hcju/isabelleps huggingface dataset:
Inf_eqI: 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 set"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y i) \<Longrightarrow> less_eq y x" shows "Inf A = x"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'var option"   and x3 :: "('pname \<times> bool) option"   and x4 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'var option \<Rightarrow> ('pname \<times> bool) option \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "SDG_edge x1 x2 x3 x4"   and "\<And>n m n' m'. \<lbrakk>n = CFG_node m; n' = CFG_node m'; m controls m'\<rbrakk> \<Longrightarrow> P n None None n'"   and "\<And>a Q r p fs n a' n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; n = CFG_node (targetnode a); a' \<in> get_return_edges a; n' = CFG_node (sourcenode a')\<rbrakk> \<Longrightarrow> P n None None n'"   and "\<And>n' m n. \<lbrakk>valid_SDG_node n'; m = parent_node n'; n = CFG_node m; n \<noteq> n'\<rbrakk> \<Longrightarrow> P n None None n'"   and "\<And>n V n'. n influences V in n' \<Longrightarrow> P n (Some V) None n'"   and "\<And>a Q r p fs n n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; n = CFG_node (sourcenode a); n' = CFG_node (targetnode a)\<rbrakk> \<Longrightarrow> P n None (Some (p, True)) n'"   and "\<And>a Q p f n n'. \<lbrakk>valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; n = CFG_node (sourcenode a); n' = CFG_node (targetnode a)\<rbrakk> \<Longrightarrow> P n None (Some (p, False)) n'"   and "\<And>a Q r p fs ins outs V x n n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; (p, ins, outs) \<in> set procs; V = ins ! x; x < length ins; n = Actual_in (sourcenode a, x); n' = Formal_in (targetnode a, x)\<rbrakk> \<Longrightarrow> P n (Some V) (Some (p, True)) n'"   and "\<And>a Q p f ins outs V x n n'. \<lbrakk>valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; (p, ins, outs) \<in> set procs; V = outs ! x; x < length outs; n = Formal_out (sourcenode a, x); n' = Actual_out (targetnode a, x)\<rbrakk> \<Longrightarrow> P n (Some V) (Some (p, False)) n'" shows "P x1 x2 x3 x4"
simps_12: fixes f :: "'a"   and v :: "'a local.term"   and va :: "'a local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v \<^bold>\<otimes> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> v \<^bold>\<otimes> va"
prems: shows "upd_rev_cs cs (as' @ [a']) = []"
L2_set_right_distrib: fixes r :: "real"   and f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "0 \<le> r" shows "r * L2_set f A = L2_set (\<lambda>x. r * f x) A"
valid_path_base': fixes v :: "'a" assumes "v \<in> V" shows "valid_path (LCons v LNil)"
floor_unique: fixes z :: "int"   and x :: "'d" assumes "of_int z \<le> x"   and "x < of_int z + (1::'d)" shows "\<lfloor>x\<rfloor> = z"
SUP_subset_mono: 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 :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
del_def: fixes x :: "'a"   and t :: "'a tree" shows "del x t = Time_Monad.val (del_tm x t)"
power_2_mult_step_le: fixes n' :: "nat"   and n :: "nat"   and k' :: "nat"   and k :: "nat" assumes "n' \<le> n"   and "2 ^ n' * k' < 2 ^ n * k" shows "2 ^ n' * (k' + 1) \<le> 2 ^ n * k"
min_0_1_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq zero one = zero"
add_le_cancel_left: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" shows "(c + a \<subseteq># c + b) = (a \<subseteq># b)"
exact_seq_end_iff: fixes G :: "'b monoid"   and H :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and q :: "('b \<Rightarrow> 'b) list" shows "exact_seq ([G, H], g # q) = (q = [] \<and> group_hom H G g)"
Sum_any_left_distrib: fixes g :: "'a \<Rightarrow> 'b"   and r :: "'b" assumes "finite {a. g a \<noteq> (0::'b)}" shows "Sum_any g * r = (\<Sum>n. g n * r)"
fromIMor_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'b" shows "fromIMor h \<equiv> snd \<circ> h"
