hcju/isabelleps huggingface dataset:
0: shows "(\<lambda>x. m_ivl (fun S2 x)) = \<lambda>x. 0 on set (Abs_State.dom S2) - set (Abs_State.dom S1)"
circle_axioms_def: fixes d :: "real" shows "circle_axioms d \<equiv> 0 < d"
inj_on_Pair2: fixes x :: "'b"   and A :: "'a set" shows "inj_on (Pair x) A"
span_subset_spanI: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> ideal B" shows "ideal A \<subseteq> ideal B"
boundary_of_region_is_subset_of_partition_boundaries_2: shows "one_chain_typeII \<subseteq> two_chain_boundary two_chain_typeII"
r_inv_ex: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<exists>y\<in>carrier R. x \<oplus> y = \<zero>"
is_pointermap_impl_prec: shows "precise is_pointermap_impl"
if_true: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "eval\<^sub>B mem b" shows "(Stmt.If b t e, mem) \<leadsto>\<^sub>s (t, mem)"
hyps_6: fixes x :: "real" assumes "a \<le> x"   and "x < i" shows "f1 x = f x"
fold_plus_sum_list_rev: fixes xs :: "'a list" shows "fold (+) xs = (+) (sum_list (rev xs))"
enum_ex_bit0_def: fixes P :: "'a bit0 \<Rightarrow> bool" shows "enum_class.enum_ex P = (\<exists>b\<in>set enum_class.enum. P b)"
hbt_balR: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height l - m \<le> height r \<and> height r \<le> height l + m + 1" shows "hbt (balR l a r)"
ide_char: fixes t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.ide t = (t \<noteq> A_B.null \<and> functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Map t) \<and> A_B.Dom t = A_B.Map t \<and> A_B.Cod t = A_B.Map t)"
iso_iff_section_and_retraction: fixes f :: "'s" shows "local.iso f = (section f \<and> retraction f)"
rinit_throw_ext: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and e'' :: "char list exp"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and e\<^sub>t :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>RI (C,e'') ; Cs \<leftarrow> e\<^sub>0,s''\<rangle> \<Rightarrow> \<langle>throw e\<^sub>t,s'\<rangle>"   and "\<And>e' s'. P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; Cs \<leftarrow> e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>throw e\<^sub>t,s'\<rangle>"
pr_sesk_card: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and d :: "rat"   and e :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "IntAgrK (S (Card n, n, run)) = Some d"   and "ExtAgrK (S (Card n, n, run)) = Some e" shows "Key (sesK (d * e)) \<in> U"
subterms_self: fixes fa2 :: "floatarith" shows "fa2 \<in> subterms fa2"
akra_bazzi_light_aux1: fixes a :: "real"   and n0 :: "nat"   and n :: "nat"   and b :: "real" assumes "0 < a"   and "a < 1"   and "n0 < n"   and "(max 0 b + 1) / (1 - a) \<le> real n0" shows "nat \<lceil>a * real n + b\<rceil> < n"
dvd_mult_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times b c"
msb_word_eq: fixes w :: "'b word" shows "msb w = bit w (LENGTH('b) - 1)"
assms_4: shows "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>h h'"
bivl: shows "(f (real_of_float l) + f (real_of_float u) - real_of_float a * real_of_float (l + u)) / 2 \<in>\<^sub>r bivl"
outer_loop_impl_def: fixes Qi :: "'q"   and \<pi>i :: "'m" shows "outer_loop_impl Qi \<pi>i \<equiv> while (\<lambda>(Qi, \<pi>i). \<not> Q_is_empty Qi) (\<lambda>(Qi, \<pi>i). let (u, uu_) = Q_getmin Qi; adjs = G_adj g u; (Qi, \<pi>i) = foreach_impl Qi \<pi>i u adjs; Qi = Q_delete u Qi in (Qi, \<pi>i)) (Qi, \<pi>i)"
hyps: shows "length (M1 \<otimes> M2) = length M1 * length M2"
not_Inner_conv: fixes x :: "'a node" shows "(x \<notin> range Inner) = (x = SOURCE)"
approximation_preproc_int_15: fixes i :: "int"   and j :: "int" shows "(i \<le> j) = (real_of_int i \<le> real_of_int j)"
map_simp_ide: fixes a :: "'a" assumes "S.ide a" shows "\<rr>.map a = \<r>[a]"
