hcju/isabelleps huggingface dataset:
elims_2: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c \<times> 'd Filters.filter"   and xa :: "'c \<times> 'd Filters.filter" assumes "triple phi"   and "triple.pairs_less_eq x xa" obtains xb :: "'c" and y :: "'d Filters.filter" and z :: "'c" and w :: "'d Filters.filter" where "x = (xb, y)" and "xa = (z, w)" and "xb \<le> z \<and> w \<le> y"
length_aux: shows "Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j + length aux = min (Monitor.progress \<sigma> \<phi>'' j) (Monitor.progress \<sigma> \<psi>'' j)"
assoc: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "min (min a b) c = min a (min b c)"
process_mru_map_add: fixes s :: "'b v_state_scheme"   and v_f :: "process \<Rightarrow> val option" assumes "s \<in> Vinv1" shows "process_mru ((votes s)(v_state.next_round s := v_f)) = process_mru (votes s) ++ (\<lambda>p. map_option (Pair (v_state.next_round s)) (v_f p))"
mod_eq_self_iff_div_eq_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(modulo a b = a) = (divide a b = zero)"
simple_if_stuck: fixes S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>+ (S\<^sub>2, \<theta>\<^sub>2)"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>2,\<theta>\<^sub>2\<rangle>"   and "\<nexists>S\<^sub>3 \<theta>\<^sub>3. (S\<^sub>2, \<theta>\<^sub>2) \<leadsto> (S\<^sub>3, \<theta>\<^sub>3) \<and> \<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>3,\<theta>\<^sub>3\<rangle>" shows "simple S\<^sub>2"
that: fixes c :: "real" assumes "(f has_real_derivative c) F" shows "thesis"
aux3: fixes acc :: "'Q \<Rightarrow> nat set"   and num_acc :: "nat"   and q :: "'Q" assumes "F = {{q. i \<in> acc q} |i. i < num_acc}"   and "\<Union> (range acc) \<subseteq> {0..<num_acc}"   and "acc q \<noteq> {}" shows "q \<in> V"
sgnx_able_def: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" shows "f sgnx_able F = (\<exists>c. (f has_sgnx c) F)"
Invariants_points: fixes A :: "'a set"   and x :: "'a" assumes "A \<in> sets Invariants"   and "x \<in> A" shows "T x \<in> A"
slist_apply_slist_of_list: fixes xs :: "'d list"   and i :: "nat" shows "slist_apply (slist_of_list xs) i = (if i < length xs then xs ! i else (0::'d))"
lists_neq_if: fixes l :: "'a list"   and l' :: "'a list" assumes "\<exists>i. l ! i \<noteq> l' ! i" shows "l \<noteq> l'"
preserves_ide: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and a :: "'ag" assumes "functor A B F"   and "partial_magma.ide A a" shows "partial_magma.ide B (F a)"
arrow_from_functor_axioms_def: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and e :: "'c" shows "arrow_from_functor_axioms D C F x y e \<equiv> partial_magma.ide D x \<and> partial_magma.in_hom C e (F x) y"
LSC_to_SC: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n" shows "\<Gamma> \<Rightarrow> {#}"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "'fsym"   and Tl :: "'fsym trm list" shows "I.int \<xi> (Fn f Tl) = intF f (map (I.int \<xi>) Tl)"
carrSopen: shows "carrier open\<^bsub>T\<^esub>"
Eq_Zero_sf: fixes i :: "name" shows "Sigma_fm (Var i EQ Zero)"
assms_2: shows "\<exists>y. [y\<^sub>v*A]=c"
unit_rcs_in_set_rcs: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H " shows "H \<in> set_rcs G H"
hyps: fixes gpv :: "('a, 'b, 'c) gpv" shows "expectation_gpv' gpv \<le> expectation_gpv 0 \<I>' (\<lambda>a. y) gpv"
finfun_snd_def: fixes f :: "('a, 'c \<times> 'b) finfun" shows "finfun_snd f = finfun_comp snd f"
openin_path_components_of_locally_path_connected_space: fixes X :: "'c topology"   and c :: "'c set" assumes "locally_path_connected_space X"   and "c \<in> path_components_of X" shows "openin X c"
