hcju/isabelleps huggingface dataset:
prime_elem_dvd_prod_msetE: 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"   and p :: "'a"   and A :: "'a multiset" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (comm_monoid_mult.prod_mset times one A)" obtains a :: "'a" where "a \<in># A" and "dvd.dvd times p a"
assms_2: shows "\<chi> \<noteq> principal_dchar n \<and> dcharacter n \<chi> \<or> \<chi> = principal_dchar n \<and> 1 \<notin> A"
defUse_path_dominated: fixes g :: "'g"   and v :: "'val"   and ns :: "'node list"   and n :: "'node"   and n' :: "'node" assumes "g \<turnstile> defNode g v-ns\<rightarrow>n"   and "defNode g v \<notin> set (tl ns)"   and "v \<in> allUses g n"   and "n' \<in> set ns" shows "dominates g (defNode g v) n'"
simps_2: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "mapi f (x # xs) = f (length xs) x # mapi f xs"
inext_neq_iMin: fixes I :: "nat set"   and n :: "nat" assumes "\<exists>x\<in>I. n < x" shows "inext n I \<noteq> iMin I"
assms_2: shows "Deliver (i, Add i e2) \<in> set (history j)"
word_log_defs_3: fixes a :: "'b word"   and b :: "'b word" shows "a XOR b = word_of_int (uint a XOR uint b)"
LLL_invD_9: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "i \<le> m"
ord_Zp_p_nat_unit: fixes n :: "nat" assumes "int n mod p \<noteq> 0" shows "ord_Zp ([n] \<cdot> \<one>) = 0"
secret_object2_def: shows "\<alpha> = - (- (\<iota> \<squnion> \<iota> \<squnion> \<iota>) \<squnion> \<iota>)"
distr_pair_swap: shows "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M = distr (Pi\<^sub>M J M \<Otimes>\<^sub>M Pi\<^sub>M I M) (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (\<lambda>(x, y). (y, x))"
solve_loop_body_def: fixes state :: "State"   and Vbl :: "nat set" shows "solve_loop_body state Vbl = (let state' = exhaustiveUnitPropagate state in if getConflictFlag state' then if currentLevel (getM state') = 0 then state'\<lparr>getSATFlag := FALSE\<rparr> else applyBackjump (applyLearn (applyExplainUIP (applyConflict state'))) else if Vbl \<subseteq> vars (elements (getM state')) then state'\<lparr>getSATFlag := TRUE\<rparr> else applyDecide state' Vbl)"
simps_7: fixes P :: "pi"   and Q :: "pi" shows "hnf (P \<oplus> Q) = (hnf P \<and> hnf Q \<and> P \<noteq> \<zero> \<and> Q \<noteq> \<zero>)"
apply_operations_added_files: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and f :: "'b \<Rightarrow> 'a set \<times> 'a set"   and x :: "'b" assumes "es prefix of j"   and "apply_operations es = Some f" shows "snd (f x) \<subseteq> set (added_files es x)"
l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_child_nodes_locs get_parent get_parent_locs \<equiv> get_parent = l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent get_child_nodes \<and> get_parent_locs = l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent_locs get_child_nodes_locs"
isTrivialPhi_def: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" shows "isTrivialPhi g v v' = (v' \<noteq> v \<and> (case phi g v of None \<Rightarrow> False | Some vs \<Rightarrow> set vs = {v, v'} \<or> set vs = {v'}))"
set_less_aux_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubset>' B"   and "B \<sqsubset>' A" shows "False"
subcls_leq_path: fixes C :: "char list"   and D :: "char 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 wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C \<preceq>\<^sup>* D"   and "wf_prog wf_md P" shows "\<exists>Cs. P,C \<turnstile> [C] \<sqsubseteq> Cs @ [D]"
matrix_to_iarray_echelon_form_of_det: fixes A :: "(('c, 'd) vec, 'd) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" shows "matrix_to_iarray (snd (echelon_form_of_det A bezout)) = snd (echelon_form_of_det_iarrays (matrix_to_iarray A) bezout)"
multi_inverse_conv_div: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> inverse b = a / b"
vfinite_vcard_vinsert_nin: fixes A :: "V"   and a :: "V" assumes "vfinite A"   and "a \<notin>\<^sub>\<circ> A" shows "vcard (vinsert a A) = csucc (vcard A)"
