hcju/isabelleps huggingface dataset:
enc_respects_barbs_iff_source_target_rel: shows "((\<forall>S x. (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<and> (\<forall>S x. (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB)))"
less_le_trans: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x < y"   and "y \<le> z" shows "x < z"
sasp_op_to_strips_set_delete_effects_is: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "set (delete_effects_of (\<phi>\<^sub>O \<Psi> op)) = (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
coalesce_eq_OK1_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [+f] ys) = OK zs" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
some_geodesic_in_subset: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> S"   and "y \<in> S" shows "{x--S--y} \<subseteq> S"
fun_sum_commute: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "f (0::'b) = (0::'a)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
neg: shows "negligible (S - (\<Union>i\<in>C. cball (a i) (r i)))"
elims: fixes x :: "pnat"   and xa :: "pnat"   and y :: "pnat" assumes "pnat_plus x xa = y" obtains x :: "nat" and y :: "nat" where "xa__ = pnat x" and "xa = pnat y" and "ya__ = pnat (x + y)"   | uv_ :: "pnat" where "xa__ = pnat.PInfty" and "xa = uv_" and "ya__ = pnat.PInfty"   | uu_ :: "pnat" where "xa__ = uu_" and "xa = pnat.PInfty" and "ya__ = pnat.PInfty"
p_sn: shows "(\<lambda>n. real (nth_prime (Suc n)) / (real (Suc n) * ln (real (Suc n)))) \<longlonglongrightarrow> 1"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "(f a = f b) = (a \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> b \<in> H)"
cl: shows "[A,e,B]\<^bsub>T\<^esub> = [A,d,B]\<^bsub>T\<^esub>"
degree_mult_le: fixes p :: "'a poly"   and q :: "'a poly" shows "degree (p * q) \<le> degree p + degree q"
E_IH_prop: shows "generate_valid_topology M \<lparr>nodes = V, edges = E\<rparr> = \<lparr>nodes = V, edges = E_IH\<rparr>"
invfun_r: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and b :: "'f" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "b \<in> B" shows "f ((f\<inverse>\<^bsub>B,A\<^esub>) b) = b"
Inf_finite_3_def: fixes A :: "finite_3 set" shows "\<Sqinter> A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"
rs_1: shows "rs \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
g_def: shows "g = (\<lambda>x. p * sgn x * \<bar>x\<bar> powr (p - 1))"
ca: fixes x :: "'a"   and s :: "'s" assumes "x \<in> S s" shows "bd_cts (wp (a x))"
idx_intro_ext: fixes i :: "nat" assumes "i < length (ext_ids s)" shows "idx s (ext_ids s ! i) = Inr i"
lt_in_keys: fixes p :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "extended_ord.lpp p \<in> keys p"
length_tl: fixes xs :: "'e list" shows "length (tl xs) = length xs - 1"
is_natural_1: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "F.FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi> (C.VV.dom f) = \<Phi> f"
call_return_nodes_in_slice_1: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and a' :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r' :: "'ret"   and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and S :: "'node SDG_node set" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'"   and "a \<in> get_return_edges a'"   and "CFG_node (targetnode a) \<in> HRB_slice S" shows "CFG_node (sourcenode a) \<in> HRB_slice S"
succ_tran_def: fixes G :: "'b graph"   and v :: "'b" shows "succ_tran G v = {e2. (v, e2) \<in> (edges G)\<^sup>+}"
self_factor_not_simple: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "\<not> simple_group (G Mod carrier G)"
arity_generalize_fundef: fixes fd2 :: "('dyn, 'c, 'd, 'e, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" shows "arity (Subx.generalize_fundef fd2) = arity fd2"
