hcju/isabelleps huggingface dataset:
oddE: 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 a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'a" where "a = plus (times (numeral.numeral one plus (num.Bit0 num.One)) b) one"
e_def: shows "e = M / (4 + 2 / (b - a))"
refl_onP_Un: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "refl_onP A r"   and "refl_onP B s" shows "refl_onP (A \<union> B) (\<lambda>x y. r x y \<or> s x y)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "g \<cdot> f = (if Arr f \<and> Arr g \<and> B.cod f = B.dom g then g \<cdot>\<^sub>B f else B.null)"
while_option_rule: fixes P :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "while_option b c s = Some t"   and "P s" shows "P t"
simps_2: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_ZObis (c1 ;; c2) = (SC_siso c1 \<and> SC_ZObis c2 \<or> SC_ZObis c1 \<and> SC_discr c2 \<or> SC_discr (c1 ;; c2) \<or> SC_siso (c1 ;; c2))"
rlex2_not_sym: fixes x :: "nat list \<times> nat list"   and y :: "nat list \<times> nat list" assumes "x <\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2 y" shows "\<not> y <\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2 x"
Un_ac_3: fixes A :: "'b set"   and B :: "'b set" shows "(A::'b set) \<union> (B::'b set) = B \<union> A"
subst_atm_mset_list_comp_subst: fixes AAs :: "'a multiset list"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "AAs \<cdot>aml (\<tau> \<odot> \<sigma>) = AAs \<cdot>aml \<tau> \<cdot>aml \<sigma>"
finite_dom_graph: fixes f :: "'a \<Rightarrow> 'b option" assumes "finite (dom f)" shows "finite (map_graph f)"
rewrite_iter_fast_def: fixes \<phi> :: "'b ltln" shows "rewrite_iter_fast \<phi> \<equiv> iterate (rewrite_modal \<circ> rewrite_X) \<phi> (size \<phi>)"
useful_tautologies_5: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> (\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi>) in v]"
inject_2: fixes x4 :: "nat"   and y4 :: "nat" shows "(PubK x4 = PubK y4) = (x4 = y4)"
check_no_overlap_def: fixes \<gamma> :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and ft :: "('e, 'g) flow_entry_match list" shows "check_no_overlap \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. \<forall>p\<in>UNIV. ofe_prio a = ofe_prio b \<and> \<gamma> (ofe_fields a) p \<and> a \<noteq> b \<longrightarrow> \<not> \<gamma> (ofe_fields b) p)"
Collect_conv_if: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "{x. x = a \<and> P x} = (if P a then {a} else {})"
read_mrexps: fixes r :: "(bool \<times> 'b) rexp"   and s :: "'b rexp"   and a :: "'b" assumes "r \<in> mrexps s" shows "read a r \<in> mrexps s"
r_rs_4: shows "x = r \<cdot> m + rs \<bullet>\<cdot> ms"
call_only_one_intra_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>!a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<and> intra_kind (kind a')"
filt_equiv_def: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure" shows "filt_equiv F M N = (sets M = sets N \<and> Filtration.filtration M F \<and> (\<forall>t A. A \<in> sets (F t) \<longrightarrow> (emeasure M A = 0) = (emeasure N A = 0)))"
tc_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "tc (monom_mult c t p) = c * tc p"
Map_comp: fixes f :: "('o, 'a) arr"   and g :: "('o, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Map (COMP g f) = Comp (Cod g) (Dom g) (Dom f) (Map g) (Map f)"
assms_3: shows "\<forall>x. (\<forall>y. y \<in> P \<longrightarrow> x \<le> y) \<and> x \<in> P \<longrightarrow> Q x"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "hmultiset" assumes "local.encode x xa = y" obtains e :: "nat" and n :: "nat" where "x = e" and "xa = n" and "y = (if n = 0 then 0 else of_nat (n mod base) * \<omega>^ (local.encode 0 e) + local.encode (e + 1) (n div base))"
