hcju/isabelleps huggingface dataset:
SUP_constant: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and c :: "'e"   and A :: "'f set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
assms_2: shows "A \<subseteq> U\<^sub>3"
permutation_lift_signed_lconjperm_eq_signed_list_lconjaction': fixes xs :: "('w \<times> bool) list" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "(\<rightarrow>) (freeword_funlift signed_lconjpermutation (Abs_freeword xs)) = signed_list_lconjaction (map fst xs)"
elims: fixes x :: "'h option"   and y :: "'h exception" assumes "exception_of_option x = y" obtains "x = None" and "y = Fault"   | xa :: "'h" where "x = Some xa" and "y = OK xa"
insertI_2_2: shows "s = 0 \<or> Re (g s - z) = 0"
tabulate_def: fixes ks :: "'c odlist"   and f :: "'c \<Rightarrow> 'd" shows "tabulate ks f = ODList (map (\<lambda>k. (k, f k)) (toList ks))"
compl_mono: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
length_dropWhile_le: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (dropWhile P xs) \<le> length xs"
path_connected_space_iff_path_component: fixes X :: "'a topology" shows "path_connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. path_component_of X x y)"
mem_in_image2: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> A" shows "f a \<in> f ` A"
sorted_auxlist': shows "sorted_wrt (\<lambda>x y. fst y < fst x) auxlist'"
prems: shows "is_cnf (F1 \<^bold>\<or> F2)"
moebius: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and z :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc"   and "z \<in> unit_disc"   and "P (moebius_pt M z) (moebius_pt M u) (moebius_pt M v) (moebius_pt M w)" shows "P z u v w"
\<alpha>: shows "\<alpha> \<in> elts (ordertype M VWF)"
last_remdups_adj: fixes xs :: "'a list" shows "last (remdups_adj xs) = last xs"
basis_in_indep_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> X" shows "\<E>.indep_in \<E> X"
\<gamma>': shows "\<gamma>' = drop (length \<delta>) \<gamma>"
intensional_simps_24: fixes P :: "'b \<Rightarrow> bool" shows "#True \<or> P \<equiv> #True"
index_uminus_vec_2: fixes v :: "'b vec" shows "dim_vec (- v) = dim_vec v"
pcr_dual_ord_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_dual_ord T \<equiv> T OO cr_dual_ord"
setdist_Lipschitz: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "\<bar>setdist {x} S - setdist {y} S\<bar> \<le> dist x y"
map_tl: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map f (tl xs) = tl (map f xs)"
sorted_wrt_distinct_linorder_nth: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "distinct xs"   and "i < length xs"   and "j < length xs" shows "((xs ! i, xs ! j) \<in> R) = (i \<le> j)"
hyps: fixes s' :: "nat"   and r' :: "nat \<Rightarrow>f value option" assumes "obtains s r e s' r' t" shows "visits s e s' r' t"
append_Cons_eq_iff: fixes x :: "'e"   and xs :: "'e list"   and ys :: "'e list"   and xs' :: "'e list"   and ys' :: "'e list" assumes "x \<notin> set xs"   and "x \<notin> set ys" shows "(xs @ x # ys = xs' @ x # ys') = (xs = xs' \<and> ys = ys')"
llist_le_finT: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" assumes "r \<le> s"   and "s \<in> A\<^sup>\<star>" shows "r \<in> A\<^sup>\<star>"
seq: shows "{HaddP k1 k2 k, x IN k, OrdP k1} \<turnstile> x IN k1 OR SyntaxN.Ex i (HaddP k1 (Var i) x AND Var i IN k2)"
proj_append_2: fixes l :: "'b"   and A :: "('b strand_label \<times> 'a) list"   and B :: "('b strand_label \<times> 'a) list" shows "proj_unl l (A @ B) = proj_unl l A @ proj_unl l B"
