hcju/isabelleps huggingface dataset:
pasting_lemma_exists_closed: fixes I :: "'a set"   and X :: "'b topology"   and T :: "'a \<Rightarrow> 'b set"   and Y :: "'c topology"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite I"   and "topspace X \<subseteq> \<Union> (T ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x" obtains g :: "'b \<Rightarrow> 'c" where "continuous_map X Y g" and "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
weak_composition_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "weak_composition V H \<equiv> (category V \<and> product_category V V) \<and> subcategory (product_category.comp V V) (\<lambda>\<mu>\<nu>. H (fst \<mu>\<nu>) (snd \<mu>\<nu>) \<noteq> partial_magma.null V) \<and> functor (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. H (fst \<mu>\<nu>) (snd \<mu>\<nu>) \<noteq> partial_magma.null V)) V (\<lambda>\<mu>\<nu>. H (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<and> weak_composition_axioms V H"
set_plus_closed: fixes A :: "'m set"   and B :: "'m set" assumes "A \<subseteq> M"   and "B \<subseteq> M" shows "A + B \<subseteq> M"
igAbsIPresIGWlsSTR_imp_igAbsIPresIGWls: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igAbsIPresIGWlsSTR MOD" shows "igAbsIPresIGWls MOD"
eval_simps_4: fixes f :: "'c local.term" assumes "Arr f" shows "\<lbrace>Dom f\<rbrace> = local.dom \<lbrace>f\<rbrace>"
hyps: assumes "1 < length l"   and "strict_sorted l"   and "\<not> List.member l x"   and "l ! 0 < x"   and "x < l ! (length l - 1)" shows "\<exists>n<length l - 1. l ! n < x \<and> x < l ! (n + 1)"
subgroup_Int: fixes I :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and J :: "'c set" assumes "subgroup I G"   and "subgroup J G" shows "subgroup (I \<inter> J) G"
rel_pred_neg_distr_imp: fixes B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "right_unique B \<and> left_total B \<or> left_unique A \<and> right_total A" shows "rel_pred_neg_distr_cond A B"
sp_optim_instr: fixes n :: "nat"   and instr :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and \<Sigma>1 :: "Unboxed.type option list"   and instr' :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'fun \<Rightarrow> 'd fundef option" assumes "optim_instr n instr \<Sigma>1 = Ok (instr', \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "Subx.sp_instr F instr' \<Sigma>1 = Ok \<Sigma>2"
explicit_subset_rel_interior_convex_hull_minimal: fixes S :: "'a set" assumes "finite S" shows "{y. \<exists>u. (\<forall>x\<in>S. 0 < u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y} \<subseteq> rel_interior (convex hull S)"
hom_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
LPO_S_subset_LPO_NS: shows "LPO_S \<subseteq> LPO_NS"
q''_def: shows "q'' = (\<lambda>g. q g + mu * q' g)(g1 := punit.tail (q g1) + mu * q' g1, g2 := q g2 + mu * q' g2 + mu * mu2)"
sep_disj_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x y = sep_disj y x"
assms_3: shows "\<forall>l. M l \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>))"
mono_aval': fixes S :: "char list \<Rightarrow> 'a"   and S' :: "char list \<Rightarrow> 'a"   and e :: "aexp" assumes "S \<le> S'" shows "aval' e S \<le> aval' e S'"
2: shows "continuous_on (cbox (- 1) 1) (negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w))"
RETURN_def: fixes x :: "'a" shows "RETURN x \<equiv> RES {x}"
top_2: shows "top_on\<^sub>c C2 (- vars (strip C2))"
valid_frac_add: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" assumes "valid_frac X"   and "valid_frac Y" shows "valid_frac (add_rel X Y)"
