hcju/isabelleps huggingface dataset:
linear_continuous_within: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "bounded_linear f" shows "continuous (at x within s) f"
simps_3: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and z :: "'b"   and zs :: "'b list" shows "x # xs \<cong> {[], z # zs, P} = (\<not> P x xs \<and> x = z \<and> xs \<cong> {[], zs, P})"
unprot_ntsem: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "unprot \<sigma> \<or> protFw \<sigma>" shows "ntsem \<sigma> a = a"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and tbq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.insertList (t # bq) tbq = SkewBinomialHeapStruc.insertList bq (SkewBinomialHeapStruc.insert (SkewBinomialHeapStruc.val t) (SkewBinomialHeapStruc.prio t) tbq)"
collapse_6: fixes formula :: "formula" assumes "is_Neg formula" shows "Neg (un_Neg formula) = formula"
remove_empty_implies_subset: fixes M :: "'a multiset"   and S :: "'a set" assumes "M -s S = {#}" shows "set_mset M \<subseteq> S"
substitute_remove_anyways_aux: fixes f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and xs :: "'b list"   and T :: "'b set"   and t :: "'b ttree" assumes "repeatable (f x)"   and "xs \<in> paths (substitute f T t)"   and "t \<otimes>\<otimes> f x = t" shows "xs \<in> paths (substitute (f(x := TTree.empty)) T t)"
distinct_take: fixes xs :: "'g list"   and i :: "nat" assumes "distinct xs" shows "distinct (take i xs)"
map_of_map: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "('d \<times> 'f) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
prem: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau> s $ Fv x \<tau>) (Abs_fv x \<tau> t $ Fv x \<tau>)"
norm_def: fixes v :: "(nat \<Rightarrow> real) \<times> nat" shows "\<parallel>v\<parallel> = sqrt (\<Sum>j = 1..vlen v. (v\<^bsub>j\<^esub>)\<^sup>2)"
x2: shows "x\<^sub>2 = (- b - sqrt D) / (2 * a)"
word_less_nowrapI': fixes x :: "'b word"   and z :: "'b word"   and k :: "'b word" assumes "x \<le> z - k"   and "k \<le> z"   and "0 < k" shows "x < x + k"
ccpo_Sup_least: fixes A :: "('a \<times> 'b) set"   and z :: "'a \<times> 'b" assumes "Complete_Partial_Order.chain (rel_prod orda ordb) A"   and "\<And>x. x \<in> A \<Longrightarrow> rel_prod orda ordb x z" shows "rel_prod orda ordb (prod_lub luba lubb A) z"
simps_3: shows "interpolation_poly Neville_Aitken = neville_aitken_interpolation_poly"
init_tableau_id: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<T> (init t) = t"
inject_2: fixes x2 :: "char list"   and y2 :: "char list" shows "(B x2 = B y2) = (x2 = y2)"
le': shows "d_OUT (h_plus i) x < d_OUT g x"
assoc: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "times_prod' (times_prod' a b) c = times_prod' a (times_prod' b c)"
assms: shows "x \<in>\<^sub>\<circ> fpairs r"
fst_conv: fixes x1 :: "'d"   and x2 :: "'e" shows "fst (x1, x2) = x1"
vblcase_nextX: fixes a :: "'a"   and b :: "vbl \<Rightarrow> 'a"   and n :: "vbl" shows "(case nextX n of zeroX \<Rightarrow> a | nextX x \<Rightarrow> b x) = b n"
shiftl_shiftr1: fixes c :: "nat"   and b :: "nat"   and a :: "'a word" assumes "c \<le> b" shows "a << b >> c = a && mask (size a - b) << b - c"
Y_arr_is_transformation: fixes f :: "'c" assumes "C.arr f" shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y (C.dom f)) (Y (C.cod f)) (Y f)"
t: shows "s\<^sup>2 + 1 = (4 * k + 1) * t"
sub: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> Translations" shows "a \<in> G \<rightarrow>\<^sub>E G"