elims_1: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list"   and y :: "bool" assumes "binrelchain x xa = y" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = P__" and "xa = []" and "y"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" where "x = Pa__" and "xa = [xb__]" and "y"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" and ya :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # ya # xs" and "y = (P xb ya \<and> binrelchain P (ya # xs))"
l2_inv3I: fixes x :: "l2_state" assumes "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz (generators x)) \<and> chan x \<subseteq> faked_chan_msgs x \<union> chan_generators" shows "x \<in> l2_inv3"
factorization_is_sound: fixes C :: "'a eclause"   and D :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl C)"   and "factorization C D \<sigma> k C'" shows "clause_entails_clause (cl_ecl C) (cl_ecl D)"
MittenTauschA: fixes \<Delta> :: "'b formula list"   and \<DD> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Delta> @ \<DD> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<DD> # \<Delta> @ \<Gamma> \<Rightarrow> \<Theta>"
f_xi_lim: fixes xi :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "xi \<longlonglongrightarrow> x"   and "\<And>i. xi i \<in> S" shows "(\<lambda>i. f (xi i)) \<longlonglongrightarrow> g x"
uses_in_\<alpha>n: fixes v :: "'var"   and g :: "'g"   and n :: "'node" assumes "v \<in> oldUses g n" shows "n \<in> set (\<alpha>n g)"
simps: 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 a1 :: "ty list"   and a2 :: "val list"   and a3 :: "val list" shows "(P \<turnstile> a1 Casts a2 to a3 ) = (a1 = [] \<and> a2 = [] \<and> a3 = [] \<or> (\<exists>T v v' Ts vs vs'. a1 = T # Ts \<and> a2 = v # vs \<and> a3 = v' # vs' \<and> P \<turnstile> T casts v to v'  \<and> P \<turnstile> Ts Casts vs to vs' ))"
inv: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W" shows "FGModuleIso G smult' W smult (the_inv_into V T \<down> W) V"
indRelRp_indRelR_eq: shows "indRelRp = indRelRinfix"
floor_fl: fixes x :: "float" shows "real_of_float (floor_fl x) \<le> real_of_float x"
cod_char: fixes f :: "'a" shows "Left.cod f = (if Left.arr f then cod f else null)"
index_smult_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and a :: "'b" assumes "i < dim_row A"   and "j < dim_col A" shows "(a \<cdot>\<^sub>m A) $$ (i, j) = a * A $$ (i, j)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) CC.comp Diag Diag Diag"
frequently_bex_finite_distrib: fixes A :: "'b set"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c filter" assumes "finite A" shows "(\<exists>\<^sub>F x in F. \<exists>y\<in>A. P x y) = (\<exists>y\<in>A. \<exists>\<^sub>F x in F. P x y)"
mult_cancel_left: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](g1)"   and "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'b)" shows "((\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)) = (f2 \<in> o[F](g2))"
hyps_2: shows "n \<in> set (predecessors g (hd ns))"
is_fl_dghm_axioms_def: fixes \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "is_fl_dghm_axioms \<AA> \<BB> \<FF> \<equiv> \<forall>a b. a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<longrightarrow> \<FF>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> Hom \<AA> a b = Hom \<BB> (\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) (\<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>)"
FF_index: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j < n" shows "mat_of_rows n fs $$ (i, j) = fs ! i $ j"
distincts_in_all_maps_list: fixes X :: "'b list \<times> ('b \<times> 'b) list"   and xss :: "('b \<times> 'b) list list" assumes "distinct (snd X)"   and "xss \<in> set (all_maps_list X)" shows "distincts xss"
extf_irrefl_from_trans: fixes xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "\<forall>z\<in>set xs. \<forall>y\<in>set xs. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> extf f gt xs xs"
map_add_None: fixes m :: "'e \<Rightarrow> 'd option"   and n :: "'e \<Rightarrow> 'd option"   and k :: "'e" shows "((m ++ n) k = None) = (n k = None \<and> m k = None)"