left_uniqueD: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and z :: "'e"   and y :: "'d" assumes "left_unique A"   and "A x z"   and "A y z" shows "x = y"
finfun_clearjunk_def: shows "finfun_clearjunk = id"
polypoly_def: fixes bs :: "'b list"   and p :: "'b poly" shows "polypoly bs p = map (Ipoly bs) (coefficients p)"
list_all2_list_all: fixes f :: "'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list" shows "list_all2 (\<lambda>x. f) xs ys = (length xs = length ys \<and> list_all f ys)"
module_def: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" shows "module R M \<equiv> cring R \<and> abelian_group M \<and> module_axioms R M"
subst_apply_bvars_disj_NegChecks: fixes X :: "'c list"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "set X \<inter> subst_domain \<theta> = {}" shows "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) \<or>\<notin>: G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>\<rangle>"
generate_subgroup_incl: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H (G\<lparr>carrier := generate G {a}\<rparr>)" shows "generate (G\<lparr>carrier := generate G {a}\<rparr>) K \<subseteq> H"
sup_mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "a \<squnion> b \<le> c \<squnion> d"
P13_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b l\<rightarrow> c \<le> a l\<rightarrow> c"
assms_3: fixes i :: "'a" assumes "i \<in> I" shows "0 < R i"
cyclic_group_axioms: shows "Finite_And_Cyclic_Groups.cyclic_group G gen"
proj_setD_2: fixes y :: "'b"   and x :: "'b"   and A :: "'b set" assumes "y \<in> proj_set x A" shows "dist x y = infdist x A"
list_all2_append2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and zs :: "'d list" shows "list_all2 P xs (ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)"
l_get_dom_component_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes get_disconnected_nodes_locs to_tree_order heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name"
add_arc_in: fixes a :: "'b" assumes "a \<in> arcs G" shows "G.add_arc a = G"
intros_38: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and err_v :: "v" shows "evaluate_match ck env s v1 [] err_v (s, Rerr (Rraise err_v))"
a0n_agreement_trans: shows "{a0n_agreement} TS.trans a0n {> a0n_agreement}"
similar_mat_wit_sym: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit B A Q P"
Rep_float_inject: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" shows "(Rep_float x = Rep_float y) = (x = y)"
Valid_def: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" shows "{p}   c   {q} \<equiv> \<forall>s s'. Sem c s s' \<longrightarrow> s \<in> p \<longrightarrow> s' \<in> q"
rtranclp_converseD: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "r\<inverse>\<inverse>\<^sup>*\<^sup>* x y" shows "r\<^sup>*\<^sup>* y x"
prv_eql_imp_trans: fixes t1 :: "'trm"   and t2 :: "'trm"   and t3 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "t3 \<in> trm" shows "prv (imp (eql t1 t2) (imp (eql t2 t3) (eql t1 t3)))"
wf_acc_iff: fixes r :: "('a \<times> 'a) set" shows "wf r = (\<forall>x. x \<in> Wellfounded.acc r)"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<sqsubseteq> b" obtains "a = a \<sqinter> b"
hmsetmset_def: fixes hmultiset :: "hmultiset" shows "hmsetmset hmultiset = (case hmultiset of HMSet x \<Rightarrow> x)"
finite_subset: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A"   and "finite B" shows "finite A"
elims_2: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c stream" assumes "holds x xa" obtains P :: "'c \<Rightarrow> bool" and xs :: "'c stream" where "x = P" and "xa = xs" and "P (shd xs)"
exec1s_impl_exec: fixes c :: "com"   and s :: "state"   and t :: "state" assumes "([c], s) \<rightarrow>\<^sup>* ([], t)" shows "s -c\<rightarrow> t"
simps_3: fixes p :: "('a, 'b, 'c) formula" shows "SIGF (! p) = SIGF p"
