hcju/isabelleps huggingface dataset:
cod_char: fixes f :: "'a \<times> 'a" shows "VoV.cod f = (if VoV.arr f then VxV.cod f else VxV.null)"
complement_intro: fixes z :: "i"   and x :: "i" assumes "\<forall>w. P w z = (\<not> O w x)" shows "\<midarrow> x = z"
Scan_\<pi>_fix: fixes T :: "(symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" shows "Scan T k (\<pi> k T I) = \<pi> k T I"
retraction_strict: fixes f :: "'h \<rightarrow> 'g"   and g :: "'g \<rightarrow> 'h" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "f\<cdot>\<bottom> = \<bottom>"
antisym_conv1: fixes x :: "'b"   and y :: "'b" assumes "\<not> x < y" shows "(x \<le> y) = (x = y)"
interval_extension1D: fixes F :: "float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real"   and ivl :: "float interval"   and ivl' :: "float interval"   and x :: "real" assumes "interval_extension1 F f"   and "F ivl = Some ivl'"   and "x \<in>\<^sub>r ivl" shows "f x \<in>\<^sub>r ivl'"
exec_step_ind_New_Called: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and frs :: "frame list"   and sh :: "sheap" assumes "new_Addr h = \<lfloor>a\<rfloor>" shows "exec_step_ind (StepI (New C)) P h stk loc C\<^sub>0 M\<^sub>0 pc (Called Cs) frs sh (None, h(a \<mapsto> blank P C), (Addr a # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, No_ics) # frs, sh)"
thetaSeqTT_sym: shows "sym thetaSeqTT"
final_notinI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and T :: "('d, 'e) xstate set" assumes "\<And>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<Longrightarrow> t \<notin> T" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T"
ldrop_simps_3: fixes n :: "enat"   and x :: "'c"   and xs :: "'c llist" shows "ldrop (eSuc n) (LCons x xs) = ldrop n xs"
kth_obs_not_none: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and i :: "nat" assumes "is_kth_obs (path \<sigma>) k i" obtains a :: "'out" where "obsp \<sigma> i = Some a"
assms: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>x\<rparr> = r\<lparr>y\<rparr>" shows "x = y"
quot_Ex: fixes i :: "name"   and A :: "fm" shows "\<guillemotleft>SyntaxN.Ex i A\<guillemotright> = Q_Ex (quot_dbfm (trans_fm [i] A))"
N_def: shows "N = wN_of_wstate (llast Sts)"
shadow_root_ptrs_def: fixes heap :: "(_) heap" shows "shadow_root_ptrs heap = ffilter is_shadow_root_ptr (shadow_root_ptr_kinds heap)"
tdev_pdevs_of_ivl: fixes l :: "'b"   and u :: "'b" shows "tdev (pdevs_of_ivl l u) = \<bar>u - l\<bar> /\<^sub>R 2"
assms: shows "\<forall>X. 0 \<le> b (n, X)"
ide_dom: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.ide (Left_a.dom f)"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c set['e]"   and y :: "'d set['e]"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_bset R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_bset x; yb \<in> set_bset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_bset Ra x y"
contrapos_np: fixes Q :: "bool"   and P :: "bool" assumes "\<not> Q"   and "\<not> P \<Longrightarrow> Q" shows "P"
formulas_mono: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "formulas S \<subseteq> formulas T"
pathfinish_join: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" shows "pathfinish (g1 +++ g2) = pathfinish g2"
lex_take_index: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(xs, ys) \<in> lex r" obtains i :: "nat" where "i < length xs" and "i < length ys" and "take i xs = take i ys" and "(xs ! i, ys ! i) \<in> r"
componentsI: fixes x :: "'c"   and U :: "'c set" assumes "x \<in> U" shows "connected_component_set U x \<in> components U"
bij_swap: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "bijective z"   and "bijective x" shows "(z \<le> y\<^sup>T ; x) = (x \<le> y ; z)"
flatten_minus: fixes p :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" shows "flatten (p - q) = flatten p - flatten q"
