hcju/isabelleps huggingface dataset:
divide_nonneg_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less zero y" shows "less_eq zero (divide x y)"
f1: fixes x :: "atom" assumes "x \<in> ba xs" shows "x \<sharp> f xs T"
abc_ac_neq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "a \<noteq> c"
insert_remove: fixes A :: "'d set"   and x :: "'d" assumes "finite A" shows "Max (insert x A) = (if A - {x} = {} then x else max x (Max (A - {x})))"
vcomp_ide_cod: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B \<tau> G = \<tau>"
coinduct: fixes f :: "'a \<Rightarrow> 'a"   and X :: "'a" assumes "mono f"   and "X \<le> f (sup X (gfp f))" shows "X \<le> gfp f"
Liminf_upto_llist_0: fixes Xs :: "'a set llist" shows "Liminf_upto_llist Xs 0 = (if lnull Xs then {} else lhd Xs)"
simps: fixes N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "Q_of_state (N, P, Q) = Q"
simps_8: fixes \<nu> :: "'a ltln"   and \<nu>' :: "'a ltln" shows "pure_universal (\<nu> M\<^sub>n \<nu>') = (pure_universal \<nu> \<and> pure_universal \<nu>')"
le_m1_iff_lt: fixes x :: "'b word"   and y :: "'b word" shows "(0 < x) = ((y \<le> x - 1) = (y < x))"
eo: shows "1 - (signs ! a)\<^sup>2 = 1 \<or> 1 - (signs ! a)\<^sup>2 = 0"
gen_cond_def: fixes gds :: "('a, 'b) gen_dfs_struct"   and V0 :: "'a set"   and s :: "'b" shows "gen_dfs_defs.gen_cond gds V0 s \<equiv> (V0 \<subseteq> {v. gds_is_discovered gds v s} \<longrightarrow> \<not> gds_is_empty_stack gds s) \<and> \<not> gds_is_break gds s"
normalize_mult_unit_factor: fixes a :: "'a" shows "normalize a * unit_factor a = a"
finite_PUpdate: fixes S :: "(('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) set" assumes "finite S" shows "finite (Expr.PUpdate S)"
recvmsg_simps_8: fixes P :: "'a \<Rightarrow> bool" shows "recvmsg P \<tau>\<^sub>s = True"
degree_neville_aitken_main: fixes i :: "nat"   and j :: "nat" shows "degree (local.neville_aitken_main i j) \<le> j - i"
fds_perzeta_int: fixes q :: "real" assumes "q \<in> \<int>" shows "fds_perzeta q = fds_zeta"
antisym: fixes b :: "'c list"   and a :: "'c list" assumes "suffix b a"   and "suffix a b" shows "a = b"
reindex_bij_betw: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and h :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "bij_betw h S T" shows "abel_semigroup_set.F f z (\<lambda>x. g (h x)) S = abel_semigroup_set.F f z g T"
eventuallyI: fixes P :: "'m \<Rightarrow> bool"   and F :: "'m filter" assumes "\<And>x. P x" shows "eventually P F"
range: fixes A :: "'a set" shows "MaxR_opt A \<in> Some ` (A \<inter> Field r) \<union> {None}"
quo_zero: fixes x :: "nat" shows "quo [x, 0] = 0"
prems_2: fixes m :: "'a match_expr"   and m' :: "'a match_expr" assumes "normalized_nnf_match m"   and "f m = Some m'" shows "normalized_nnf_match m'"
evaluate_clock_mono_2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list"   and s' :: "'b state"   and r2 :: "(v list, v) result" assumes "evaluate_list ck env s es (s', r2)" shows "clock s' \<le> clock s"
add_right_imp_eq: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b + a = c + a" shows "b = c"
deriv_c2: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>2 UNIV,UNIV"
pw_ERETURN_3: fixes x :: "'d"   and y :: "'d" shows "inres (ERETURN x) (Inr y) = (x = y)"
simps_2: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e"   and xs :: "'e list" shows "filter P (Cons x xs) = (if P x then Cons x (filter P xs) else filter P xs)"
opt_bst2_dom: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "Optimal_BST2 w" shows "\<forall>args. Wellfounded.accp (Optimal_BST2.opt_bst2_rel w) args"
that: fixes p :: "'a set set" assumes "insert (cbox a b) p division_of cbox a b \<union> cbox c d" shows "thesis"