inverse_arrowsE: fixes f :: "'s"   and g :: "'s" assumes "inverse_arrows f g" obtains "ide (g \<cdot> f)" and "ide (f \<cdot> g)"
know2knows_k: fixes A :: "specID"   and m :: "Keys" assumes "know A (kKS m)" shows "knows A [kE m]"
r_unit_add_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> \<zero>\<^bsub>rec_rng_of_frac\<^esub> = (r |\<^bsub>rel\<^esub> s)"
np_spec_new_pairs_sorted: fixes rel :: "nat \<times> 'g \<Rightarrow> bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool" shows "punit.np_spec (punit.new_pairs_sorted rel)"
sum_monom_0_iff: fixes S :: "'b set"   and g :: "'b \<Rightarrow> nat"   and f :: "'b \<Rightarrow> 'c" assumes "finite S"   and "\<And>i j. g i = g j \<Longrightarrow> i = j" shows "((\<Sum>i\<in>S. monom (f i) (g i)) = 0) = (\<forall>i\<in>S. f i = (0::'c))"
cr_option_gpv_def: fixes x :: "'d option"   and gpv :: "('d, 'e, 'f) gpv" shows "cr_option_gpv x gpv = (gpv = (lift_spmf \<circ> return_pmf) x)"
ereal_SUP_uminus_eq: fixes f :: "'a \<Rightarrow> ereal"   and S :: "'a set" shows "(SUP x\<in>S. - f x) = - Inf (f ` S)"
eq_cong_ax_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e eq_cong_ax"
TR2_sub_carr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR2 R M N P \<subseteq> carrier P"
tree_rank_estimate_upper: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a r ts)" shows "size (SkewBinomialHeapStruc.tree_to_multiset (SkewBinomialHeapStruc.Node e a r ts)) \<le> 2 ^ Suc r - 1"
right_idem: fixes a :: "'e"   and b :: "'e" shows "sup (sup a b) b = sup a b"
dnso3: fixes x :: "'a" shows "ans_d x + (1::'a) = (1::'a)"
circ_sup_sub: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y)\<^sup>\<star> * x\<^sup>\<star> \<le> (x \<squnion> y)\<^sup>\<star>"
set_in_convex_hull: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "A \<subseteq> convex_hull A"
sccs_verts_conv_scc_of: shows "H.sccs_verts = H.scc_of ` verts H"
fds_ln_def: fixes l :: "'b"   and f :: "'b fds" shows "fds_ln l f = fds_integral l (fds_deriv f / f)"
distinct_1: fixes x2 :: "'a" shows "None \<noteq> \<lfloor>x2\<rfloor>"
rel_spmfE: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf" assumes "rel_spmf P p q" obtains pq :: "('d \<times> 'e) spmf" where "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y" and "p = map_spmf fst pq" and "q = map_spmf snd pq"
universally_influential_def: fixes A :: "'a"   and \<C> :: "'a set" shows "universally_influential (A, \<C>) = (A \<in> \<C> \<and> (\<forall>B. B \<in> \<C> \<and> B \<noteq> A \<longrightarrow> A \<rightarrow>\<^sub>\<S>\<^sup>+ B))"
resCasesB': fixes x' :: "name"   and P :: "pi"   and a :: "subject"   and y' :: "name"   and P' :: "pi" assumes "<\<nu>x'>P \<longmapsto> a\<guillemotleft>y'\<guillemotright> \<prec> P'" obtains Pa :: "pi" and aa :: "name" and b :: "name" and P'a :: "pi" where "<\<nu>x'>P = <\<nu>b>Pa__" and "a\<guillemotleft>y'\<guillemotright> \<prec> P' = aa__<\<nu>b> \<prec> P'a__" and "Pa__ \<longmapsto> aa__[b] \<prec> P'a__" and "aa__ \<noteq> b"   | Pa :: "pi" and aa :: "subject" and x :: "name" and P'a :: "pi" and y :: "name" where "<\<nu>x'>P = <\<nu>y>Pa" and "a\<guillemotleft>y'\<guillemotright> \<prec> P' = aa\<guillemotleft>x\<guillemotright> \<prec> <\<nu>y>P'a" and "Pa \<longmapsto> aa\<guillemotleft>x\<guillemotright> \<prec> P'a" and "y \<sharp> aa" and "y \<noteq> x" and "x \<sharp> Pa" and "x \<sharp> aa"
bounded_iff: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
simps_3: fixes p :: "nat"   and a :: "nat"   and x :: "nat"   and xs :: "pratt list" shows "valid_cert (Triple p a x # xs) = (1 < p \<and> 0 < x \<and> valid_cert xs \<and> (x = 1 \<or> (\<exists>q y. x = q * y \<and> Prime q \<in> set xs \<and> Triple p a y \<in> set xs \<and> [a ^ ((p - 1) div q) \<noteq> 1] (mod p))))"
pr_index_enumerator_mono1: fixes n1 :: "nat"   and n2 :: "nat"   and m :: "nat" assumes "n1 < n2" shows "pr_index_enumerator n1 m < pr_index_enumerator n2 m"
