hcju/isabelleps huggingface dataset:
real_cond_exp_nested_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and G :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "subalgebra M G"   and "subalgebra G F"   and "integrable M f" shows "AEeq M (real_cond_exp M F (real_cond_exp M G f)) (real_cond_exp M F f)"
suicide_aux_l: fixes Y :: "'a list set"   and X :: "'a list set"   and n :: "nat" assumes "\<forall>w\<in>Y. 0 \<le> |w|"   and "\<forall>w\<in>X\<^bsup>Suc n\<^esup>. n \<le> |w|" shows "\<forall>w\<in>X\<^bsup>Suc n\<^esup> \<cdot> Y. n \<le> |w|"
jordan_block_index_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and a :: "'b" assumes "i < n"   and "j < n" shows "jordan_block n a $$ (i, j) = (if i = j then a else if Suc i = j then 1::'b else (0::'b))"
subset_UNIV: fixes A :: "'f set" shows "A \<subseteq> UNIV"
\<eta>: shows "\<guillemotleft>\<eta> : trg f\<^sub>a \<Rightarrow> f\<^sub>a \<star> f\<guillemotright> \<and> \<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g] = g \<star> \<eta> \<and> (\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sub>a, f]) \<cdot> (f \<star> \<eta>) = \<r>[f]"
T_size_tree_Leaf: shows "T_size_tree \<langle>\<rangle> = 1"
one_closed: shows "\<one>\<^bsub>DirProds (f \<circ> Gs) I\<^esub> \<in> carrier (DirProds (f \<circ> Gs) I)"
SUP_absorb: 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 k :: "'f"   and I :: "'f set"   and A :: "'f \<Rightarrow> 'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "k \<in> I" shows "sup (A k) (Sup (A ` I)) = Sup (A ` I)"
symmetry_functor_axioms: shows "symmetry_functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>A\<^sub>1)"
generate_incl: fixes H :: "int set" assumes "H \<subseteq> carrier (Z 1)" shows "generate (Z 1) H \<subseteq> carrier (Z 1)"
inj_on_diff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f A" shows "inj_on f (A - B)"
is_dyn_operand_eq_typeof_Dyn: fixes x :: "('a, 'b, 'c) unboxed" shows "is_dyn_operand x = (typeof x = None)"
is_sup_uniq: fixes x :: "'a"   and y :: "'a"   and s :: "'a"   and s' :: "'a" assumes "is_sup x y s"   and "is_sup x y s'" shows "s = s'"
Min_disc_is_scc_root: fixes v :: "'v"   and scc :: "'v set" assumes "v \<in> scc"   and "v \<in> dom (discovered s')"   and "is_scc E scc"   and "\<delta> s' v = Min (\<delta> s' ` (scc \<inter> dom (discovered s')))" shows "scc_root s' v scc"
zeroes_ni_tl_gen2: fixes m :: "nat"   and n :: "nat"   and A :: "nat"   and B :: "nat" shows "(zeroes m, zeroes n) \<notin> set (tl (gen2 A B m n))"
poly_mapping_eq_zeroI: fixes s :: "'c \<Rightarrow>\<^sub>0 'd" assumes "keys s = {}" shows "s = 0"
guard_simps_2: fixes b :: "bool" shows "(guard b = None) = (\<not> b)"
degree_gt: fixes x :: "'a pdevs"   and j :: "nat" assumes "pdevs_apply x j \<noteq> (0::'a)" shows "j < degree x"
C_eq_All_untilSorted: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = C p"
usubst_I: fixes p :: "fm"   and n :: "int"   and t :: "Ferrack.num"   and x :: "real"   and bs :: "real list" assumes "isrlfm p"   and "0 < real_of_int n"   and "numbound0 t" shows "Ifm (x # bs) (usubst p (t, n)) = Ifm (Inum (x # bs) t / real_of_int n # bs) p \<and> bound0 (usubst p (t, n))"
lincomb_x: shows "Berlekamp_subspace.lincomb a B = x"
dismantle_simp: fixes s :: "trm"   and y :: "name"   and n :: "trm"   and L :: "stack" shows "s \<star> [y]n\<ggreater>L = (s to y in n) \<star> L"
bits_mod_div_trivial: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (modulo a b) b = zero"
finprod_closed_subgroup: fixes H :: "'a set"   and f :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "subgroup H G"   and "f \<in> A \<rightarrow> H" shows "finprod G f A \<in> H"