induct: fixes x1 :: "x \<Rightarrow> x option"   and x2 :: "s"   and x3 :: "s"   and P :: "(x \<Rightarrow> x option) \<Rightarrow> s \<Rightarrow> s \<Rightarrow> bool" assumes "tr_s x1 x2 x3"   and "\<And>T s_s'_list. list_all (\<lambda>f. f) (map (\<lambda>(s_XXX, s_'). tr_s T s_XXX s_' \<and> P T s_XXX s_') s_s'_list) \<Longrightarrow> P T (s_block (map (\<lambda>(s_XXX, s_'). s_XXX) s_s'_list)) (s_block (map (\<lambda>(s_XXX, s_'). s_') s_s'_list))"   and "\<And>T var var' x x'. \<lbrakk>(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'; (case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'\<rbrakk> \<Longrightarrow> P T (s_ass var x) (s_ass var' x')"   and "\<And>T var var' x x' f. \<lbrakk>(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'; (case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'\<rbrakk> \<Longrightarrow> P T (s_read var x f) (s_read var' x' f)"   and "\<And>T x x' y y' f. \<lbrakk>(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'; (case T y of None \<Rightarrow> y | Some x' \<Rightarrow> x') = y'\<rbrakk> \<Longrightarrow> P T (s_write x f y) (s_write x' f y')"   and "\<And>T x x' y y' s1 s1' s2 s2'. \<lbrakk>(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'; (case T y of None \<Rightarrow> y | Some x' \<Rightarrow> x') = y'; tr_s T s1 s1'; P T s1 s1'; tr_s T s2 s2'; P T s2 s2'\<rbrakk> \<Longrightarrow> P T (s_if x y s1 s2) (s_if x' y' s1' s2')"   and "\<And>T var var' ctx cl. (case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var' \<Longrightarrow> P T (s_new var ctx cl) (s_new var' ctx cl)"   and "\<And>T var var' x x' y_y'_list meth. \<lbrakk>(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'; (case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'; list_all (\<lambda>f. f) (map (\<lambda>(y_XXX, y_'). (case T y_XXX of None \<Rightarrow> y_XXX | Some x' \<Rightarrow> x') = y_') y_y'_list)\<rbrakk> \<Longrightarrow> P T (s_call var x meth (map (\<lambda>(y_XXX, y_'). y_XXX) y_y'_list)) (s_call var' x' meth (map (\<lambda>(y_XXX, y_'). y_') y_y'_list))" shows "P x1 x2 x3"
holf: shows "f holomorphic_on interior (convex hull {a, b, c})"
a: shows "[d2 = e2 * b + q * b - x * b + (q - r)] (mod q)"
ts_def: fixes s :: "nat \<Rightarrow> 'a list" shows "ts s \<equiv> \<forall>i. length (s i) = 1"
cases: fixes a1 :: "'b set"   and a2 :: "'b list"   and \<Sigma>i :: "'b set" assumes "(a1, a2) \<in> dfs_initial \<Sigma>i" obtains W :: "'b list" where "a1 = \<Sigma>i" and "a2 = W" and "distinct W" and "set W = \<Sigma>i"
similar_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "similar_mat A B = (\<exists>P Q. similar_mat_wit A B P Q)"
simps_3: fixes s :: "'state_t"   and d :: "'dom_t"   and as' :: "'action_t list"   and execs' :: "'action_t list list" shows "CISK_control s d ([] # as' # execs') = (None, as' # execs', s)"
nat_to_pr_at_0: shows "nat_to_pr 0 = (\<lambda>x. 0)"
le_word_or2: fixes x :: "'b word"   and y :: "'b word" shows "x \<le> x OR y"
countable_nat_UN: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> M" shows "\<Union> (range A) \<in> M"
prj_simps_2: shows "C.seq f_src.prj\<^sub>0 \<p>\<^sub>1[src.leg0 \<cdot> f_src.prj\<^sub>0, src.leg1]"
simps_3: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and bs :: "('a \<times> ('b, 'a) Term.term) list" shows "unify ((Var x, t) # E) bs = (if t = Var x then unify E bs else if x \<in> fv t then None else unify (subst_list (subst x t) E) ((x, t) # bs))"
finite_set_of_finite_funs: fixes A :: "'b set"   and B :: "'c set"   and d :: "'c" assumes "finite A"   and "finite B" shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}"
ltl_WeakUntil_StrongRelease_con_1: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> W\<^sub>c \<psi> = (\<not> \<xi> \<Turnstile>\<^sub>c not\<^sub>c \<phi> M\<^sub>c not\<^sub>c \<psi>)"