id_1: shows "mat\<^sub>h (C * D) = mat\<^sub>h C * mat\<^sub>h D"
hconf_heap_write_mono: fixes h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and v :: "'addr val"   and h' :: "'heap"   and T :: "ty" assumes "heap_write h a al v h'"   and "hconf h"   and "P,h \<turnstile> a@al : T"   and "P,h \<turnstile> v :\<le> T" shows "hconf h'"
left_commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "abel_semigroup f" shows "f b (f a c) = f a (f b c)"
cr_vreal_transfer_domain_rule: shows "Domainp cr_vreal = (\<lambda>x. x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>)"
eucl_rel_poly_smult_right: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "a \<noteq> (0::'b)"   and "eucl_rel_poly x y (q, r)" shows "eucl_rel_poly x (smult a y) (smult (inverse a) q, r)"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
lossless_iff_set_pmf_None: fixes p :: "'b spmf" shows "lossless_spmf p = (None \<notin> set_pmf p)"
not_in_iff: fixes x :: "'e"   and M :: "'e multiset" shows "(x \<notin># M) = (count M x = 0)"
ortho: fixes u :: "'a vec" assumes "u \<in> U" shows "u \<bullet>c v = (0::'a)"
affine_Inter: fixes f :: "'b set set" assumes "\<And>s. s \<in> f \<Longrightarrow> affine s" shows "affine (\<Inter> f)"
simps: fixes C :: "'b literal set" shows "subterms_of_cl C = {x. \<exists>L. x \<in> subterms_of_lit L \<and> L \<in> C}"
self_finance_def: fixes Mkt :: "('d, 'c) discrete_market"   and v :: "real"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and asset :: "'c" shows "self_finance Mkt v pf asset = qty_sum pf (qty_single asset (remaining_qty Mkt v pf asset))"
nth_root_nat_power_le: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "nth_root_nat k n ^ k \<le> n"
semiring_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semiring_ow U plus times \<equiv> ab_semigroup_add_ow U plus \<and> semigroup_mult_ow U times \<and> semiring_ow_axioms U plus times"
r_def: shows "r \<equiv> \<lambda>x. if x \<in> W0 then r0 x else x"
bigomegaI_tendsto_norm: fixes c :: "real"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "c \<noteq> 0"   and "((\<lambda>x. norm (f x / g x)) \<longlongrightarrow> c) F" shows "f \<in> \<Omega>[F](g)"
mset_le_add_mset: fixes x :: "'c"   and B :: "'c multiset"   and C :: "'c multiset" assumes "add_mset x B \<subseteq># C" shows "{#x#} \<subseteq># C \<and> B \<subseteq># C"
finsum_zero: fixes A :: "'b set" shows "(\<Sum>i\<in>A. (0::'a)) = (0::'a)"
Mem_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (t IN u)"
parts_snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "Y \<in> parts H"
has_trace_def: fixes ioa :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and b :: "nat \<Rightarrow> 'e option" shows "has_trace ioa b \<equiv> \<exists>ex\<in>executions ioa. b = mk_trace ioa (fst ex)"
abbd: fixes a :: "real"   and b :: "real pdevs" assumes "(a, b) \<in> set XS" shows "degree b \<le> degree_aforms XS"
code: fixes eff :: "'b \<Rightarrow> 'a \<Rightarrow> 'a fset \<Rightarrow> bool"   and rs :: "'b stream"   and s :: "'a" shows "RuleSystem_Defs.mkTree eff rs s = Node (s, shd (RuleSystem_Defs.trim eff rs s)) (RuleSystem_Defs.mkTree eff (stl (RuleSystem_Defs.trim eff rs s)) |`| RuleSystem_Defs.pickEff eff (shd (RuleSystem_Defs.trim eff rs s)) s)"
mult_nonneg_nonpos: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "b \<le> (0::'a)" shows "a * b \<le> (0::'a)"
ec': shows "simAbs ec' \<in> simAbs ` set (k_succs x)"
oldUses_def: fixes n :: "'b"   and g :: "'a" assumes "n \<in> set (\<alpha>n g)" shows "oldUses g n = var g ` lookup_multimap (uses g) n"