intro: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open"   and "class.second_countable_topology_axioms open" shows "class.second_countable_topology open"
hisPrev_same: fixes n'' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat"   and nid' :: "'nid"   and nid'' :: "'nid" assumes "\<exists>n'<n''. \<exists>nid'. (n', nid') \<in> his t n nid"   and "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid"   and "(n', nid') \<in> his t n nid"   and "(n'', nid'') \<in> his t n nid"   and "hisPred t n nid n' = hisPred t n nid n''" shows "n' = n''"
create_factor_tree_def: fixes xs :: "int poly list" shows "create_factor_tree xs = (let ys = map (\<lambda>f. (f, degree f)) xs in Let (rev (sort_key snd ys)) create_factor_tree_balanced)"
m'_geq: fixes i :: "nat" assumes "\<phi> 0 \<le> i" shows "m' i = t (\<phi> (i - \<phi> 0))"
inj_on_pairself: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "('c \<times> 'c) set" assumes "inj_on f S"   and "T \<subseteq> S \<times> S" shows "inj_on (pairself f) T"
discrete_Taylor: shows "((\<lambda>s. (1 - s) *\<^sub>R (h ^ 3 * p / 2) *\<^sub>R f'' (t + s * (h * p), x t + (s * (h * p)) *\<^sub>R f (t, x t)) $ (1, f (t, x t)) $ (1, f (t, x t))) has_integral discrete_evolution (rk2_increment p (\<lambda>t x. f (t, x))) (t + h) t (x t) - x t - h *\<^sub>R f (t, x t) - (h\<^sup>2 / 2) *\<^sub>R f' (t, x t) $ (1, f (t, x t))) {0..1}"
has_subprob_density_def: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "has_subprob_density M N f \<equiv> has_density M N f \<and> subprob_space M"
loose_quotient_web: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. weight \<Gamma> x \<noteq> \<top>"   and "current \<Gamma> f"   and "wave \<Gamma> f"   and "\<And>w. \<lbrakk>current \<Gamma> w; wave \<Gamma> w; f \<le> w\<rbrakk> \<Longrightarrow> f = w" shows "loose (quotient_web \<Gamma> f)"
wf_constr_bvars_disj'_1: fixes S :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S" shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
fresh_subst_fmla_if: fixes j :: "atom"   and A :: "fmla"   and i :: "name"   and x :: "trm" shows "j \<sharp> A(i::=x) = (atom i \<sharp> A \<and> j \<sharp> A \<or> j \<sharp> x \<and> (j \<sharp> A \<or> j = atom i))"
support: shows "set_pmf p' \<subseteq> set_pmf p"
circline_set_unit_circle: shows "circline_set unit_circle = of_complex ` {z. cmod z = 1}"
Qg_top: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set" assumes "Group G"   and "G \<triangleright> N" shows "c_top G N \<in> set_rcs G N \<rightarrow> set_rcs G N \<rightarrow> set_rcs G N"
monadic_WHILEIT_unfold: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool nres"   and f :: "'a \<Rightarrow> 'a nres"   and s :: "'a" shows "monadic_WHILEIT I b f s = ASSERT (I s) \<bind> (\<lambda>_. b s \<bind> (\<lambda>bb. if bb then f s \<bind> monadic_WHILEIT I b f else RETURN s))"
one_time_pad_units: fixes f :: "nat \<Rightarrow> nat"   and q :: "nat" assumes "inj_on f ({..<q} - {0})"   and "f ` ({..<q} - {0}) = {..<q} - {0}" shows "map_spmf f (sample_uniform_units q) = sample_uniform_units q"
inv_gorder_inv: fixes L :: "'a gorder" shows "inv_gorder (inv_gorder L) = L"
read_only_reads_read_only: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "sharing_consistent \<S> \<O> sb" shows "read_only_reads \<O> sb \<subseteq> \<O> \<union> all_acquired sb \<union> read_only \<S>"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "\<phi>.B.arr f"   and "\<phi>.B.cod f = b" shows "b \<cdot>\<^sub>B f = f"