wf_mstate_minit: fixes \<phi> :: "'b MFOTL.formula"   and R :: "'b list set" assumes "safe_formula \<phi>" shows "wf_mstate \<phi> pnil R (minit \<phi>)"
1: shows "[a] @- v\<^sub>2 =\<^sub>I l @- u"
n_less_equal_power_2: fixes n :: "nat" shows "n < 2 ^ n"
not_diff_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not (minus a b) = plus (not a) b"
prod_wb_lens: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'e" assumes "wb_lens X"   and "wb_lens Y" shows "wb_lens (X \<times>\<^sub>L Y)"
sum_ideals_la1: fixes R :: "('c, 'd) Ring_scheme"   and I1 :: "'c set"   and I2 :: "'c set" assumes "Ring R"   and "ideal R I1"   and "ideal R I2" shows "I1 \<subseteq> I1 \<minusplus>\<^bsub>R\<^esub> I2"
add_commute: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b = b \<sqinter> a"
add_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(b = plus b a) = (a = zero)"
attach_parallel2: fixes conv1 :: "('y, 'aa, 'ac, 'ae) converter"   and conv2 :: "('z, 'ab, 'ad, 'af) converter"   and res1 :: "('ac, 'ae) resource"   and res2 :: "('ad, 'af) resource" shows "conv1 |\<^sub>= conv2 \<rhd> res1 \<parallel> res2 = (conv1 \<rhd> res1) \<parallel> (conv2 \<rhd> res2)"
solve_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "solve A b = (if consistent A b then Some (solve_consistent_rref (fst (solve_system A b)) (snd (solve_system A b)), basis_null_space A) else None)"
FWdelay_bisimulation_final_base_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_final_base final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_final_base final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) \<tau>move2 \<tau>move1"
eq: shows "A * f + B * g = A' * f + B' * g"
the_elem_hom: shows "(\<lambda>x. the_elem (f ` x)) \<in> ring_hom (R Quot I) (image_ring f R)"
finite_succeed_t': fixes i :: "nat" assumes "q\<^sub>0 \<notin> F" shows "finite (succeed i) = finite (succeed_t i)"
closure_linear_image_subset: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f" shows "f ` closure S \<subseteq> closure (f ` S)"
exponential_order_integral: fixes a :: "real"   and f :: "real \<Rightarrow> 'a"   and I :: "real \<Rightarrow> 'a"   and M :: "real"   and c :: "real" assumes "\<And>t. a \<le> t \<Longrightarrow> (f has_integral I t) {a..t}"   and "exponential_order M c f"   and "0 < c" obtains M' :: "real" where "exponential_order M' c I"
SubstAtomicP_cong: fixes H :: "fm set"   and v :: "tm"   and v' :: "tm"   and tm :: "tm"   and tm' :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" assumes "H \<turnstile> v EQ v'"   and "H \<turnstile> tm EQ tm'"   and "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> SubstAtomicP v tm x y IFF SubstAtomicP v' tm' x' y'"
dvd_imp_mod_0: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "b mod a = (0::'a)"
ordertype_eq_ordertype: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('b \<times> 'b) set"   and B :: "'b set" assumes "wf r"   and "total_on A r"   and "trans r"   and "small A"   and "wf s"   and "total_on B s"   and "trans s"   and "small B" shows "(ordertype A r = ordertype B s) = (\<exists>f. bij_betw f A B \<and> (\<forall>x\<in>A. \<forall>y\<in>A. ((f x, f y) \<in> s) = ((x, y) \<in> r)))"
input_sizes_deep_model': fixes rs :: "nat list" assumes "1 \<le> length rs" shows "input_sizes (deep_model'_l rs) = replicate (2 ^ (length rs - 1)) (last rs)"
inv_solve_right: fixes a :: "'a vec"   and b :: "'a vec"   and c :: "'a vec" assumes "a \<in> carrier_vec n"   and "b \<in> carrier_vec n"   and "c \<in> carrier_vec n" shows "(a = b + \<ominus>\<^bsub>V\<^esub> c) = (b = a + c)"
preserves_cod: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah"   and f :: "'ag" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.cod B (\<tau> f) = G (partial_magma.cod A f)"
