hcju/isabelleps huggingface dataset:
EVEN_intersect_ODD: shows "EVEN \<inter> ODD = {}"
const_onD: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'b"   and y :: "'a" assumes "const_on f S x"   and "y \<in> S" shows "f y = x"
vp_not_whole: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "vp K v \<noteq> carrier (Vr K v)"
rho_le_x_Ax_imp_rho_le_rx: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec"   and \<rho> :: "real" assumes "x \<in> X"   and "le_vec (\<rho> *s x) (A *v x)" shows "\<rho> \<le> r x"
inv: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "inverse_transformations A B F G \<tau> \<tau>'"   and "partial_magma.ide A a" shows "category.inverse_arrows B (\<tau> a) (\<tau>' a)"
proj_ik_append: fixes l :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) strand_step) list"   and B :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "ik\<^sub>s\<^sub>t (proj_unl l (A @ B)) = ik\<^sub>s\<^sub>t (proj_unl l A) \<union> ik\<^sub>s\<^sub>t (proj_unl l B)"
assms_1: shows "n' \<in> set (\<alpha>n g)"
dg_Obj_if_Dom_vrange: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
hoare_conseq_right: fixes P :: "'e set"   and R :: "('e \<times> 'e) set"   and Q' :: "'e set"   and Q :: "'e set" assumes "{P} R {> Q'}"   and "Q' \<subseteq> Q" shows "{P} R {> Q}"
qring_carrier: fixes I :: "'a set" assumes "ideal R I" shows "carrier (R /\<^sub>r I) = {X. \<exists>a\<in>carrier R. a \<uplus>\<^bsub>R\<^esub> I = X}"
mset_replicate_mono: fixes m1 :: "'a multiset"   and m2 :: "'a multiset"   and n :: "nat" assumes "m1 \<subseteq># m2" shows "\<Sum>\<^sub># (mset (replicate n m1)) \<subseteq># \<Sum>\<^sub># (mset (replicate n m2))"
vc_enter_foreach: fixes v :: "'v"   and V :: "'v set"   and S :: "'v set" assumes "v \<notin> V"   and "v \<noteq> tgt"   and "rpre S (V, v)" shows "fe_inv S V v (E `` {v}) (insert v V, False)"
Diag_TensorE_5: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "u \<noteq> \<^bold>\<I>"
l10_6_uniqueness_spec: fixes P :: "'p"   and P1 :: "'p"   and A :: "'p"   and B :: "'p"   and P2 :: "'p" assumes "P P1 ReflectL A B"   and "P P2 ReflectL A B" shows "P1 = P2"
sup_ty_opt_refl: fixes P :: "'b prog"   and T :: "ty err" shows "P \<turnstile> T \<le>\<^sub>\<top> T"
affected_fixate: fixes a :: "'b"   and f :: "'b perm" shows "affected (fixate a f) \<subseteq> affected f - {a}"
unforwarded_non_volatile_reads_antimono_in: fixes x :: "nat"   and xs :: "'b memref list"   and W' :: "nat set"   and W :: "nat set" assumes "x \<in> unforwarded_non_volatile_reads xs W'"   and "W \<subseteq> W'" shows "x \<in> unforwarded_non_volatile_reads xs W"
types_preserved_tee_local: fixes v :: "e"   and i :: "nat"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[v, $Tee_local i]\<rparr> \<leadsto> \<lparr>[v, v, $Set_local i]\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [v, $Tee_local i] : ts _> ts'"   and "is_const v" shows "\<S>\<bullet>\<C> \<turnstile> [v, v, $Set_local i] : ts _> ts'"
transpose_triangle: fixes w :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<nu> :: "'a"   and v :: "'a" assumes "ide w"   and "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<nu> : v \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright>" shows "k.trnl\<^sub>\<epsilon> (h \<star> u) (\<a>[k\<^sup>*, h, u] \<cdot> ((k\<^sup>* \<star> h) \<star> \<theta>) \<cdot> \<a>[k\<^sup>* \<star> h, p\<^sub>0, w] \<cdot> (tab \<star> w) \<cdot> \<nu>) = (h \<star> \<theta>) \<cdot> \<a>[h, p\<^sub>0, w] \<cdot> (\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w] \<cdot> (k \<star> \<nu>)"
map_prod_def: fixes f :: "'r \<Rightarrow> 't"   and g :: "'s \<Rightarrow> 'u" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