strict_trans2: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "lt b a"   and "le c b" shows "lt c a"
hide_interleave: fixes A :: "'a event set"   and C :: "'a event set"   and r :: "'a event list"   and t :: "'a event list"   and u :: "'a event list" assumes "A \<inter> C = {}"   and "r setinterleaves ((t, u), C)" shows "trace_hide r A setinterleaves ((trace_hide t A, trace_hide u A), C)"
distinct_257: shows "BinOr \<noteq> ShiftRightZeros"
le_zero_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
len_comp: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and a :: "nat"   and c :: "nat"   and xs :: "nat list"   and b :: "nat"   and ys :: "nat list" shows "len m a c (xs @ b # ys) = len m a b xs + len m b c ys"
basic_just_sp_pred_def: fixes G :: "('c, 'd) pre_digraph"   and dist :: "'c \<Rightarrow> ereal"   and c :: "'d \<Rightarrow> real"   and s :: "'c"   and num :: "'c \<Rightarrow> enat"   and pred :: "'c \<Rightarrow> 'd option" shows "basic_just_sp_pred G dist c s num pred \<equiv> basic_sp G dist c s \<and> basic_just_sp_pred_axioms G dist c s num pred"
free_vars_cexpr_subst: fixes e :: "cexpr"   and e' :: "cexpr" shows "free_vars (cexpr_subst 0 e e') \<subseteq> Suc -` free_vars e' \<union> free_vars e"
extend_1_evals: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es'' :: "char list exp list"   and s'' :: "heap \<times> sfields \<times> sheap"   and b'' :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es'',s'',b''\<rangle>"   and "P,shp s \<turnstile>\<^sub>b (es,b) \<surd>"   and "P \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
rotate_diff: fixes m :: "nat"   and xs :: "'a list"   and n :: "nat" assumes "rotate m xs = rotate n xs" shows "rotate (m - n) xs = xs"
ord_spmf_expand: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "NO_MATCH (=) R" shows "ord_spmf R = rel_spmf R OO ord_spmf (=)"
spectral_radius_def: fixes A :: "complex mat" shows "Spectral_Radius.spectral_radius A = Max (cmod ` spectrum A)"
symD: fixes r :: "('c \<times> 'c) set"   and b :: "'c"   and a :: "'c" assumes "sym r"   and "(b, a) \<in> r" shows "(a, b) \<in> r"
dist_fun_def: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd" shows "dist x y = (\<Sum>n. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
set_Cons_def: fixes A :: "'c set"   and XS :: "'c list set" shows "set_Cons A XS = {x # xs |x xs. x \<in> A \<and> xs \<in> XS}"
right_hcomp_closed: fixes \<mu> :: "'a"   and x :: "'a"   and y :: "'a"   and \<nu> :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>\<mu> : x \<Rightarrow>\<^sub>S y\<guillemotright>"   and "\<guillemotleft>\<nu> : c \<Rightarrow> d\<guillemotright>"   and "\<mu> \<star> \<nu> \<noteq> null" shows "\<guillemotleft>\<mu> \<star> \<nu> : x \<star> c \<Rightarrow>\<^sub>S y \<star> d\<guillemotright>"
tab_simps_4: shows "D.dom (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho>) = F g"
Quotient3_rel_rep: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and a :: "'c"   and b :: "'c" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"
m_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "x \<oplus>\<^bsub>UP INTEG\<^esub> y \<in> carrier (UP INTEG)"
map_of_is_SomeI: fixes xys :: "('e \<times> 'f) list"   and x :: "'e"   and y :: "'f" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = \<lfloor>y\<rfloor>"
lemma_1_reachability_s_i: fixes s :: "('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "s \<in> valid_states PROB" shows "s \<in> reachable_s PROB s"
wf_below_mult2_s_local: fixes s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "wf_below s (set_mset X)"   and "s O ns \<subseteq> s"   and "refl ns"   and "trans ns" shows "wf_below (mult2_s ns s) {X}"