append: fixes xs :: "'c list"   and ys :: "'c list" shows "prod_list (xs @ ys) = prod_list xs * prod_list ys"
IN_h'_outside: fixes x :: "'v" assumes "x \<notin> \<^bold>V" shows "d_IN h' x = 0"
Limit_eq_Sup_self: fixes i :: "V" assumes "Limit i" shows "\<Squnion> (elts i) = i"
backflows_minus_backflows: fixes X :: "('b \<times> 'b) set"   and Y :: "('b \<times> 'b) set" shows "backflows (X - backflows Y) = backflows X - Y"
A: fixes n :: "nat" shows "\<bar>real n * additive_strength f xi + real n * additive_strength (inv f) xi\<bar> \<le> 6 * deltaG TYPE('a)"
assms_1: shows "\<exists>y. (f \<longlongrightarrow> y) F"
f'_def: shows "f' = (\<lambda>x. (1 - 4 * ln x) / x\<^sup>2 / ln 2)"
length_aforms_of_ivls: fixes a :: "real list"   and b :: "real list" shows "length (aforms_of_ivls a b) = min (length a) (length b)"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd"   and xa :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup (x - xa) = (\<lambda>k. lookup x k - lookup xa k)"
icount_impl_def: shows "icount_impl \<equiv> \<lambda>xi. Array.new (xi + 1) 0 \<bind> (\<lambda>x'. heap_array_set x' 0 1 \<bind> (\<lambda>x. heap_array_set x 1 1 \<bind> (\<lambda>x. heap_array_set x 2 1 \<bind> (\<lambda>x. heap_array_set x 3 2 \<bind> (\<lambda>x'b. imp_for' 4 (xi + 1) (\<lambda>xf \<sigma>. imp_for' 3 xf (\<lambda>xj \<sigma>'''. Array.nth \<sigma> (xf - xj - 1) \<bind> (\<lambda>x'i. return (\<sigma>''' + x'i))) 0 \<bind> (\<lambda>x'f. Array.nth \<sigma> (xf - 1) \<bind> (\<lambda>x. heap_array_set \<sigma> xf (x + 1 + x'f)))) x'b \<bind> (\<lambda>x'd. Array.nth x'd xi))))))"
map_cong0: fixes x :: "'g list"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "\<And>z. z \<in> set x \<Longrightarrow> f z = g z" shows "map f x = map g x"
symmetric: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). x \<sim> y}" shows "(b, a) \<in> {(x, y). x \<sim> y}"
assms_3: shows "(u, t) \<in> r\<^sup>\<leftrightarrow>\<^sup>*"
m_c_def: fixes m :: "'b \<Rightarrow> 'a"   and c :: "'b acom" shows "m_c m c = (let as = annos c in \<Sum>i = 0..<length as. m (as ! i))"
in_pdevs_domain: fixes i :: "nat"   and x :: "'b pdevs" shows "(i \<in> pdevs_domain x) = (pdevs_apply x i \<noteq> (0::'b))"
set_fold_remove1_distinct: fixes xs :: "'b list"   and ys :: "'b list" assumes "distinct xs" shows "set (fold remove1 ys xs) = set xs - set ys"
countableE: fixes S :: "'f set" assumes "countable S" obtains f :: "'f \<Rightarrow> nat" where "inj_on f S"
induct: fixes G :: "prog"   and x1 :: "state"   and x2 :: "state"   and P :: "state \<Rightarrow> state \<Rightarrow> bool" assumes "G\<turnstile>x1 \<midarrow>sxalloc\<rightarrow> x2"   and "\<And>s. P (Norm s) (Norm s)"   and "\<And>j s. P (Some (Jump j), s) (Some (Jump j), s)"   and "\<And>e s. P (Some (Error e), s) (Some (Error e), s)"   and "\<And>a s. P (Some (Xcpt (Loc a)), s) (Some (Xcpt (Loc a)), s)"   and "\<And>s0 xn a x s1. G\<turnstile>Norm s0 \<midarrow>halloc CInst (SXcpt xn)\<succ>a\<rightarrow> (x, s1) \<Longrightarrow> P (Some (Xcpt (Std xn)), s0) (Some (Xcpt (Loc a)), s1)" shows "P x1 x2"
unlabel_mem_has_label: fixes x :: "'a"   and A :: "('b \<times> 'a) list" assumes "x \<in> set (unlabel A)" shows "\<exists>l. (l, x) \<in> set A"
\<Delta>\<Phi>_pass1: fixes hs :: "'b tree" shows "\<Phi> (pass\<^sub>1 hs) - \<Phi> hs \<le> 2 * log 2 (real (size hs + 1)) - real (len hs) + 2"
eq_setFinal_iff: fixes f :: "face" shows "(f = setFinal f) = final f"