bigstep_det: fixes c1 :: "com"   and s :: "char list \<Rightarrow> int"   and p1 :: "nat"   and t1 :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(c1, s) \<Rightarrow> p1 \<Down> t1"   and "(c1, s) \<Rightarrow> p \<Down> t" shows "p1 = p \<and> t1 = t"
gso_times_self_is_norm: fixes j :: "nat" assumes "j < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! j \<bullet> fs.gs.gso j = \<parallel>fs.gs.gso j\<parallel>\<^sup>2"
cross_ratio_not_zero: fixes z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> u"   and "v \<noteq> w" shows "cross_ratio z u v w \<noteq> 0\<^sub>h"
some_case: fixes l :: "'a"   and v :: "('b, 'a, 'c) val" assumes "\<sigma> l = Some v" shows "\<sigma>' (\<beta> l) = Some (\<R>\<^sub>V \<alpha> \<beta> v)"
vwalk_join_hd: fixes p :: "'a list"   and q :: "'a list" assumes "joinable p q" shows "hd (p \<oplus> q) = hd p"
split0_stupid: fixes p :: "tm" shows "\<exists>x y. (x, y) = split0 p"
B_trusts_OR3: fixes S :: "agent"   and B :: "agent"   and X :: "msg"   and NB :: "msg"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Says S B \<lbrace>X, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "A \<noteq> B"   and "evs \<in> otway" shows "\<exists>NA. Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"
card_distinct: fixes xs :: "'a list" assumes "card (list.set xs) = length xs" shows "distinct xs"
mono: fixes x :: "('b, 'call, 'ret) gpv \<times> 's" shows "spmf.mono_body (\<lambda>inline1. case x of (gpv, s) \<Rightarrow> the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))"
chine_assoc_props_4: shows "doms.Prj\<^sub>0\<^sub>0 \<cdot> doms.chine_assoc = doms.Prj\<^sub>0"
lfp_trans_lemma3: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_trans t u; \<And>P. sound P \<Longrightarrow> sound (t P); \<And>P. sound P \<Longrightarrow> sound (u P)\<rbrakk> \<Longrightarrow> le_trans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. sound Q \<Longrightarrow> sound (t Q); sound P\<rbrakk> \<Longrightarrow> sound (T t P)"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)" shows "le_trans (lfp_trans T) (T (lfp_trans T))"
event_ex_aux_a: fixes a :: "nat \<Rightarrow> int"   and xa :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "a 0 = 0"   and "\<forall>n. xa (Suc n) = (\<not> xa n)"   and "\<forall>n. xa n \<and> a (Suc n) = a n + 1 \<or> \<not> xa n \<and> a (Suc n) = a n - 1 \<or> a (Suc n) = 0" shows "(a n = - 1 \<longrightarrow> xa n) \<and> (a n = 1 \<longrightarrow> \<not> xa n) \<and> - 1 \<le> a n \<and> a n \<le> 1"
setOfPairsEquality: fixes X :: "('c \<times> 'd set) set" shows "{(fst pair, {y}) |y pair. y \<in> snd pair \<and> pair \<in> X} = {(fst pair, Y) |Y pair. Y \<in> finestpart (snd pair) \<and> pair \<in> X}"
fresh_2: fixes a :: "name"   and x3 :: "'d"   and x2 :: "name list"   and x1 :: "'d" shows "a \<sharp> x3\<lparr>\<nu>*x2\<rparr>\<langle>x1\<rangle> = (a \<sharp> x3 \<and> a \<sharp> x2 \<and> a \<sharp> x1)"
Collect_neg_eq: fixes P :: "'d \<Rightarrow> bool" shows "{x. \<not> P x} = - {x. P x}"
rel_gpv_def: shows "rel_gpv \<equiv> \<lambda>R1 R2. (BNF_Def.Grp {x. results'_gpv x \<subseteq> {(x, y). R1 x y} \<and> outs'_gpv x \<subseteq> {(x, y). R2 x y}} (map_gpv fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. results'_gpv x \<subseteq> {(x, y). R1 x y} \<and> outs'_gpv x \<subseteq> {(x, y). R2 x y}} (map_gpv snd snd)"