get_disconnected_nodes_new_document_different_pointers: fixes new_document_ptr :: "(_) document_ptr"   and ptr' :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "new_document_ptr \<noteq> ptr'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
exhaust_sel: fixes sctn :: "'a sctn" obtains "sctn = Sctn (normal sctn) (pstn sctn)"
simps_1: fixes s :: "'a" shows "tLast (tInit s) = s"
dtor_set: fixes z :: "'b tree" shows "set_tree z = set1_pre_tree (dtor_tree z) \<union> \<Union> (set_tree ` set2_pre_tree (dtor_tree z))"
I: fixes n :: "nat" assumes "0 < n" shows "B n \<le> P * 2 ^ (2 * k) * real (n ^ (2 * k))"
tab'_vn: shows "(tab'(x # xs [\<mapsto>] ys)) vn = None"
mod_by_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"
det_identical_columns: fixes j :: "'a"   and k :: "'a"   and A :: "(('b, 'a) vec, 'a) vec" assumes "j \<noteq> k"   and "column j A = column k A" shows "det A = (0::'b)"
not_G_isolated_a: shows "{tail G a, head G a} \<inter> G.isolated_verts = {}"
ker: shows "a_kernel (K [X]) R (\<lambda>p. local.eval p x) = PIdl\<^bsub>K [X]\<^esub> q"
sim_rfunc_usr: fixes p :: "'s_rest spmf"   and q :: "'s_rest' spmf"   and iu :: "'iusr_rest"   and s1 :: "'s_rest"   and s2 :: "'s_rest'"   and ou :: "'ousr_rest \<times> 'event list"   and s1' :: "'s_rest"   and s2' :: "'s_rest'" assumes "S p q"   and "iu \<in> IU"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(ou, s1') \<in> set_spmf (rfunc_usr rest1 s1 iu)"   and "(ou, s2') \<in> set_spmf (rfunc_usr rest2 s2 iu)" shows "S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_usr rest2 s2 iu)) ou)"
WTrtNil: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "P,E,h \<turnstile> [] [:] []"
induct: fixes x1 :: "exp"   and x2 :: "exp"   and P :: "exp \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 \<in> isubexprs(x2)"   and "\<And>e fi. P e (FieldProj e fi)"   and "\<And>e m es. P e (MethodInvk e m es)"   and "\<And>ei es e m. ei \<in> set es \<Longrightarrow> P ei (MethodInvk e m es)"   and "\<And>ei es C. ei \<in> set es \<Longrightarrow> P ei (New C es)"   and "\<And>e C. P e (Cast C e)" shows "P x1 x2"
minus_eq: fixes x :: "'a set"   and y :: "'a set" shows "x \<ominus>\<^bsub>R Quot I\<^esub> y = x \<oplus>\<^bsub>R Quot I\<^esub> \<ominus>\<^bsub>R Quot I\<^esub> y"
N: shows "decides_subsets (f -` {0} \<inter> \<F>) N"
plus_left_id: fixes x :: "'a" shows "zer + x = x"
assms_2: shows "insert (All2 i t A) (insert (A(i::=t)) H) \<turnstile> B"
outer_measure_of_eq: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "outer_measure_of M A = emeasure M A"
word_less_def: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a < b) = (uint a < uint b)"
assms_3: shows "Inr (Insert m) = create_insert s n \<sigma> new_id"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_real1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta>) \<turnstile>g absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>)))) \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>))))) (obsf_resource (sim1 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal1 \<eta>)) (obsf_resource (real1 \<eta>)))"
finite_UNIV_bit1_def: shows "finite_UNIV = Phantom('a bit1) True"
abs_eq: shows "(\<le>) = (\<lambda>x y. Rep_atoms x \<subseteq> Rep_atoms y)"
mult_minus1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus one) z = uminus z"
of_nat_le_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
invol_dual_var: fixes dual :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.dual dual" shows "dual (dual x) = x"
convergence_G: fixes x :: "'s"   and y :: "'s"   and z :: "'a" assumes "convergence_G x y z" shows "summable (\<lambda>n. p x y n *\<^sub>R z ^ n)"
simps_1: fixes a :: "'a" shows "a\<^bsup>0\<^esup> = (1::'a)"
