hcju/isabelleps huggingface dataset:
pdivides_imp_is_root: fixes p :: "'a list"   and x :: "'a" assumes "p \<noteq> []"   and "x \<in> carrier R"   and "[\<one>, \<ominus> x] pdivides p" shows "is_root p x"
bop_ueval: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "('c, 'b) uexpr"   and y :: "('d, 'b) uexpr"   and b :: "'b" shows "\<lbrakk>bop f x y\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b)"
in_restricted_global_in_updated_global_1: fixes r' :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" assumes "r' \<in> RID\<^sub>G (s(r := None))" shows "r' \<in> RID\<^sub>G (s(r \<mapsto> ls))"
elims: fixes x :: "'d"   and xa :: "('c, 'e) step"   and y :: "('c, 'd \<times> 'e) step" assumes "fix_step x xa = y" obtains a :: "'d" where "x = a__" and "xa = Done" and "y = Done"   | a :: "'d" and s :: "'e" where "x = aa__" and "xa = Skip sa__" and "y = Skip (aa__, sa__)"   | a :: "'d" and xb :: "'c" and s :: "'e" where "x = a" and "xa = Yield xb s" and "y = Yield xb (a, s)"
P'_2: shows "path_conforms_with_strategy p P' \<sigma>"
perp__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "Coplanar A B C D"
bb': fixes i :: "'a" assumes "i \<in> Basis" shows "b (b' i) = i"
ix_gt: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat"   and d :: "nat" assumes "p \<in> sparsegrid dm lm"   and "d < dm" shows "0 < ix p d"
cmp_components_are_iso: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi>\<^sub>F (f, g))"
prove_nonneg_simps_1: fixes prnt :: "(nat \<times> nat \<times> char list) list"   and p :: "nat"   and slp :: "floatarith list"   and X :: "(real \<times> real pdevs) list list" shows "prove_nonneg prnt 0 p slp X = False"
le_INF_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and u :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq u (Inf (f ` A)) = (\<forall>i\<in>A. less_eq u (f i))"
elements_mat: fixes A :: "'b mat" shows "elements_mat A = {A $$ (i, j) |i j. i < dim_row A \<and> j < dim_col A}"
nat_mult_div_cancel_disj: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k * m div (k * n) = (if k = 0 then 0 else m div n)"
simps_11: fixes f1 :: "'a"   and f2 :: "('b, 'c) Node \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> ('b, 'c) Digit \<Rightarrow> ('b, 'c) FingerTreeStruc \<Rightarrow> ('b, 'c) Digit \<Rightarrow> 'a"   and x31 :: "'c"   and x32 :: "('b, 'c) Digit"   and x33 :: "('b, 'c) FingerTreeStruc"   and x34 :: "('b, 'c) Digit" shows "(case Deep x31 x32 x33 x34 of Empty \<Rightarrow> f1 | Single x \<Rightarrow> f2 x | Deep x xa xb xc \<Rightarrow> f3 x xa xb xc) = f3 x31 x32 x33 x34"
vec_subtensor: fixes A :: "'b tensor"   and i :: "nat" assumes "dims A \<noteq> []"   and "i < hd (dims A)" shows "vec (subtensor A i) = fixed_length_sublist (vec A) (prod_list (tl (dims A))) i"
orderE: fixes a :: "'a word"   and b :: "'a word" assumes "a \<le>s b" obtains "a = signed.min a b"
assign_elim: fixes x :: "'Var"   and e :: "'AExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>x \<leftarrow> e, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "c' = Stop \<and> mds = mds' \<and> mem' = mem(x := eval\<^sub>A mem e)"
eventually_prod_same: fixes P :: "'a \<times> 'a \<Rightarrow> bool"   and F :: "'a filter" shows "eventually P (F \<times>\<^sub>F F) = (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
inv: fixes a :: "'a" assumes "A.ide a" shows "inv.B.inverse_arrows (\<sigma> a) (\<sigma>' a)"
span_redundant: fixes x :: "'d"   and S :: "'d set" assumes "x \<in> cspan S" shows "cspan (insert x S) = cspan S"
less_numeral_extra_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