less_Sup_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and a :: "'a"   and S :: "'a set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup S) = (\<exists>x\<in>S. less a x)"
thm_relation_negation_1_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and x :: "\<nu>"   and y :: "\<nu>" shows "[\<lparr>F\<^sup>-,x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F,x\<^sup>P,y\<^sup>P\<rparr> in v]"
local_lipschitz_def: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "local_lipschitz T X f \<equiv> \<forall>x\<in>X. \<forall>t\<in>T. \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
assms_4: shows "(\<Sum>x\<in>X. of_rat (\<beta> x) * exp x) = 0"
equivalence_in_bicategory_axioms: shows "internal_equivalence f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (local.inv \<phi> \<star> g))"
transp_on_emb: fixes P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool" assumes "transp_on P F" shows "transp_on (emb P) trees"
zeroprimeideal: shows "primeideal {\<zero>\<^bsub>poly_ring R\<^esub>} (poly_ring R)"
Gauss_Jordan_in_ij_PA_def: fixes A' :: "(('d, 'e) vec, 'e) vec \<times> (('d, 'f) vec, 'e) vec"   and i :: "'e"   and j :: "'f" shows "Gauss_Jordan_in_ij_PA A' i j = (let P = fst A'; A = snd A'; n = LEAST n. A $ n $ j \<noteq> (0::'d) \<and> i \<le> n; interchange_A = interchange_rows A i n; interchange_P = interchange_rows P i n; P' = mult_row interchange_P i ((1::'d) / interchange_A $ i $ j) in (\<chi>s. if s = i then P' $ s else row_add P' s i (- interchange_A $ s $ j) $ s, Gauss_Jordan_in_ij A i j))"
fold_graph_fold: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and B :: "'e set"   and z :: "'e" assumes "comp_fun_commute_on f A"   and "finite B"   and "B \<subseteq> A"   and "z \<in> A" shows "fold_graph f z B (Finite_Set.fold f z B)"
abs_triangle_half_r: fixes y :: "'c"   and x1 :: "'c"   and e :: "'c"   and x2 :: "'c" assumes "\<bar>y - x1\<bar> < e / (2::'c)"   and "\<bar>y - x2\<bar> < e / (2::'c)" shows "\<bar>x1 - x2\<bar> < e"
segment_step_2: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> segment f a b"   and "b \<noteq> a"   and "inj f" shows "x \<in> segment f a (f b)"
assms: shows "match (t\<^sub>1 $ t\<^sub>2) u = Some env"
BOresChainSupp: fixes xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "supp (\<lparr>\<nu>*xvec\<rparr>B) = supp B - supp xvec"
closed_buffon_set': fixes l :: "real"   and d :: "real" shows "closed (buffon_set' l d)"
bounded_linear_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'd" shows "bounded_linear (blinfun_apply a)"
antecedent: shows "\<exists>z. PP z x \<or> PP z y"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 (src f) (src f)"
card_of_Func_UNIV: fixes B :: "'g set" shows "|Func UNIV B| =o |{f. range f \<subseteq> B}|"
B: shows "(\<lambda>p. Busemann_function_at xi ((f ^^ p) basepoint) basepoint / real p - (2 * deltaG TYPE('a) + 1) * (1 / real p)) \<longlonglongrightarrow> additive_strength f xi - (2 * deltaG TYPE('a) + 1) * 0"
otherI: fixes I :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> \<sigma>' i = \<sigma> i"   and "\<And>j. j \<notin> I \<Longrightarrow> P (\<sigma> j) (\<sigma>' j)" shows "other P I \<sigma> \<sigma>'"
non_volatile_owned_or_read_only_outstanding_non_volatile_writes: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<subseteq> \<O> \<union> all_acquired sb"
mcont_bind_spmf: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd spmf"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'e spmf" assumes "mcont luba orda lub_spmf (ord_spmf (=)) f"   and "\<And>y. mcont luba orda lub_spmf (ord_spmf (=)) (g y)" shows "mcont luba orda lub_spmf (ord_spmf (=)) (\<lambda>x. f x \<bind> (\<lambda>y. g y x))"
