hcju/isabelleps huggingface dataset:
var_ueval: fixes x :: "'e \<Longrightarrow> 'f"   and b :: "'f" shows "\<lbrakk>x\<rbrakk>\<^sub>e b = get\<^bsub>x\<^esub> b"
VSLEGBlocks_def: shows "VSLEGBlocks = VSGeneralBlocks \<circ> VSEqualityBlocks \<circ> VSLuckyBlocks"
non_wildcard_ifaces: fixes n :: "nat" shows "set (non_wildcard_ifaces n) = {s. length s = n \<and> \<not> iface_name_is_wildcard s}"
power_down_fl: fixes x :: "float"   and p :: "nat"   and n :: "nat" assumes "0 \<le> x" shows "power_down_fl p x n \<le> x ^ n"
top_greatest: fixes a :: "'b" shows "\<bottom> \<le> a"
relcompE: fixes xz :: "'a \<times> 'c"   and r :: "('a \<times> 'd) set"   and s :: "('d \<times> 'c) set" assumes "xz \<in> r O s" obtains x :: "'a" and y :: "'d" and z :: "'c" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "'b prog"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M, b :  Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
T_ndet: fixes P :: "'c process"   and Q :: "'c process" shows "T (P |-| Q) = T P \<union> T Q"
InitCfgStart: shows "\<forall>c\<in>set initCfgList. states c = start"
h_outside_AB: fixes x :: "'v"   and y :: "'v"   and n :: "nat" assumes "x \<notin> A \<Gamma> \<or> y \<notin> B \<Gamma>" shows "h n (x, y) = 0"
cotrace_trace_cos: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event llist"   and i :: "nat" assumes "cotrace c t" shows "trace c (Co_Snapshot.ltake i t) (local.cos c t i)"
gallery_def: fixes xs :: "'b set list" shows "codomain.gallery xs \<equiv> Ball (set xs) codomain.chamber \<and> adjacentchain xs"
diff: fixes f :: "'q \<Rightarrow> 'r"   and x :: "'q"   and y :: "'q" assumes "additive f" shows "f (x - y) = f x - f y"
f_Exec_Stream_nth_gr0_calc: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n < length xs"   and "0 < n" shows "f_Exec_Comp_Stream trans_fun xs c ! n = trans_fun (xs ! n) (f_Exec_Comp_Stream trans_fun xs c ! (n - 1))"
simps_1: fixes lc :: "'b"   and q :: "'b list"   and r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "divide_poly_main_list lc q r d (Suc n) = (let cr = hd r in if cr = (0::'b) then divide_poly_main_list lc (cr ## q) (tl r) d n else let a = cr div lc; qq = a ## q; rr = minus_poly_rev_list r (map ((*) a) d) in if hd rr = (0::'b) then divide_poly_main_list lc qq (tl rr) d n else [])"
a_subid_aux1: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> y \<le> y"
less_imp_le_nat: fixes m :: "nat"   and n :: "nat" assumes "order.greater n m" shows "order.greater_eq n m"
h_compose_smooth_on: fixes k :: "enat"   and S :: "'b set"   and g :: "'b \<Rightarrow> real" assumes "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. h (g x))"
map_snd_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "map snd (xs || ys) = ys"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "ide (e \<cdot> m)" shows "ide (R e \<cdot> R m)"
map_bset_bempty: fixes f :: "'e \<Rightarrow> 'd" shows "map_bset f bempty = bempty"
norm_V_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<subseteq> V"   and "canonical M n" shows "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> \<subseteq> V"
prepend: fixes xs :: "'a llist" assumes "lfinite xs"   and "\<And>zs. \<exists>ys. xs = prepend ys zs \<and> ys \<noteq> [] \<Longrightarrow> P zs" shows "P xs"
iffs: fixes CatDom :: "('a, 'c, 'e) Category_scheme"   and CatCod :: "('b, 'd, 'f) Category_scheme"   and MapM :: "'c \<Rightarrow> 'd"   and more :: "'g"   and CatDom' :: "('a, 'c, 'e) Category_scheme"   and CatCod' :: "('b, 'd, 'f) Category_scheme"   and MapM' :: "'c \<Rightarrow> 'd"   and more' :: "'g" shows "(\<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = \<lparr>CatDom = CatDom', CatCod = CatCod', MapM = MapM', \<dots> = more'\<rparr>) = (CatDom = CatDom' \<and> CatCod = CatCod' \<and> MapM = MapM' \<and> more = more')"
finite_insert: fixes a :: "'b"   and A :: "'b set" shows "finite ((a::'b) \<triangleright> (A::'b set)) = finite A"