cut_le_Max_all: fixes I :: "'b set"   and t :: "'b" assumes "finite I"   and "Max I \<le> t" shows "I \<down>\<le> t = I"
rep_eq: fixes x :: "'a zmultiset" shows "count (mset_neg x) = (case rep_zmultiset x of (Mp, Mn) \<Rightarrow> count (Mn - Mp))"
elims_3: fixes x :: "('a, 'e, 'f, 'g) SLin_state_scheme \<times> ('a, 'h, 'i, 'j) SLin_state_scheme" assumes "\<not> P9 x" obtains s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme" and s2 :: "('a, 'h, 'i, 'j) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (initialized s2 \<longrightarrow> dstate s1 \<preceq> dstate s2)"
interchange: fixes f :: "('c, 'e \<Rightarrow> 'd) state"   and x :: "'e" shows "f \<diamondop> pure x = pure (\<lambda>f. f x) \<diamondop> f"
coprime_def: fixes a :: "'a"   and b :: "'a" shows "algebraic_semidom_class.coprime a b = (\<forall>c. c dvd a \<longrightarrow> c dvd b \<longrightarrow> is_unit c)"
simps_2: fixes x :: "quat" shows "quat_proj x (Suc 0) = Im1 x"
ffb_surj: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "\<exists>f. fb\<^sub>\<F> f = \<phi>"
lm07: fixes f :: "'d \<Rightarrow> 'c"   and l :: "'d list" shows "maxpositions (map f l) = concat (map (\<lambda>n. if Max (f ` set l) \<le> f (l ! n) then [n] else []) [0..<length l])"
k: fixes x :: "'a"   and e :: "real" assumes "x \<notin> N"   and "0 < e" shows "\<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow> norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<AA>\<lparr>Comp\<rparr>) = []\<^sub>\<circ>" shows "\<AA>\<lparr>Comp\<rparr> = []\<^sub>\<circ>"
mult2_ns_conv: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_ns ns s = mult2_s ns s \<union> multpw ns"
c1: shows "((\<lambda>y. infdist y (\<omega>_limit_set x)) \<circ> (flow0 x \<circ> s \<circ> r)) \<longlonglongrightarrow> 0"
sorted: fixes xs :: "'a llist" assumes "xs \<in> Y" shows "lsorted xs"
covering_space_lift_unique: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and g1 :: "'e \<Rightarrow> 'c"   and a :: "'e"   and g2 :: "'e \<Rightarrow> 'c"   and T :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "covering_space c p S"   and "g1 a = g2 a"   and "continuous_on T f"   and "f ` T \<subseteq> S"   and "continuous_on T g1"   and "g1 ` T \<subseteq> c"   and "\<And>x. x \<in> T \<Longrightarrow> f x = p (g1 x)"   and "continuous_on T g2"   and "g2 ` T \<subseteq> c"   and "\<And>x. x \<in> T \<Longrightarrow> f x = p (g2 x)"   and "connected T"   and "a \<in> T"   and "x \<in> T" shows "g1 x = g2 x"
not_prime_eq_prod_nat: fixes n :: "nat" assumes "1 < n"   and "\<not> prime n" shows "\<exists>m k. n = m * k \<and> 1 < m \<and> m < n \<and> 1 < k \<and> k < n"
wf_list_graph_stateful_list_policy_to_list_graph: fixes G :: "'b list_graph"   and E :: "('b \<times> 'b) list" assumes "wf_list_graph G"   and "distinct E"   and "set E \<subseteq> set (edgesL G)" shows "wf_list_graph (stateful_list_policy_to_list_graph \<lparr>hostsL = nodesL G, flows_fixL = edgesL G, flows_stateL = E\<rparr>)"
scast_scast_id_2: fixes y :: "'b signed word" shows "SCAST('b \<rightarrow> 'b signed) (SCAST('b signed \<rightarrow> 'b) y) = y"
elims_3: fixes x :: "'b \<times> 'c"   and xa :: "'b \<times> 'c" assumes "\<not> Psi.subset x xa" obtains l :: "'b" and u :: "'c" and l' :: "'b" and u' :: "'c" where "x = (l, u)" and "xa = (l', u')" and "\<not> (l' \<le> l \<and> u \<le> u')"
f: shows "f \<in> Ar\<^bsub>set_cat U\<^esub>"
subst_of_simps_3: fixes b :: "'a \<times> ('b, 'a) Term.term"   and ss :: "('a \<times> ('b, 'a) Term.term) list" shows "subst_of (b # ss) = subst_of ss \<circ>\<^sub>s subst (fst b) (snd b)"
