hcju/isabelleps huggingface dataset:
HPhase1or2Write_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv5_inner s p"   and "HInv4c s p" shows "HInv5_inner s' p"
continuous_map_alt: fixes T1 :: "'c topology"   and T2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map T1 T2 f = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> topspace T2)"
undirected: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G" shows "v \<rightarrow>\<^bsub>G\<^esub> w = w \<rightarrow>\<^bsub>G\<^esub> v"
simps: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv" shows "interaction_bound consider gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound consider (c input)) else \<Squnion>input. interaction_bound consider (c input))"
trans_le_add1: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i \<le> j" shows "i \<le> j + m"
has_typ_iff_typ_of: fixes t :: "Core.term"   and ty :: "typ" shows "\<turnstile>\<^sub>\<tau> t : ty = (typ_of t = Some ty)"
cases: fixes x :: "('a, 'b) rbt" obtains "x = Leaf"   | l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r"
inter_guards_sym: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" assumes "(c1 \<inter>\<^sub>g c2) = Some c" shows "(c2 \<inter>\<^sub>g c1) = Some c"
bal_ins: fixes t :: "'b tree234"   and a :: "'b" assumes "bal t" shows "bal (tree\<^sub>i (ins a t)) \<and> height (ins a t) = height t"
representation_of_functor_axioms: shows "representation_of_functor (\<cdot>) (\<cdot>\<^sub>S) \<phi> Cones.map a \<Phi>"
thetaIf_Sretr: shows "thetaIf \<subseteq> Sretr (thetaIf \<union> Sbis)"
await: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "ev\<^sub>B mem b"   and "no_await c\<^sub>1"   and "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>2, mds', mem'\<rangle>"   and "is_final c\<^sub>2" shows "\<langle>Await b c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>2, mds', mem'\<rangle>"
con_rews_1: fixes tree1 :: "'r tree"   and tree2 :: "'r tree" shows "Node\<cdot>\<bottom>\<cdot>tree1\<cdot>tree2 = \<bottom>"
sF: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and s :: "(char list \<Rightarrow> int option) \<times> nat" shows "wp\<^sub>3 (WHILE b DO c) Q s = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) s"
Pair_inject: fixes a :: "'c"   and b :: "'d"   and a' :: "'c"   and b' :: "'d" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
lang_pow_code_def: shows "Regular_Set.lang_pow = compow"
rep_eq: fixes x :: "'c"   and xa :: "'c poly" shows "coeff (smult x xa) = (\<lambda>n. x * coeff xa n)"
either_contains_arg1: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "paths t \<subseteq> paths (t \<oplus>\<oplus> t')"
a1: shows "A \<or>\<^sub>A C \<Longrightarrow>\<^sub>t E"
less_eq_of_match_field_def: fixes a :: "of_match_field"   and b :: "of_match_field" shows "(a \<le> b) = less_eq_of_match_field1 a b"
ord_spmf_eq_pmf_None_eq: fixes Y :: "'c spmf set"   and p :: "'d spmf"   and q :: "'d spmf" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "ord_spmf (=) p q"   and "pmf p None = pmf q None" shows "p = q"
congr: fixes S :: "nat \<Rightarrow> bool option" shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"
empty_upd_none: fixes x :: "'c" shows "Map.empty(x := None) = Map.empty"
simps_3: fixes uw :: "typ list"   and ux :: "variable"   and T :: "typ" shows "typ_of1 uw (Fv ux T) = Some T"
Eq: shows "\<lparr>\<nu>*xvec\<rparr>N \<prec>' P = \<lparr>\<nu>*yvec\<rparr>M \<prec>' Q"