arith_extra_simps_5: fixes a :: "'a" shows "(0::'a) + a = a"
inext_mono2_fin: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "finite I"   and "n \<noteq> Max I" shows "n < inext n I"
Collect_conj_eq: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
quad_form_hermitean_real: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" assumes "hermitean H" shows "is_real (quad_form z H)"
matroid_axioms: shows "matroid (insert x X) (indep_in (insert x X))"
step_12: shows "InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
dom_kildall_entry: shows "dom_kildall start ! 0 = []"
One_not_eq_j1: shows "quiver.One \<noteq> parallel_pair.j1"
set_attribute_locs_impl: shows "set_attribute_locs = a_set_attribute_locs"
S_intersection_coeffs_distinct: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_abs p \<in> hyp2" shows "S_intersection_coeff1 p q \<noteq> S_intersection_coeff2 p q"
diff_add: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "b - a + a = b"
rref_condition2: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'d)) = (1::'d)"
nth_list_update: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
pdevs_val_eqI: fixes y :: "'a pdevs"   and x :: "'a pdevs"   and e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "\<And>i. \<lbrakk>i \<in> pdevs_domain y; i \<in> pdevs_domain x\<rbrakk> \<Longrightarrow> e i *\<^sub>R pdevs_apply x i = f i *\<^sub>R pdevs_apply y i"   and "\<And>i. \<lbrakk>i \<in> pdevs_domain y; i \<notin> pdevs_domain x\<rbrakk> \<Longrightarrow> f i *\<^sub>R pdevs_apply y i = (0::'a)"   and "\<And>i. \<lbrakk>i \<in> pdevs_domain x; i \<notin> pdevs_domain y\<rbrakk> \<Longrightarrow> e i *\<^sub>R pdevs_apply x i = (0::'a)" shows "pdevs_val e x = pdevs_val f y"
Ext_foo_stat_override: fixes sig :: "sig"   and old :: "qtname \<times> methd" assumes "tprg \<turnstile> qmdecl sig (Ext, snd Ext_foo) overrides\<^sub>S qmdecl sig old" shows "old = (Base, snd Base_foo)"
wf_trms_subst_compose_Var_range_1: fixes \<theta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "subst_range \<delta> \<subseteq> range Var" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (\<delta> \<circ>\<^sub>s \<theta>))"
elims_3: fixes x :: "sat_plan_variable" assumes "\<not> valid_state_var x" obtains t :: "nat" and k :: "nat" where "x = Operator t__ k__" and "\<not> (t__ < Suc h \<and> k__ < Suc (length ast\<delta>))"   | t :: "nat" and k :: "nat" where "x = State t k" and "\<not> t < Suc h"
rep_eq: fixes x :: "literal" shows "asciis_of_literal x = map of_char (explode x)"
wp: shows "P \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"
XX_def: shows "XX \<equiv> \<lambda>j. BB j j ` \<K> j j"
geodesic_segment_dist_unique: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "'b" assumes "geodesic_segment_between G x y"   and "a \<in> G"   and "b \<in> G"   and "dist x a = dist x b" shows "a = b"
widen_trans: fixes P :: "'c prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "P \<turnstile> S \<le> U"   and "P \<turnstile> U \<le> T" shows "P \<turnstile> S \<le> T"
monotoneI: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
is_faithful: fixes f :: "'d"   and f' :: "'d" assumes "D.par f f'"   and "D.cod \<iota>\<^sub>D \<otimes>\<^sub>D f = D.cod \<iota>\<^sub>D \<otimes>\<^sub>D f'" shows "f = f'"
Bag_ns_mul_ext: fixes xs :: "('b, nat) alist"   and ys :: "('b, nat) alist"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" shows "((Bag xs, Bag ys) \<in> ns_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = snd (mul_ext f (ass_list_to_single_list (alist.impl_of xs)) (ass_list_to_single_list (alist.impl_of ys)))"