hide_set_bot: fixes A :: "'b set" shows "\<bottom> \ A = \<bottom>"
N: shows "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (rep_metric_completion y n) (rep_metric_completion y m) < e / 2"
assms_2: fixes \<rho> :: "'a OCL_Types.type" assumes "\<tau> = Bag \<rho>"   and "\<rho> < \<sigma>" shows "P"
proper_it'I: fixes it :: "'b \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it' :: "'b \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<And>s. proper_it (it s) (it' s)" shows "proper_it' it it'"
set_disconnected_nodes_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"   and "h \<turnstile> set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
varFalse: fixes b :: "BExpr"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "\<not> evalB b (fst t)"   and "s \<equiv>\<^sub>\<beta> t" shows "(b, \<Phi>, \<beta>, s, t) \<in> var"
lower_dim: shows "\<exists>a b c. \<not> Bet a b c \<and> \<not> Bet b c a \<and> \<not> Bet c a b"
CFG_wf_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set" shows "CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses \<equiv> \<forall>g. \<forall>m\<in>set (\<alpha>n g). \<forall>v\<in>uses g m. CFG_base.defAss' \<alpha>n invar inEdges' Entry defs g m v"
Rats_power: fixes a :: "'a"   and n :: "nat" assumes "a \<in> \<rat>" shows "a ^ n \<in> \<rat>"
not_finite_nat_set_iff2: fixes N :: "nat set" shows "infinite N = (\<forall>n. \<exists>i\<in>N. n < i)"
mem_fsps_fst_sub1: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and a :: "'a set"   and b :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "(a, b) \<in> fsps R M N f H" shows "a \<subseteq> carrier M"
prod_rel_def: fixes R1 :: "('e \<times> 'g) set"   and R2 :: "('f \<times> 'h) set" shows "R1 \<times>\<^sub>r R2 \<equiv> {((a, b), a', b'). (a, a') \<in> R1 \<and> (b, b') \<in> R2}"
lt_monom_mult: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)"   and "p \<noteq> 0" shows "lpp (local.punit.monom_mult c t p) = local.punit.splus t (lpp p)"
vs_1: shows "vs \<in> F \<rightarrow> carrier_vec n"
step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and target :: "nat"   and t :: "(String.literal \<times> value list) list" assumes "\<exists>(s', T)|\<in>|possible_steps e s r l i. visits target e s' (evaluate_updates T i r) t" shows "visits target e s r ((l, i) # t)"
diag_block_mat_append: fixes As :: "'b mat list"   and Bs :: "'b mat list" shows "diag_block_mat (As @ Bs) = (let A = diag_block_mat As; B = diag_block_mat Bs in four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)"
bell10_index_2: shows "|\<beta>\<^sub>1\<^sub>0\<rangle> $$ (1, 0) = 0"
mono_closed_ereal: fixes S :: "real set" assumes "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"   and "closed S" shows "\<exists>a. S = {x. a \<le> ereal x}"
linearize_t_Nil: fixes q :: "'a list \<times> 'a list" shows "((case q of (fs, ls) \<Rightarrow> fs @ rev ls) = []) = (q = ([], []))"
fBex_cong: fixes A :: "'b fset"   and B :: "'b fset"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x" shows "fBex A P = fBex B Q"
fds_one_abs_converges: fixes s :: "'b" shows "fds_abs_converges 1 s"
disjoint_union: fixes C :: "'c set set"   and B :: "'c set set" assumes "disjoint C"   and "disjoint B"   and "\<Union> C \<inter> \<Union> B = {}" shows "disjoint (C \<union> B)"
auth_prikey_asset_i: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, Crypt (Auth_ShaKey m) (Auth_PriKey n)) \<in> s" shows "False"
insert_preserve_indices': fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c option"   and ys :: "('c \<times> 'd \<times> bool) list" assumes "insert xs e i = Some ys" shows "fst ` set (the (insert xs e i)) = fst ` set xs \<union> {fst e}"