commute: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a +\<^sub>o\<^sub>w b = b +\<^sub>o\<^sub>w a"
resumption_ord_simps_2: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and a :: "'f option" shows "resumption_ord r (Resumption.resumption.Done a) = (\<exists>aa. r = Resumption.resumption.Done aa \<and> option_ord aa a)"
g_trivial_D0: fixes v :: "'a" assumes "v \<in> D0" shows "g v = v"
angle_actrans_def: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "\<langle>P\<rangle>_v \<equiv> \<not> _actrans (\<not> P) v"
rcoset_replist_replacehd: fixes H :: "'g set"   and g' :: "'g"   and g :: "'g"   and gs :: "'g list" assumes "Subgroup H"   and "g' \<in> rcoset_rel H `` {g}"   and "is_rcoset_replist H (g # gs)" shows "is_rcoset_replist H (g' # gs)"
proved_terms_well_formed_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and p :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> p" shows "typ_of p = Some propT \<and> term_ok \<Theta> p"
wfvarsoccs\<^sub>s\<^sub>s\<^sub>t_append: fixes S :: "('f, 'e) stateful_strand_step list"   and S' :: "('f, 'e) stateful_strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>s\<^sub>t (S @ S') = wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<union> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S'"
IH_1: fixes x :: "nat"   and xa :: "nat"   and xaa :: "'a" assumes "x = dim_row A"   and "xa = dim_col A"   and "i < x \<and> j < xa"   and "xaa = A $$ (i, j)"   and "xaa = (0::'a)"   and "concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'a) then [i'] else []) [Suc i..<x]) = []"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc2" shows "gauss_jordan_main (mat\<^sub>h A) (mat\<^sub>h B) i (Suc j) = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan_main A B i (Suc j))"
padic_zero_simp_2: fixes p :: "int"   and n :: "nat" shows "padic_zero p n = 0"
return_None_bind_spmf: fixes f :: "'e \<Rightarrow> 'd spmf" shows "return_pmf None \<bind> f = return_pmf None"
recv_marker_means_snapshotted_1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and ev :: "('d, 'e, 'f) event"   and i :: "nat"   and p :: "'d"   and q :: "'d"   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 c :: "('d, 'e, 'f) configuration"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "ev = RecvMarker i p q"   and "distributed_system.next channel trans send recv c ev c'" shows "ps c' p \<noteq> None"
norm_v_eq_0: fixes v :: "(complex, 'a) Finite_Cartesian_Product.vec" shows "(map_vector cmod v = 0) = (v = 0)"
assms_2: shows "X \<subseteq> f\<inverse> `` Y"
weaken_proves_set: fixes \<Gamma>2 :: "Core.term set"   and \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term" assumes "finite \<Gamma>2"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "\<forall>A\<in>\<Gamma>2. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>2. typ_of A = Some propT"   and "finite \<Gamma>" shows "\<Theta>,\<Gamma> \<union> \<Gamma>2 \<turnstile> B"
M\<^sub>p_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "M\<^sub>p P = (\<lambda>\<sigma>. \<forall>i. \<exists>\<beta>. P (stake i \<sigma> @- \<beta>))"
rel_refl_strong: fixes x :: "'a pmf"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> Ra z z" shows "rel_pmf Ra x x"
length_prefixes: fixes xs :: "'c list" shows "length (prefixes xs) = length xs + 1"
Cn_conj_bis: fixes p :: "'a"   and q :: "'a" shows "Cn {p .\<and>. q} = Cn {p, q}"
