 'd \<times> 'e) set set" assumes "range w \<subseteq> \<Sigma>" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, \<I>) w = accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F \<inter> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0, (\<lambda>I. I \<inter> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0) ` \<I>) w"
nat_le_add_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "nat"   and m :: "nat"   and n :: "nat" assumes "i \<le> j" shows "(i * u + m \<le> j * u + n) = (m \<le> (j - i) * u + n)"
substitutes_onI: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "\<And>B a b. \<lbrakk>B \<subseteq> A; a \<in> A; b \<in> A; b \<notin> f (insert b B)\<rbrakk> \<Longrightarrow> b \<notin> f (insert a (insert b B))" shows "substitutes_on A f"
identity_rules_def: fixes L :: "'b Standard_Constant set" shows "identity_rules L \<equiv> {reflexivity_rule S_Idt, transitive_rule S_Idt, symmetry_rule S_Idt} \<union> congruence_rules S_Idt L"
unx_none_simps_3: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c"   and u :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unconst (app t u) = None"
suf_comp_sym: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie>\<^sub>s v" shows "v \<bowtie>\<^sub>s u"
star_if_lang_pow: fixes w :: "'b list"   and n :: "nat"   and A :: "'b list set" assumes "w \<in> A ^^ n" shows "w \<in> star A"
mea: shows "emeasure M {x \<in> space M. f x \<noteq> g x} = 0"
not_inf_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" shows "\<not> AlternativeSmallStep.inf \<Gamma> cs css Stuck"
P2D2Q2: shows "(P2, D2, Q2) = Smith_2x2 D"
rs_filter_impl: shows "set_filter rs.\<alpha> rs.invar rs.\<alpha> rs.invar Benchmark_ICF.rs_filter"
sup_bot_left: fixes x :: "'b multiset" shows "{#} \<union># x = x"
mono: shows "\<forall>n m. n < m \<longrightarrow> k + n < k + m"
partition_coverture: fixes A :: "'d set"   and B :: "'d set set" assumes "Congruence.partition A B" shows "\<Union> B = A"
assms_3: shows "length ls \<noteq> Suc (hd xs)"
Max_add_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and k :: "'b" assumes "class.linordered_ab_semigroup_add plus less_eq less"   and "finite S"   and "S \<noteq> {}" shows "linorder.Max less_eq ((\<lambda>x. plus (f x) k) ` S) = plus (linorder.Max less_eq (f ` S)) k"
cases: fixes a :: "'a stream" assumes "run a" obtains x :: "'a" and y :: "'a" and xs :: "'a stream" where "a = x ## y ## xs" and "x \<rightarrow> y" and "run (y ## xs)"
equiv_trans_refl: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "equiv_trans t t"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "- MPoly x = MPoly (- x)"
Cauchy_product: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>k. norm (a k))"   and "summable (\<lambda>k. norm (b k))" shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
asymmetric_iff_irreflexive_antisymmetric: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "asymmetric A less = (irreflexive A less \<and> antisymmetric A less)"
poly_Nil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.semiring_0 plus zero times" shows "semiring_0.poly plus zero times [] x = zero"
simps_2: fixes k :: "'b"   and n :: "nat" shows "cbnd k (Suc n) = cbnd (cbnd k n) n"
subset_PiE: fixes I :: "'c set"   and S :: "'c \<Rightarrow> 'd set"   and T :: "'c \<Rightarrow> 'd set" shows "(Pi\<^sub>E I S \<subseteq> Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. S i \<subseteq> T i))"
domino_singleton: fixes d :: "(nat \<times> nat) set"   and b :: "nat" assumes "d \<in> domino"   and "b < 2" shows "\<exists>i j. evnodd d b = {(i, j)}"
b'_def: shows "b' = (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i)"
simps: fixes A :: "'a Matrix.mat" shows "square_mat A = (dim_col A = dim_row A)"
cr_prod2_Grp: fixes x :: "'i"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'j" shows "cr_prod2 x (BNF_Def.Grp A f) = BNF_Def.Grp A (\<lambda>b. (x, f b))"
comm_rootE: fixes x :: "'b list"   and y :: "'b list" assumes "x \<cdot> y = y \<cdot> x" obtains t :: "'b list" where "x \<in> t*" and "y \<in> t*"