k_3: shows "k \<notin> Sec - \<Union> {GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) |l1 l2. l1 \<noteq> l2}"
rel_U_neg_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q) itself" shows "rel_U_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1'. rel_U (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_U L1 Co1 Co2 Contra1 Contra2 OO rel_U L1' Co1' Co2' Contra1' Contra2')"
P1: shows "A \<noteq> B \<and> A' \<noteq> B' \<and> P \<noteq> B \<and> P' \<noteq> B' \<and> C \<noteq> B \<and> C' \<noteq> B'"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'b set" assumes "A \<in> sets N"   and "B \<in> Y.events" shows "emeasure (N \<Otimes>\<^sub>M distr M T Y) (A \<times> B) = emeasure N A * emeasure (distr M T Y) B"
cancel: fixes x :: "real"   and c :: "real"   and y :: "real"   and z :: "real"   and u :: "real"   and v :: "real" assumes "0 < x"   and "c * x\<^sup>2 * y * z \<le> u * v" shows "c * x ^ 3 * y * z \<le> x * u * v"
st_mat_mult_st_vec: fixes A :: "'a st_mat"   and X :: "'a st_vec" shows "st_mat A *v st_vec X = st_vec (A *st X)"
sets_K: fixes x :: "'a" assumes "x \<in> space M" shows "sets (K x) = sets M"
N_dvd_mono: fixes m :: "int"   and n :: "int" assumes "m dvd n" shows "N n \<le> N m"
foldl_LTLOr_prop_entailment: fixes S :: "'c ltl set"   and i :: "'c ltl"   and xs :: "'c ltl list" shows "S \<Turnstile>\<^sub>P foldl LTLOr i xs = (S \<Turnstile>\<^sub>P i \<or> (\<exists>y\<in>set xs. S \<Turnstile>\<^sub>P y))"
SemCall: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes " s , body \<rightarrow>\<^sub>n  t " shows " s , Call \<rightarrow>\<^sub>n + 1  t "
simps_4: fixes v :: "val" shows "eval_unop UNot v = Bool (\<not> the_Bool v)"
subsets_smash_R_def: fixes n :: "nat"   and s1 :: "(nat list \<times> nat list) list"   and s2 :: "(nat list \<times> nat list) list" shows "subsets_smash_R n s1 s2 = concat (map (\<lambda>l1. map (\<lambda>l2. (fst l1 @ map ((+) n) (fst l2), snd l1 @ map ((+) n) (snd l2))) s2) s1)"
under_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "under r a \<equiv> {b. (b, a) \<in> r}"
cring_simprules_21: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
have_the_same_form_sound: fixes v1 :: "db"   and \<rho> :: "nat \<rightarrow> ValD"   and v2 :: "db" assumes "evalDdb v1\<cdot>\<rho> = evalDdb v2\<cdot>\<rho>"   and "val v1"   and "val v2" shows "v1 \<sim> v2"
is_scc_unique: fixes E :: "('c \<times> 'c) set"   and scc :: "'c set"   and scc' :: "'c set"   and v :: "'c" assumes "is_scc E scc"   and "is_scc E scc'"   and "v \<in> scc"   and "v \<in> scc'" shows "scc = scc'"
prime_id: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "primerec (recf.id m n) m"
assms: fixes f :: "'a" shows "n \<cdot> f = n \<and> f \<cdot> n = n"
simps_3: fixes y :: "'a" shows "flat_1 (deny (allow y)) = deny y"
zp: assumes "0 = w1 \<and> w2 = POS_INF" shows "P w1 w2"
markedElementsToAreMarkedElements: fixes m :: "'b"   and e :: "'b"   and M :: "('b \<times> bool) list" assumes "m \<in> set (markedElementsTo e M)" shows "m \<in> set (markedElements M)"
tc_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tc p \<equiv> lookup p (punit.tt p)"
contract_conj_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> K \<div> (\<phi> .\<and>. \<psi>)" shows "K \<div> (\<phi> .\<and>. \<psi>) \<subseteq> K \<div> \<phi>"
subset_insertI: fixes B :: "'a set"   and a :: "'a" shows "B \<subseteq> insert a B"
distinct_1: fixes x2 :: "'f" shows "None \<noteq> \<lfloor>x2\<rfloor>"
F1set3_bd: fixes x :: "('e, 'f, 'd) F1" shows "|F1set3 x| \<le>o IFbd"
preorder_on_axioms: shows "Order_Predicates.preorder_on (\<Union> (set As)) (of_weak_ranking As)"
Inf_lower2: fixes A :: "'al set"   and v :: "'al"   and u :: "'al" assumes "A \<subseteq> U"   and "v \<in> U"   and "u \<in> A"   and "u \<le>\<^sub>o\<^sub>w v" shows "\<Sqinter>\<^sub>o\<^sub>w A \<le>\<^sub>o\<^sub>w v"