eq3: shows "fpxs_compose_power (fpxs_shift (- (fpxs_val f * rat_of_nat q)) (fpxs_of_fls (fps_to_fls f'))) (1 / rat_of_nat q) = f"
sign_bl_bin: fixes bs :: "bool list" shows "bin_sign (bl_to_bin bs) = 0"
in_carrier_def: fixes xs :: "'a list" shows "in_carrier xs = (\<forall>x\<in>set xs. x \<in> carrier R)"
infinite_idx_SomeE: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<omega>" obtains x :: "'b" where "t !! i = Some x"
summable_sums_iff: fixes f :: "nat \<Rightarrow> 'b" shows "summable f = f sums suminf f"
simps_1: fixes uu :: "'e"   and p :: "'f" shows "pol uu (Oldp p) = Cext"
ACI_assoc: fixes r :: "'b rexp"   and s :: "'b rexp"   and t :: "'b rexp" shows "ACI (Plus (Plus r s) t) (Plus r (Plus s t))"
hyps: fixes kv :: "'a \<times> 'b" assumes "(0::'b) \<notin> snd ` set xs" shows "(0::'b) \<notin> snd ` set (update_by_pair kv xs)"
simps: fixes k :: "natural"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a"   and x :: "'b" shows "Random.iterate k f x = (if k = 0 then Pair x else scomp (f x) (Random.iterate (k - 1) f))"
intros: fixes operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and \<tau> :: "'a"   and name :: "String.literal"   and \<pi> :: "'a list"   and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" assumes "unique_static_operation operations \<tau> name \<pi> oper" shows "static_operation_defined' operations \<tau> name \<pi>"
assms: shows "mat (row_length (a # M)) (length (a # M)) (a # M)"
set_integrable_divide: fixes a :: "'a"   and M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "a \<noteq> (0::'a) \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. f t / a)"
le_approx_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq> Q \<equiv> D Q \<subseteq> D P \<and> (\<forall>s. s \<notin> D P \<longrightarrow> Ra P s = Ra Q s) \<and> min_elems (D P) \<subseteq> T Q"
DynT_prop_def: fixes G :: "prog"   and mode :: "inv_mode"   and D :: "qtname"   and t :: "ref_ty" shows "G\<turnstile>mode\<rightarrow>D\<preceq>t = (mode = IntVir \<longrightarrow> is_class G D \<and> (if \<exists>T. t = ArrayT T then D = Object else G\<turnstile>Class D\<preceq>RefT t))"
fsub_def: fixes m :: "roundmode"   and a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "fsub m a b = (if is_nan a \<or> is_nan b \<or> is_infinity a \<and> is_infinity b \<and> sign a = sign b then some_nan else if is_infinity a then a else if is_infinity b then - b else zerosign (if is_zero a \<and> is_zero b \<and> sign a \<noteq> sign b then sign a else if m = To_ninfinity then 1 else 0) (IEEE.round m (valof a - valof b)))"
pseudo_proj_True_stake: fixes n :: "nat"   and w :: "bool stream" shows "stake n (pseudo_proj_True n w) = stake n w"
prime_order: fixes \<G> :: "'a cyclic_group" assumes "schnorr_base \<G>" shows "prime (order \<G>)"
divide_self_if: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a a = (if a = zero then zero else one)"
simps_2: fixes n :: "nat"   and ns :: "nat list"   and c :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and c' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "conc.meval_sched_abv c (n # ns) c' = (\<exists>c''. conc.meval_abv c n c'' \<and> conc.meval_sched_abv c'' ns c')"
Ints_of_int: 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 :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus z \<in> ring_1.Ints one plus zero minus"
assms_2: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto><(X \<union> weakBisim)> Q \<and> (Q, P) \<in> X \<union> weakBisim"
linorder_le_less_linear: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<or> y < x"
assms_2: assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1" shows "P rt1 rt2 dip"
vars_st: shows "vars_mset s \<subseteq># vars_mset (App s t)"
r: shows "yun_gcd.yun_factorization_main gcd (f div gcd f (pderiv f)) (pderiv f div gcd f (pderiv f)) 0 [] = r"