goedel_second: assumes "consistent" shows "\<not> {} \<turnstile> neg (PP \<guillemotleft>Fls\<guillemotright>)"
intro: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
pls_cong1: fixes t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP pls t u EQ pls t' u'"
init: shows "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"
connf: shows "continuous_on (closure S) (cmod \<circ> f)"
subspace_def: fixes S :: "('c \<Rightarrow>\<^sub>0 'b) set" shows "phull.subspace S = (0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c \<cdot> x \<in> S))"
sep_impl_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_impl plus sep_disj P Q \<equiv> \<lambda>h. \<forall>h'. sep_disj h h' \<and> P h' \<longrightarrow> Q (plus h h')"
imageE: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "b \<in> f ` A" obtains x :: "'b" where "b = f x" and "x \<in> A"
det_linear_rows_sum: fixes S :: "nat set"   and a :: "nat \<Rightarrow> nat \<Rightarrow> 'b vec"   and n :: "nat" assumes "finite S"   and "a \<in> {0..<n} \<rightarrow> S \<rightarrow> carrier_vec n" shows "det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('b) n (a i) S)) = (\<Sum>f | (\<forall>i\<in>{0..<n}. f i \<in> S) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i). det (mat\<^sub>r n n (\<lambda>i. a i (f i))))"
v1: fixes h :: "'a" shows "h - Proj M *\<^sub>V h \<in> orthogonal_complement (space_as_set M)"
mod_lemma: fixes c :: "nat"   and a :: "nat"   and b :: "nat"   and n :: "nat" assumes "c \<le> a"   and "a < b"   and "b - c < n" shows "b mod n \<noteq> a mod n"
irreducible_imp_irreducible_hom: fixes a :: "'a" assumes "irreducible a" shows "irreducible (hom a)"
one_final_subdivFace': fixes g :: "graph"   and f :: "face"   and u :: "nat"   and v :: "nat"   and n :: "nat"   and ovs :: "nat option list" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g"   and "one_final_but g (if u = v then {} else Edges (u # between (vertices f) u v @ [v]))" shows "one_final (subdivFace' g f v n ovs)"
keys_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "keys (higher p v) = {u \<in> keys p. v \<prec>\<^sub>t u}"
fun_graph_def: fixes f :: "'c \<Rightarrow> 'd" shows "fun_graph f = {(x, f x) |x. True}"
proj_def: fixes r :: "('c \<times> 'b) set"   and x :: "'c" shows "proj r x = r `` {x}"
FP_2: shows "part {..<brn (fst cf2)} (F ` P)"
seq_if_composable: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "src \<nu> = trg \<mu>"
trg_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (B.cod \<mu>) = trg \<mu>"
simps_4: fixes b :: "nat"   and t1 :: "trm"   and t2 :: "trm" shows "fvi b (Less t1 t2) = fvi_trm b t1 \<union> fvi_trm b t2"
A_def: fixes Q :: "'a \<Rightarrow> 'c"   and \<pi> :: "'a \<Rightarrow> 'b option" shows "A Q \<pi> \<equiv> {(u, v). \<pi> u = Some v \<and> Q u = \<infinity>}"
finite_smult_distr: fixes fs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> rat"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> rat poly" assumes "distinct fs" shows "(\<Prod>(x, y)\<in>set fs. Polynomial.smult (f x y) (g x y)) = Polynomial.smult (\<Prod>(x, y)\<in>set fs. f x y) (\<Prod>(x, y)\<in>set fs. g x y)"
lincomb_sum: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K"   and "b \<in> A \<rightarrow> carrier K" shows "lincomb (\<lambda>v. a v \<oplus>\<^bsub>K\<^esub> b v) A = lincomb a A \<oplus>\<^bsub>V\<^esub> lincomb b A"
less_infI1: fixes a :: "'b multiset"   and x :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># x" shows "a \<inter># b \<subset># x"
S: shows "mi.case_ifexi fti ffi fii ni bdd = Some r"
distinct_3: fixes x1 :: "'c" shows "Le x1 \<noteq> \<infinity>"
path_eq: fixes p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "path p"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "path q"
rSim: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "R \<leadsto>[(bisim O X O bisim)] S"
invertible_inverse_inverse: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "sub.inverse (sub.inverse u) = u"