gauss_jordan_single_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat"   and x :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C"   and "x \<in> carrier_vec nc" shows "(A *\<^sub>v x = 0\<^sub>v nr) = (C *\<^sub>v x = 0\<^sub>v nr)"
inf_idem: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
diff_diff_eq2: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
sameDom_lift1: fixes inp :: "'a \<Rightarrow> 'b option"   and f :: "'b \<Rightarrow> 'c" shows "sameDom inp (lift f inp)"
legal_imp_weakly_legal_execution: fixes P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set"   and Ews :: "('b \<times> ('c, 'b) obs_event action) llist \<times> (nat \<Rightarrow> nat)" assumes "legal_execution P \<E> Ews" shows "weakly_legal_execution P \<E> Ews"
CallStatic\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and ps :: "nat exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and ls\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = body in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"
assms_1: shows "i < dim_row (Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)))"
prod_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less_eq (f i) (g i)" shows "less_eq (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
Spair_fundfoldpair: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "Spair s \<in> fundfoldpairs"
c_2: shows "monotone (\<le>) (\<sqsubseteq>) c"
alw_ev_shift: fixes P :: "'b stream \<Rightarrow> bool"   and u :: "'b list"   and v :: "'b stream" shows "alw (ev P) (u @- v) = alw (ev P) v"
sep_conj_sep_true_left: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "(P ** Q) s" shows "(sep_true ** Q) s"
filter_compl1: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(filter P xs = []) = (filter (\<lambda>x. \<not> P x) xs = xs)"
redT_updTs_nta_bisim_inv: fixes tas1 :: "('t, 'x1, 'm1) new_thread_action list"   and tas2 :: "('t, 'x2, 'm2) new_thread_action list"   and ts1 :: "'t \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option"   and T :: "'t"   and ts2 :: "'t \<Rightarrow> ('x2 \<times> 'b \<Rightarrow>f nat) option" assumes "list_all2 (nta_bisim bisim) tas1 tas2"   and "(ts1 T = None) = (ts2 T = None)" shows "(redT_updTs ts1 tas1 T = None) = (redT_updTs ts2 tas2 T = None)"
add_stalk_def: shows "add_stalk \<equiv> add_rel"
red_diff_in_pmdl: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" shows "p - q \<in> local.punit.pmdl F"
semiring_norm_9: fixes m :: "num"   and n :: "num" shows "num.Bit1 m + num.Bit0 n = num.Bit1 (m + n)"
mset_remove1: fixes a :: "'b"   and xs :: "'b list" shows "mset (remove1 a xs) = mset xs - {#a#}"
update_convs_3: fixes lcl' :: "lenv \<Rightarrow> lenv"   and prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "lcl_update lcl' \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = \<lparr>prg = prg, cls = cls, lcl = lcl' lcl, \<dots> = more\<rparr>"
topology: fixes A :: "'b set" shows "topology (topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s}))"
stern_brocot_fractions_not_repeated: fixes path :: "dir list"   and path' :: "dir list" assumes "root (traverse_tree path stern_brocot_iterate) = root (traverse_tree path' stern_brocot_iterate)" shows "path = path'"
distinct_19: fixes x41 :: "'b"   and x42 :: "value list"   and x51 :: "'b gexp"   and x52 :: "'b gexp" shows "In x41 x42 \<noteq> Nor x51 x52"
mul_ext_impl_sound: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and xs :: "'b list"   and ys :: "'b list" shows "mul_ext_impl f xs ys = mul_ext f xs ys"
c: shows "h \<turnstile> local.a_get_dom_component ptr \<rightarrow>\<^sub>r c"