abs_sgn: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'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 a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn (abs a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
wf_ineq_append'': fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t V (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
rec_swap_def: fixes f :: "recf" shows "rec_swap f = CN f [recf.Id 2 1, recf.Id 2 0]"
p_def: fixes x :: "'a" shows "p x = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
combine_options_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b option"   and y :: "'b option" shows "combine_options f x y = (case x of None \<Rightarrow> y | Some x \<Rightarrow> case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y))"
tdev_nonneg: fixes X :: "'a pdevs" shows "(0::'a) \<le> tdev X"
eqn: shows "lst \<tau>3'' = lst \<sigma>3''"
complete_ht_update_rehash: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and k :: "'a"   and v :: "'b" assumes "length l * load_factor \<le> the_size ht * 100" shows "<is_hashtable l ht> hm_update k v ht <\<lambda>r. is_hashtable l ht * is_hashtable (abs_update k v (ls_rehash l)) r>"
antisym_conv: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "(y \<le> x) = (x = y)"
interval_not_empty: fixes a :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "cbox a b \<noteq> {}"
Cauchy_convergent: fixes X :: "nat \<Rightarrow> 'b" assumes "Cauchy X" shows "convergent X"
residual_eq_iff_perm: fixes \<alpha>1 :: "'a"   and P1 :: "'b"   and \<alpha>2 :: "'a"   and P2 :: "'b" shows "(\<langle>\<alpha>1,P1\<rangle> = \<langle>\<alpha>2,P2\<rangle>) = (\<exists>p. supp (\<alpha>1, P1) - bn \<alpha>1 = supp (\<alpha>2, P2) - bn \<alpha>2 \<and> (supp (\<alpha>1, P1) - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> (\<alpha>1, P1) = (\<alpha>2, P2) \<and> p \<bullet> bn \<alpha>1 = bn \<alpha>2)"
contf: shows "continuous_on ({x. norm x = 1 \<and> b \<bullet> x \<noteq> 1} \<inter> T) f"
assms: assumes "c \<noteq> (0::'a)" shows "f abs_summable_on A"
permChainFresh_1: fixes x :: "name"   and xvec :: "name list"   and yvec :: "name list"   and p :: "'a" assumes "x \<sharp> xvec"   and "x \<sharp> yvec"   and "length xvec = length yvec" shows "x \<sharp> [xvec yvec] \<bullet>\<^sub>v p = x \<sharp> p"
index_map_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and f :: "'a \<Rightarrow> 'b" assumes "i < dim_row A"   and "j < dim_col A" shows "map_mat f A $$ (i, j) = f (A $$ (i, j))"
it_to_it_rev_linord_correct: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and S :: "'a set" assumes "set_iterator_rev_linord it S" shows "set_iterator_rev_linord (it_to_it it) S"
productive_imp_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "ground_resolution_with_selection.Interp S N C \<Turnstile> C"
DirProd_iso_set_trans: fixes g :: "'a \<Rightarrow> 'i"   and G2 :: "('i, 'j) monoid_scheme"   and h :: "'k \<Rightarrow> 'l"   and H :: "('k, 'm) monoid_scheme"   and I :: "('l, 'n) monoid_scheme" assumes "g \<in> Group.iso G G2"   and "h \<in> Group.iso H I" shows "(\<lambda>(x, y). (g x, h y)) \<in> Group.iso (G \<times>\<times> H) (G2 \<times>\<times> I)"
rcos_module_imp: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and x' :: "'c" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> H #>\<^bsub>G\<^esub> x" shows "x' \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H"
thetaParWL1_Wretr: shows "thetaParWL1 \<subseteq> Wretr (thetaParWL1 \<union> Wbis)"
rel_mset_via_perm: fixes rel :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list" shows "rel_mset rel (mset xs) (mset ys) = (\<exists>zs. xs <~~> zs \<and> list_all2 rel zs ys)"