le_d2: fixes k :: "nat" shows "AE x in lebesgue_on {a..b}. norm ((if x \<le> a + 1 / real (Suc k) then real (Suc k) * d * (a + 1 / real (Suc k) - x) else 0)\<^sup>2) \<le> d\<^sup>2"
vsv_vdomain_vempty_vrange_vempty: fixes r :: "V" assumes "vsv r"   and "\<D>\<^sub>\<circ> r \<noteq> 0" shows "\<R>\<^sub>\<circ> r \<noteq> 0"
atms_nonempty_progress: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "atms r \<noteq> {}" shows "(\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) ` atms r = (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) ` regex.atms r"
add_comm: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2" shows "ell_field.add R a p\<^sub>1 p\<^sub>2 = ell_field.add R a p\<^sub>2 p\<^sub>1"
split_option_ex: fixes P :: "'a option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (pure x)))"
rank_eq_state_rank: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "rank x n = state_rank (token_run x n) n"
int_intv_frac_gt0: fixes a :: "int"   and b :: "int" assumes "a < b"   and "b < a + 1" shows "0 < frac (real_of_int b)"
that_2: fixes x :: "'a"   and ys :: "'a fset" assumes "x |\<notin>| ys"   and "xs = finsert x ys" shows "thesis"
mul_add_mul_eq_imp_mul: fixes M :: "'a multiset"   and N :: "'a multiset"   and r :: "('a \<times> 'a) set"   and P :: "'a multiset"   and Q :: "'a multiset" assumes "(M, N) \<in> mul r"   and "(P, Q) \<in> mul_eq r" shows "(M + P, N + Q) \<in> mul r"
NI_indirect_sources_total_def: shows "int.NI_indirect_sources_total \<equiv> \<forall>execs a n. int.realistic_executions execs \<longrightarrow> (let s_f = int.run_total n s0 execs in output_f (int.run_total n s0 (int.ipurge_l execs (current s_f))) a = output_f (int.run_total n s0 (int.ipurge_r execs (current s_f))) a)"
vector_component_5: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(x - y) $ i = x $ i - y $ i"
finite_subcls1: fixes G :: "prog" shows "finite (subcls1 G)"
eqvt_lambda: fixes p :: "perm"   and f :: "'a \<Rightarrow> 'b" shows "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> f (unpermute p x)"
cloT: shows "closedin (top_of_set (S \<union> T)) T"
nth_list_update': fixes l :: "'a list"   and i :: "nat"   and x :: "'a"   and j :: "nat" shows "l[i := x] ! j = (if i = j \<and> i < length l then x else l ! j)"
Map_cod: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Map (cod f) = []"
swap_def: fixes i :: "'a"   and j :: "'a"   and x :: "'a" shows "swap i j x = (if x = i then j else if x = j then i else x)"
a': shows "( \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>, n) \<in> derivable (ext R R2 M1 M2)"
prefix_snocD: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "prefix (xs @ [x]) ys" shows "strict_prefix xs ys"
distr_singleton: fixes i :: "'i" shows "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i"
distinct_20: fixes ccs1' :: "ccs"   and ccs2' :: "ccs"   and ccs1 :: "ccs"   and ccs2 :: "ccs" shows "ccs1' \<parallel> ccs2' \<noteq> ccs1 \<oplus> ccs2"
chain_def: fixes Y :: "nat \<Rightarrow> 'b" shows "chain Y = (\<forall>i. Y i \<sqsubseteq> Y (Suc i))"
ir_commit_finite: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat" assumes "finite (dom runz)" shows "finite (ir_commit runz A B Kab Ts Ta)"
insertAssertionEqvt: fixes p :: "(name \<times> name) list"   and F :: "'b frame"   and \<Psi> :: "'b" shows "p \<bullet> insertAssertion F \<Psi> = insertAssertion (p \<bullet> F) (p \<bullet> \<Psi>)"
