hcju/isabelleps huggingface dataset:
inorder2_inorder: fixes t :: "'b tree"   and xs :: "'b list" shows "Tree.inorder2 t xs = inorder t @ xs"
implies_entails: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>s. P s \<Longrightarrow> Q s" shows "\<guillemotleft> P \<guillemotright> \<tturnstile> \<guillemotleft> Q \<guillemotright>"
Seq1: fixes c :: "com"   and s :: "char list \<Rightarrow> val" shows "(SKIP;; c, s) \<rightarrow> (c, s)"
Cons_3: shows "map_pmf (fst \<circ> snd) init = bv (length s0)"
pow_ex_1: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "pow_cond t n (length (bc (\<sigma>\<^bsub>MBC t n\<^esub>t n)))"
idem: fixes a :: "'g" shows "a \<squnion> a = a"
simps_11: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and x22 :: "'c list"   and y21 :: "'d"   and y22 :: "'d list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
finlsts_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A\<^sup>\<star> \<subseteq> B\<^sup>\<star>"
real_root_mult_exp: fixes m :: "nat"   and n :: "nat"   and x :: "real" shows "root (m * n) x = root m (root n x)"
le_less: fixes x :: "'b list"   and y :: "'b list" shows "prefix x y = (strict_prefix x y \<or> x = y)"
pa_invar_add_def: shows "pa_invar_add \<equiv> {(Q, W, \<delta>d). hs.invar Q \<and> ls.invar \<delta>d}"
face_cycle_pred_succ: fixes a :: "'b" shows "face_cycle_pred (face_cycle_succ a) = a"
bnd0: shows "\<forall>x\<in>set (remdups (uset (simpfm p))). bound0 (case x of (c, t) \<Rightarrow> msubst2 (simpfm p) (c *\<^sub>p (- 2)\<^sub>p) t)"
hconf_def: fixes P :: "'a prog"   and h :: "heap" shows "P \<turnstile> h \<surd> \<equiv> (\<forall>a obj. h a = \<lfloor>obj\<rfloor> \<longrightarrow> P,h \<turnstile> obj \<surd>) \<and> preallocated h"
ord_eq_less_subst: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a = f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
split_01: shows "{0..1} = {0..1 / 2} \<union> {1 / 2..1}"
FAccRedsNone: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"   and "hp s' a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>THROW NoSuchFieldError,s',b'\<rangle>"
rcp_specD2: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and h :: "('t \<Rightarrow>\<^sub>0 'e) \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and b :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "rcp_spec rcp"   and "h \<in> set (fst (rcp gs bs ps sps data))"   and "b \<in> set gs \<union> set bs"   and "fst b \<noteq> 0" shows "\<not> lt (fst b) adds\<^sub>t lt (fst h)"
evaln_max2: fixes G :: "prog"   and s1 :: "state"   and t1 :: "Term.term"   and n1 :: "nat"   and w1 :: "vals"   and s1' :: "state"   and s2 :: "state"   and t2 :: "Term.term"   and n2 :: "nat"   and w2 :: "vals"   and s2' :: "state" assumes "G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> (w1, s1')"   and "G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> (w2, s2')" shows "G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max n1 n2\<rightarrow> (w1, s1') \<and> G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max n1 n2\<rightarrow> (w2, s2')"
weakBisimE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "Q \<approx> P"
simps_3: fixes x31 :: "'a convnet"   and x32 :: "'a convnet"   and y31 :: "'a convnet"   and y32 :: "'a convnet" shows "(Pool x31 x32 = Pool y31 y32) = (x31 = y31 \<and> x32 = y32)"
eval_final_def: shows "eval_final \<equiv> \<Lambda> e. eval_work_final\<cdot>e\<cdot>Just\<cdot>Nothing"
backwards_c_res_stab: fixes ts :: "traffic"   and d :: "cars"   and n :: "nat"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "(ts \<^bold>\<midarrow>c( d, n ) \<^bold>\<rightarrow> ts') \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1 \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