dickson_less_vI: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and v :: "'c"   and m :: "nat"   and u :: "'c" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "d (term_powerprod.pp_of_term pair_of_term v) \<le> m"   and "d (term_powerprod.pp_of_term pair_of_term u) \<le> m"   and "ord_term_strict v u" shows "gd_term.dickson_less_v pair_of_term ord_term_strict d m v u"
sums: fixes I :: "(nat \<Rightarrow>\<^sub>0 nat) set"   and a' :: "nat \<Rightarrow>\<^sub>0 nat"   and c :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'a" assumes "finite I" shows "(\<Sum>x\<in>I. MPoly_Type.monom (x + a') (c x)) = (\<Sum>x\<in>(\<lambda>x. x + a') ` I. MPoly_Type.monom x (c (x - a')))"
nxt_def: shows "nxt \<equiv> \<lambda>N n. N \<inter> {n<..}"
lub_spmf_def: shows "local.lub_spmf = embed_spmf (\<lambda>x. enn2real (\<Squnion>p\<in>Y. ennreal (spmf p x)))"
iIN_inter: fixes n :: "nat"   and n' :: "nat"   and d :: "nat"   and d' :: "nat" assumes "n \<le> n'"   and "n' \<le> n + d"   and "n + d \<le> n' + d'" shows "[n\<dots>,d] \<inter> [n'\<dots>,d'] = [n'\<dots>,n + d - n']"
simps_1: fixes f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term Unit f = Unit"
ps_not_Nil: shows "ps\<^sub>+ \<noteq> []"
abs_eq: fixes xa :: "nat"   and x :: "nat" shows "(to_Arity xa \<sqsubseteq> to_Arity x) = (x \<le> xa)"
in_arcs_imp_in_arcs_ends: fixes e :: "'b" assumes "e \<in> arcs H" shows "tail H e \<rightarrow>\<^bsub>H\<^esub> head H e"
enf_not_fulfilled_if_in_offending: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "wf_graph G"   and "\<forall>m\<in>set M. \<exists>P. \<forall>G. c_sinvar m G = (\<forall>e\<in>edges G. P e)" shows "\<forall>x\<in>\<Union>m\<in>set M. \<Union> (c_offending_flows m (fully_connected G)). \<not> all_security_requirements_fulfilled M \<lparr>nodes = V, edges = insert x E\<rparr>"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "C.CCC.dom f = (if C.arr (fst f) \<and> C.CC.arr (snd f) then (C.dom (fst f), C.CC.dom (snd f)) else (C.null, C.CC.null))"
Quote_def: fixes x :: "hf"   and x' :: "hf" shows "Quote x x' \<equiv> \<exists>s k. SeqQuote x x' s k"
nat_pow_eq_0_iff: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "(a [^] n = \<zero>) = (a = \<zero> \<and> n \<noteq> 0)"
WTrt_elim_cases_6: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e\<^sub>1 :: "'addr expr"   and C :: "String.literal"   and V :: "String.literal"   and e\<^sub>2 :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> try e\<^sub>1 catch(C V) e\<^sub>2 : T" obtains T1 :: "ty" where "P,E,h \<turnstile> e\<^sub>1 : T1" and "P,E(V \<mapsto> Class C),h \<turnstile> e\<^sub>2 : T" and "P \<turnstile> T1 \<le> T"
sub: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> G" shows "a \<in> Sym"
ideal_of_def: fixes A :: "('c \<Rightarrow> 'd) set" shows "\<I> A = {f. \<forall>a\<in>A. poly_eval a f = (0::'d)}"
blockmat_tensor_distrib: fixes a :: "brick list"   and b :: "brick list" shows "blockmat (a \<otimes> b) = blockmat a \<otimes> blockmat b"
normal_subgroup_singular_relboundary_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "singular_relboundary_set p X S \<lhd> relcycle_group p X S"
V: shows "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> \<subseteq> V"
known_ptr_not_character_data_ptr: fixes ptr :: "(_) object_ptr" assumes "\<not> is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"   and "CharacterDataClass.known_ptr ptr" shows "ElementClass.known_ptr ptr"
Aia: shows "A $ i $ a \<noteq> (0::'a)"
assms_3: shows "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>B c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>B f : a \<rightarrow>\<^sub>B c\<guillemotright>"
Cauchy_Schwarz_ineq: fixes x :: "'a"   and y :: "'a" shows "cinner x y * cinner y x \<le> cinner x x * cinner y y"
pairwise_alt: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S - {x}. R x y)"
chine_simps_2: shows "C.dom \<mu>.chine = \<mu>.dom.apex"