atLeastAtMost_iff: fixes i :: "'e word"   and l :: "'e word"   and u :: "'e word" shows "(i \<in> signed.atLeastAtMost l u) = (l \<le>s i \<and> i \<le>s u)"
While': fixes b :: "bexp"   and c :: "com" assumes "sec b = 0"   and "0 \<turnstile>' c" shows "0 \<turnstile>' WHILE b DO c"
add_vert_simps_2: fixes u :: "'a" shows "arcs (add_vert u) = arcs G"
gen_gba_impl_rel_eext_def: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i \<Rightarrow> bool)) set" shows "\<langle>Rm, Rl\<rangle>gen_gba_impl_rel_eext = {(\<lparr>gbai_L = Li, \<dots> = mi\<rparr>, \<lparr>gba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
preserves_reflects_arr: fixes f :: "'b" shows "B.arr (trg\<^sub>B f) = B.arr f"
fconverse_small: fixes r :: "V" shows "small {[b, a]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
option_rel_simp_1: fixes a :: "'b option"   and R :: "('a \<times> 'b) set" shows "((None, a) \<in> \<langle>R\<rangle>option_rel) = (a = None)"
order_trans_rules_22: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x < y"   and "y \<le> z" shows "x < z"
MkArr_Map: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "MkArr (local.Dom f) (local.Cod f) (Map f) = f"
abs_real_le_2_powr_bitlen: fixes m2 :: "int" shows "\<bar>real_of_int m2\<bar> < 2 powr real_of_int (bitlen \<bar>m2\<bar>)"
simps_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
snoc_eq_iff_butlast: fixes xs :: "'e list"   and x :: "'e"   and ys :: "'e list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
shows_prec_list_def: fixes d :: "nat"   and l :: "'a list" shows "shows_prec d l = shows_list l"
snoc: shows "set (xs::('a \<times> 'c) list) O set (ys::('c \<times> 'b) list) = fold (\<lambda>a::'a \<times> 'c. case a of (x::'a, y::'c) \<Rightarrow> fold (\<lambda>a::'c \<times> 'b. case a of (y'::'c, z::'b) \<Rightarrow> \<lambda>A::('a \<times> 'b) set. if y = y' then insert (x, z) A else A) ys) xs {}"
comparison_impl_1: fixes x :: "real"   and y :: "real" shows "(x \<le> y) = ge_0 (y - x)"
A_def: shows "A = lang_rhs (rhs - {Trn X r |r. Trn X r \<in> rhs})"
sop_assoc: fixes s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and i :: "'d \<Rightarrow> 'd"   and z :: "'d"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_z R s z"   and "sop_one R s (aug_pm_set z i (s_set R s A))" shows "\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>addition_set f (aug_pm_set z i (s_set R s A)). a \<^sub>s\<cdot> (b \<^sub>s\<cdot> x) = a \<cdot>\<^sub>r b \<^sub>s\<cdot> x"
B3: shows "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel"
ctxt_imp_supteq: fixes C :: "('d, 'e) ctxt"   and t :: "('d, 'e) Term.term" shows "C\<langle>t\<rangle> \<unrhd> t"
path_conforms_with_strategy_start: fixes p :: "Player"   and v :: "'a"   and w :: "'a"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p (LCons v (LCons w P)) \<sigma>"   and "v \<in> VV p" shows "\<sigma> v = w"
funpow_Field_leq: fixes k :: "nat" shows "(f ^^ k) a \<in> Field leq"
simps_4: fixes I :: "('b, 'c, 'a) interp"   and \<phi> :: "('b, 'c, 'a) formula" shows "prog_sem I (? \<phi>) = {(\<nu>, \<nu>) |\<nu>. \<nu> \<in> fml_sem I \<phi>}"
graph_homo: fixes G :: "('a, 'b) labeled_graph"   and f :: "'b \<Rightarrow> 'c" assumes "graph G" shows "graph_homomorphism G (map_graph_fn G f) (on_graph G f)"
nospoof_setbydecision: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list" shows "nospoof TYPE('a) iface ipassmt rs = (setbydecision TYPE('a) iface rs FinalAllow \<subseteq> ipcidr_union_set (set (the (ipassmt iface))))"