honest_never_says_newer_timestamp_in_auth: fixes evs :: "event list"   and T :: "nat"   and X :: "msg"   and A :: "agent"   and B :: "agent"   and Y :: "msg" assumes "CT evs \<le> T"   and "Number T \<in> parts {X}"   and "A \<notin> bad"   and "evs \<in> kerbV" shows "Says A B \<lbrace>Y, X\<rbrace> \<notin> set evs"
rstar_increasing: fixes y :: "'a" assumes "has_least_prefixpoint (Rf y)" shows "y \<le> rstar y"
inject_4: fixes x41 :: "'d regex"   and x42 :: "'d regex"   and y41 :: "'d regex"   and y42 :: "'d regex" shows "(Times x41 x42 = Times y41 y42) = (x41 = y41 \<and> x42 = y42)"
vcons_vcpower1E: fixes xs :: "V"   and A :: "V" assumes "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> 1\<^sub>\<nat>" obtains x :: "V" where "xs = [x]\<^sub>\<circ>" and "x \<in>\<^sub>\<circ> A"
chop_rule: fixes splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list"   and xs :: "'b list" assumes "reducing splitf" shows "chop splitf xs = (let (pre, post) = splitf xs in if pre = [] then ([], xs) else let (xss, zs) = chop splitf post in (pre # xss, zs))"
biessentially_surjective_on_objects: fixes a' :: "'b" assumes "D.obj a'" shows "\<exists>a. C.obj a \<and> D.equivalent_objects (map\<^sub>0 a) a'"
rel_cases: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e list"   and b :: "'f list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'e" and x2 :: "'e list" and y1 :: "'f" and y2 :: "'f list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
drop_rev: fixes n :: "nat"   and xs :: "'g list" shows "drop n (rev xs) = rev (take (length xs - n) xs)"
minus_eq_not_minus_1: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "uminus a = not (minus a one)"
nat_to_string_simps_1: fixes n :: "nat" assumes "is_digit n" shows "nat_to_string n = [digit_to_char n]"
simps_1: shows "FingerTreeStruc.is_measured_ftree Empty = True"
bij_Rep_perm: fixes p :: "perm" shows "bij (Rep_perm p)"
closure_closed: fixes S :: "'d set" assumes "closed S" shows "closure S = S"
Contingent_expl: shows "\<lfloor>\<lambda>w. \<forall>x. Contingent x w = (\<^bold>\<not>Necessary x) w\<rfloor>"
diamond_cube_is_type_I: shows "typeI_twoCube diamond_cube"
elims_3: fixes x :: "'b graph"   and xa :: "'b \<Rightarrow> nat" assumes "\<not> SINVAR_BLPbasic.sinvar x xa" obtains G :: "'b graph" and nP :: "'b \<Rightarrow> nat" where "x = G" and "xa = nP" and "\<not> (\<forall>(e1, e2)\<in>edges G. nP e1 \<le> nP e2)"
t_sum_le_a_sum: fixes ot :: "'op rose_tree" assumes "local.wf ot" shows "cost_sum ot \<le> acost_sum ot"
set_tag_name_def: shows "set_tag_name \<equiv> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name"
matrix_vector_mult_add_rdistrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and x :: "('a, 'c) Finite_Cartesian_Product.vec" shows "(A + B) *v x = A *v x + B *v x"
phull_row_echelon: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ps :: "('f \<Rightarrow>\<^sub>0 'g) list"   and ts :: "'f list" assumes "term_powerprod pair_of_term term_of_pair"   and "Keys (set ps) \<subseteq> set ts"   and "distinct ts" shows "phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)"
elims: fixes x :: "cell list"   and y :: "nat" assumes "bl2wc x = y" obtains xs :: "cell list" where "x = xs" and "y = bl2nat xs 0"
ordLess_omax: fixes R :: "('c \<times> 'c) set set"   and r :: "('c \<times> 'c) set"   and p :: "('d \<times> 'd) set" assumes "finite R"   and "\<forall>r\<in>R. Well_order r"   and "r \<in> R"   and "p <o r" shows "p <o omax R"