OAlist_tc_update_by_fun_gr_def: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('a, 'b) oalist_tc" shows "OAlist_tc_update_by_fun_gr k f xs = OAlist_tc (update_by_fun_gr_pair_tc k f (list_of_oalist_tc xs))"
divide_less_eq_numeral1_2: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / - numeral w < a) = (a * - numeral w < b)"
subset_divisors_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_mult times one" shows "({c. dvd.dvd times c a} \<subseteq> {c. dvd.dvd times c b}) = dvd.dvd times a b"
apply_affected: fixes f :: "'a perm"   and a :: "'a" shows "(f \<langle>$\<rangle> a \<in> affected f) = (a \<in> affected f)"
rep_eq: fixes x :: "'c" shows "Rep_formula (Pred x) = Pred\<^sub>\<alpha> x"
sup_top_left: fixes x :: "'a upred" shows "(true \<or> x) = true"
p\<^sub>0_simps: shows "trg p\<^sub>0 = src s\<^sub>1"
chine_hcomp_def: fixes \<nu> :: "('a, 'b) arrow_of_spans_data_scheme"   and \<mu> :: "('a, 'c) arrow_of_spans_data_scheme" shows "chine_hcomp \<nu> \<mu> \<equiv> \<langle>Chn \<nu> \<cdot> \<p>\<^sub>1[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)] \<lbrakk>Leg0 (Cod \<nu>), Leg1 (Cod \<mu>)\<rbrakk> Chn \<mu> \<cdot> \<p>\<^sub>0[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]\<rangle>"
zeta_neg_of_nat: shows "zeta (- of_nat n) = - complex_of_real (bernoulli' (Suc n)) / of_nat (Suc n)"
xmin_2: fixes x :: "nat" assumes "x < xmin" shows "\<not> (0 < x \<and> \<not> (s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)))"
splay_bstL: fixes t :: "'b tree"   and a :: "'b"   and l :: "'b tree"   and e :: "'b"   and r :: "'b tree"   and x :: "'b" assumes "bst t"   and "splay a t = \<langle>l, e, r\<rangle>"   and "x \<in> set_tree l" shows "x < a"
vars_llist_def: fixes xs :: "(char list list \<times> int) list" shows "vars_llist xs = \<Union> (set ` mononoms xs)"
SYSTEM_def: shows "SYSTEM \<equiv> (SEND \<lbrakk>SYN\<rbrakk> REC) \ SYN"
Bisimulation_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Bisimulation A B sim \<equiv> (\<forall>a b a'. A a b \<longrightarrow> sim a a' \<longrightarrow> (\<exists>b'. B a' b' \<and> sim b b')) \<and> (\<forall>a a' b'. B a' b' \<longrightarrow> sim a a' \<longrightarrow> (\<exists>b. A a b \<and> sim b b'))"
is_GB_map_key_PP: fixes G :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) set" assumes "finite G" shows "pp_pm.punit.is_Groebner_basis (Poly_Mapping.map_key PP ` G) = punit.is_Groebner_basis G"
zip_map1: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and ys :: "'b list" shows "zip (map f xs) ys = map2 (\<lambda>x. Pair (f x)) xs ys"
divide_eq_eq_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide b a = one) = (a \<noteq> zero \<and> a = b)"
eq_fmem_trans: fixes a :: "'c"   and b :: "'c"   and A :: "'c fset" assumes "a = b"   and "b |\<in>| A" shows "a |\<in>| A"
fact_div_fact_le_pow: fixes r :: "nat"   and n :: "nat" assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
vrrestriction_atD_1: fixes Aa :: "V"   and A :: "V"   and a :: "V"   and b :: "V" assumes "((r \<restriction>\<^sub>\<circ> Aa) \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> (r \<restriction>\<^sub>\<circ> Aa) -`\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> A"
prefixes_def: shows "prefixes \<equiv> rec_list [[]] (\<lambda>x xs xsa. [] # map ((#) x) xsa)"
horner_bounds_1: fixes x :: "float"   and f :: "nat \<Rightarrow> nat"   and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and F :: "nat \<Rightarrow> nat"   and s :: "nat"   and lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and prec :: "nat"   and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and n :: "nat"   and j' :: "nat" assumes "0 \<le> real_of_float x"   and "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"   and "\<And>i k x. lb 0 i k x = 0"   and "\<And>n i k x. lb (Suc n) i k x = float_plus_down prec (lapprox_rat prec 1 (int k)) (- float_round_up prec (x * ub n (F i) (G i k) x))"   and "\<And>i k x. ub 0 i k x = 0"   and "\<And>n i k x. ub (Suc n) i k x = float_plus_up prec (rapprox_rat prec 1 (int k)) (- float_round_down prec (x * lb n (F i) (G i k) x))" shows "real_of_float (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (f (j' + j))) * real_of_float (x ^ j))"