do_I_def: fixes c :: "'d \<Rightarrow> 'c"   and S :: "'c set" shows "do_I c S \<equiv> (\<lambda>(A, A'). tr A = tr A' \<and> S \<inter> ref A' = {}) \<triangleleft> wait \<circ> snd \<triangleright>  (\<lambda>(A, A'). hd (tr A' - tr A) \<in> S \<and> c (alpha_rp.more A) = last (tr A'))"
bisimResPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<sim> Q"   and "x \<sharp> \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<sim> \<lparr>\<nu>x\<rparr>Q"
pred_intros_imp': fixes K :: "bool"   and P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "K \<Longrightarrow> pred M P" shows "pred M (\<lambda>x. K \<longrightarrow> P x)"
symI: fixes r :: "('e \<times> 'e) set" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r" shows "sym r"
subprob_measurableD_2: fixes N :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and S :: "'c measure"   and x :: "'b" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "sets (N x) = sets S"
l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove"
nabla_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "\<nabla>x +  |x\<^sup>\<star>\<rangle>  |y\<rangle> \<nabla>(x + y) = \<nabla>(x + y)"
mHom_func: fixes f :: "'f \<Rightarrow> 'g"   and M :: "('f, 'a, 'h) Module_scheme"   and N :: "('g, 'a, 'i) Module_scheme" assumes "f \<in> mHom R M N" shows "f \<in> carrier M \<rightarrow> carrier N"
set_mset_single: fixes b :: "'a" shows "set_mset (b, {#}) = {b}"
carrier_conv_generator: shows "carrier \<G> = ([^]) \<^bold>g ` {..<order \<G>}"
return_def: fixes R :: "'c measure"   and x :: "'c" shows "return R x = measure_of (space R) (sets R) (\<lambda>A. indicator A x)"
VA': shows "Vagree \<nu> \<omega> (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP a)))"
finite_UNIV_natural_def: shows "finite_UNIV = Phantom(natural) False"
pr_var_vwb_lens: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "vwb_lens (&x)"
satisfy_equation_def: fixes p :: "real poly"   and qs :: "real poly list"   and subset_list :: "nat list list"   and sign_list :: "rat list list" shows "satisfy_equation p qs subset_list sign_list = (matrix_A sign_list subset_list *\<^sub>v construct_lhs_vector p qs sign_list = construct_rhs_vector p qs subset_list)"
Inl_Inr_False: fixes x :: "'d"   and y :: "'e" shows "(Inl x = Inr y) = False"
iWeakUntil_imp: fixes I :: "nat set"   and P1 :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool"   and P2 :: "nat \<Rightarrow> bool" assumes "P1 t1. t1 \<W> t2 I. Q t2"   and "\<box> t I. P1 t \<longrightarrow> P2 t" shows "P2 t1. t1 \<W> t2 I. Q t2"
list_it_correct: fixes s :: "'s" assumes "invar s" shows "set_iterator (list_it s) (\<alpha> s)"
abs_fun_supp_approx: fixes a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "supp ([a].x) \<subseteq> supp (x, a)"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and s :: "'b stream" shows "(x # xs) @- s = x ## xs @- s"
category_with_binary_products_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "category_with_binary_products C \<equiv> category C \<and> category_with_binary_products_axioms C"
lemma_2_4_7_a: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "a l-n\<rightarrow> b = a ^ n l\<rightarrow> b"
fd_def: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> y = ad (ad (x \<cdot> y))"
geodesic_segmentI2_1: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> 'b" assumes "isometry_on {a..b} g"   and "a \<le> b" shows "geodesic_segment_between (g ` {a..b}) (g a) (g b)"
nb\<^sub>1\<^sub>7: shows "dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op')) = dom (\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op')))"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "R.in_hom f a b = (R.arr a \<and> R.arr b \<and> R.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