cdf_bounded: fixes x :: "real" shows "cdf M x \<le> prob (space M)"
scale_zero_left: fixes x :: "('b, 'c) vec" shows "(0::'b) *s x = 0"
bal_dblk_q: shows "bal (dblock s q) \<in> Ballot q"
fd_map_split': fixes list :: "fd list" shows "map (\<lambda>(x, y). fd_def x y) (map (case_fd Pair) list) = list"
homr_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "homr G H = {h \<in> hom G H. h \<in> extensional (carrier G)}"
distinct_25: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list"   and x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "Cn x41 x42 x43 \<noteq> Pr x51 x52 x53"
vars_in_instance_in_range_cls: fixes C :: "('b, 'a) Term.term literal multiset"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "vars_clause (subst_cls C \<sigma>) \<subseteq> \<Union> (vars_term ` range \<sigma>)"
intros: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelR Enc"
assms_6: fixes t :: "'a"   and v :: "'t" shows "t \<oplus> v = splus (term_of_pair (t, the_min)) v"
cr: shows "length rr = card {x. poly (map_poly complex_of_real q) x = 0 \<and> x \<in> \<real>}"
case_1: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
Fourier_products_integrable_sin: fixes f :: "real \<Rightarrow> real"   and k :: "real" assumes "f absolutely_integrable_on {- pi..pi}" shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. sin (k * x) * f x)"
maxr_lg: fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "Maxr lgR [x, y] x = lg x y"
hypsubst: fixes A :: "'e set"   and B :: "'e set"   and x :: "'e" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
obs_finite: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and backward_slice :: "'g set \<Rightarrow> 'g set"   and n :: "'g"   and S :: "'g set" assumes "BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice" shows "finite (CFG.obs sourcenode targetnode valid_edge n (backward_slice S))"
D: shows "\<langle>n, Pwd n\<rangle> \<in> spied s'"
vector_mul_rcancel: fixes a :: "'a"   and x :: "('a, 'b) vec"   and b :: "'a" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
simps_3: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (Eq (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
big_union_def: fixes f :: "'d \<Rightarrow> 'c odlist"   and X :: "'d list" shows "big_union f X \<equiv> foldr (\<lambda>a. ODList.union (f a)) X ODList.empty"
link0_def: fixes i :: "Index" shows "link0 i = (if i = none then left0 else right0)"
bounded_by_bound_of_alt: fixes P :: "'a \<Rightarrow> real"   and a :: "real" assumes "Expectations.bounded P"   and "bound_of P = a" shows "bounded_by a P"
lim_proc_def: fixes X :: "'a process set" shows "lim_proc X = Abs_process (\<Inter> (F ` X), \<Inter> (D ` X))"
induct: fixes x1 :: "recf"   and x2 :: "nat list"   and P :: "recf \<Rightarrow> nat list \<Rightarrow> bool" assumes "terminate x1 x2"   and "\<And>n. P z [n]"   and "\<And>n. P s [n]"   and "\<And>n m xs. \<lbrakk>n < m; length xs = m\<rbrakk> \<Longrightarrow> P (recf.id m n) xs"   and "\<And>f xs gs n. \<lbrakk>terminate f (map (\<lambda>g. rec_exec g xs) gs); P f (map (\<lambda>g. rec_exec g xs) gs); \<forall>g\<in>set gs. terminate g xs \<and> P g xs; length xs = n\<rbrakk> \<Longrightarrow> P (Cn n f gs) xs"   and "\<And>x g xs n f. \<lbrakk>\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \<and> P g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]); terminate f xs; P f xs; length xs = n\<rbrakk> \<Longrightarrow> P (Pr n f g) (xs @ [x])"   and "\<And>xs n f r. \<lbrakk>length xs = n; terminate f (xs @ [r]); P f (xs @ [r]); rec_exec f (xs @ [r]) = 0; \<forall>i<r. (terminate f (xs @ [i]) \<and> P f (xs @ [i])) \<and> 0 < rec_exec f (xs @ [i])\<rbrakk> \<Longrightarrow> P (Mn n f) xs" shows "P x1 x2"