sym: fixes x :: "'a"   and y :: "'a" assumes "\<not> x \<bowtie> y"   and "x \<in> A"   and "y \<in> A" shows "\<not> y \<bowtie> x"
map2_val_compat'D2: fixes f :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'f) list \<times> 'o"   and zs :: "('a \<times> 'e) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "snd (f zs) = snd zs"
natLeq_on_ofilter_iff: fixes m :: "nat"   and A :: "nat set" shows "ofilter (natLeq_on m) A = (\<exists>n\<le>m. A = {0..<n})"
cons_post_rule: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and Q' :: "'d \<Rightarrow> assn" assumes "<P> c <Q>"   and "\<And>x. Q x \<Longrightarrow>\<^sub>A Q' x" shows "<P> c <Q'>"
le_imp_add: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.semi_canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" shows "\<exists>c. b = plus a c"
utility_le: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "u x \<le> u y"
seq_INF_distrib_UNIV: fixes c :: "'a::refinement_lattice"   and d :: "'d::type \<Rightarrow> 'a::refinement_lattice" shows "(c::'a::refinement_lattice) ; \<Sqinter> range (d::'d::type \<Rightarrow> 'a::refinement_lattice) = (\<Sqinter>x::'d::type. c ; d x)"
arctan_upper_41_def: shows "arctan_upper_41 \<equiv> \<lambda>x. - (pi / 2) - (256 + 5943 * x\<^sup>2 + 19250 * x ^ 4 + 15015 * x ^ 6) / (35 * x * (35 + 315 * x\<^sup>2 + 693 * x ^ 4 + 429 * x ^ 6))"
consistent_sign_vec_def: fixes qs :: "real poly list"   and x :: "real" shows "consistent_sign_vec qs x \<equiv> map (\<lambda>q. if 0 < poly q x then 1 else if poly q x = 0 then 0 else - 1) qs"
rep_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "partial_magma.in_hom V (bicategory_of_spans.rep V H \<a> \<i> src trg r) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg r) (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg r))) r"
exhaust_sel: fixes dual :: "'b dual" obtains "dual = \<partial> (\<partial>\<^sup>- dual)"
arr_char: fixes f :: "'d \<times> 'd" shows "D'.VV.arr f = (D'.arr (fst f) \<and> D'.arr (snd f) \<and> D'.src (fst f) = D'.trg (snd f))"
setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst'': fixes t :: "('c, 'd) Term.term \<times> ('c, 'd) Term.term"   and b :: "('c, 'd) stateful_strand_step"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "t \<in> setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)" shows "\<exists>s\<in>setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p b. t = s \<cdot>\<^sub>p rm_vars (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b)) \<delta>"
simps_2: fixes x :: "'a" shows "merge_option (Some x, None) = Some x"
cl_inv_to_cl_inv': fixes t :: "('a, 'b) rbt" assumes "cl_inv t" shows "cl_inv' t"
assms_2: fixes x :: "real" assumes "b \<le> x" shows "u \<le> f x"
mini_to_mini: fixes F :: "'b formula" assumes "is_mini_formula F" shows "to_mini_formula F = F"
cond_def: fixes p :: "bool Heap"   and h :: "heap" shows "cond p h = fst (the (execute p h))"
InitState_notmem_States: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto"   and T :: "'a" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)"   and "T \<in> HAInitStates A"   and "T \<noteq> InitState SA" shows "T \<notin> States SA"
thetaSeqZOT_def: shows "thetaSeqZOT \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>01T d1 \<and> c2 \<approx>01T d2}"
ListRedsVal: fixes e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list"   and es' :: "char list exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<rightarrow>]* \<langle>Val v # es',s\<^sub>2\<rangle>"
algebraic_plus: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "algebraic y" shows "algebraic (x + y)"
is_scalar_prod_def: shows "is_scalar = (\<lambda>_. False)"
sup_bot_right: fixes x :: "'a" shows "sup_class.sup x bot_class.bot = x"