MWLsSteps_det_cases_2: fixes x :: "'id"   and \<iota> :: "nat"   and e :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>x:=\<^bsub>\<iota>\<^esub> e,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "\<alpha> = []" and "p = None" and "m' = m(x := E e m)"
proj_sing1: fixes \<theta> :: "('b, 'sz) trm" shows "singleton \<theta> vid1 = \<theta>"
i_drop_Suc_append: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" shows "((x # xs) \<frown> f) \<Up> Suc n = (xs \<frown> f) \<Up> n"
exhaust: fixes y :: "'c flowtable_behavior" obtains x1 :: "'c" where "y = Action x1"   | "y = NoAction"   | "y = Undefined"
CondF\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and e\<^sub>1 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
fract_poly_dvd: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "map_poly to_fract p dvd map_poly to_fract q"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "monad_fail return bind fail" shows "monad_fail_axioms bind fail"
diag_2: shows "\<not> {} \<turnstile> \<delta>"
trivial_DD: fixes \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set" assumes "\<sigma> \<in> seq ars" shows "DD ars r (\<sigma>, (fst \<sigma>, []), (lst \<sigma>, []), \<sigma>)"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat"   and a2 :: "Core.term" assumes "\<And>i n u. P (Bv i) n u"   and "\<And>T body n u. P body (n + 1) (lift u 0) \<Longrightarrow> P (Abs T body) n u"   and "\<And>f t n u. \<lbrakk>P f n u; P t n u\<rbrakk> \<Longrightarrow> P (f $ t) n u"   and "\<And>v va uu_ uv_. P (Ct v va) uu_ uv_"   and "\<And>v va uu_ uv_. P (Fv v va) uu_ uv_" shows "P a0 a1 a2"
add_closed_converse_left: fixes G :: "'a set"   and g :: "'a"   and x :: "'a" assumes "Group G"   and "g + x \<in> G"   and "x \<in> G" shows "g \<in> G"
l_distr: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
matrix_to_lpolies_vec_of_row: fixes i :: "nat"   and A :: "rat mat" assumes "i < dim_row A" shows "matrix_to_lpolies A ! i = vec_to_lpoly (row A i)"
WatchInvariantsAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = assertLiteral literal decision state in InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state') \<and> InvariantWatchListsUniq (getWatchList state') \<and> InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
taut_conj_elim: fixes A :: "'b upred"   and B :: "'b upred" assumes "`A \<and> B`" obtains "`A`" and "`B`"
mann9: fixes x :: "'a" shows "x = - (- (- 3 \<times> x \<squnion> x) \<squnion> - 3 \<times> x)"
unsat: shows "(set I \<subseteq> indices_state (assert_all_state t as) \<and> (\<nexists>v. (set I, v) \<Turnstile>\<^sub>i\<^sub>s assert_all_state t as)) \<and> (distinct_indices_state (assert_all_state t as) \<longrightarrow> subsets_sat_core (assert_all_state t as))"
pre_splitFace_def: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and nvs :: "nat list" shows "pre_splitFace g ram1 ram2 oldF nvs \<equiv> oldF \<in> \<F> g \<and> \<not> final oldF \<and> distinct (vertices oldF) \<and> distinct nvs \<and> \<V> g \<inter> set nvs = {} \<and> \<V> oldF \<inter> set nvs = {} \<and> ram1 \<in> \<V> oldF \<and> ram2 \<in> \<V> oldF \<and> ram1 \<noteq> ram2 \<and> ((ram1, ram2) \<notin> \<E> oldF \<and> (ram2, ram1) \<notin> \<E> oldF \<and> (ram1, ram2) \<notin> \<E> g \<and> (ram2, ram1) \<notin> \<E> g \<or> nvs \<noteq> [])"
d1: shows "Sup ({norm (f x) / norm x |x. x \<noteq> (0::'a)} \<union> {norm (f x) / norm x |x. x = (0::'a)}) = max (Sup {norm (f x) / norm x |x. x \<noteq> (0::'a)}) (Sup {norm (f x) / norm x |x. x = (0::'a)})"