coord_eq: fixes X :: "(('a, 'b) vec, 'b) vec"   and v :: "('a, 'b) vec"   and w :: "('a, 'b) vec" assumes "is_basis (set_of_vector X)"   and "coord X v = coord X w" shows "v = w"
a: assumes "AE w in M. f w = g w" shows "AE w in N. f w = g w"
any_closest_point_unique: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b" assumes "convex S"   and "closed S"   and "x \<in> S"   and "y \<in> S"   and "\<forall>z\<in>S. dist a x \<le> dist a z"   and "\<forall>z\<in>S. dist a y \<le> dist a z" shows "x = y"
complex_inner_i_left: fixes x :: "complex" shows "\<i> \<bullet> x = Im x"
assms_4: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c"   and "x \<in> lvars (\<T> s)"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)" shows "P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update\<B>\<I> (UBI_upd dir) i x c s)"
termi_c2: shows "\<Gamma>\<turnstile>c2 \<down> s"
prop_def: fixes A :: "prop" shows "(PROP A) \<equiv> PROP A"
map_poly_Re_coeffs: shows "coeffs (map_poly Re p) = map Re (coeffs p)"
po_on_imp_qo_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "po_on P A" shows "qo_on P\<^sup>=\<^sup>= A"
simps_4: fixes uw :: "'a" shows "(Val uw \<le> Top) = True"
minus_zero: shows "\<ominus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub> = \<zero>\<^bsub>S\<^esub>"
nodes_def: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "nodes A \<equiv> {x. transition_system_initial.nodesp (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) (\<lambda>p. p = initial A) x}"
closedin_imp_subset: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "closedin (subtopology U S) T" shows "T \<subseteq> S"
assms_1: shows "validTrans (Trans s1 a1 ou1 s1')"
seqd: shows "matrix_seq d (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>))"
hyps: fixes ls :: "'a list" assumes "0 < |ls|" shows "ls ! ((n + m) mod |ls|) = rotate m ls ! (n mod |ls|)"
mdl_is_module: fixes H :: "'a set" assumes "submodule R M H" shows "R module mdl M H"
rreq_rrep_sn_simps_1: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "rreq_rrep_sn (Rreq hops dip dsn dsk oip osn sip) = (1 \<le> osn)"
nn_integral_count_space_eq: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = 0"   and "\<And>x. x \<in> B - A \<Longrightarrow> f x = 0" shows "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) f"
C5_a: fixes x :: "'a"   and y :: "'a" shows "-r x l\<rightarrow> -r y = y r\<rightarrow> x"
sup_absorb1: fixes y :: "'a multiset"   and x :: "'a multiset" assumes "y \<subseteq># x" shows "x \<union># y = x"
comp_null_2: fixes f :: "'s" shows "f \<cdot> S.null = S.null"
simps_1: fixes x :: "'d"   and t1 :: "'d ifex"   and t2 :: "'d ifex"   and X :: "'d set" shows "reduced (IF x t1 t2) X = (x \<notin> X \<and> t1 \<noteq> t2 \<and> reduced t1 (insert x X) \<and> reduced t2 (insert x X))"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a set" shows "lookup (except x xa) = except_fun (lookup x) xa"
delete1_def: shows "delete1 \<equiv> \<lambda>(l, r, buf). if (0::'b) < l then RETURN (l - (1::'b), r, buf) else RETURN (l, r, buf)"
disc_17: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "\<not> isSnapshot (Send x21 x22 x23 x24 x25 x26)"
simps_4: fixes uv :: "'b list"   and F :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "trms\<^sub>s\<^sub>t\<^sub>p \<forall>uv\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
ap_set_def: fixes F :: "('b \<Rightarrow> 'a) set"   and X :: "'b set" shows "F \<diamondop> X = {f x |f x. f \<in> F \<and> x \<in> X}"
psubset_imp_subset: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subset> B" shows "A \<subseteq> B"
Subst: fixes H :: "fm set"   and A :: "fm"   and i :: "name"   and x :: "tm" assumes "H \<turnstile> A"   and "\<forall>B\<in>H. atom i \<sharp> B" shows "H \<turnstile> A(i::=x)"