natural_transformation_axioms: shows "natural_transformation J.COMP (\<cdot>) \<chi>.A.map local.map \<chi>"
rels: fixes x :: "'a" assumes "x \<in> rel_interior S" shows "x \<in> S"
states_def: fixes P :: "'b prog"   and mxs :: "nat"   and mxl :: "nat" shows "states P mxs mxl \<equiv> fst (JVM_SemiType.sl P mxs mxl)"
streams_stl: fixes s :: "'b stream"   and A :: "'b set" assumes "s \<in> streams A" shows "stl s \<in> streams A"
induct: fixes P :: "natural \<Rightarrow> bool"   and natural :: "natural" assumes "P 0"   and "\<And>natural. P natural \<Longrightarrow> P (Code_Numeral.Suc natural)" shows "P natural"
assms_3: fixes n :: "nat" shows "\<exists>N. \<forall>k>N. s k n = a n"
in_fdom_alt: fixes k :: "'e"   and m :: "('e, 'f) fmap" shows "(k \<in># dom_m m) = (\<not> is_None (fmlookup m k))"
connected_iff4: fixes x :: "'a" shows "connected x = (x\<^sup>T ; 1 ; x\<^sup>T \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>)"
outputPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "(P, Q) \<in> Rel" shows "a{b}.P \<leadsto>[Rel] a{b}.Q"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {..<Suc n} = prod g {..<n} * g n"
alphaAll_trans: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> (\<forall>X' X''. X #= X' \<and> X' #= X'' \<longrightarrow> X #= X'')) \<and> (qGoodAbs A \<longrightarrow> (\<forall>A' A''. A $= A' \<and> A' $= A'' \<longrightarrow> A $= A''))"
T1a: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> x \<^bold>\<rightarrow> \<^bold>\<diamond>(\<lambda>v. \<exists>xa. existsAt xa v \<and> x xa)) w)\<rfloor>"
proj2_lines_define_point: fixes l :: "proj2_line"   and m :: "proj2_line" shows "\<exists>p. proj2_incident p l \<and> proj2_incident p m"
oalist_inv_raw_Nil: shows "cmp2.oalist_inv_raw []"
assms_1: shows "\<forall>G\<in>set Gs. AbGroup G"
collect_clki_def: fixes I :: "'f \<Rightarrow> ('d, 'e) cconstraint" shows "collect_clki I = \<Union> {collect_clock_pairs (I x) |x. True}"
fmfilter_of_list: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c \<times> 'd) list" shows "fmfilter P (fmap_of_list m) = fmap_of_list (filter (\<lambda>(k, uu_). P k) m)"
list_slice2_last: fixes xs :: "'b list"   and k :: "nat"   and m :: "nat" assumes "0 < length xs mod k"   and "m = length xs div k" shows "list_slice2 xs k ! m = xs \<up> (length xs div k * k)"
cscc_outer_invar_loc_axioms: shows "cscc_outer_invar_loc G it l D"
induct: fixes P :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list" assumes "\<And>Pa. P Pa []"   and "\<And>Pa x ys. P Pa ys \<Longrightarrow> P Pa (x # ys)" shows "P a0 a1"
rcos_module_rev: fixes x :: "'a"   and x' :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "x' \<otimes> inv x \<in> H" shows "x' \<in> H #> x"
out_bet_out_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A C"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Out Bet P A B"
a_in: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a" shows "a \<in> arcs G"
y: shows "\<not> 1 << LENGTH('a) - 1 \<le> y"
assms_3: shows "(\<Sum>(c, \<alpha>)\<leftarrow>xs. c * exp \<alpha>) = 0"
simps: fixes v :: "'d option"   and m :: "('c \<times> ('c, 'd) trie) list" shows "is_empty_trie (Trie v m) = (v = None \<and> m = [])"
RS: fixes f :: "'a \<Rightarrow> 'b nres"   and f' :: "'c \<Rightarrow> 'd nres"   and x :: "'a"   and x' :: "'c" assumes "\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> \<Down> S (f' x')"   and "(x, x') \<in> R" shows "body f x \<le> \<Down> S (body' f' x')"
complex_nonneg_Reals_iff: fixes z :: "complex" shows "(z \<in> \<real>\<^sub>\<ge>\<^sub>0) = (0 \<le> Re z \<and> Im z = 0)"
normal_form_diff_in_ideal: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]" shows "p - normal_form F p \<in> ideal F"
assms_2: shows "Vx : A \<turnstile> e\<equiv>d : B \<in> Axioms.Theory T"