denote_measure_positive_trace_dim: fixes x1 :: "nat"   and x2a :: "nat \<Rightarrow> complex mat"   and x3a :: "com list"   and x4 :: "complex mat" assumes "well_com (Measure x1 x2a x3a)"   and "x4 \<in> carrier_mat d d"   and "partial_density_operator x4"   and "\<And>x3aa \<rho>. \<lbrakk>x3aa \<in> set x3a; well_com x3aa; \<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (denote x3aa \<rho>) \<and> trace (denote x3aa \<rho>) \<le> trace \<rho> \<and> denote x3aa \<rho> \<in> carrier_mat d d" shows "positive (denote (Measure x1 x2a x3a) x4) \<and> trace (denote (Measure x1 x2a x3a) x4) \<le> trace x4 \<and> denote (Measure x1 x2a x3a) x4 \<in> carrier_mat d d"
map_poly_reciprocal: fixes P :: "'a poly"   and p :: "nat"   and f :: "'a \<Rightarrow> 'b" assumes "degree P \<le> p"   and "f (0::'a) = (0::'b)" shows "map_poly f (reciprocal_poly p P) = reciprocal_poly p (map_poly f P)"
A_def: shows "A = {norm (f x) / norm x |x. x \<noteq> (0::'a)}"
pmf_of_set_singleton: fixes x :: "'c" shows "pmf_of_set {x} = return_pmf x"
AnalBinds_delete_lookup: fixes exp :: "exp \<Rightarrow> 'c \<rightarrow> 'b"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'c\<^sub>\<bottom>" shows "(ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae) x = \<bottom>"
onorm_eq_0: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "(onorm f = 0) = (\<forall>x. f x = (0::'d))"
prm_inv_def: shows "prm_inv \<equiv> map_fun rep_prm abs_prm preprm_inv"
simps_1: shows "ht \<langle>\<rangle> = 0"
Pd'_d'_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and ds :: "'e set"   and Pd' :: "'e \<Rightarrow> ('d \<times> 'd) set" assumes "TruePrefs Xd Xh Pd Ch x X ds Pd'" shows "Field (Pd' (Xd x)) \<subseteq> {y. Xd y = Xd x}"
Amax_ge: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat"   and j :: "nat" assumes "f \<in> {j. j \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity>"   and "j \<in> {j. j \<le> n}" shows "f j \<le> Amax n f"
chine_assoc_def: shows "SPN_fgh.chine_assoc \<equiv> tuple_ABC"
gram_schmidt_def: fixes n :: "nat"   and ws :: "'b vec list" shows "gram_schmidt n ws = rev (gram_schmidt_sub n [] ws)"
btran_spec: fixes x :: "'b"   and y :: "'b" shows "x \<le> \<lfloor>y\<rfloor> (x \<sqinter> y)"
succ_of_list_correct_aux: fixes l :: "(nat \<times> nat) list" shows "(succ_of_list l, set l) \<in> br (\<lambda>succs. {(u, v). v \<in> succs u}) (\<lambda>_. True)"
that: fixes x :: "'a" assumes "x \<in> set xs" shows "coprime x a"
simps_4: fixes v :: "'b val" shows "flat (val.Right v) = flat v"
simps_2: fixes atm :: "'atom" shows "brn (Atm atm) = 1"
IH: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "k < n"   and "(lb, l) \<in> set (identify_blocks A k)"   and "local.inv_upto local.jb A k"   and "local.inv_from local.uppert A k"   and "local.inv_at local.one_zero A k"   and "local.ev_block n A"   and "set bs \<subseteq> set (identify_blocks A k)"   and "\<And>be. \<lbrakk>be \<notin> snd ` set bs; be \<notin> {l, k}; be < n\<rbrakk> \<Longrightarrow> A $$ (be, k) = (0::'a)"   and "\<And>bb be. (bb, be) \<in> set bs \<Longrightarrow> be - bb \<le> l - lb"   and "x = A $$ (l, k)"   and "x \<noteq> (0::'a)" shows "local.inv_all local.uppert (step_3_c x l k bs A) \<and> local.same_diag A (step_3_c x l k bs A) \<and> local.same_upto k A (step_3_c x l k bs A) \<and> local.inv_at (local.single_non_zero l k x) (step_3_c x l k bs A) k"
gfp_F_code: fixes ds :: "'b set" shows "gfp_F ds = map_prod id undual (while (\<lambda>A. F ds A \<noteq> A) (F ds) top)"
l2_inv8D: fixes x :: "l2_state" assumes "x \<in> l2_inv8" shows "secret x \<subseteq> {Exp (Exp Gen (NonceF (R' $ N'))) (NonceF (test $ N)) |N N' R'. R' \<in> partners}"
4: shows "\<forall>S\<in>\<S>. \<forall>S'\<in>\<S>. fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S' = {}"
star_outer_increasing: fixes x :: "'a"   and y :: "'a" shows "x \<le> y\<^sup>\<star> * x * y\<^sup>\<star>"