solutions_linorder_strict: fixes D :: "nat"   and a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "pell D"   and "pell.solution D (a, b)"   and "pell.solution D (x, y)" shows "(a, b) = (x, y) \<or> a < x \<and> b < y \<or> x < a \<and> y < b"
typeof_conf: fixes h :: "heap"   and v :: "val"   and T :: "ty"   and P :: "'a prog" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
cpx_rng_def: shows "cpx_rng \<equiv> \<lparr>carrier = UNIV, monoid.mult = (*), one = 1, zero = 0, add = (+)\<rparr>"
zsgn_def: fixes i :: "int" shows "sgn i = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
in_Sup_llist_in_Sup_state: fixes C :: "'a literal multiset" assumes "C \<in> Sup_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)" shows "\<exists>D \<sigma>. D \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts) \<and> D \<cdot> \<sigma> = C \<and> is_ground_subst \<sigma>"
prj_simps_16: shows "C.cod f_src_src.Prj\<^sub>1 = f.apex"
if_distribR: fixes b :: "bool"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(if b then f else g) x = (if b then f x else g x)"
dropWhile_cong: fixes l :: "'a list"   and k :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "dropWhile P l = dropWhile Q k"
reflection_of_barbs_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b CWB)" shows "rel_reflects_binary_pred (Rel\<^sup>*) (\<lambda>b. hasBarb b CWB)"
write_def: fixes c :: "'c \<Rightarrow> 'b"   and a :: "'c"   and P :: "'b process" shows "c`!`a \<rightarrow> P \<equiv> Mprefix {c a} (\<lambda>x. P)"
f_Equiv_Exec_expand_shrink_equiv_state_set: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "'h list" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "\<And>input0 m. \<lbrakk>set input0 \<subseteq> A; m \<in> A\<rbrakk> \<Longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input0 \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input0 \<odot> k2) c2)"   and "set input \<subseteq> A" shows "equiv_states (localState1 (f_Exec_Comp trans_fun1 (map input_fun1 input \<odot> k1) c1)) (localState2 (f_Exec_Comp trans_fun2 (map input_fun2 input \<odot> k2) c2))"
WHILE_rule_gen_le: fixes m0 :: "'c nres"   and I :: "'c nres"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" assumes "m0 \<le> I"   and "\<And>s. \<lbrakk>RETURN s \<le> I; b s\<rbrakk> \<Longrightarrow> f s \<le> I" shows "m0 \<bind> WHILE b f \<le> filter_nb b I"
sp: shows "automaton.trans A = oseqp_sos \<Gamma> i"
B: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqinter> z = y \<sqinter> z \<and> x \<squnion> z = y \<squnion> z" shows "x = y"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> H" shows "inv a \<in> H"
simps_130: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x9 :: "8 word" shows "(case IPv4Proto x9 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f9 x9"
main: assumes "p \<noteq> 0"   and "ipoly p x = (0::'a)"   and "degree p \<noteq> 0" shows "thesis"
interaction_bounded_by'_exec_gpv_count: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "nat"   and x :: "'b"   and s' :: "'d"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'd) spmf"   and s :: "'d"   and count :: "'d \<Rightarrow> nat" assumes "interaction_bounded_by' consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); True; consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); True; \<not> consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "True" shows "count s' \<le> n + count s"
Pxy_nn: fixes x :: "'b"   and y :: "'c" assumes "x \<in> space S"   and "y \<in> space T" shows "0 \<le> Pxy (x, y)"
relpow_Suc: fixes n :: "nat"   and r :: "('a \<times> 'a) set" shows "r ^^ Suc n = r O r ^^ n"
G_nested_propos_Only_G: fixes \<phi> :: "'b ltl" shows "Only_G (\<^bold>G \<phi>)"