j: shows "j < dim_col (mat_of_rows (dim_col A) (map (Matrix.row A) [0..<k]) @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row A) [k..<dim_row A]))"
N_2: fixes n :: "nat" assumes "N \<le> n" shows "Sigma_Algebra.measure (distr M borel (\<lambda>x. birkhoff_sum f n x / B n)) {x. C0 \<le> \<bar>x\<bar>} < alpha"
rowvector_row_def: fixes v :: "('d, 'e) vec"   and i :: "'f" shows "rowvector_row v i = (\<chi>j. v $ j)"
get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok: fixes h :: "(_) heap"   and ptr :: "(_) character_data_ptr"   and getter :: "'s option RCharacterData_scheme \<Rightarrow> 'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter"
induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "'d"   and a2 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>x y. P x y \<langle>\<rangle>"   and "\<And>x y l a b uu_ r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y l; cmp x a = GT \<Longrightarrow> P x y r\<rbrakk> \<Longrightarrow> P x y (B uu_ l (a, b) r)"   and "\<And>x y l a b uv_ r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y l; cmp x a = GT \<Longrightarrow> P x y r\<rbrakk> \<Longrightarrow> P x y (R uv_ l (a, b) r)" shows "P a0 a1 a2"
prime_aux: fixes p :: "nat" assumes "prime p" shows "f_prime_power p 1 = f_prime p"
mult_commute_imp_mult_inverse_commute: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "times y x = times x y" shows "times (inverse y) x = times x (inverse y)"
remdups_fwd_acc_filter: fixes Acc :: "'b set"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "remdups_fwd_acc Acc (filter P xs) = filter P (remdups_fwd_acc Acc xs)"
eq_divide_eq_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(one = divide b a) = (a \<noteq> zero \<and> a = b)"
intro: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ordered_powerprod ord ord_strict" shows "gd_powerprod ord ord_strict"
scale_cancel_right: fixes a :: "rat"   and x :: "'a"   and b :: "rat" shows "(a *R x = b *R x) = (a = b \<or> x = (0::'a))"
hom_graph_imp_formulas: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b" assumes "hom_graph f S x a" shows "x \<in> formulas S"
newpkt_def: fixes p :: "('i, 'pkt_ext) tagged_packet_scheme" shows "(newpkt::('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool) (p::('i, 'pkt_ext) tagged_packet_scheme) \<equiv> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New"
compose_const_1: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c" shows "f \<circ> (\<lambda>x. a) = (\<lambda>x. f a)"
space_sup_measure': fixes B :: "'b measure"   and A :: "'b measure" assumes "sets B = sets A" shows "space (sup_measure' A B) = space A"
interior_interior: fixes S :: "'c set" shows "interior (interior S) = interior S"
simps_85: fixes x9 :: "Term.var"   and x31 :: "ty"   and x32 :: "expr" shows "Acc x9 \<noteq> Cast x31 x32"
mbody_is_def: fixes C :: "Class"   and m :: "Method"   and M :: "Var list \<times> (int \<times> Instr) list \<times> int" shows "mbody_is C m M = (\<exists>CD. P\<down>C = Some CD \<and> CD\<down>m = Some M)"
budan_fourier_gt: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "int (proots_count p {x. a < x}) \<le> changes_gt_der a p \<and> even (changes_gt_der a p - int (proots_count p {x. a < x}))"
dghm_cn_comp_ArrMap_vsv: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr>)"
is_finite_weak_ranking_singleton: fixes x :: "'b set" shows "is_finite_weak_ranking [x] = (x \<noteq> {} \<and> finite x)"
prod_list_coprime_left: fixes xs :: "'a list"   and a :: "'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> coprime x a" shows "coprime (prod_list xs) a"
E: shows "set us - {us ! i} \<subseteq> carrier_vec n"
downset_set_idem: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "ord.downset_set less_eq \<circ> ord.downset_set less_eq = ord.downset_set less_eq"
logD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Log]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Undecided"