open_segment_separate_left: fixes u :: "'a"   and a :: "'a"   and b :: "'a"   and v :: "'a"   and x :: "'a" assumes "u \<in> {a<--<b}"   and "v \<in> {u<--<b}"   and "dist x u < dist u v"   and "x \<in> {a<--<b}" shows "x \<in> {a<--<v}"
less_bit0_def: fixes a :: "'a bit0"   and b :: "'a bit0" shows "(a < b) = (Rep_bit0 a < Rep_bit0 b)"
intros_1: fixes p :: "'c"   and A :: "('d, 'c) nba" assumes "p \<in> initial A" shows "p \<in> nodes A"
R16_3: shows "pmf (sds R16) d = 1 - pmf (sds R16) a"
min_lvar_not_in_bounds_lvars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> x\<^sub>i \<in> lvars (\<T> s)"
da_good_approx_evalnE: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and n :: "nat"   and v :: "vals"   and s1 :: "state"   and C :: "qtname"   and L :: "lenv"   and T :: "tys"   and A :: "assigned" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog G" obtains "normal s1 \<Longrightarrow> nrm A \<subseteq> dom (locals (snd s1))" and "\<And>l. \<lbrakk>fst s1 = Some (Jump (Break l)); normal s0\<rbrakk> \<Longrightarrow> brk A l \<subseteq> dom (locals (snd s1))" and "\<lbrakk>fst s1 = Some (Jump Ret); normal s0\<rbrakk> \<Longrightarrow> Result \<in> dom (locals (snd s1))"
FV_EqOnGL: fixes p :: "pvar"   and p' :: "pvar" shows "FV (EqOnGL p p') \<subseteq> {p, p'}"
borderD_suf: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x \<le>s w"
simps_2: fixes x :: "'r"   and xs :: "[:'r:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "slength\<cdot>(x :# xs) = slength\<cdot>xs + 1"
uint64_shiftl_def: fixes x :: "uint64"   and n :: "integer" shows "uint64_shiftl x n = (if n < 0 \<or> 64 \<le> n then undefined push_bit x n else push_bit (nat_of_integer n) x)"
WTrt_binop_i_i_i_i_oI: fixes x :: "'b prog"   and xa :: "ty"   and xb :: "bop"   and xc :: "ty"   and xd :: "ty" assumes "x \<turnstile> xa\<guillemotleft>xb\<guillemotright>xc : xd" shows "pred.eval (WTrt_binop_i_i_i_i_o x xa xb xc) xd"
countable_Un_Int_1: fixes I :: "'b set"   and N :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Union> (N ` I) \<in> sets M"
False: shows "\<not> (n = 0 \<or> d = (0::'a))"
rep_eq: fixes x :: "real" shows "pmf (bernoulli_pmf x) = (\<lambda>b. if b then min 1 (max 0 x) else 1 - min 1 (max 0 x))"
simps_1: fixes x2 :: "int"   and y2 :: "int" shows "(Num_arc x2 = Num_arc y2) = (x2 = y2)"
UR_def: shows "UR = ((1, 0), 1, 1)"
simps_8: fixes z :: "octo" shows "Im7 (Octonions.cnj z) = - Im7 z"
wf_chain_def: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list" shows "wf_chain \<Gamma> rs \<equiv> \<forall>r\<in>set rs. \<forall>chain. get_action r = Call chain \<longrightarrow> \<Gamma> chain \<noteq> None"
rt_fresh_as_inc_invalidate: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "\<forall>rip\<in>dom dests. rip \<in> vD rt \<and> the (dests rip) = inc (sqn rt rip)" shows "rt \<approx>\<^bsub>dip\<^esub> invalidate rt dests"
Keys_bad_insert_payload: fixes Ik :: "msg set"   and Bad :: "agent set"   and x :: "msg" assumes "Keys_bad Ik Bad"   and "x \<in> payload" shows "Keys_bad (insert x Ik) Bad"
\<sigma>'_1: shows "\<sigma>' \<in> \<sigma>_set"
Abs_mini_alg_unique_inverse: fixes y :: "mini_alg" assumes "y \<in> {r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r)}" shows "Rep_mini_alg_unique (Abs_mini_alg_unique y) = y"
function_of_sequence_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "function_of A enum (sequence_of A enum f) = f"