sep_empty_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_empty zero zero"
Can_red2_1: fixes a :: "'a local.term"   and b :: "'a local.term" assumes "Ide a"   and "Nml a"   and "Ide b"   and "Nml b"   and "Src a = Trg b" shows "Can (a \<^bold>\<Down> b)"
non_terminating_def: fixes P :: "'b process" shows "non_terminating P \<equiv> RUN UNIV \<sqsubseteq>\<^sub>T P"
zero_le_power_eq_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and w :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a (numeral w)) = (even (numeral w) \<or> odd (numeral w) \<and> less_eq zero a)"
compact_negations: fixes s :: "'a set" assumes "compact s" shows "compact (uminus ` s)"
fres_canc1: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" shows "less_eq_prod (times_prod' (prod.fres y x) x) y"
simps_6: fixes r :: "real"   and x :: "octo" shows "Im5 (r *\<^sub>R x) = r * Im5 x"
mset_single_iff_right: fixes x :: "'k"   and xs :: "'k list" shows "({#x#} = mset xs) = (xs = [x])"
0: shows "\<not> real_of_int (T_A n) \<le> real (n * length s0)"
f: shows "mcont lub ord \<Union> (\<subseteq>) f"
raw_match_term_imp_raw_match_term_on_map_le: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and lesubs :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'"   and "lesubs \<subseteq>\<^sub>m subs" shows "\<exists>lesubs'. raw_match_term t u lesubs = Some lesubs' \<and> lesubs' \<subseteq>\<^sub>m subs'"
denote_while_dim_positive: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "denote_while M0 M1 DS \<rho> \<in> carrier_mat d d \<and> positive (denote_while M0 M1 DS \<rho>) \<and> trace (denote_while M0 M1 DS \<rho>) \<le> trace \<rho>"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. \<lbrakk>x \<le> y \<Longrightarrow> P xs (y # ys); \<not> x \<le> y \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
l2norm_diff: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "l2norm S (\<lambda>x. f x - g x) = l2norm S (\<lambda>x. g x - f x)"
assms_3: fixes a :: "V" assumes "P a"   and "\<forall>b. Ord b \<and> P b \<longrightarrow> a \<le> b" shows "Q a"
not_fic_crename: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname" assumes "\<not> fic (M[a\<turnstile>c>b]) c"   and "c \<sharp> (a, b)" shows "\<not> fic M c"
zero_le_scaleR_iff: fixes a :: "real"   and b :: "'b" shows "((0::'b) \<le> a *\<^sub>R b) = (0 < a \<and> (0::'b) \<le> b \<or> a < 0 \<and> b \<le> (0::'b) \<or> a = 0)"
leD: fixes y :: "'k multiset"   and x :: "'k multiset" assumes "y \<subseteq># x" shows "\<not> x \<subset># y"
continuous_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
OtherTrue: assumes "\<not> P \<and> Q" shows "R"
dec_emp: fixes G :: "'b list set" shows "Dec G \<epsilon> = \<epsilon>"
max_PInf: fixes x :: "ereal" shows "max \<infinity> x = \<infinity>"
star_unfold_left_Nil: fixes A :: "'a list set" shows "A\<star> = (A - {[]}) \<cdot> A\<star> \<union> {[]}"
field_differentiable_derivI: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "f field_differentiable at x" shows "(f has_field_derivative deriv f x) (at x)"
one_less_floor: fixes x :: "'a" shows "(1 < \<lfloor>x\<rfloor>) = ((2::'a) \<le> x)"
Inf_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "\<Sqinter> A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
integrable_count_space: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite X" shows "integrable (count_space X) f"
simps_10: fixes i :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (ExN i \<phi>) \<Gamma> = (\<exists>l. length l = i \<and> eval \<phi> (l @ \<Gamma>))"