simps_2: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list"   and r :: "nat option" shows "correct ((f, xs, ls) # rest, r) = (if eval f xs \<down> then reachable ((f, xs, ls) # rest, r) (rest, eval f xs) else nonterminating ((f, xs, ls) # rest, None))"
abs_fun_eq: fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
invar_update: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and a :: "'c"   and b :: "'d" assumes "invar t" shows "invar (update a b t)"
Cons_eq_filter_iff: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" shows "(x # xs = filter P ys) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
elims: fixes x :: "'b tree"   and y :: "'b tree" assumes "pass\<^sub>1 x = y" obtains hsx :: "'b tree" and xa :: "'b" and hsy :: "'b tree" and ya :: "'b" and hs :: "'b tree" where "x = \<langle>hsx, xa, \<langle>hsy, ya, hs\<rangle>\<rangle>" and "y = link \<langle>hsx, xa, \<langle>hsy, ya, pass\<^sub>1 hs\<rangle>\<rangle>"   | "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | v :: "'b tree" and va :: "'b" where "x = \<langle>v, va, \<langle>\<rangle>\<rangle>" and "y = \<langle>v, va, \<langle>\<rangle>\<rangle>"
inj_dual_iff: fixes x :: "'b"   and y :: "'b" shows "(- x = - y) = (x = y)"
prob_compl: fixes M :: "'a measure"   and A :: "'a set" assumes "prob_space M"   and "A \<in> sets M" shows "Sigma_Algebra.measure M (space M - A) = 1 - Sigma_Algebra.measure M A"
a: shows "\<not> fic (M[x\<turnstile>n>y]) c"
npref_rev_suf_iff: fixes s :: "'b list"   and w :: "'b list" shows "(s \<le>np w) = rev s \<le>ns rev w"
valid_cfgI: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "s \<in> S"   and "cfg \<in> MDP.cfg_on s" shows "cfg \<in> MDP.valid_cfg"
neg_in_map_literalD: fixes A :: "'b"   and f :: "'c \<Rightarrow> 'b"   and D :: "'c literal set" assumes "Neg A \<in> map_literal f ` D" shows "\<exists>A0. Neg A0 \<in> D \<and> A = f A0"
bex_triv_one_point2: fixes A :: "'a set"   and a :: "'a" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
hyp_BinOp: fixes e1 :: "expr"   and e2 :: "expr"   and binop :: "binop" assumes "P e1"   and "P e2" shows "P (BinOp binop e1 e2)"
rbt_fold1_keys_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "rbt_fold1_keys f t = fold f (tl (RBT_Impl.keys t)) (hd (RBT_Impl.keys t))"
sub3_closed: fixes k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "local.sub3 k l (r, A) = (r', A')"   and "A \<in> carrier_mat m n" shows "A' \<in> carrier_mat m n"
dot_cross_det: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "x \<bullet> y \<times> z = det (vector [x, y, z])"
skl1_obs_iagreement_Init: shows "oreach skl1 \<subseteq> l2_iagreement_Init"
assms: shows "P \<subseteq> A\<^sup>\<infinity>"
col_space_eq': fixes A :: "(('b, 'd) vec, 'c) vec" shows "col_space A = range ((*v) A)"
InvariantVarsF_def: fixes F :: "Literal list list"   and F0 :: "Literal list list"   and Vbl :: "nat set" shows "InvariantVarsF F F0 Vbl \<equiv> vars F \<subseteq> vars F0 \<union> Vbl"
len_gt_0: fixes len_of :: "'a itself \<Rightarrow> nat" assumes "class.len len_of" shows "0 < len_of TYPE('a)"
that: fixes C' :: "String.literal" assumes "class_type_of' T' = \<lfloor>C'\<rfloor>"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "thesis"
nat_mirror_add: fixes n :: "nat"   and k :: "nat"   and l :: "nat"   and r :: "nat" shows "nat_mirror (n + k) l r = nat_mirror n l r - k"
simps_6: fixes \<sigma> :: "'a basic_type" shows "Integer \<squnion> \<sigma> = (if Integer \<le> \<sigma> then \<sigma> else if \<sigma> \<le> Integer then Integer else OclAny)"