subprob_measurableD_4: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and S :: "'d measure"   and x :: "'c"   and K :: "'e measure" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "K \<rightarrow>\<^sub>M N x = K \<rightarrow>\<^sub>M S"
Suc_unat_minus_one: fixes x :: "'b word" assumes "x \<noteq> 0" shows "Suc (unat (x - 1)) = unat x"
prod_unit_l: fixes a :: "'a list"   and b :: "'a list" assumes "a \<otimes>\<^bsub>K [X]\<^esub> b \<in> Units (K [X])"   and "a \<in> Units (K [X])"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "b \<in> Units (K [X])"
converse_UNIV: shows "UNIV\<inverse> = UNIV"
simul_Union: fixes Retr :: "'b set \<Rightarrow> 'b set"   and Theta :: "'b set set" assumes "Bisim.mono Retr"   and "\<And>theta. theta \<in> Theta \<Longrightarrow> simul Retr theta" shows "simul Retr (\<Union> Theta)"
iso_refines_def: fixes G :: "'b program"   and F :: "'b program"   and X :: "'b program set" shows "G iso_refines F wrt X \<equiv> F \<in> welldef \<inter> X \<longrightarrow> G \<in> welldef \<inter> X"
append_self_conv2: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs @ ys = ys) = (xs = [])"
frac_integrable: fixes t :: "real" assumes "t \<in> {a..b}" shows "(\<lambda>t. g' t / g t) integrable_on {a..t}"
id: shows "(\<Prod>(x, i)\<leftarrow>(y, j) # yjs. [:- x, 1:] ^ Suc i) = [:- y, 1:] ^ Suc j * (\<Prod>(x, i)\<leftarrow>yjs. [:- x, 1:] ^ Suc i)"
fgroup_by'_def: fixes f :: "'f \<Rightarrow> 'd \<times> 'e"   and s :: "'f fset" shows "fgroup_by' f s = fcompress ((\<lambda>a. let (fa, uu_) = f a in (fa, (snd \<circ> f) |`| ffilter (\<lambda>a'. fa = fst (f a')) s)) |`| s)"
SUP_le_iff: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and A :: "'c set"   and u :: "'a \<times> 'b" shows "less_eq_prod (Sup_prod (\<P> f A)) u = (\<forall>i\<in>A. less_eq_prod (f i) u)"
prime_elem_dict: shows "prime_elem \<equiv> comm_semiring_1.prime_elem (*) (1::'a) (0::'a)"
intro: fixes M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> composition x y = composition y x" shows "commutative_monoid_axioms M composition"
intros_2: fixes f :: "'a \<Rightarrow> 'b" shows "hom_graph f {} \<top> \<top>"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_end0 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "0 < n \<and> (l, r) = ([Bk], Oc \<up> n @ Bk # Oc \<up> n)"
xx: fixes s' :: "('i, 'a) state" shows "A \<Turnstile>\<^sub>i \<B>\<I> s' = Q A (\<lambda>x y. y < x) \<B>\<^sub>i\<^sub>l \<B>\<^sub>i\<^sub>u \<B>\<^sub>l \<B>\<^sub>u \<B>\<^sub>i\<^sub>l_update \<I>\<^sub>l \<I>\<^sub>u Geq Leq s'"
signed_take_bit_decr_length_iff: fixes k :: "'b"   and l :: "'b" shows "(signed_take_bit (LENGTH('c) - Suc 0) k = signed_take_bit (LENGTH('c) - Suc 0) l) = (take_bit LENGTH('c) k = take_bit LENGTH('c) l)"
simps_2: fixes \<mu> :: "'a" shows "Cod \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>cod \<mu>\<^bold>\<rangle>"
deg_mr_uq: shows "degree (Polynomial.smult m r) < degree (q * u)"
Red_preserves_type: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> e : T" shows "\<exists>T'. P \<turnstile> T' \<le> T \<and> P,E,hp s' \<turnstile> e' : T'"
swap_2: assumes "remove1 c (u @ [a] @ [b] @ v) = remove1 c (u @ [b] @ [a] @ v)" shows "thesis"
l1: shows "llength (ltake (enat (lfind P lst)) lst) = enat (lfind P lst)"
lsubgroup_def: shows "lsubgroup = {A \<in> subgroup. \<forall>a b. a \<in> A \<and> b \<in> A \<longrightarrow> a \<sqinter> b \<in> A \<and> a \<squnion> b \<in> A}"
WTrtSeq: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h,sh \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h,sh \<turnstile> e\<^sub>1;; e\<^sub>2 : T\<^sub>2"