split: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> ('d, 'e) bintree list \<Rightarrow> 'c"   and bintree :: "('d, 'e) bintree" shows "P (case bintree of Node x xa xb \<Rightarrow> f x xa xb) = (\<forall>x1 x2 x3. bintree = Node x1 x2 x3 \<longrightarrow> P (f x1 x2 x3))"
E_ProjRcd: fixes fs :: "(char list \<times> trm) list"   and l :: "char list"   and v :: "trm" assumes "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>v\<rfloor>"   and "v \<in> value" shows "Rcd fs..l \<longmapsto> v"
E: shows "\<not> (\<exists>v\<in>sinks_aux I D (D ` set ys) zs. (v, D z) \<in> I) \<longrightarrow> ipurge_tr_rev_aux I D (insert (D z) U) (xs @ zs) = ipurge_tr_rev_aux I D (insert (D z) U) (xs @ ys @ ipurge_tr_aux I D (D ` set ys) zs)"
len_vFrom: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "|verticesFrom f v| = |vertices f|"
Seq_decomp_relpow: fixes n :: "nat"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and p' :: "('d, 'e, 'f) com"   and s' :: "'d" assumes "\<Gamma>\<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>nn (p', Normal s')"   and "final (p', Normal s')" shows "(\<exists>n1<n. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Throw, Normal s')) \<and> p' = Throw \<or> (\<exists>t n1 n2. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Skip, Normal t) \<and> n1 < n \<and> n2 < n \<and> \<Gamma>\<turnstile> (p\<^sub>2, Normal t) \<rightarrow>\<^sup>nn2 (p', Normal s'))"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "regex.pred_regex P = (\<lambda>x. Ball (regex.atms x) P)"
A_hom_mult: fixes x :: "nat discr"   and y :: "nat discr" shows "n2a\<cdot>(x *\<^sub># y) = n2a\<cdot>x >=> n2a\<cdot>y"
separable_impl_main_def: fixes p :: "int"   and ff_ops :: "'b arith_ops_record"   and f :: "int poly" shows "separable_impl_main p ff_ops f = separable_i ff_ops (of_int_poly_i ff_ops (poly_mod.Mp p f))"
insert_inter_insert: fixes a :: "'d"   and A :: "'d set"   and B :: "'d set" shows "insert a A \<inter> insert a B = insert a (A \<inter> B)"
is_finalI: fixes tp :: "cell list \<times> cell list" shows "is_final (0, tp)"
suc_greater: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}"   and "b \<in> B" shows "local.suc B \<noteq> b \<and> (b, local.suc B) \<in> r"
preserves_arr: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "B.arr (uncurry F f)"
simps_1: fixes uu :: "'b multiset" shows "repeat_mset 0 uu = {#}"
ge_def: fixes c :: "int"   and t :: "Ferrack.num" shows "ge c t = (if c = 0 then Ge t else if 0 < c then Ge (CN 0 c t) else Le (CN 0 (- c) (Neg t)))"
AE_iff_null_sets: fixes N :: "'b set"   and M :: "'b measure" assumes "N \<in> sets M" shows "(N \<in> null_sets M) = (AE x in M. x \<notin> N)"
preserves_modes_mem_def2: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and \<R> :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of" shows "sifum_refinement.preserves_modes_mem var\<^sub>C_of \<R> = (\<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (((c\<^sub>A, mds\<^sub>A), mem\<^sub>A), (c\<^sub>C, mds\<^sub>C), mem\<^sub>C) \<in> \<R> \<longrightarrow> mem\<^sub>A = sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>C \<and> mds\<^sub>A = sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C)"
image_subset_iff_subset_vimage: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" shows "(f ` A \<subseteq> B) = (A \<subseteq> f -` B)"
step_6: shows "InvariantWatchListsUniq (getWatchList state')"
scaleR2_rel_def: fixes A :: "('a \<times> ('b \<times> 'c) set) set" shows "\<langle>A\<rangle>scaleR2_rel = ((ereal_rel \<times>\<^sub>r ereal_rel) \<times>\<^sub>r A) O br (\<lambda>((l, u), X). scaleR2 l u X) (\<lambda>((l, u), uu_). ereal -` {l..u} \<noteq> {})"