minus_of_int: fixes x :: "complex"   and n :: "int" shows "g (x - of_int n) = g x"
module_hom_ident: shows "linear (*s) (*s) (\<lambda>x. x)"
simps_1: fixes c :: "int \<times> int" shows "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
periodic_orbitI: fixes t :: "real"   and t' :: "real"   and y :: "'a" assumes "t \<noteq> t'"   and "t \<in> existence_ivl0 y"   and "t' \<in> existence_ivl0 y"   and "flow0 y t = flow0 y t'"   and "f y \<noteq> (0::'a)" shows "periodic_orbit y"
DisjunctiveD: fixes x :: "'e \<Rightarrow> 'f"   and X :: "'e set" assumes "x \<in> Apply.Disjunctive" shows "x (\<Squnion>X) = \<Squnion>(x ` X)"
stableco: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> stable A" shows "F \<in> A - B co A"
left_idem: fixes a :: "'d"   and b :: "'d" shows "a \<sqinter> (a \<sqinter> b) = a \<sqinter> b"
insertionNegative: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly" shows "insertion f p = - insertion f (- p)"
minBound_def: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "enum.minBound enum \<equiv> hd enum"
GEN_ALGO_tag_def: fixes P :: "'g" shows "GEN_ALGO_tag P \<equiv> P"
one_closed: shows "h \<one>\<^bsub>G\<^esub> \<in> carrier G"
preserves_arr: fixes f :: "'a" assumes "B.arr f" shows "B.arr (B.L f)"
build_first: fixes w :: "nat \<Rightarrow> 'b" shows "w 0 ## suffix (Suc 0) w = w"
lookup_lower: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and u :: "'a"   and v :: "'a" shows "lookup (punit.lower p u) v = (if v \<prec> u then lookup p v else (0::'c))"
line_incident_point_not_in_S: fixes l :: "proj2_line" shows "\<exists>p. p \<notin> S \<and> proj2_incident p l"
sum_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "\<Sum> (f ` A) = sum f A"
assms_1: shows "lideal I B addB multB zeroB oneB"
not_subseq_length: fixes ys :: "'a list"   and xs :: "'a list" assumes "length ys < length xs" shows "\<not> subseq xs ys"
prod_dfs_mono: fixes z :: "nat option list list \<times> (nat \<times> nat) list"   and xs :: "(nat \<times> nat) list"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "prod_invariant A B z"   and "list_all (prod_is_node A B) xs"   and "fst z ! i ! j = Some k" shows "fst (dfs z xs) ! i ! j = Some k"
compress_def: fixes x :: "wall"   and y :: "wall" shows "compress x y = (compress_top x y \<or> compress_bottom x y)"
all_info_hyp: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set (snd (snd (reduce_system p (qs, M_mat signs subsets, subsets, signs))))"
elementary_cartesian_monoidal_category_axioms: shows "elementary_cartesian_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit runit assoc \<tau> \<delta>"
incs_filter: fixes x :: "nat"   and a :: "nat" assumes "x \<le> B" shows "incs a x = (\<lambda>(xs, s). filter (cond_cons C) (map (\<lambda>x. (x # xs, s + a * x)) [x..<B + 1]))"
distinct_lists_children: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "local.a_distinct_lists h"   and "known_ptr ptr"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
has_field_derivative_transform_within: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and a :: "'a"   and S :: "'a set"   and d :: "real"   and g :: "'a \<Rightarrow> 'a" assumes "(f has_field_derivative f') (at a within S)"   and "0 < d"   and "a \<in> S"   and "\<And>x. \<lbrakk>x \<in> S; dist x a < d\<rbrakk> \<Longrightarrow> f x = g x" shows "(g has_field_derivative f') (at a within S)"
mk_trace_def: fixes ioa :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set" shows "mk_trace ioa \<equiv> filter_oseq (\<lambda>a. a \<in> externals (asig_of ioa))"