conga_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'" shows "A' B' C' CongA A B C"
distrib_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
freshs_floatariths_max_Var_floatarithsI: fixes xs :: "nat list"   and f :: "floatarith list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> max_Var_floatariths f \<le> x" shows "freshs_floatariths f xs"
get_tag_name_locs_def: shows "get_tag_name_locs \<equiv> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs"
fls_X_pow_conv_fps_X_pow: fixes n :: "nat" shows "fls_regpart (fls_X ^ n) = fps_X ^ n"
in_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c pmf"   and b :: "'d pmf" shows "rel_pmf R a b = (\<exists>z. z \<in> {x. set_pmf x \<subseteq> {(x, y). R x y}} \<and> map_pmf fst z = a \<and> map_pmf snd z = b)"
coprime_MP_Rel: shows "(MP_Rel ===> MP_Rel ===> (=)) coprime_m comm_monoid_mult_class.coprime"
WTrtBinOpAdd: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P,E,h \<turnstile> e\<^sub>1 : Integer"   and "P,E,h \<turnstile> e\<^sub>2 : Integer" shows "P,E,h \<turnstile> e\<^sub>1 \<guillemotleft>Add\<guillemotright> e\<^sub>2 : Integer"
sorted_append: fixes xs :: "'a word list"   and ys :: "'a word list" shows "signed.sorted (xs @ ys) = (signed.sorted xs \<and> signed.sorted ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le>s y))"
sub: fixes a :: "'a" assumes "a \<in> K" shows "a \<in> N"
stream_pred_shift: fixes P :: "'b \<Rightarrow> bool"   and u :: "'b list"   and v :: "'b stream" shows "pred_stream P (u @- v) = (list_all P u \<and> pred_stream P v)"
map2fun_def': fixes v :: "(nat, 'a) mapping"   and x :: "nat" shows "\<langle>v\<rangle> x \<equiv> case Mapping.lookup v x of None \<Rightarrow> 0::'a | Some y \<Rightarrow> y"
ide_compE: fixes g :: "'c"   and f :: "'c" assumes "Cop.ide (g \<cdot>\<^sup>o\<^sup>p f)" obtains "Cop.seq g f" and "Cop.seq f g" and "g \<cdot>\<^sup>o\<^sup>p f = Cop.dom f" and "g \<cdot>\<^sup>o\<^sup>p f = Cop.cod g"
dist: shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>i = {}"
retraction_maps_to_retract_maps: fixes X :: "'b topology"   and Y :: "'c topology"   and r :: "'b \<Rightarrow> 'c"   and s :: "'c \<Rightarrow> 'b" assumes "retraction_maps X Y r s" shows "retraction_maps X (subtopology X (s ` topspace Y)) (s \<circ> r) id"
locals_def2: fixes g :: "oref \<Rightarrow> obj option"   and l :: "lname \<Rightarrow> val option" shows "locals (st g l) = l"
safe_length: fixes x :: "'d" assumes "x \<in> Csafe" shows "length ode_e = DIM('d)"
solves_ode_domainD: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
existence_ivl_empty2: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and x0 :: "'b"   and t0 :: "real" assumes "ll_on_open_it T f X"   and "x0 \<notin> X" shows "ll_on_open.existence_ivl T f X t0 x0 = {}"
DERIV_pos_inc_left: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "(f has_real_derivative l) (at x)"   and "0 < l" shows "\<exists>d>0. \<forall>h>0. h < d \<longrightarrow> f (x - h) < f x"
H_halt_inv: fixes M :: "(action \<times> nat) list"   and ns :: "nat list" assumes "\<not> halts M ns" shows "{\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)}"
select_convs_1: fixes tg_mut :: "mut"   and tg_t :: "t"   and more :: "'a" shows "tg_mut \<lparr>tg_mut = tg_mut, tg_t = tg_t, \<dots> = more\<rparr> = tg_mut"
embedding_map_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "embedding_map X Y f \<equiv> homeomorphic_map X (subtopology Y (f ` topspace X)) f"
simps_2: fixes x31 :: "jump"   and x32 :: "stmt"   and y31 :: "jump"   and y32 :: "stmt" shows "(x31\<bullet> x32 = y31\<bullet> y32) = (x31 = y31 \<and> x32 = y32)"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and p :: "float poly"   and e :: "float interval" shows "tm_round_floats_of_normed prec I a (TaylorModel p e) = (let x = split_by_prec prec p in case case x of (x, y) \<Rightarrow> (x, map_poly interval_of y) of (l, r) \<Rightarrow> TaylorModel l (round_interval prec (e + compute_bound_poly prec r I a)))"