reverse_iterateoi_rule_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> '\<sigma> \<Rightarrow> bool"   and \<sigma>0 :: "'\<sigma>"   and c :: "'\<sigma> \<Rightarrow> bool"   and f :: "'u \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>"   and P :: "'\<sigma> \<Rightarrow> bool" assumes "invar m"   and "I (\<alpha> m) \<sigma>0"   and "\<And>k it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; \<forall>j\<in>it. j \<le> k; \<forall>j\<in>\<alpha> m - it. k \<le> j; it \<subseteq> \<alpha> m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f k \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> \<alpha> m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. k \<le> j\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (reverse_iterateoi m c f \<sigma>0)"
content_eq_zero_iff: fixes p :: "'b poly" shows "(content p = (0::'b)) = (p = 0)"
dining_state_update_def: shows "Up\<^sub>D = (\<lambda>(ps, fs) e. (Up\<^sub>P ps e, Up\<^sub>F fs e))"
hyps_2: fixes B :: "'a fset" assumes "\<And>x. (x \<in> fset A) = (x \<in> fset B)" shows "A = B"
extend_constrains_slice: fixes F :: "'a program"   and A :: "'c set"   and B :: "'c set"   and y :: "'b" assumes "extend h F \<in> A co B" shows "F \<in> slice A y co slice B y"
fin: fixes n :: "nat" shows "f\<cdot>x = (\<Union>i\<le>n. g\<cdot>(r\<^bsup>i\<^esup>\<cdot>x)) \<union> f\<cdot>(r\<^bsup>Suc n\<^esup>\<cdot>x)"
abel_semigroup_axioms: shows "abel_semigroup (\<parallel>)"
normal_set_mult_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" assumes "second_isomorphism_grp H G S" shows "subgroup (H <#>\<^bsub>G\<^esub> S) G"
\<tau>move2_sync1D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) pc xcp"   and "pc < length (compE2 o')" shows "\<tau>move2 P h stk o' pc xcp"
wf_until_aux_Cons: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and a :: "nat \<times> 'a option list set \<times> 'a option list set"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ne :: "nat" assumes "wf_until_aux \<sigma> n R pos \<phi> I \<psi> (a # aux) ne" shows "wf_until_aux \<sigma> n R pos \<phi> I \<psi> aux (Suc ne)"
Gramian_determinant_ge1: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs)" shows "1 \<le> gs.d k"
split_lv_rel_def: fixes X :: "'c \<times> 'd" shows "split_lv_rel X = (fst X, snd X)"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "F x = F' x"
inL3': fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, G, H, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "H, F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
intro: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subcring H R"   and "subdomain_axioms H R" shows "subdomain H R"
atom_step1: fixes r :: "'a"   and b :: "'a"   and a :: "'a" assumes "r \<cdot> b \<le> b \<cdot> r" shows "(a + b + r)\<^sup>\<infinity> = b\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity> \<cdot> (a \<cdot> b\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity>)\<^sup>\<infinity>"
stat_lemma: fixes c :: "nat" shows "rec_exec rec_stat [c] = stat c"
point_construction_different: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<exists>C. Bet A B C \<and> B \<noteq> C"
univ_asAbs_alphaAbsGood: fixes fAbs :: "('e, 'f, 'varSort, 'var, 'g) qAbs \<Rightarrow> 'h"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "fAbs respectsP alphaAbsGood"   and "qGoodAbs A" shows "Equiv_Relation2.univ fAbs (asAbs A) = fAbs A"
IH_1: fixes inputs :: "real vec list"   and j :: "nat"   and a :: "nat" assumes "map dim_vec inputs = input_sizes m1"   and "valid_net m1"   and "j < output_size m1" shows "polyfun {..<a + count_weights s m1} (\<lambda>f. evaluate_net (insert_weights s m1 (\<lambda>i. f (i + a))) inputs $ j)"
conservation_const_pointwise: fixes u :: "nat" assumes "u \<in> cf.V - {s, t}" shows "(\<Sum>v\<in>cf.E `` {u}. f' (u, v)) = (\<Sum>v\<in>cf.E\<inverse> `` {u}. f' (v, u))"