strict_mono_def: fixes f :: "'a \<Rightarrow> 'b" shows "OrdinalInduct.strict_mono f \<equiv> \<forall>A B. A < B \<longrightarrow> f A < f B"
simps_959: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x51 :: "tf"   and x52 :: "b_e list" shows "(case Block x51 x52 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f5 x51 x52"
inj_of_ocircline: fixes H :: "ocircline"   and H' :: "ocircline" assumes "of_ocircline H = of_ocircline H'" shows "H = H' \<or> H = opposite_ocircline H'"
index_of_c_snd_def: shows "index_of_c_snd = sch_to_nat Base_snd"
divide_divide_eq_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide a b) c = divide a (times b c)"
intros_7: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A >> A'" shows "Fst (SimplyTyped.Pair A B) >> A'"
invc2_baldL: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc2 l"   and "invc r" shows "invc2 (baldL l a r)"
longer_falsifies: fixes Cs :: "fterm literal set set"   and ds :: "bool list"   and d :: "bool list" assumes "falsifies\<^sub>c\<^sub>s ds Cs" shows "falsifies\<^sub>c\<^sub>s (ds @ d) Cs"
ex_redchainTr6: fixes n :: "nat" shows "\<forall>f. d_gchain G n f \<longrightarrow> (\<exists>g. D_gchain G (card (f ` {i. i \<le> n}) - 1) g \<and> g ` {i. i \<le> card (f ` {i. i \<le> n}) - 1} = f ` {i. i \<le> n})"
simps_2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" shows "orbits_list f (x # xs) = orbit_list f x # orbits_list f (fold remove1 (orbit_list f x) xs)"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (a \<preceq> b \<and> b \<preceq> a)"
implies_dist_implies: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(if x \<le> (if y \<le> z then top else z) then top else if y \<le> z then top else z) \<le> (if (if x \<le> y then top else y) \<le> (if x \<le> z then top else z) then top else if x \<le> z then top else z)"
order_iff_strict: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "(a \<preceq> b) = (a \<prec> b \<or> a = b)"
Tinv: fixes x :: "(real, 'n) vec" shows "T (inv T x) = x"
delta_pos_neg_least_2: fixes rm :: "('d, 'a) synlr"   and rp :: "('d, 'a) synlr" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_pos \<le> rp"
cases: fixes a1 :: "com"   and a2 :: "state" assumes "a1 \<down> a2" obtains f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = Do f__" and "a2 = s__" and "f__ s__ \<noteq> {}"   | c1 :: "com" and s0 :: "state" and c2 :: "com" where "a1 = (c1__; c2__)" and "a2 = s0" and "c1__ \<down> s0" and "\<And>s1. s0 -c1__\<rightarrow> s1 \<Longrightarrow> c2__ \<down> s1"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" where "a1 = IF ba__ THEN c1a__ ELSE c2a__" and "a2 = sa__" and "ba__ sa__" and "c1a__ \<down> sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and c1 :: "com" where "a1 = IF bb__ THEN c1 ELSE c2" and "a2 = sb__" and "\<not> bb__ sb__" and "c2 \<down> sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE bc__ DO ca__" and "a2 = sc__" and "\<not> bc__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE b DO cb__" and "a2 = sd__" and "b sd__" and "cb__ \<down> sd__" and "\<And>t. sd__ -cb__\<rightarrow> t \<Longrightarrow> WHILE b DO cb__ \<down> t"   | s :: "state" where "a1 = CALL" and "a2 = se__" and "body \<down> se__"   | c :: "com" and f :: "state \<Rightarrow> state" and s :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = LOCAL f; c; g" and "a2 = s" and "c \<down> f s"
substitutionts: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and \<sigma> :: "char list \<Rightarrow> fterm"   and ts :: "fterm list" shows "eval\<^sub>t\<^sub>s E F (ts \<cdot>\<^sub>t\<^sub>s \<sigma>) = eval\<^sub>t\<^sub>s (evalsub E F \<sigma>) F ts"