rel_funE: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
Sigma_Int_distrib1: fixes I :: "'c set"   and J :: "'c set"   and C :: "'c \<Rightarrow> 'd set" shows "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<sqsubseteq> a"   and "d \<sqsubseteq> b" shows "c \<squnion> d \<sqsubseteq> a \<squnion> b"
strict_coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c < a" shows "c < max a b"
preserves_inv: fixes f :: "'a" assumes "S.iso f" shows "S.trg (S.inv f) = S.inv (S.trg f)"
fps_binomial_nth: fixes a :: "'a"   and n :: "nat" shows "fps_nth (fps_binomial a) n = a gchoose n"
bigo_sum1: fixes h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "'c \<Rightarrow> 'd set" assumes "\<forall>x y. (0::'e) \<le> h x y"   and "\<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y" shows "(\<lambda>x. sum (f x) (A x)) \<in> O(\<lambda>x. sum (h x) (A x))"
linorder_inj_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x < y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> f x \<noteq> f y"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x \<le> y \<or> y \<le> x" shows "inj_on f A"
D: shows "f = (\<lambda>t2. {n. \<exists>t1. x = t1 @ t2 \<and> t1 @ [tick] \<in> T S \<and> (t2, X) \<in> F (Y n)})"
concurrent_def: fixes s1 :: "'a"   and s2 :: "'a" shows "s1 \<parallel> s2 \<equiv> \<not> s1 \<prec> s2 \<and> \<not> s2 \<prec> s1"
some_betw_xor: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q"   and "a \<noteq> b"   and "a \<noteq> c"   and "b \<noteq> c" shows "betw a b c \<and> \<not> betw b c a \<and> \<not> betw c a b \<or> betw b c a \<and> \<not> betw a b c \<and> \<not> betw c a b \<or> betw c a b \<and> \<not> betw a b c \<and> \<not> betw b c a"
quotient_of_split: fixes P :: "int \<times> int \<Rightarrow> bool"   and q :: "rat" shows "P (quotient_of q) = (\<forall>a b. 0 < b \<longrightarrow> coprime a b \<longrightarrow> q = Rat.Fract a b \<longrightarrow> P (a, b))"
simps_1: fixes R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "deriv_result R [] = R"
xs_min: fixes xs' :: "'a list" assumes "v \<leadsto>xs'\<leadsto> w" shows "length xs \<le> length xs'"
not_in_netmap: fixes i :: "nat"   and ns :: "'d net_state" assumes "i \<notin> net_ips ns" shows "netmap ns i = None"
ih2: fixes x :: "name"   and M :: "trm"   and M' :: "trm" assumes "x:M \<in> \<parallel>(A)\<parallel>"   and "M \<longrightarrow>\<^sub>a* M'" shows "x:M' \<in> \<parallel>(A)\<parallel>"
borel_def: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.borel open = sigma UNIV {S. open S}"
LAnd1: fixes b :: "coname"   and a1 :: "coname"   and M1 :: "trm"   and a2 :: "coname"   and M2 :: "trm"   and N :: "trm"   and y :: "name"   and x :: "name" assumes "b \<sharp> ([a1].M1, [a2].M2, N, a1, a2)"   and "y \<sharp> ([x].N, M1, M2, x)"   and "x \<sharp> (M1, M2)"   and "a1 \<sharp> (M2, N)"   and "a2 \<sharp> (M1, N)"   and "a1 \<noteq> a2" shows "Cut <b>.AndR <a1>.M1 <a2>.M2 b y.AndL1 x.N y \<longrightarrow>\<^sub>l Cut <a1>.M1 x.N"
rrsmult_range: fixes r :: "('f, 'g) aezfun"   and f :: "('f, 'g) aezfun \<Rightarrow> 'a" shows "range (rightreg_scalar_mult r f) \<subseteq> {0::'a} \<union> range f"
svar_value: fixes C_info :: "(name, dt_def) fmap"   and name :: "name" shows "\<not> constructors.is_value C_info (Svar name)"
ufa_\<alpha>_memD: fixes x :: "nat"   and y :: "nat"   and l :: "nat list" assumes "(x, y) \<in> ufa_\<alpha> l" shows "x < length l \<and> y < length l \<and> rep_of l x = rep_of l y"
OF_eq_sort: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "('a, 'c) flow_entry_match list"   and p :: "'b" assumes "no_overlaps \<gamma> f" shows "OF_priority_match \<gamma> f p = OF_match_linear \<gamma> (sort_descending_key ofe_prio f) p"
lim2: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. g n $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub g $$ (i, j)"
1: shows "Inv xs ys \<inter> {(y, x) |y. y < x in xs \<and> y < x in ys} = {}"