derivF: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_vector_derivative f (g x)) (at (g x) within g ` {a..b})"
sup_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<le> sup x y"
atU_xchange_stack: fixes U :: "'a set"   and u :: "'a"   and r :: "'a list"   and c :: "'a list multiset"   and r' :: "'a list" assumes "atU U (add_mset (u # r) c)" shows "atU U (add_mset (u # r') c)"
ln_float_bounds_1: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 \<le> real_of_float x"   and "real_of_float x < 1" shows "real_of_float (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (real_of_float x + 1)"
SeqQuoteP_lemma: fixes m :: "name"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name" assumes "atom m \<sharp> (t, u, s, k, n, sm, sm', sn, sn')"   and "atom n \<sharp> (t, u, s, k, sm, sm', sn, sn')"   and "atom sm \<sharp> (t, u, s, k, sm', sn, sn')"   and "atom sm' \<sharp> (t, u, s, k, sn, sn')"   and "atom sn \<sharp> (t, u, s, k, sn')"   and "atom sn' \<sharp> (t, u, s, k)" shows "{SeqQuoteP t u s k} \<turnstile> t EQ Zero AND u EQ Zero OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN k AND Var n IN k AND SeqQuoteP (Var sm) (Var sm') s (Var m) AND SeqQuoteP (Var sn) (Var sn') s (Var n) AND t EQ Eats (Var sm) (Var sn) AND u EQ Q_Eats (Var sm') (Var sn')))))))"
restrict_nonempty_less_eq: fixes k :: "'c"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and l :: "'c"   and ls :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "((k # ks) \<langle>f\<rangle> (l # ls) \<preceq> (k # ks) \<langle>g\<rangle> (l # ls)) = ([k] \<langle>f\<rangle> [l] \<preceq> [k] \<langle>g\<rangle> [l] \<and> [k] \<langle>f\<rangle> ls \<preceq> [k] \<langle>g\<rangle> ls \<and> ks \<langle>f\<rangle> [l] \<preceq> ks \<langle>g\<rangle> [l] \<and> ks \<langle>f\<rangle> ls \<preceq> ks \<langle>g\<rangle> ls)"
prj_chine_assoc'_2: shows "Prj\<^sub>0\<^sub>1 \<cdot> chine_assoc' = Prj\<^sub>1\<^sub>0"
assert_all_state_def: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and assert :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list"   and as :: "('a \<times> 'b atom) list" assumes "AssertAllState' init assert" shows "AssertAllState'.assert_all_state init assert t as \<equiv> AssertAllState'.assert_loop assert as (init t)"
ideD_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "local.dom a = a"
is_median_def: fixes x :: "'b"   and xs :: "'b list" shows "is_median x xs = (length (filter (\<lambda>y. y < x) xs) \<le> length xs div 2 \<and> length (filter ((<) x) xs) \<le> length xs div 2)"
c: shows "Partial_Cost_Model.Step (rTS h0) ([x, y], h) x = ([x, y], x # h)"
arctan_lower_14_def: shows "arctan_lower_14 \<equiv> \<lambda>x. pi / 2 - 1 / x"
interval_integral_endpoints_same: fixes a :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integral lborel a a f = (0::'a)"
pred_plus_impl: fixes P :: "'BExp set"   and mem :: "'Var \<Rightarrow> 'Val"   and e :: "'BExp"   and S :: "'Var set \<times> 'Var set" assumes "pred P mem"   and "ev\<^sub>B mem e" shows "pred P +\<^sub>S e mem"
intros_2: fixes m :: "'n"   and n :: "'n" assumes "m \<in> slice"   and "(m, n) \<in> pdg" shows "n \<in> slice"
DIAndSubst341_def: shows "DIAndSubst341 = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Pc (Inl pid3) && Pc (Inl pid4)) else if C = pid2 then Some (Pc (Inl pid3)) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
errMOD_igWlsDisj: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igWlsDisj MOD" shows "igWlsDisj (errMOD MOD)"