IdNatTransNatTrans: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" assumes "Functor F" shows "NatTrans (IdNatTrans F)"
intyness_simps_3: fixes a :: "nat"   and n :: "nat" shows "real a ^ n = real (a ^ n)"
fps_of_poly_monom: fixes c :: "'b"   and n :: "nat" shows "fps_of_poly (monom c n) = fps_const c * fps_X ^ n"
lead_coeff_min_int_poly_pos: fixes x :: "'b" shows "0 < Polynomial.lead_coeff (min_int_poly x)"
Lxy_def: fixes xs :: "'c list"   and S :: "'c set" shows "Lxy xs S = filter (\<lambda>z. z \<in> S) xs"
no_prems_init_active: shows "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd Ns)"
ts_bit_stream_One: fixes x :: "nat \<Rightarrow> bit list"   and i :: "nat" assumes "ts x"   and "x i \<noteq> [Zero]" shows "x i = [One]"
wf_TVar: fixes \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmark.type" assumes "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f TVar i"
rel_interior_convex_Int_affine: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "affine T"   and "interior S \<inter> T \<noteq> {}" shows "rel_interior (S \<inter> T) = interior S \<inter> T"
intro: fixes M :: "'c \<Rightarrow> 'd measure" assumes "\<And>i. sigma_finite_measure (M i)" shows "product_sigma_finite M"
postrecording_event: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "computation channel trans send recv init final" shows "computation.postrecording_event channel trans send recv init t i \<equiv> i < length t \<and> regular_event (t ! i) \<and> ps (distributed_system.s channel trans send recv init t i) (occurs_on (t ! i)) \<noteq> None"
inner_prod_with_itself_Re_non0: fixes u :: "complex vec" assumes "u \<noteq> 0\<^sub>v (dim_vec u)" shows "0 < Re \<langle>u|u\<rangle>"
beta_C_meta_1: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and v :: "i"   and x :: "\<nu>" assumes "IsProperInX \<phi>" shows "[\<lparr>\<^bold>\<lambda>y. \<phi> (y\<^sup>P),x\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) in v]"
dg_prod_3_ObjI: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "c \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "[a, b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Obj\<rparr>"
LeftDerivation_implies_Derivation: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "LeftDerivation a D b" shows "Derivation a D b"
finite_lists_length_eq: fixes A :: "'g set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
NewInitOOM: fixes sh :: "sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap" assumes "\<nexists>sfs. sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v',(h', l', sh')\<rangle>"   and "new_Addr h' = None"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>THROW OutOfMemory,(h', l', sh')\<rangle>"
merge_inefficient_as_efficient: fixes as :: "'a list"   and bs :: "'a list"   and accs :: "'a list" shows "merge_tailrec_inefficient as bs (rev accs) = merge_tailrec as bs accs"
indep_remove: fixes x :: "'a" assumes "x \<in> X" shows "rank_of (X - {x}) = card (X - {x})"
ys_def: shows "ys = \<Oplus>Gs\<leftarrow> y"
verit_comp_simplify_2: fixes a :: "'y" shows "a \<le> a"
int_of_integer_inverse: fixes x :: "integer" shows "integer_of_int (int_of_integer x) = x"
elims_3: fixes x :: "'b hp" assumes "\<not> php x" obtains xa :: "'b" and hs :: "'b hp list" where "x = Hp xa hs" and "\<not> (\<forall>h\<in>set hs. Ball (set_hp h) ((\<le>) xa) \<and> php h)"
exec_redex_Stuck: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>redex c,s\<rangle> \<Rightarrow> Stuck" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck"
observable_set_stack_in_slice: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')" shows "\<forall>mx\<in>set (tl ms'). \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"