elims: fixes x :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)"   and xa :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and y :: "nat" assumes "abs_steps x xa = y" obtains c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt" and uu_ :: "Mode \<Rightarrow> Eg1.var set" and uv_ :: "Eg1.var \<Rightarrow> nat" and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt" and uw_ :: "Mode \<Rightarrow> var\<^sub>C set" and ux_ :: "var\<^sub>C \<Rightarrow> nat" where "x = \<langle>c\<^sub>A, uu_, uv_\<rangle>\<^sub>A" and "xa = \<langle>c\<^sub>C, uw_, ux_\<rangle>\<^sub>C" and "y = abs_steps' c\<^sub>A c\<^sub>C"
OclNot_contrapos_nn: fixes \<tau> :: "'d state \<times> 'd state"   and A :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> A"   and "\<tau> \<Turnstile> not B"   and "\<tau> \<Turnstile> A \<Longrightarrow> \<tau> \<Turnstile> B" shows "\<tau> \<Turnstile> not A"
comp_arr_dom: fixes f :: "'c"   and a :: "'c" assumes "C.arr f"   and "C.dom f = a" shows "f \<cdot>\<^sub>C a = f"
a_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
hyps: shows "\<Gamma>\<turnstile> (cs @ xs, css, Stuck) \<rightarrow>\<^sup>* (xs, css, Stuck)"
row_base: shows "signof (permutation_insert n j p) = (- (1::'a)) ^ (n + j) * signof p"
induct: fixes x1 :: "('a, 'b, 'c) subst"   and x2 :: "('a, 'c) ODE"   and x3 :: "('c + 'c) set"   and P :: "('a, 'b, 'c) subst \<Rightarrow> ('a, 'c) ODE \<Rightarrow> ('c + 'c) set \<Rightarrow> bool" assumes "Oadmit x1 x2 x3"   and "\<And>\<sigma> c U. P \<sigma> (OVar c) U"   and "\<And>\<sigma> \<theta> U x. \<lbrakk>TUadmit \<sigma> \<theta> U; TadmitF \<sigma> \<theta>\<rbrakk> \<Longrightarrow> P \<sigma> (OSing x \<theta>) U"   and "\<And>\<sigma> ODE1 U ODE2. \<lbrakk>Oadmit \<sigma> ODE1 U; P \<sigma> ODE1 U; Oadmit \<sigma> ODE2 U; P \<sigma> ODE2 U; ODE_dom (Osubst ODE1 \<sigma>) \<inter> ODE_dom (Osubst ODE2 \<sigma>) = {}\<rbrakk> \<Longrightarrow> P \<sigma> (OProd ODE1 ODE2) U" shows "P x1 x2 x3"
bin_rest_numeral_simps_3: shows "- 1 div 2 = - 1"
rel_reflp: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "reflp R1"   and "reflp R2" shows "reflp (tllist_all2 R1 R2)"
triangle_left: shows "Cop.H f' \<epsilon>' \<cdot> \<a>\<^sup>-\<^sup>1[f', g, f'] \<cdot> Cop.H \<eta>' f' = \<l>\<^sup>-\<^sup>1[f'] \<cdot> \<r>[f']"
head: fixes e :: "nat \<times> nat"   and u :: "nat"   and v :: "nat" assumes "e = (u, v)" shows "head G e = v"
Radj_repv1: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" shows "Radj x y (repv \<nu> y r) = repv (Radj x y \<nu>) x r"
sup_continuous_mult_left_ennreal': fixes c :: "ennreal" shows "sup_continuous ((*) c)"
ODnods_less: fixes x :: "'c Order set"   and y :: "'c Order set" assumes "x \<in> carrier ODnods"   and "y \<in> carrier ODnods" shows "x \<prec>\<^bsub>ODnods\<^esub> y = x \<sqsubset> y"
in\<T>: fixes K :: "'a set" assumes "K \<in> \<T>" shows "\<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
inf_eq_top_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = 1) = (x = 1 \<and> y = 1)"
distinct_1: fixes x2 :: "'c" shows "None \<noteq> \<lfloor>x2\<rfloor>"
mv_box_inv_halt: fixes m :: "nat"   and n :: "nat"   and initlm :: "nat list" assumes "max m n < length initlm"   and "m \<noteq> n" shows "\<exists>stp. case abc_steps_l (0, initlm) (mv_box m n) stp of (as, lm) \<Rightarrow> as = 3 \<and> mv_box_inv (as, lm) m n initlm"