step: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<lessapprox><weakBisim> Q"   and "\<Psi> \<rhd> P \<leadsto><weakBisim> Q"   and "\<forall>\<Psi>''. \<Psi> \<otimes> \<Psi>'' \<rhd> P \<approx> Q"   and "\<Psi> \<rhd> Q \<approx> P" shows "\<Psi> \<rhd> P \<approx> Q"
LVar: fixes G :: "prog"   and s :: "st"   and vn :: "lname" shows "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<rightarrow> Norm s"
inject_2: fixes b :: "'e"   and b' :: "'e" shows "(Inr b = Inr b') = (b = b')"
\<alpha>_is_revrev\<alpha>: shows "\<alpha> = rev rev\<alpha>"
zero_less_iff_neq_zero: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
1: fixes \<phi> :: "'b ltln" shows "atoms_ltln (rewrite_syn_imp (rewrite_modal (rewrite_X \<phi>))) \<subseteq> atoms_ltln \<phi>"
rep_eq: fixes x :: "'b fset" shows "fMin x = Min (fset x)"
zero_le_dist: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "0 \<le> dist x y"
ord_Field: shows "Field order_of_orders = {r. strict_ext base_r r}"
list_update_same_conv: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < length xs" shows "(xs[i := x] = xs) = (xs ! i = x)"
evenperm_inv: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "evenperm (inv p) = evenperm p"
assms_9: shows "start = None \<or> (\<exists>r. start = Some r \<and> r \<in> set (interp_ins ops))"
invar_def: fixes q :: "'b queue" shows "invar q = (lenf q = length (front_list q) \<and> lenr q = length (rear_list q) \<and> lenr q \<le> lenf q \<and> (case status q of Rev ok f f' r r' \<Rightarrow> 2 * lenr q \<le> length f' \<and> ok \<noteq> 0 | App ok f r \<Rightarrow> 2 * lenr q \<le> length r | _ \<Rightarrow> True) \<and> rem_steps (status q) \<le> 2 * length (front q) \<and> (\<exists>rest. front_list q = front q @ rest) \<and> (\<forall>x. status q \<noteq> Done x) \<and> st_inv (status q))"
allocationReverseInjective: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" shows "(a \<in> allAllocations N G) = (a\<inverse> \<in> injections (Range a) N \<and> Range a \<in> all_partitions G)"
fake_mono_chan: fixes chan :: "chan set"   and chan' :: "chan set"   and bad :: "agent set"   and IK :: "msg set" assumes "chan \<subseteq> chan'" shows "fake bad IK chan \<subseteq> fake bad IK chan'"
wp_arrayinit_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (CLEAR x[]) Q s = Q (s(x := \<lambda>_. 0))"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 t \<tau> t\<^sub>0 t\<^sub>1 (\<pi> \<cdot> \<mu>)"
tagged_division_ofD_5: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set"   and x1 :: "'b"   and K1 :: "'b set"   and x2 :: "'b"   and K2 :: "'b set" assumes "s tagged_division_of i"   and "(x1, K1) \<in> s"   and "(x2, K2) \<in> s"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
star_divide_def: shows "(div) \<equiv> *f2* (div)"
Unit_eq_dividesone: fixes u :: "'a" assumes "u \<in> carrier (Ring_Divisibility.mult_of R)" shows "(u \<in> Units (Ring_Divisibility.mult_of R)) = u divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> \<one>"
even_add: 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
that: fixes x :: "'a" assumes "x \<in> R" shows "flowsto (X0 x) T CX X1"
remove_singleton_from_tagged_division_of: fixes A :: "(real \<times> real set) set"   and a :: "real"   and b :: "real"   and x :: "real" assumes "A tagged_division_of {a..b}"   and "a < b"   and "x \<in> {a..b}" shows "\<exists>k c d. c < d \<and> (k, {c..d}) \<in> A \<and> x \<in> {c..d}"
init_exp_switch: fixes e\<^sub>0 :: "'a exp"   and e :: "'a exp"   and e' :: "'a exp" assumes "init_exp_of e\<^sub>0 = \<lfloor>e\<rfloor>" shows "init_exp_of (init_switch e\<^sub>0 e') = \<lfloor>e'\<rfloor>"
boundedI: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd b"   and "a dvd c" shows "a dvd gcd b c"
NegChecks: shows "a = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>"