HPhase1or2Write_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2b s"   and "HPhase1or2Write s s' p d" shows "Inv2b s'"
LatticeDisjunctionElim2: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> (G \<leadsto> H)"
elims: fixes x :: "'o"   and xa :: "('a \<times> 'c) list \<times> 'o"   and y :: "('a \<times> 'c) list" assumes "sort_oalist_aux x xa = y" obtains ko :: "'o" and xs :: "('a \<times> 'c) list" and ox :: "'o" where "x = ko" and "xa = (xs, ox)" and "y = (if ko = ox then xs else sort_oalist ko xs)"
extensional_def: fixes A :: "'h set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
\<tau>move2CondThenExit': fixes pc :: "nat"   and e :: "'addr expr1"   and e1 :: "'addr expr1"   and P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e2 :: "'addr expr1" assumes "pc = Suc (length (compE2 e) + length (compE2 e1))" shows "\<tau>move2 P h stk (if (e) e1 else e2) pc None"
inf_sup_distrib1: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
trace_callee_eq_complete: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf" assumes "trace_callee_eq callee1 callee2 A p q" obtains S :: "'d spmf \<Rightarrow> 'g spmf \<Rightarrow> bool" where "S p q" and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))" and "\<And>p q a s b s'. \<lbrakk>S p q; a \<in> A; s \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 s a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)"
runit_simps_3: fixes f :: "'b" assumes "D.ide f" shows "trg\<^sub>D \<r>\<^sub>D[f] = trg\<^sub>D f"
le_cases: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" obtains "a \<preceq> b"   | "b \<preceq> a"
bind_simps_2: fixes x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b list" shows "List.bind (x # xs) f = f x @ List.bind xs f"
hyps: shows "(poly (sum_list (map power2 I1)) x = 0) = (\<forall>i\<in>set I1. poly i x = 0)"
islimpt_closure_open: fixes s :: "'a set"   and t :: "'a set"   and x :: "'a" assumes "open s"   and "t = closure s"   and "x \<in> t" shows "x islimpt t"
invariant_reaches: fixes a :: "'a"   and b :: "'a" assumes "A.reaches a b"   and "PA a" shows "PA b"
llist_all2_cases: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys" obtains "xs = LNil" and "ys = LNil"   | x :: "'c" and xs' :: "'c llist" and y :: "'d" and ys' :: "'d llist" where "xs = LCons x xs'" and "ys = LCons y ys'" and "P x y" and "llist_all2 P xs' ys'"
vm_path_axioms_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" shows "vm_path_axioms G P v0 \<equiv> (\<not> lnull P \<and> Digraph.valid_path G P) \<and> Digraph.maximal_path G P \<and> lhd P = v0"
primroot_root: fixes u :: "'b list"   and q :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> q*" shows "\<rho> q = \<rho> u"
tm_def: shows "tm = Max {s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}}"
assms_1: shows "X = atom_var ` snd ` set as"
sa_impl_rel_eext_internal_def: fixes Rm :: "('c \<times> 'f) set"   and Rv :: "('a \<times> 'd) set"   and Rl :: "('b \<times> 'e) set" shows "sa_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl\<rangle>gen_sa_impl_rel_eext"
simplify_preserves_strict_redundancy: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "all_fulfill finite S"   and "strictly_redundant C S" shows "strictly_redundant C (simplify S)"
ring_record_simps_11: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and zero :: "'e"   and add :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
lm024: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "f \<circ> g = id" shows "inj g"
has_derivativeI_sandwich: fixes e :: "real"   and f' :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> real" assumes "0 < e"   and "bounded_linear f'"   and "\<And>y. \<lbrakk>y \<in> s; y \<noteq> x; dist y x < e\<rbrakk> \<Longrightarrow> norm (f y - f x - f' (y - x)) / norm (y - x) \<le> H y"   and "(H \<longlongrightarrow> 0) (at x within s)" shows "(f has_derivative f') (at x within s)"
max_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "maximal t" shows "maximal (wp (Embed t))"