complex_inner_1_right: fixes x :: "complex" shows "\<langle>x, 1\<rangle> = cnj x"
smult_spvec_empty: fixes y :: "'b" shows "smult_spvec y [] = []"
res_units_eq: fixes m :: "int" assumes "residues m" shows "Units (residue_ring m) = {x. 0 < x \<and> x < m \<and> coprime x m}"
B_2: shows "A \<Delta> B \<in> null_sets M"
less_multiset_def: fixes M' :: "'k multiset"   and M :: "'k multiset" shows "(M' < M) = ((M', M) \<in> mult {(x', x). x' < x})"
subset_remdups_wrt: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "set (remdups_wrt f xs) \<subseteq> set xs"
nodeToList_empty: fixes nd :: "('c, 'd) Node" shows "nodeToList nd \<noteq> []"
simps_1: fixes ys :: "udom\<cdot>maybe" shows "plusU\<cdot>Nothing\<cdot>ys = ys"
ic_not_closed: fixes a :: "real"   and b :: "real" assumes "a < b" shows "\<not> closed {a<..b}"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d set \<Rightarrow> 'c"   and nres :: "'d nres" shows "h (case nres of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = (case nres of FAILi \<Rightarrow> h f1 | RES x \<Rightarrow> h (f2 x))"
uniformly_convergent_on_def: fixes X :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" shows "uniformly_convergent_on X f = (\<exists>l. uniform_limit X f l sequentially)"
rev_fimage_eqI: fixes x :: "'a"   and A :: "'a fset"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "x |\<in>| A"   and "b = f x" shows "b |\<in>| f |`| A"
assms_2: shows "P \<Longrightarrow>\<^sub>t F * P'"
set_val_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_val_locs character_data_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h = type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"
in_edges_in_vertices: fixes x :: "nat"   and y :: "nat"   and f :: "face" assumes "(x, y) \<in> \<E> f" shows "x \<in> \<V> f \<and> y \<in> \<V> f"
antisym_conv: fixes y :: "'a word"   and x :: "'a word" assumes "y \<le>s x" shows "(x \<le>s y) = (x = y)"
simps_1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option" shows "process_call \<Gamma> [] = []"
assms_4: shows "fst (dominant_term_ms_aux F) < 0"
element_put_get_preserved: fixes getter :: "('x, 'y, 'z, 'aa, 'ab option) RElement_scheme \<Rightarrow> 'ac"   and setter :: "('ad \<Rightarrow> 'ad) \<Rightarrow> ('x, 'y, 'z, 'aa, 'ab option) RElement_scheme \<Rightarrow> ('x, 'y, 'z, 'aa, 'ab option) RElement_scheme"   and v :: "'ad"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and element_ptr' :: "(_) element_ptr" assumes "\<And>x. getter (setter (\<lambda>_. v) x) = getter x"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr' getter) h h'"
that: fixes \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "((\<sigma>, \<zeta>), i:deliver(d), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"   and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')" shows "thesis"
uncurry_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
soundness_mat_leqb_eqc1: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and X :: "(nat, rat) mapping" assumes "dim_row A = dim_vec b"   and "simplex (mat_leqb_eqc A b c) = Inr X" shows "\<exists>x. [A*\<^sub>vx]\<le>b"
residue_field_cd: fixes R :: "('e, 'f) Ring_scheme"   and mx :: "'e set" assumes "Ring R"   and "maximal_ideal R mx" shows "Corps (R /\<^sub>r mx)"
simps_24: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Skip"
add_point_sub_sys: fixes p :: "'a" shows "sub_incidence_system \<V> \<B> (add_point p) \<B>"
hoareQ_complete': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<Turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"