P2_ideal_model_end_def: fixes M :: "'grp \<times> 'grp"   and l :: "'grp"   and bs :: "('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'e"   and \<A>3 :: "'grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> 'd spmf" shows "P2_ideal_model_end M l bs \<A>3 = (let (x0, x1) = M; ((h0, h1, a, b0, b1), s) = bs; \<sigma>' = if l = \<one> then False else True in mal_def.trusted_party (x0, x1) \<sigma>' \<bind> (\<lambda>(uu_, x\<sigma>). sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> (if \<sigma>' then \<one> else x\<sigma>); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> (if \<sigma>' then x\<sigma> else \<one>); e0 = (w0, z0); e1 = (w1, z1) in \<A>3 e0 e1 s \<bind> (\<lambda>out. return_spmf ((), out))))))))"
repr'_def: fixes \<psi> :: "'b" shows "repr' \<psi> = Abs_euclidean_space (repr \<psi> \<circ> rep)"
tllist_all2_prs: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs1 :: "'g \<Rightarrow> 'h"   and Rep1 :: "'h \<Rightarrow> 'g"   and R2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and Abs2 :: "'i \<Rightarrow> 'j"   and Rep2 :: "'j \<Rightarrow> 'i"   and P :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and ts :: "('h, 'j) tllist"   and ts' :: "('h, 'j) tllist" assumes "Quotient3 R1 Abs1 Rep1"   and "Quotient3 R2 Abs2 Rep2" shows "tllist_all2 ((Abs1 ---> Abs1 ---> id) P) ((Abs2 ---> Abs2 ---> id) Q) (tmap Rep1 Rep2 ts) (tmap Rep1 Rep2 ts') = tllist_all2 P Q ts ts'"
sel_5: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.UB (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x5"
hmsetmset_plus: fixes M :: "hmultiset"   and N :: "hmultiset" shows "hmsetmset (M + N) = hmsetmset M + hmsetmset N"
ap_cong: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and f :: "'a itrm"   and f' :: "'a itrm"   and x :: "'a itrm"   and x' :: "'a itrm" assumes "itrm_cong R f f'"   and "itrm_cong R x x'" shows "itrm_cong R (f \<diamondop> x) (f' \<diamondop> x')"
hypreal_add_nonneg_eq_0_iff: fixes x :: "real star"   and y :: "real star" assumes "0 \<le> x"   and "0 \<le> y" shows "(x + y = 0) = (x = 0 \<and> y = 0)"
prod_assn_pair_conv: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a1 :: "'a"   and b1 :: "'c"   and a2 :: "'b"   and b2 :: "'d" shows "(A \<times>\<^sub>a B) (a1, b1) (a2, b2) = A a1 a2 * B b1 b2"
transC_MtransT: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and s'' :: "'state" assumes "cf \<rightarrow>c cf'"   and "cf' \<rightarrow>*t s''" shows "cf \<rightarrow>*t s''"
neg_equal_iff_equal: fixes a :: "'b"   and b :: "'b" shows "(- a = - b) = (a = b)"
F: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "(F has_real_derivative f x) (at x)"
s_ss_4: shows "m = s \<cdot> n + ss \<bullet>\<cdot> ms"
Abs_ord_homset_cases: fixes x :: "('b, 'c) ord_homset" obtains y :: "'b \<Rightarrow> 'c" where "x = Abs_ord_homset y" and "y \<in> ord_homset"
rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
blaschke_inv_a_inf: fixes a :: "complex" assumes "cmod a \<noteq> 1" shows "moebius_pt (blaschke a) (inversion (of_complex a)) = \<infinity>\<^sub>h"
summable_imp_bounded: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "bounded (range f)"
compseriesTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "compseries G n f" shows "tW_cmpser G n f"
true_iff: fixes P :: "'a upred" shows "(P \<Leftrightarrow> true) = P"
set_isin: fixes s :: "trie"   and x :: "bool list" assumes "True" shows "isin s x = (x \<in> set_trie s)"
M5exist: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a" assumes "class.interval meets \<I>"   and "meets p q" shows "\<exists>r s t. meets r p \<and> meets p q \<and> meets q s \<and> meets r t \<and> meets t s"
min_bot2: fixes x :: "'c list" shows "suffix_order.min x [] = []"