case_prodE: fixes c :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b \<times> 'c" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'b" and y :: "'c" where "p = (x, y)" and "c x y"
simps_1: fixes A :: "X5 set" shows "CX5h H1 A = (if {X5.Xd1', X5.Xd2} \<subseteq> A then {X5.Xd1', X5.Xd2} else if {Xd2''} \<subseteq> A then {Xd2''} else if {X5.Xd1} \<subseteq> A then {X5.Xd1} else if {X5.Xd1'} \<subseteq> A then {X5.Xd1'} else if {X5.Xd2} \<subseteq> A then {X5.Xd2} else {})"
strict_boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b \<sqinter> c" obtains "a < b" and "a < c"
disc_1: fixes x1 :: "'c" shows "isl (Inl x1)"
assms: shows "FGModuleIso G (\<star>) W smult'' S X"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "less_eq_option x xa" obtains uu_ :: "'a option" where "x = None" and "xa = uu_"   | a :: "'a" and b :: "'a" where "x = Some a" and "xa = Some b" and "less_eq0 a b"
currentLevelPrefixToLevelEq_aux: fixes i :: "nat"   and l :: "nat"   and M :: "('b \<times> bool) list" assumes "i \<le> l"   and "l - i \<le> currentLevel M" shows "currentLevel (prefixToLevel_aux M l i) = l - i"
uminus_carrier_vec: fixes v :: "'a vec"   and n :: "nat" shows "(- v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
RID\<^sub>SE: fixes r :: "'d"   and \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option" assumes "r \<in> RID\<^sub>S \<sigma>" obtains l :: "'e" and v :: "('d, 'e, 'f) val" where "\<sigma> l = Some v" and "r \<in> RID\<^sub>V v"
raw_match_term_assoc_match_term: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" shows "raw_match_term t u (map_of subs) = map_option map_of (assoc_match_term t u subs)"
assms: shows "(a, b) \<in> A\<^sup>\<down>"
from_on_left_moving_to_check_left_moving: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and list :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and s' :: "nat" assumes "inv_on_left_moving_in_middle_B (as, lm) (s, Bk # list, Bk # r) ires" shows "inv_check_left_moving_on_leftmost (as, lm) (s', list, Bk # Bk # r) ires"
dp_total2: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_total y"
Subset_Mem_sf_lemma: fixes t :: "tm"   and u :: "tm"   and n :: "nat" assumes "size t + size u < n" shows "Sigma_fm (t SUBS u) \<and> Sigma_fm (t IN u)"
length_cons_code: fixes s :: "'s" shows "local.length_cons s = (case generator g s of Done \<Rightarrow> 0 | Skip x \<Rightarrow> local.length_cons x | Yield a s' \<Rightarrow> 1 + local.length_cons s')"
gen_dfs_simps_1: fixes succs :: "'e \<Rightarrow> 'e list"   and ins :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d" shows "gen_dfs succs ins memb S [] = S"
IntD2: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
unclocked_determ_2: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3a :: "'b state \<times> (v, v) result"   and r3b :: "'b state \<times> (v, v) result" assumes "Big_Step_Unclocked.evaluate env s e r3a"   and "Big_Step_Unclocked.evaluate env s e r3b" shows "r3a = r3b"
less_not_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "size x < size y"   and "x = y" shows "False"
constr: fixes name :: "name"   and \<Gamma> :: "(name, value) fmap"   and ts :: "sterm list"   and us :: "value list" assumes "name |\<in>| C"   and "list_all2 (veval' \<Gamma>) ts us" shows "\<Gamma> \<turnstile>\<^sub>v name $$ ts \<down> Vconstr name us"
trcl_empty_simp: fixes c :: "'a"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" shows "((c, [], c') \<in> trcl t) = (c = c')"