dense_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and z :: "'b" assumes "class.dense_linorder less_eq less"   and "\<And>x. less x y \<Longrightarrow> less_eq x z" shows "less_eq y z"
xss: shows "list_all3 list_all2 (map (\<lambda>\<phi> i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s) (map2 upt (map (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) \<phi>s) (map (\<lambda>\<phi>. Monitor.progress \<sigma> P' \<phi> j') \<phi>s)) xss"
appliedExplain_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedExplain stateA stateB \<equiv> \<exists>l reason. getConflictFlag stateA = True \<and> l el getC stateA \<and> formulaEntailsClause (getF stateA) reason \<and> isReason reason (opposite l) (elements (getM stateA)) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA \<and> getConflictFlag stateB = True \<and> getC stateB = resolve (getC stateA) reason l"
steps: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (Throw, Normal t)"
size_mset_sum_mset_distrib: fixes A :: "'a multiset multiset" shows "size (\<Sum>\<^sub># A) = \<Sum>\<^sub># (image_mset size A)"
wf_wellorderI: assumes "wf {(x, y). x < y}"   and "OFCLASS('a, linorder_class)" shows "OFCLASS('a, wellorder_class)"
right_idem: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<inter># b \<inter># b = a \<inter># b"
refl_on_restrictPI': fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and C :: "'b set" assumes "refl_onP A r"   and "B = A \<inter> C" shows "refl_onP B (r |` C)"
at_within_open_subset: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "a \<in> S"   and "open S"   and "S \<subseteq> T" shows "topological_space.at_within open a T = topological_space.at_within open a UNIV"
bisim1While4: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and c :: "'addr expr1" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,while (c) e,h \<turnstile> (e';; while (c) e, xs) \<leftrightarrow> (stk, loc, Suc (length (compE2 c) + pc), xcp)"
max_absorb2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "max x y = y"
getParts_def: fixes rs :: "'b simple_rule list" shows "getParts rs = partitioningIps (extract_IPSets rs) [wordinterval_UNIV]"
rwf_query_def: fixes n :: "nat"   and V :: "nat set"   and Qp :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set" shows "rwf_query n V Qp Qn = (wf_query n V Qp Qn \<and> covering V Qp \<and> included V Qp \<and> included V Qn \<and> non_empty_query Qp \<and> non_empty_query Qn)"
dim_row: fixes n :: "nat"   and v :: "complex mat" assumes "state n v" shows "dim_row v = 2 ^ n"
minus_p_mod_def: fixes x :: "int"   and y :: "int" assumes "x \<in> {0..<p}"   and "y \<in> {0..<p}" shows "minus_p p x y = (x - y) mod p"
minus_in_fset: fixes xs :: "'b fset"   and ys :: "'b fset"   and x :: "'b" shows "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)"
both_mono1': fixes t :: "'b ttree"   and t' :: "'b ttree"   and t'' :: "'b ttree" assumes "t \<sqsubseteq> t'" shows "t \<otimes>\<otimes> t'' \<sqsubseteq> t' \<otimes>\<otimes> t''"
is_binqueue_children: fixes l :: "nat"   and ts :: "('c, 'd) bintree list"   and t :: "'c" assumes "is_bintree_list l ts"   and "is_heap_list t ts" shows "is_binqueue 0 (map Some (rev ts))"
supp_sum_subset_union_supp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "supp (f + g) \<subseteq> supp f \<union> supp g"
Ord_linear2: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k \<subset>\<^sub>\<circ> l"   | "l \<subseteq>\<^sub>\<circ> k"
contj: shows "continuous_on (ball (0::'a) 1) j"
the_elem_def: fixes X :: "'a set" shows "the_elem X = (THE x. X = {x})"
silent_moves_equal_length_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "length ms = length s"
IdD: fixes a :: "'e"   and b :: "'e" assumes "(a, b) \<in> Id" shows "a = b"
le_imp_less_or_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "less_eq x y" shows "less x y \<or> x = y"
allAllocationsUniverse: fixes G :: "'b set"   and N :: "'a set" shows "allAllocations N G \<subseteq> allocationsUniverse"