simps_4: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (fm.Atom (Neq p)) = (var \<notin> vars p)"
vertices_graph_code: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "vertices_graph (Graph fs n f h) = [0..<n]"
mvt: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> D f \<mapsto> f' x at x" obtains \<xi> :: "real" where "a < \<xi>" and "\<xi> < b" and "f b - f a = f' \<xi> (b - a)"
not_QuadRes_ex: fixes p :: "int" assumes "1 < p"   and "odd p" shows "\<exists>x. \<not> QuadRes p x"
allBlocks_def: fixes s :: "state" shows "allBlocks s = \<Union> (range (blocksOf s))"
msubst2_nb: fixes p :: "fm"   and t :: "tm"   and c :: "poly" assumes "islin p"   and "tmbound0 t" shows "bound0 (msubst2 p c t)"
L3_5ii: fixes c1 :: "'b com"   and c2 :: "'b com"   and X :: "'b set" shows "SEM (Seq c1 c2) X = SEM c2 (SEM c1 X)"
app_mono: fixes p :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and \<tau> :: "(ty list \<times> ty err list) option"   and \<tau>' :: "(ty list \<times> ty err list) option"   and i :: "instr"   and m :: "nat"   and rT :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>' \<tau>'"   and "app i P m rT pc mpc xt \<tau>'" shows "app i P m rT pc mpc xt \<tau>"
add: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "additive f" shows "f (x + y) = f x + f y"
assms: shows "infsetsum'_converges (\<lambda>x. norm (f x)) A"
cond_def: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_antidomain_kleene_algebra.cond p x y = rdom p ; x \<union> rel_ad p ; y"
jumpF_polyL_0_1: fixes p :: "real poly"   and a :: "real" shows "jumpF_polyL 0 p a = 0"
wt_kildall_def: fixes P :: "jvm_prog"   and C' :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table" shows "wt_kildall P C' b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<equiv> 0 < length is \<and> (let first = \<lfloor>([], (case b of Static \<Rightarrow> [] | NonStatic \<Rightarrow> [OK (Class C')]) @ map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor>; start = OK first # replicate (length is - 1) (OK None); result = kiljvm P mxs ((case b of Static \<Rightarrow> 0 | NonStatic \<Rightarrow> 1) + length Ts + mxl\<^sub>0) T\<^sub>r is xt start in \<forall>n<length is. result ! n \<noteq> Err)"
sat_MatchP_rec: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex" shows "Formula.sat \<sigma> V v i (formula.MatchP I r) = (mem 0 I \<and> Regex.eps (Formula.sat \<sigma> V v) i r \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> (\<exists>s\<in>Regex.rpd (Formula.sat \<sigma> V v) i r. Formula.sat \<sigma> V v (i - 1) (formula.MatchP (subtract (\<Delta> \<sigma> i) I) s)))"
vec_of_list_alt: fixes xs :: "'c list" shows "vec_of_list xs = vec (length xs) ((!) xs)"
RepFun_iff: fixes v :: "hf"   and A :: "hf"   and f :: "hf \<Rightarrow> hf" shows "(v \<^bold>\<in> RepFun A f) = (\<exists>u. u \<^bold>\<in> A \<and> v = f u)"
complex_roots_c: fixes c :: "complex" shows "complex_roots_complex [:c:] = []"
not_Ici_le_Icc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and l' :: "'a"   and h' :: "'a" assumes "class.no_top less_eq less" shows "\<not> ord.atLeast less_eq l \<subseteq> ord.atLeastAtMost less_eq l' h'"
solve_construct: shows "w_vec p qs signs = solve_for_lhs p qs subsets (M_mat signs subsets)"
sig_red_lt: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q" shows "lt q \<preceq>\<^sub>t lt p"
if_elim': fixes b :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(Stmt.If b p q, mem) \<leadsto>\<^sub>s (c', mem')" obtains "c' = p" and "mem' = mem" and "eval\<^sub>B mem b"   | "c' = q" and "mem' = mem" and "\<not> eval\<^sub>B mem b"
nat_add_eq_self_zero: fixes m :: "nat"   and n :: "nat" assumes "m = m + n" shows "n = 0"