inf2D2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "(A \<sqinter> B) x y" shows "B x y"
simps: fixes \<phi> :: "'b ltlc" shows "ltlc_to_ltln \<phi> = ltlc_to_ltln' False \<phi>"
jipsen2l: fixes y :: "'a"   and x :: "'a" shows "inv_resr (inv_resl y (inv_resr x y)) y = inv_resr x y"
nomatch': fixes rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and s :: "state" assumes "\<And>r. r \<in> set rs \<Longrightarrow> \<not> matches \<gamma> (get_match r) p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> s"
k: shows "((\<lambda>s. (1 - s) *\<^sub>R (h\<^sup>2 * p\<^sup>2) *\<^sub>R f'' (t + s * (h * p), x t + (s * (h * p)) *\<^sub>R f (t, x t)) $ (1, f (t, x t)) $ (1, f (t, x t))) has_integral f ((t, x t) + (h * p) *\<^sub>R (1, f (t, x t))) - f (t, x t) - f' (t, x t) $ (h * p, (h * p) *\<^sub>R f (t, x t))) {0..1}"
inject_5: fixes x6 :: "'b ptrm"   and y6 :: "'b ptrm" shows "(PFst x6 = PFst y6) = (x6 = y6)"
span_UNIV: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module.span scale UNIV = UNIV"
asTerm_def: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "asTerm qX \<equiv> proj alphaGood qX"
inject_4: fixes x41 :: "typ"   and x42 :: "Core.term"   and y41 :: "typ"   and y42 :: "Core.term" shows "(Abs x41 x42 = Abs y41 y42) = (x41 = y41 \<and> x42 = y42)"
assms_2: shows "A *\<^sub>m\<^sub>v v = A *\<^sub>m\<^sub>v v'"
6: shows "finite {e. f e \<noteq> bot \<and> f e + bot = fe + bot}"
top_def: shows "\<top> = (1::'a)\<^sup>\<omega>"
assms_3: fixes s :: "'a" assumes "I s"   and "b s" shows "f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> V)"
subgraph_def: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" shows "subgraph G\<^sub>1 G\<^sub>2 = (graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2)"
vbrelation_def: fixes r :: "V" shows "vbrelation r \<equiv> vpairs r = r"
that: fixes n' :: "'node"   and ns' :: "'node list" assumes "g \<turnstile> n'-ns'\<rightarrow>m"   and "P n'"   and "\<forall>x\<in>set (tl ns'). \<not> P x"   and "suffix ns' ns" shows "thesis"
red2_Diag: fixes a :: "'a local.term"   and b :: "'a local.term" assumes "Diag (a \<^bold>\<otimes> b)" shows "a \<^bold>\<Down> b = a \<^bold>\<otimes> b"
shiftr_div_2n: fixes w :: "'a word"   and n :: "nat" shows "uint (w >> n) = uint w div 2 ^ n"
subcls1_def2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "subcls1 P = (SIGMA C:{C. is_class P C}. {D. C \<noteq> Object \<and> fst (the (class P C)) = D})"
eq: shows "{..<z} \<union> {z<..} = - {z}"
funpow_decreasing: fixes m :: "nat"   and n :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "m \<le> n"   and "mono f" shows "(f ^^ m) bot \<le> (f ^^ n) bot"
fmdom_fmmerge: fixes g :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and xm :: "'e \<rightharpoonup>\<^sub>f 'g"   and ym :: "'e \<rightharpoonup>\<^sub>f 'h" shows "fmdom (fmmerge g xm ym) = fmdom xm |\<inter>| fmdom ym"
acomplete_Node_if_wbal2: fixes l :: "'b tree"   and r :: "'b tree"   and x :: "'b" assumes "acomplete l"   and "acomplete r"   and "\<bar>int (size l) - int (size r)\<bar> \<le> 1" shows "acomplete \<langle>l, x, r\<rangle>"
HEndPhase1_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "HEndPhase1 s s' p = (EndPhase1 s s' p \<and> HNextPart s s')"
ex_poincare_line_points: fixes H :: "circline" assumes "is_poincare_line H" shows "\<exists>u v. u \<in> unit_disc \<and> v \<in> unit_disc \<and> u \<noteq> v \<and> H = poincare_line u v"
F: fixes i :: "'b"   and x :: "'a" assumes "i \<in> Basis" shows "\<bar>f' x \<bullet> i\<bar> \<le> norm x * F i"
continuous_compose_quotient_map: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and X'' :: "'f topology"   and g :: "'e \<Rightarrow> 'f" assumes "quotient_map X X' f"   and "continuous_map X X'' (g \<circ> f)" shows "continuous_map X' X'' g"