take_Suc0_1: fixes l :: "'c list" assumes "l \<noteq> []" shows "take (Suc 0) l = [l ! 0]"
sub_complement': fixes x :: "'a"   and y :: "'a" shows "- x = - (- - x \<squnion> - y) \<squnion> - (- - x \<squnion> - - y)"
prob_space_distr: fixes f :: "'a \<Rightarrow> 'c"   and M' :: "'c measure" assumes "random_variable M' f" shows "prob_space (distr M M' f)"
eSwap_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "eSwap MOD zs z1 z2 ERR = ERR"
fresh_def: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "atom"   and x :: "'c" assumes "class.pt permute" shows "pt.fresh permute a x \<equiv> a \<notin> pt.supp permute x"
dis_log_adv3: fixes \<G> :: "'a cyclic_group"   and x :: "nat"   and \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" assumes "dis_log_alt_reductions \<G>" shows "Discrete_Log.dis_log_alt.advantage3 \<G> x \<A> = Discrete_Log.dis_log.advantage \<G> (Discrete_Log.dis_log_alt.adversary3 \<G> x \<A>)"
splitFace_incr_faces: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list" assumes "pre_splitFace g u v f vs" shows "finals (snd (snd (splitFace g u v f vs))) = finals g \<and> |nonFinals (snd (snd (splitFace g u v f vs)))| = Suc |nonFinals g|"
store_assign_def: fixes n :: "nat"   and v2 :: "'b store_v"   and st :: "'b store_v list" shows "store_assign n v2 st = (if n < length st \<and> store_v_same_type (st ! n) v2 then Some (st[n := v2]) else None)"
viewLn_list: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node"   and s :: "('c, 'd) FingerTreeStruc" assumes "viewLn t = Some (nd, s)" shows "toList t = nodeToList nd @ toList s"
ex_moebius: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and w1 :: "complex_homo"   and w2 :: "complex_homo"   and w3 :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "w1 \<noteq> w2"   and "w1 \<noteq> w3"   and "w2 \<noteq> w3" shows "\<exists>M. moebius_pt M z1 = w1 \<and> moebius_pt M z2 = w2 \<and> moebius_pt M z3 = w3"
rsquarefree: fixes q :: "int poly" assumes "q \<in> P" shows "rsquarefree (Ring_Hom_Poly.of_int_poly q)"
converse_tranclp_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r y b \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>+\<^sup>+ z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
a_in: shows "a \<in> read_only (\<S> \<ominus>\<^bsub>A\<^esub> L)"
run_heap: fixes f :: "heap \<Rightarrow> 'b \<times> heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and res :: "'b" assumes "run (Heap_Monad.heap f) \<sigma> \<sigma>' res"   and "\<not> is_exn \<sigma>" obtains "\<sigma>' = Some (snd (f (the_state \<sigma>)))" and "res = fst (f (the_state \<sigma>))"
inj_on_rotate_min_iff: fixes A :: "nat list set" assumes "\<forall>vs\<in>A. distinct vs"   and "[] \<notin> A" shows "inj_on rotate_min A = inj_on (\<lambda>vs. {vs} // {\<cong>}) A"
simps_3: fixes K :: "nat"   and X :: "freemsg" shows "freeright (CRYPT K X) = freeright X"
geseq_iff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "((f, g) \<in> geseq) = (f \<in> SEQ A \<and> g \<in> SEQ A \<and> (f = g \<or> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j))))"
Domainp_mod_ring_rel64: fixes p :: "int"   and pp :: "uint64" assumes "mod_ring_locale TYPE('b) p"   and "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "Domainp (mod_ring_locale.mod_ring_rel64 p) = (\<lambda>x. 0 \<le> x \<and> x < pp)"
Ln_times_of_nat: fixes r :: "nat"   and z :: "complex" assumes "0 < r"   and "z \<noteq> 0" shows "Ln (of_nat r * z) = Ln (of_nat r) + Ln z"
square_eq_1_iff: fixes x :: "'a" shows "(x * x = (1::'a)) = (x = (1::'a) \<or> x = - (1::'a))"
1: shows "\<forall>x\<in>MPLS PROB. x < k + 1"
simps: fixes b :: "bool"   and h :: "'b"   and as :: "'c set" shows "pure_assn_raw b (h, as) = (as = {} \<and> b)"
1: fixes s' :: "'state" assumes "(c, s) \<rightarrow>*t s'"   and "s' \<approx> t'" shows "thesis"
ordIso_refl: fixes r :: "('d \<times> 'd) set" assumes "Card_order r" shows "r =o r"