supchamberD_1: fixes X :: "'b set set"   and v :: "'b" assumes "ChamberComplex X"   and "v \<in> \<Union> X" shows "SimplicialComplex.maxsimp X (SOME C. SimplicialComplex.maxsimp X C \<and> v \<in> C)"
sublens_pres_vwb: fixes Y :: "'e \<Longrightarrow> 'f"   and X :: "'g \<Longrightarrow> 'f" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "vwb_lens X"
gt_trans: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and default :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "order_pair gt default"   and "gt x y"   and "gt y z" shows "gt x z"
that: fixes T :: "'a set" assumes "T \<subseteq> S"   and "closed T"   and "\<phi> T"   and "\<And>U. \<lbrakk>U \<subseteq> S; closed U; \<phi> U\<rbrakk> \<Longrightarrow> \<not> U \<subset> T" shows "thesis"
sup2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> k \<squnion> b" shows "a \<squnion> B \<equiv> k \<squnion> (a \<squnion> b)"
return_bind_spmf: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'd spmf" shows "return_spmf x \<bind> f = f x"
linear_strip_comb_cases: fixes pat :: "term" assumes "linear pat" obtains s :: "name" and args :: "term list" where "strip_comb pat = (Const s__, args)" and "pat = s__ $$ args"   | s :: "name" where "strip_comb pat = (Free s, [])" and "pat = Free s"
Invariants_in_sets: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets (qmpt.Invariants M T)" shows "A \<in> sets M"
lnth_Suc_LCons: fixes x :: "'g"   and xs :: "'g llist"   and n :: "nat" shows "lnth (LCons x xs) (Suc n) = lnth xs n"
intro: fixes ESi :: "'b ES_rec"   and \<V>i :: "'b V_rec"   and \<V> :: "'b V_rec" assumes "ES_valid ESi"   and "isViewOn \<V>i E\<^bsub>ESi\<^esub>"   and "V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> = V\<^bsub>\<V>i\<^esub>"   and "C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> \<subseteq> C\<^bsub>\<V>i\<^esub>" shows "CompositionSupport ESi \<V> \<V>i"
fSup_unfold: fixes f :: "nat \<Rightarrow> 'a" shows "f 0 \<squnion> (\<Squnion>n. f (Suc n)) = \<Squnion> range f"
disjunctive_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask"   and "\<And>n. \<not> bit a n \<or> \<not> bit b n" shows "plus a b = or a b"
m_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
field_differentiable_diff: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable F"   and "g field_differentiable F" shows "(\<lambda>z. f z - g z) field_differentiable F"
gt_hd_def: fixes ys :: "'a list"   and xs :: "'a list" shows "gt_hd ys xs = gt (hd ys) (hd xs)"
fmsubset_drop_set: fixes S :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdrop_set S m \<subseteq>\<^sub>f m"
quadratic_shape2b: fixes a :: "real"   and x :: "real"   and y :: "real"   and b :: "real"   and c :: "real"   and z :: "real" assumes "a < 0"   and "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"   and "y < z" shows "a * z\<^sup>2 + b * z + c < 0"
nrename_ImpL_aux: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and M :: "trm"   and w :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = ImpL <c>.M w.N u" shows "x = u \<and> x = y \<or> x \<noteq> u"
complete_sc_def: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" shows "complete_sc s vs = unfold_llist (\<lambda>(s, vs). \<forall>t ta s'. \<not> s -t\<triangleright>ta\<rightarrow> s') (\<lambda>(s, vs). fst (SOME ((t, ta), s'). s -t\<triangleright>ta\<rightarrow> s' \<and> ta_seq_consist P vs (llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))) (\<lambda>(s, vs). let ((t, ta), s') = SOME ((t, ta), s'). s -t\<triangleright>ta\<rightarrow> s' \<and> ta_seq_consist P vs (llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) in (s', mrw_values P vs \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (s, vs)"