length_vec_to_iarray: fixes xa :: "('b, 'c) vec" shows "IArray.length (vec_to_iarray xa) = CARD('c)"
inversions_trivial: fixes xs :: "'a list" assumes "length xs \<le> Suc 0" shows "inversions xs = {}"
list_emb_code_3: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "list_emb P (x # xs) (y # ys) = (if P x y then list_emb P xs ys else list_emb P (x # xs) ys)"
set_unsat_bounds_id: fixes I :: "'b list"   and s :: "('b, 'a) state" shows "\<B> (set_unsat I s) = \<B> s"
path_cons_simp': fixes v :: "nat"   and v' :: "nat"   and e :: "(nat \<times> ((char list, prop_funs) pform, char list) out_port) \<times> nat \<times> ((char list, prop_funs) pform, char list) antecedent"   and pth :: "((nat \<times> ((char list, prop_funs) pform, char list) out_port) \<times> nat \<times> ((char list, prop_funs) pform, char list) antecedent) list" shows "task1_1.path v v' (e # pth) = (\<exists>b a ba. e = ((v, b), a, ba) \<and> ((v, b), a, ba) \<in> {((0, Reg A), 1, plain_ant A)} \<and> task1_1.path a v' pth)"
toArr_fromArr: fixes x :: "bool" assumes "x \<in> {False, True}" shows "toArr (fromArr x) = x"
is_interval_box: fixes a :: "'b"   and b :: "'b" shows "is_interval (box a b)"
iv: shows "vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x within {0..1}) = (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u))"
hyps_2: assumes "wt t \<ge>\<^sub>p wt s2" shows "vars s2 \<subseteq> vars t"
rep_eq: fixes x :: "String.literal set" shows "literal.explode (fresh_name' x) = fresh_name (literal.explode ` x)"
elims_2: fixes x :: "('b \<times> 'c) tree" assumes "Leftist_Heap.linorder_class.heap x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> 'c) tree" and m :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (m, uu_), r\<rangle>" and "Ball (Tree2.set_tree l \<union> Tree2.set_tree r) ((\<le>) m) \<and> Leftist_Heap.linorder_class.heap l \<and> Leftist_Heap.linorder_class.heap r"
sets_Q: shows "Q.events = sets (Pi\<^sub>M (\<Union> (range J)) M)"
unwrapExist'': fixes i :: "nat"   and j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (ExN (i + j) A) L = eval (ExN i (ExN j A)) L"
simps_4: fixes \<sigma> :: "nat \<Rightarrow> exp option"   and C :: "nat"   and es :: "exp list" shows "substs \<sigma> (New C es) = New C (subst_list2 \<sigma> es)"
AR_from_Un_Int: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "closed T"   and "AR (S \<union> T)"   and "AR (S \<inter> T)" shows "AR S"
interior_halfspace_le: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
bytes_replicate_def: shows "bytes_replicate \<equiv> map_fun id (map_fun Rep_uint8 Abs_bytes) (\<lambda>n b. replicate n (Abs_uint8 b))"
aae: shows "open_segment a (a + complex_of_real e) \<subseteq> inside (path_image p)"
borel_measurable_implies_simple_function_sequence': fixes u :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "u \<in> borel_measurable M" obtains f :: "nat \<Rightarrow> 'c \<Rightarrow> ennreal" where "\<And>i. simple_function M (f i)" and "incseq f" and "\<And>i x. f i x < top" and "\<And>x. (SUP i. f i x) = u x"
lossless_return_spmf: fixes x :: "'c" shows "lossless_spmf (return_spmf x)"
trs_def: fixes r :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "trs r = {. inpt r .} \<circ> [: r :]"
ln_Gamma_complex_of_real: fixes x :: "real" assumes "0 < x" shows "ln_Gamma (complex_of_real x) = complex_of_real (ln_Gamma x)"
generator_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme"   and H :: "'j set" shows "generator R M H \<equiv> H \<subseteq> carrier M \<and> linear_span R M (carrier R) H = carrier M"
linepath_in_path: fixes x :: "real"   and a :: "'a"   and b :: "'a" assumes "x \<in> {0..1}" shows "linepath a b x \<in> closed_segment a b"
less_eq_mset_zmsetE: fixes M :: "'a zmultiset"   and N :: "'a zmultiset" assumes "M \<le> N" obtains A :: "'a multiset" and B :: "'a multiset" and C :: "'a zmultiset" where "M = zmset_of A + C" and "N = zmset_of B + C" and "A \<le> B"