Fg': shows "\<F> g' = {f\<^sub>1} \<union> set (replace f [f\<^sub>2] (faces g))"
mult_imp_div_pos_less: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and y :: "'e"   and x :: "'e"   and z :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less x (times z y)" shows "less (divide x y) z"
openin_PiE_gen: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "openin (product_topology X I) (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> finite {i \<in> I. S i \<noteq> topspace (X i)} \<and> (\<forall>i\<in>I. openin (X i) (S i)))"
induct: fixes P :: "'c \<Rightarrow> ('c \<times> nat) list \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "('c \<times> nat) list" assumes "\<And>x y n. P x [(y, n)]"   and "\<And>uu_. P uu_ []"   and "\<And>uu_ v vb vc. P uu_ (v # vb # vc)" shows "P a0 a1"
teng: shows "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x)) \<longlonglongrightarrow> 0"
plus_eq_infty_iff_enat: fixes m :: "enat"   and n :: "enat" shows "(m + n = \<infinity>) = (m = \<infinity> \<or> n = \<infinity>)"
assms: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "ps \<in> set pss" shows "standard_decomp k ps"
pair_fin_digraph_axioms: shows "pair_fin_digraph (subdivide G e w)"
tautology_monotonous: fixes C :: "'b Literal set"   and D :: "'b Literal set" assumes "tautology C"   and "C \<subseteq> D" shows "tautology D"
basis_def: fixes A :: "'a vec set" shows "KA.basis A = (KA.lin_indpt A \<and> KA.span A = mat_kernel A \<and> A \<subseteq> mat_kernel A)"
left_unique_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
assms_4: shows "token_run x (Suc n) \<notin> F"
ub1': fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "('c \<times> 'b) list"   and y :: "'b"   and a :: "'c"   and b :: "'b" assumes "Semilat A r f"   and "\<forall>(p, s)\<in>set S. s \<in> A"   and "y \<in> A"   and "(a, b) \<in> set S" shows "b \<sqsubseteq>\<^bsub>r\<^esub> map snd (filter (\<lambda>(p', t'). p' = a) S) \<Squnion>\<^bsub>f\<^esub> y"
set_mset_single: fixes b :: "'a" shows "set_mset {#b#} = {b}"
degree_def: fixes p :: "'a list" shows "degree p = length (pnormalize p) - 1"
S6: shows "mod7 (c_fst (Suc (Suc 3))) = 2"
strand_subst_hom_2: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "x # S \<cdot>\<^sub>s\<^sub>t \<theta> = (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) # (S \<cdot>\<^sub>s\<^sub>t \<theta>)"
SNF_first_divides: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and i :: "nat" assumes "Smith_normal_form_mat A"   and "A \<in> carrier_mat n m"   and "i < min (dim_row A) (dim_col A)" shows "A $$ (0, 0) dvd A $$ (i, i)"
classes_above_has_field: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and F :: "char list"   and b :: "staticb"   and t :: "ty"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C has F,b:t in C'" shows "P' \<turnstile> C has F,b:t in C'"
diff_frac_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "minus (divide x y) (divide w z) = divide (minus (times x z) (times w y)) (times y z)"
rel_gpv_conv_rel_gpv'': fixes A :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'d \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
simps_1: fixes A :: "agent" shows "invKey (pubK A) = priK A"
simps_2: fixes x31 :: "('a, 'b, 'c) cmd"   and x32 :: "('a, 'b, 'c) cmd"   and y31 :: "('a, 'b, 'c) cmd"   and y32 :: "('a, 'b, 'c) cmd" shows "(x31 ;; x32 = y31 ;; y32) = (x31 = y31 \<and> x32 = y32)"
mult_rel_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> R \<times> S"   and "(r', s') \<in> R \<times> S" shows "mult_rel (r / s) (r' / s') = (r \<cdot> r') / (s \<cdot> s')"
is_maxsplitter_reducing: fixes P :: "'b list \<Rightarrow> bool"   and splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list" assumes "is_maxsplitter P splitf" shows "reducing splitf"