poly_mult_l_distr': fixes R :: "('c, 'd) ring_scheme"   and p1 :: "'c list"   and p2 :: "'c list"   and p3 :: "'c list" assumes "ring R"   and "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R"   and "set p3 \<subseteq> carrier R" shows "ring.poly_mult R (ring.poly_add R p1 p2) p3 = ring.poly_add R (ring.poly_mult R p1 p3) (ring.poly_mult R p2 p3)"
fac_body_final_def: shows "fac_body_final \<equiv> \<Lambda> r n. if n = 0 then up\<cdot>1 else r\<cdot>(n -\<^sub># 1) >>= (\<Lambda> b. up\<cdot>(n *\<^sub># b))"
degree_one: fixes v :: "nat" shows "MPoly_Type.degree 1 v = 0"
vdw_lemma_imp_vdw: fixes k :: "nat"   and l :: "nat"   and N :: "nat" assumes "vdw_lemma k k l N" shows "vdw k (Suc l) N"
small_sup_iff: fixes X :: "V set"   and Y :: "V set" shows "small (X \<union> Y) = (small X \<and> small Y)"
l_props: fixes x :: "real" assumes "x \<le> l" shows "poly p x \<noteq> 0"
s': shows "\<sigma>' = (None, h, ([], rev (take n stk) @ replicate mxl' undefined, D', M', 0, No_ics) # (stk, loc, C, M, pc, No_ics) # frs, sh)"
nn_integral_return: fixes x :: "'a"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (return M x) g = g x"
size_mset: fixes xs :: "'a list" shows "size (mset xs) = length xs"
no_ofail_is_ovalidNF: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option" shows "no_ofail P f \<equiv> ovalidNF P f (\<lambda>_ _. True)"
poincare_between_x_axis_intersection: fixes u :: "complex_homo"   and v :: "complex_homo"   and z :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<noteq> v"   and "u \<notin> circline_set x_axis"   and "v \<notin> circline_set x_axis"   and "z \<in> circline_set (poincare_line u v) \<inter> circline_set x_axis" shows "poincare_between u z v = (arg (to_complex u) * arg (to_complex v) < 0)"
filter_sup_left_upper_bound: fixes G :: "'a set"   and F :: "'a set" assumes "ord_class.filter G" shows "F \<subseteq> filter_sup F G"
assm2: shows "SPEC (\<lambda>c. case c of (uu_, nds) \<Rightarrow> old_next_pair ` snd (a, b) \<subseteq> old_next_pair ` nds \<and> expand_inv_result \<phi> nds) \<le> SPEC (\<lambda>r. (case r of (nm, nds) \<Rightarrow> f (a\<lparr>name := nm, new := new2 xa \<union> (new a - {xa}), old := insert xa (old a)\<rparr>, nds)) \<le> SPEC (\<lambda>c. case c of (uu_, nds) \<Rightarrow> old_next_pair ` snd (a, b) \<subseteq> old_next_pair ` nds \<and> expand_inv_result \<phi> nds))"
fmeasurable_compact: fixes S :: "'b set" assumes "compact S" shows "S \<in> fmeasurable lborel"
subst_lemma: fixes x :: "name"   and y :: "name"   and L :: "lam"   and N :: "lam"   and M :: "lam" assumes "x \<noteq> y"   and "x \<sharp> L" shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
Assign: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> s[a/x]"
nn_integral_mono: fixes M :: "'h measure"   and u :: "'h \<Rightarrow> ennreal"   and v :: "'h \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
latestAct_prop_2: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'<n. active nid (t n')" shows "dynamic_component.latestAct active nid t n < n"
hb_flat_def: fixes xqq :: "'a option lazy_sequence option lazy_sequence" shows "hb_flat xqq = lazy_sequence_of_list (concat (list.map ((\<lambda>x. case x of None \<Rightarrow> [None] | Some xs \<Rightarrow> xs) \<circ> map_option list_of_lazy_sequence) (list_of_lazy_sequence xqq)))"
Un_Diff_cancel2: fixes B :: "'e set"   and A :: "'e set" shows "B - A \<union> A = B \<union> A"
return_None_bind_spmf: fixes f :: "'d \<Rightarrow> 'b spmf" shows "return_pmf None \<bind> f = return_pmf None"
A2': fixes a :: "hyp2"   and b :: "hyp2"   and p :: "hyp2"   and q :: "hyp2"   and r :: "hyp2"   and s :: "hyp2" assumes "a b \<congruent>\<^sub>K p q"   and "a b \<congruent>\<^sub>K r s" shows "p q \<congruent>\<^sub>K r s"
sums_even: fixes f :: "nat \<Rightarrow> real" shows "(sums) f = (sums) (\<lambda>n. if even n then f (n div 2) else 0)"
length_def: fixes r :: "real_int" shows "\<parallel>r\<parallel> \<equiv> right r - left r"