div_mult_swap: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c dvd b" shows "a * (b div c) = a * b div c"
Refl: fixes x :: "'a language"   and y :: "'a language"   and R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "x = y" shows "regular_cong R x y"
cr_ell2_def: shows "cr_ell2 \<equiv> \<lambda>x y. x = Rep_ell2 y"
sign_def: fixes x :: "'a" shows "Sturm_Tarski.sign x \<equiv> if (0::'a) < x then 1 else if x = (0::'a) then 0 else - 1"
some_unit_def: fixes a :: "'a" shows "some_unit a \<equiv> SOME \<iota>. local.iso \<iota> \<and> \<guillemotleft>\<iota> : a \<star> a \<Rightarrow> a\<guillemotright>"
lift_Suc_mono_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f n) (f (Suc n))"   and "n \<le> n'" shows "less_eq (f n) (f n')"
Dag_unique1: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag p l r t" shows "\<exists>!t. Dag p l r t"
abs_dist_diff_le: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.metric_space dist uniformity open" shows "\<bar>dist a b - dist b c\<bar> \<le> dist a c"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "to_ord r x y = (\<not> r\<^sup>*\<^sup>* y x)"
simps_6: fixes v :: "'dyn" shows "generalize_instr (IPush v) = IPush v"
attach_converter_of_resource_conv_parallel_resource2: fixes res :: "('z, 'ab) resource"   and res' :: "('y, 'aa) resource" shows "1\<^sub>C |\<^sub>\<propto> converter_of_resource res \<rhd> res' = res' \<parallel> res"
inject_2: fixes x1 :: "name"   and y1 :: "name" shows "(BoundOutputS x1 = BoundOutputS y1) = (x1 = y1)"
frp1_2: fixes p :: "int poly"   and qs :: "int poly list"   and x :: "'a" assumes "factors_of_int_poly p = qs"   and "p \<noteq> 0" shows "(ipoly p x = (0::'a)) = (\<exists>q\<in>set qs. ipoly q x = (0::'a))"
a_prop: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> F \<^bold>\<noteq> F) in v]"
rHom_one: fixes A :: "('e, 'f) Ring_scheme"   and R :: "('g, 'h) Ring_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>"
React_def: fixes f :: "'f \<Rightarrow> 'e \<times> ('d, 'e, 'f) rpv"   and input :: "'f" shows "React f input = (case f input of (x, xa) \<Rightarrow> Generative_Probabilistic_Value.Pause x xa)"
ffb_temp_dyn: fixes a :: "real"   and t :: "real"   and L :: "real"   and G :: "(real, 4) vec \<Rightarrow> bool"   and Q :: "(real, 4) vec set" assumes "0 < a"   and "0 \<le> t" shows "fb\<^sub>\<F> x\<acute>=\<lambda>t. temp_vec_field a L & G on \<lambda>s. {0..t} UNIV @ 0 Q = {s. s2p UNIV s \<longrightarrow> (\<forall>t\<in>{0..t}. (\<forall>\<tau>\<in>{0..t}. G (temp_flow a L \<tau> s)) \<longrightarrow> s2p Q (temp_flow a L t s))}"
assms: shows "a_consistent (ae, a, as) (\<Gamma>, Lam [y]. e, Arg x # S)"
F_DrF: shows "F = \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> F)"
size_fsingleton: fixes f :: "'b fset" shows "(size f = 1) = (\<exists>e. f = {|e|})"
simps_12: fixes p :: "fm"   and q :: "fm" shows "bound0 (And p q) = (bound0 p \<and> bound0 q)"
bind_pmf_const: fixes M :: "'e pmf"   and c :: "'d pmf" shows "M \<bind> (\<lambda>x. c) = c"
simps_2: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.queue_to_multiset (t # q) = BinomialHeapStruc.tree_to_multiset t + BinomialHeapStruc.queue_to_multiset q"
match_tcp_flags_nomatch: fixes c :: "tcp_flag set"   and fmask :: "tcp_flag set"   and pkt :: "tcp_flag set" assumes "\<not> c \<subseteq> fmask" shows "\<not> match_tcp_flags (TCP_Flags fmask c) pkt"
sorted_insort_is_snoc: fixes xs :: "'a list"   and a :: "'a" assumes "sorted xs"   and "\<forall>x\<in>set xs. x \<le> a" shows "insort a xs = xs @ [a]"
lex_append_leftI: fixes ys :: "'g list"   and zs :: "'g list"   and r :: "('g \<times> 'g) set"   and xs :: "'g list" assumes "(ys, zs) \<in> lex r" shows "(xs @ ys, xs @ zs) \<in> lex r"
continuous_closedin_preimage_constant: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d" assumes "continuous_on S f" shows "closedin (top_of_set S) {x \<in> S. f x = a}"