WCFG_SeqSecond: fixes c\<^sub>2 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and c\<^sub>1 :: "cmd" assumes "c\<^sub>2 \<turnstile> n -et\<rightarrow> n'"   and "n \<noteq> (_Entry_)" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> n \<oplus> #:c\<^sub>1 -et\<rightarrow> n' \<oplus> #:c\<^sub>1"
int_pow_eq: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and m :: "int"   and n :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> m = x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) = (int (GE.ord x) dvd n - m)"
interval_integral_to_infinity_eq: fixes M :: "real measure"   and a :: "real"   and f :: "real \<Rightarrow> 'b" shows "interval_lebesgue_integral M (ereal a) \<infinity> f = set_lebesgue_integral M {a<..} f"
foldl_spmf_Cons': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and a :: "'c"   and x :: "'d"   and xs :: "'d list" shows "foldl_spmf f (return_spmf a) (x # xs) = f a x \<bind> (\<lambda>a'. foldl_spmf f (return_spmf a') xs)"
a1_map'_def: shows "a1_map' = (if pos then Mapping.filter (\<lambda>as _. as \<in> rel1) (upd_set a1_map (\<lambda>_. tp) (rel1 - Mapping.keys a1_map)) else upd_set a1_map (\<lambda>_. tp) rel1)"
distinct_21: fixes x31 :: "nat"   and x32 :: "nat"   and x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "recf.Id x31 x32 \<noteq> Pr x51 x52 x53"
IMPLEFT_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "x:M \<in> IMPLEFT (B IMP C) (\<parallel><B>\<parallel>) (\<parallel>(C)\<parallel>)" obtains x' :: "name" and a' :: "coname" and M' :: "trm" and N' :: "trm" where "M = ImpL <a'>.M' x'.N' x" and "fin (ImpL <a'>.M' x'.N' x) x" and "<a'>:M' \<in> \<parallel><B>\<parallel>" and "x':N' \<in> \<parallel>(C)\<parallel>"
fixed_points_0inf': fixes M :: "moebius" assumes "moebius_pt M 0\<^sub>h = 0\<^sub>h"   and "moebius_pt M \<infinity>\<^sub>h = \<infinity>\<^sub>h" shows "\<exists>k. (k \<noteq> 0\<^sub>h \<and> k \<noteq> \<infinity>\<^sub>h) \<and> (\<forall>z. moebius_pt M z = k *\<^sub>h z)"
that: fixes g' :: "'c \<Rightarrow> 'a" assumes "continuous_on U g'"   and "g' ` U \<subseteq> C"   and "\<And>y. y \<in> U \<Longrightarrow> p (g' y) = f' y" shows "thesis"
ENFnrSR_snds_weakrefl_instance: fixes P :: "'a \<Rightarrow> 'v \<Rightarrow> 'a \<Rightarrow> 'v \<Rightarrow> bool"   and default_node_properties :: "'a"   and nP :: "'v \<Rightarrow> 'a"   and f :: "('v \<times> 'v) set"   and G :: "'v graph"   and i :: "'v" assumes "sinvar_all_edges_normal_form_not_refl_SR P"   and "\<forall>s r. P default_node_properties s default_node_properties r"   and "\<forall>s r. nP s \<noteq> default_node_properties \<longrightarrow> \<not> P (nP s) s (nP r) r \<longrightarrow> \<not> P (nP s) s default_node_properties r"   and "\<forall>s r. P default_node_properties s (nP r) r"   and "f \<in> set_offending_flows G nP"   and "i \<in> snd ` f" shows "\<not> sinvar G (nP(i := default_node_properties))"
inf_distribI': fixes f :: "'g \<Rightarrow> 'h" assumes "\<And>C. f (Sup C) = Sup (f ` C)" shows "inf_distrib f"
nrename_AndL1_aux: fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL1 v.N u" shows "x = u \<and> x = y \<or> x \<noteq> u"
zero: fixes p' :: "'a multiset" assumes "p' \<in># mset_set (mset_set ` P)"   and "p' \<noteq> p" shows "count p' x = 0"
W_endomorphism: fixes w :: "'a permutation" assumes "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>" shows "ChamberComplexEndomorphism X ((\<rightarrow>) w)"