comp_assoc: fixes h :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" shows "(h \<cdot>\<^sub>S g) \<cdot>\<^sub>S f = h \<cdot>\<^sub>S g \<cdot>\<^sub>S f"
lebesgue_measurable_imp_measurable_on_nnreal_UNIV: fixes u :: "'a \<Rightarrow> real" assumes "u \<in> borel_measurable lebesgue"   and "\<And>x. 0 \<le> u x" shows "u measurable_on UNIV"
continuous_at_within_sgn: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous (at a within s) f"   and "f a \<noteq> (0::'b)" shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
Wbis_transT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.Wbis tval aval indis"   and "(s, t) \<in> indis"   and "PL.transT tval aval (c, s) s'" obtains t' :: "'a" where "PL.MtransT tval aval (d, t) t'a__" and "(s', t'a__) \<in> indis"   | d' :: "('b, 'c) com" and t' :: "'a" where "PL.MtransC tval aval (d, t) (d', t')" and "(s', t') \<in> indis" and "PL_Indis.discr tval aval indis d'"
star_rtc_least: fixes x :: "'a ppath set"   and y :: "'a ppath set" assumes "pp_one \<union> x \<union> pp_prod y y \<subseteq> y" shows "pp_star x \<subseteq> y"
Sim: fixes \<Psi>' :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>', P, Q) \<in> Rel" shows "\<Psi>' \<rhd> P \<leadsto><Rel> Q"
mem_zmset_frontier: fixes x :: "'b"   and M :: "'b zmultiset" shows "(x \<in>#\<^sub>z zmset_of (mset_set (set_antichain (frontier M)))) = (x \<in>\<^sub>A frontier M)"
prob_exp_def: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "int pmf \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b" shows "prob_exp return bind fail get put sample \<equiv> memoization return bind get put \<and> monad_state_prob return bind get put sample \<and> monad_fail return bind fail"
angle_def: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "angle a b c = vangle (a - b) (c - b)"
that: fixes r :: "'a list"   and s :: "'a list"   and i :: "nat"   and n :: "nat" assumes "(r \<cdot> s) \<^sup>@ Suc i = x"   and "(s \<cdot> r) \<^sup>@ Suc i = y"   and "(r \<cdot> s) \<^sup>@ n \<cdot> r = z"   and "s \<noteq> \<epsilon>"   and "primitive (r \<cdot> s)" shows "thesis"
sig_trd_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "spp_of (sig_trd bs p) = sig_trd_spp (map spp_of bs) (spp_of p)"
assms: shows "k\<^sub>1 \<noteq> k\<^sub>2"
small_vint_le: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> int_of_vint a \<le> int_of_vint b}"
inf_sup_ord_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
simps_8: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>n \<psi> \<and> (\<forall>j<i. suffix j \<xi> \<Turnstile>\<^sub>n \<phi>))"
index_valid_def: fixes as :: "('c \<times> 'd atom) set"   and s :: "('c, 'd) state" shows "index_valid as s = (\<forall>x b i. (Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<longrightarrow> (i, Geq x b) \<in> as) \<and> (Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> (i, Leq x b) \<in> as))"
get_mode_locs_def: shows "get_mode_locs \<equiv> l_get_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_mode_locs"
OneRootState_Conf: fixes S :: "'d"   and ST :: "('d, 'e, 'f) status" assumes "S \<in> States (HARoot (HA ST))"   and "S \<in> Conf ST" shows "S = RootState ST"
map_of_clearjunk: fixes al :: "('c \<times> 'd) list" shows "map_of (clearjunk al) = map_of al"
induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b mat" assumes "\<And>k A. (\<And>x xa xb xaa. \<lbrakk>\<not> n \<le> k; x = ??.Jordan_Normal_Form_Existence.step_3_a (k - 1) k A; xa = ??.Jordan_Normal_Form_Existence.identify_blocks x k; xb = filter (\<lambda>block. x $$ (snd block, k) \<noteq> (0::'b)) xa; xaa = (if xb = [] then x else let (l_start, l) = ??.Jordan_Normal_Form_Existence.find_largest_block (hd xb) (tl xb); xa = x $$ (l, k); C = ??.Jordan_Normal_Form_Existence.step_3_c xa l k xb x; D = mult_col_div_row (inverse xa) k C; E = ??.Jordan_Normal_Form_Existence.swap_cols_rows_block (Suc l) k D in E)\<rbrakk> \<Longrightarrow> P (Suc k) xaa) \<Longrightarrow> P k A" shows "P a0 a1"