filterlim_real_sequentially: shows "real \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
env_get_def: fixes e :: "char list \<Rightarrow> 'b option"   and x :: "char list" shows "Env e!x = e x"
tllist_ord_simps_2: fixes b :: "'f"   and b1 :: "'f"   and y :: "'g"   and ys :: "('g, 'f) tllist" shows "tllist_ord b (TNil b1) (TCons y ys) = (b1 = b)"
simps_1: fixes p :: "perm" shows "p \<bullet> [] = []"
FOREACHc_refine_genR: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "(('c set \<times> 'e) \<times> 'd set \<times> 'f) set"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres"   and R' :: "('e \<times> 'f) set" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "((S, \<sigma>0), \<alpha> ` S, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; x' = \<alpha> x; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; x \<in> it; x' \<in> it'; c \<sigma>; c' \<sigma>'\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"   and "\<And>\<sigma> \<sigma>'. (({}, \<sigma>), {}, \<sigma>') \<in> R \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; it \<noteq> {}; it' \<noteq> {}; \<not> c \<sigma>; \<not> c' \<sigma>'\<rbrakk> \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> \<Down> R' (FOREACH\<^sub>C S' c' f' \<sigma>0')"
\<psi>: shows "\<guillemotleft>\<rho>'.T0.trnr\<^sub>\<epsilon> r' \<rho>' : g \<star> f\<^sup>* \<Rightarrow> r'\<guillemotright> \<and> local.iso (\<rho>'.T0.trnr\<^sub>\<epsilon> r' \<rho>')"
exhaust: fixes y :: "protocol" obtains "y = ProtoAny"   | x2 :: "8 word" where "y = Proto x2"
elementLevelMarkedGeq1: fixes M :: "('b \<times> bool) list"   and e :: "'b" assumes "uniq (elements M)"   and "e \<in> set (markedElements M)" shows "1 \<le> elementLevel e M"
Diagonalize_Diagonalize: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
id3: shows "subresultant (n (i - 1) - 1) (F 1) (F 2) = Polynomial.smult ((- 1) ^ (\<Sum>l\<leftarrow>[3..<i]. (n (l - 2) - (n (i - 1) - 1)) * (n (l - 1) - (n (i - 1) - 1))) * (\<Prod>l\<leftarrow>[3..<i]. (\<beta> l / \<alpha> l) ^ (n (l - 1) - (n (i - 1) - 1))) * (\<Prod>l\<leftarrow>[3..<i]. f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)))) (subresultant (n (i - 1) - 1) (F (i - 2)) (F (i - 1)))"
append_eq_Cons_conv: fixes ys :: "'e list"   and zs :: "'e list"   and x :: "'e"   and xs :: "'e list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
sum_list_triv: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and r :: "'b"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
P: shows "lset P \<inter> winning_region p \<noteq> {}"
hyp: shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
zero_interval_def: shows "0 \<equiv> Interval (0::'b, 0::'b)"
scale_left_commute: fixes a :: "'a"   and b :: "'a"   and x :: "'b" shows "a *s b *s x = b *s a *s x"
hext_upd_obj: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "h a = \<lfloor>(C, fs)\<rfloor>" shows "h \<unlhd> h(a \<mapsto> (C, fs'))"
get_disconnected_nodes_pure: 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 document_ptr :: "(_) document_ptr"   and h :: "(_) heap" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "pure (get_disconnected_nodes document_ptr) h"
UN_simps_8: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set set" shows "\<Union> (B ` \<Union> A) = (\<Union>y\<in>A. \<Union> (B ` y))"
slice_kind_Call_in_slice: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set"   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 "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "sourcenode a \<in> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>SDG.cspp (targetnode a) (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S) fs"
A'_3: shows "n' \<le> funpow_dist f x y"
ideD_1: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.VV.arr a"