Y_square_unfold: shows "(\<lambda>x. (Y x)\<^sup>2) = (\<lambda>x. \<Sum>i\<in>I. \<Sum>j\<in>I. rind (A i \<inter> A j) x)"
has_typ1_arg_typ: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and U :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t $ u : R"   and "Ts \<turnstile>\<^sub>\<tau> u : U" shows "Ts \<turnstile>\<^sub>\<tau> t : U \<rightarrow> R"
ale_nat_le: fixes n :: "nat"   and m :: "nat" shows "(an n \<le> an m) = (n \<le> m)"
pred_rel: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g list" shows "list_all P x = list_all2 (eq_onp P) x x"
r_zero: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "x \<oplus> \<zero> = x"
h1_def: fixes f :: "'e \<Rightarrow> 'd"   and s :: "'e" shows "h1 f s = f s"
MPLS_nempty: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "finite PROB" shows "MPLS PROB \<noteq> {}"
pf_transpose: shows "perron_frobenius (Finite_Cartesian_Product.transpose A)"
naturality: fixes f :: "'c" assumes "C.arr f" shows "\<mu> (C.cod f) \<cdot>\<^sub>D G f = G' f \<cdot>\<^sub>D \<mu> (C.dom f)"
set_sum_sel_2: fixes s :: "'d + 'e" assumes "\<not> isl s" shows "projr s \<in> Basic_BNFs.setr s"
sublist_imp_len_filter_le: fixes l' :: "'c list"   and l :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "subseq l' l" shows "length (filter P l') \<le> length (filter P l)"
aff_dim_affine_hull: fixes S :: "'b set" shows "aff_dim (affine hull S) = aff_dim S"
loc_sl_def: fixes P :: "'b prog"   and mxl :: "nat" shows "loc_sl P mxl \<equiv> Listn.sl mxl (Err.sl (SemiType.esl P))"
write_sops_append: fixes xs :: "'a memref list"   and ys :: "'a memref list" shows "write_sops (xs @ ys) = write_sops xs \<union> write_sops ys"
add_eq_inf_sup: 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 inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "plus a b = plus (sup a b) (inf a b)"
dgrad_set_exhaust: fixes F :: "'b set"   and d :: "'b \<Rightarrow> nat" assumes "finite F" obtains m :: "nat" where "F \<subseteq> dgrad_set d m"
share_mono_in: fixes x :: "nat"   and xs :: "'b memref list"   and A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option" assumes "x \<in> dom (share xs A)"   and "dom A \<subseteq> dom B" shows "x \<in> dom (share xs B)"
decomp_F: shows "direct_decomp (ideal F \<inter> P[X]) [ideal {f} \<inter> P[X], P]"
exclude_one_is_pseudominimal: fixes S :: "'b set"   and w :: "'b"   and t :: "'b" assumes "PreCoxeterSystemWithDeletion S"   and "w \<in> \<langle>S\<rangle>"   and "t \<in> S" shows "dual_order.pseudominimal_in (PreCoxeterSystem.special_cosets S) (w +o \<langle>S - {t}\<rangle>)"
eventually_at_right: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
sets_convolution: fixes M :: "'a measure"   and N :: "'a measure" shows "sets (M \<star> N) = sets borel"
impl: shows "gcd_impl_primitive G1 G2 = normalize (primitive_part Gk)"
valid_regions_distinct: fixes X :: "'b set"   and k :: "'b \<Rightarrow> nat"   and I :: "'b \<Rightarrow> intv"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r :: "('b \<times> 'b) set"   and I' :: "'b \<Rightarrow> intv"   and J' :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r' :: "('b \<times> 'b) set"   and v :: "'b \<Rightarrow> real" assumes "valid_region X k I J r"   and "valid_region X k I' J' r'"   and "v \<in> region X I J r"   and "v \<in> region X I' J' r'" shows "region X I J r = region X I' J' r'"
power_Suc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
lookup_a2_map'_keys: shows "\<forall>tp'\<in>Mapping.keys a2_map'. case Mapping.lookup a2_map' tp' of Some m \<Rightarrow> table n R (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (nt - (left I - 1)) (tp + 1) \<and> isl tstp = (0 < left I))"
is_universal: fixes a' :: "'c"   and \<chi>' :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> C.dom e\<guillemotright> \<and> D.cones_map f (D.mkCone e) = \<chi>'"
A_greater_zero: fixes x :: "int" assumes "x \<in> A" shows "0 < x"