p2_def: shows "p2 = except p (- {u. deg_pm u < poly_deg p})"
in_set_list_format: fixes e :: "'b"   and l :: "'b list" assumes "e \<in> set l" obtains l1 :: "'b list" and l2 :: "'b list" where "l = l1 @ e # l2"
analz_symKeys_Decrypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "K \<in> symKeys"   and "Key K \<in> analz H" shows "X \<in> analz H"
base_base: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set"   and ds' :: "nat set" assumes "p \<in> sparsegrid' dm" shows "base ds (base ds' p) = base (ds \<union> ds') p"
contractible_space_subset_singleton: fixes X :: "'b topology"   and a :: "'b" assumes "topspace X \<subseteq> {a}" shows "contractible_space X"
hyps_3: shows "Abs bT s = subst_bv1 sa (length (T # Ts)) x"
shiftl_shiftr3: fixes b :: "nat"   and c :: "nat"   and a :: "'b word" assumes "b \<le> c" shows "a << b >> c = (a >> c - b) AND mask (size a - c)"
qe: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)"
set_eq_sym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier S"   and "B \<subseteq> carrier S"   and "A {.=} B" shows "B {.=} A"
oalist_inv_raw_Nil: fixes x :: "'o" shows "oalist_inv_raw x []"
dom_eval_expr: fixes t :: "nat"   and e :: "expr"   and x :: "nat" assumes "valid_sops_expr t e"   and "x \<in> fst (eval_expr t e)" shows "x \<in> {i. i < t} \<union> load_tmps (issue_expr t e)"
assms_15: shows "\<forall>a b. bf_between_arcs a b (forest_components h) d \<and> a \<le> - forest_components h \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum_class.sum (b \<sqinter> g) \<le> sum_class.sum (a \<sqinter> g)"
apfst_conv: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e"   and y :: "'d" shows "apfst f (x, y) = (f x, y)"
domain_def: fixes f :: "'b \<Rightarrow> ereal" shows "domain f = {x. f x < \<infinity>}"
emeasure_lim_emb: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> I"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>i. borel))" shows "emeasure local.lim (emb I J X) = emeasure (Pi\<^sub>M J (\<lambda>_. borel)) X"
height_balR2: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height r \<noteq> height l + 2" shows "height (balR l a r) = 1 + max (height l) (height r)"
Call_execParamD: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and obj :: "'addr expr1"   and M' :: "String.literal"   and ps :: "'addr expr1 list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and v :: "'addr val"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "exec_meth ci (compP2 P) (compE2 (obj\<bullet>M'(ps))) (compxE2 (obj\<bullet>M'(ps)) 0 0) t h (stk @ [v], loc, length (compE2 obj) + pc, xcp) ta h' (stk', loc', pc', xcp')"   and "pc < length (compEs2 ps)" shows "exec_meth ci (compP2 P) (compEs2 ps) (stack_xlift (length [v]) (compxEs2 ps 0 0)) t h (stk @ [v], loc, pc, xcp) ta h' (stk', loc', pc' - length (compE2 obj), xcp') \<and> length (compE2 obj) \<le> pc'"
paths_converge_prefix: fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and z :: "'node"   and y :: "'node"   and ys :: "'node list" assumes "g \<turnstile> x-xs\<rightarrow>z"   and "g \<turnstile> y-ys\<rightarrow>z"   and "x \<noteq> y"   and "1 < length xs"   and "1 < length ys"   and "x \<notin> set (butlast ys)"   and "y \<notin> set (butlast xs)" obtains xs' :: "'node list" and ys' :: "'node list" and z' :: "'node" where "old.pathsConverge g x xs' y ys' z'" and "prefix xs' xs" and "prefix ys' ys"
mult_right_dist_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y) \<odot> z = x \<odot> z \<oplus> y \<odot> z"