trace_core_eq_def: fixes core1 :: "('h, 'i, 'j, 'k, 'l, 'm) core"   and core2 :: "('n, 'i, 'j, 'k, 'l, 'm) core"   and E :: "'i set"   and IA :: "'j set"   and IU :: "'k set"   and p :: "'h spmf"   and q :: "'n spmf" shows "trace_core_eq core1 core2 E IA IU p q = (\<forall>tr. set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV \<longrightarrow> rel_prod (rel_fun (eq_onp (\<lambda>e. e \<in> E)) (=)) (rel_prod (rel_fun (eq_onp (\<lambda>ia. ia \<in> IA)) (=)) (rel_fun (eq_onp (\<lambda>iu. iu \<in> IU)) (=))) (trace_core' core1 p tr) (trace_core' core2 q tr))"
A': shows "OpposedThinChamberComplexFoldings X f' g' A"
i_def: shows "i \<equiv> if u \<in> Basis then u else - u"
fplus_def: shows "fplus = COERCE(udom\<cdot>'c \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c, 'b\<cdot>'c \<rightarrow> 'b\<cdot>'c \<rightarrow> 'b\<cdot>'c)\<cdot>plusU"
R23sD: fixes implem :: "chan \<Rightarrow> msg"   and s :: "l2_state"   and t :: "l3_state" assumes "dhlvl3 implem"   and "(s, t) \<in> dhlvl3.R23s implem" shows "ik s = ik t \<inter> payload \<and> secret s = secret t \<and> progress s = progress t \<and> signalsInit s = signalsInit t \<and> signalsResp s = signalsResp t \<and> chan s = semivalid_implem.abs implem (ik t) \<and> l2_state.bad s = l3_state.bad t"
subcocycle_avg_subadditive: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u" shows "Fekete.subadditive (\<lambda>n. integral\<^sup>L M (u n))"
injectively_orientable_orientable: fixes x :: "'a" assumes "injectively_orientable x" shows "orientable x"
simps: fixes k :: "nat"   and n :: "nat" shows "bools_of_nat k n = (if n = 0 then if k = 0 then [] else False # bools_of_nat (k - 1) n else (n mod 2 = 1) # bools_of_nat (k - 1) (n div 2))"
ennexp_add_mult: fixes a :: "ereal"   and b :: "ereal" assumes "\<not> (a = \<infinity> \<and> b = - \<infinity> \<or> a = - \<infinity> \<and> b = \<infinity>)" shows "ennexp (a + b) = ennexp a * ennexp b"
smc_in_Obj_in_Vset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
B4b: shows "\<^bold>\<forall>x xa. cod x \<cdot> xa \<cong> cod x \<cdot> cod xa"
insert_iff: fixes a :: "'l"   and b :: "'l"   and A :: "'l set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
less_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y < z" shows "x < z"
STL4_gen: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "\<turnstile> Init F \<longrightarrow> Init G" shows "\<turnstile> \<box>F \<longrightarrow> \<box>G"
simps: fixes \<rho>s :: "(cell \<times> interface) list" shows "com_sdia (SDiagram \<rho>s) = foldr (;;) (map (com_cell \<circ> fst) \<rho>s) Skip"
irreducible\<^sub>dE: fixes p :: "'b poly" assumes "irreducible\<^sub>d p" obtains "0 < degree p" and "\<And>q r. \<lbrakk>degree q < degree p; degree r < degree p\<rbrakk> \<Longrightarrow> p \<noteq> q * r"
inv_set_fset: fixes M :: "'c set" assumes "finite M" shows "set (inv set M) = M"
MkArr_in_hom: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'e" assumes "A \<in> {A. |A| <o \<AA>}"   and "B \<in> {A. |A| <o \<AA>}"   and "f \<in> extensional A \<inter> (A \<rightarrow> B)" shows "\<guillemotleft>MkArr A B f : MkIde A \<rightarrow> MkIde B\<guillemotright>"
main_1: fixes t :: "'a bro" assumes "t \<in> B (Suc h)" shows "fib' (Suc h) \<le> size t"
abs_eq: shows "0 = Abs_poly (\<lambda>_. 0::'b)"
iconfig_def: fixes tr :: "'a list \<times> 'b" shows "iconfig tr \<equiv> hd (fst tr)"
prod_algebra_cong: fixes I :: "'d set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and N :: "'d \<Rightarrow> 'e measure" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "prod_algebra I M = prod_algebra J N"
simps_2: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and s :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.mset_lit (Neg (Eq t s)) = {#t, t, s, s#}"
concl_i_in: shows "concl_of \<iota> \<in> N"