sign_r_pos_mult: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "sign_r_pos (p * q) x = (sign_r_pos p x = sign_r_pos q x)"
sorted_x1_l': shows "sorted_by cmp (x1 # l')"
constant_functor_axioms: shows "constant_functor (\<cdot>\<^sub>J) (\<cdot>) a'"
Notes_unique_PMS: fixes A :: "agent"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list"   and A' :: "agent"   and B' :: "agent" assumes "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "Notes A' \<lbrace>Agent B', Nonce PMS\<rbrace> \<in> set evs"   and "evs \<in> tls" shows "A = A' \<and> B = B'"
insert_dom: fixes f :: "'e \<Rightarrow> 'd option"   and x :: "'e"   and y :: "'d" assumes "f x = Some y" shows "insert x (dom f) = dom f"
validE_2: fixes x :: "vrs"   and T :: "ty"   and \<Gamma> :: "binding list" assumes "\<turnstile> (VarB x T # \<Gamma>) ok" obtains "\<turnstile> \<Gamma> ok" and "x \<sharp> trm_dom \<Gamma>" and "T closed_in \<Gamma>"
domain_eq: shows "Domainp pcr_proper_base = (\<le>) 2"
set_global: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (global \<C>)"   and "tg_t (global \<C> ! i) = t"   and "is_mut (global \<C> ! i)" shows "\<C> \<turnstile> [Set_global i] : [t] _> []"
bit_set_bit_iff: fixes m :: "nat"   and a :: "'b"   and n :: "nat" shows "bit (ring_bit_operations_class.set_bit m a) n = (bit a n \<or> m = n \<and> (2::'b) ^ n \<noteq> (0::'b))"
open_enat_def: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
that: fixes e :: "real" assumes "err = e * k"   and "e \<in> {- 1..1}" shows "thesis"
Pxy_def: fixes x :: "'b \<times> 'c" shows "Pxy x = (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M) else 0)"
e_def: fixes x :: "nat \<Rightarrow> complex"   and n :: "nat"   and k :: "nat" shows "power_sum_puzzle.e x n k = (\<Sum>Y | Y \<subseteq> {..<n} \<and> card Y = k. prod x Y)"
foldl_cong: fixes a :: "'c"   and b :: "'c"   and l :: "'d list"   and k :: "'d list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x" shows "foldl f a l = foldl g b k"
carrier_vecD: fixes v :: "'b Matrix.vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "dim_vec v = n"
inject_3: fixes x3 :: "'c"   and y3 :: "'c" shows "(In3 x3 = In3 y3) = (x3 = y3)"
rm_5: fixes x :: "int" assumes "0 < p"   and "0 \<le> x"   and "x ^ p = x" shows "y = x \<and> b"
above_sctn_nres: fixes X :: "'c set"   and sctn :: "'c sctn" shows "Inf_inner X (normal sctn) \<bind> (\<lambda>ii. RETURN (pstn sctn < ii)) \<le> above_sctn X sctn"
Lindelof_space_quotient_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and q :: "'c \<Rightarrow> 'd" assumes "quotient_map X Y q"   and "Lindelof_space X" shows "Lindelof_space Y"
connected_Union: fixes S :: "'c set set" assumes "\<And>s. s \<in> S \<Longrightarrow> connected s"   and "\<Inter> S \<noteq> {}" shows "connected (\<Union> S)"
\<psi>_def: shows "\<psi> \<equiv> \<lambda>d. card {x \<in> totatives n. ord n x = d}"
sequence_prio_emb_steps: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and prio :: "dir list \<Rightarrow> nat" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "\<exists>us. us \<noteq> [] \<and> hd us = t \<and> last us = s \<and> (\<forall>i. Suc i < length us \<longrightarrow> prio_emb_step prio (us ! i) s = us ! Suc i \<and> us ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b us ! Suc i)"
src_vcomp: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.seq \<nu> \<mu>" shows "S.src (\<nu> \<cdot>\<^sub>S \<mu>) = S.src \<nu>"
fds_scale_def: fixes c :: "nat"   and f :: "'b fds" shows "fds_scale c f = fds (\<lambda>n. if 0 < n \<and> is_nth_power c n then fds_nth f (nth_root_nat c n) else (0::'b))"
less_eq_set_def: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<le> (\<lambda>x. x \<in> B))"
SV_inv3_event_pres: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option" shows "{SV_inv3} sv_round r S v D {> SV_inv3}"