direct_decomp_direct_decomp: fixes A :: "'b set"   and s :: "'b set"   and ss :: "'b set list"   and rs :: "'b set list" assumes "direct_decomp A (s # ss)"   and "direct_decomp s rs" shows "direct_decomp A (ss @ rs)"
prod_zero_iff: fixes 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 A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.semidom plus minus zero times one"   and "finite A" shows "(comm_monoid_mult.prod times one f A = zero) = (\<exists>a\<in>A. f a = zero)"
in_nv2T: fixes x :: "Preliminaries.var"   and T :: "'b trm" shows "(x \<in> nv2T T) = (T = Var x)"
L_transform_eqvt: fixes p :: "perm"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "p \<bullet> L_transform x = L_transform (p \<bullet> x)"
gVarPresGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme" shows "gVarPresGWls MOD \<equiv> \<forall>xs x. gWls MOD (asSort xs) (gVar MOD xs x)"
path_to_def: fixes q :: "hf" shows "MN.path_to q \<equiv> SOME u. u \<in> MN.left_lang q"
simps_26: fixes v :: "'b rexp"   and va :: "'b rexp"   and vf :: "'b rexp" shows "le_rexp (Plus v va) (Star vf) = False"
measure_zero_union: fixes s :: "'a set"   and t :: "'a set" assumes "s \<in> sets M"   and "t \<in> sets M"   and "Sigma_Algebra.measure M t = 0" shows "Sigma_Algebra.measure M (s \<union> t) = Sigma_Algebra.measure M s"
inject: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d"   and y21 :: "'c"   and y22 :: "'c"   and y23 :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "(MkArr x21 x22 x23 = MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
p: shows "\<forall>asa bsa. as @ x # y # bs = asa @ bsa \<longrightarrow> (\<forall>x\<in>set asa. \<forall>y\<in>set bsa. x \<le> y)"
star_of_def: fixes x :: "'b" shows "star_of x \<equiv> star_n (\<lambda>n. x)"
nonabort_j: shows "chaos \<sqsubseteq> (j::'a::refinement_lattice)"
pfinpref_iff: fixes x :: "'b llist"   and A :: "'b set"   and s :: "'b llist" shows "(x \<in> pfinpref A s) = (x \<in> finpref A s \<and> x \<noteq> LNil)"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_State_Idle_conv2: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<diamond> t1 [0\<dots>,k - Suc 0] \<oplus> t0. output_fun (s t1) = m \<and> (State_Idle localState output_fun trans_fun (localState (s t1)) \<or> (\<circle> t2 t1 [0\<dots>]. output_fun (s t3) = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t2. output_fun (s t4) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t4)))))"
nchotomy: shows "\<forall>finite_2. finite_2 = a\<^sub>1 \<or> finite_2 = a\<^sub>2"
simps_20: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and x31 :: "com"   and x32 :: "com" shows "IF x41 THEN x42 ELSE x43 \<noteq> x31;; x32"
non_prim: fixes w :: "'b list" assumes "\<not> primitive w"   and "w \<noteq> \<epsilon>" obtains r :: "'b list" and k :: "nat" where "r \<noteq> \<epsilon>" and "1 < k" and "r \<^sup>@ k = w" and "w \<noteq> r"
less_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> \<not> x \<approx> y)"
wsorted_nth_mono: fixes L :: "'a list"   and i :: "nat"   and j :: "nat" assumes "wsorted L"   and "i \<le> j"   and "j < length L" shows "weight (L ! i) \<le> weight (L ! j)"
basic_wt_eq_wt: fixes s :: "('s, 'v) tm" shows "wt s = local.kbo_std.wt s"
assms_4: fixes P :: "pi"   and Q :: "pi"   and R :: "pi"   and T :: "pi" assumes "(P, Q) \<in> Rel"   and "(R, T) \<in> Rel'" shows "(P \<parallel> R, Q \<parallel> T) \<in> Rel'"
distinct_28: fixes x6 :: "'b ptrm"   and x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PFst x6 \<noteq> PApp x31 x32"
IH: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "\<forall>a. ListMem a as \<longrightarrow> action_needed_asses a s2 \<subseteq>\<^sub>f action_needed_asses a s1" shows "exec_plan s1 as \<subseteq>\<^sub>f exec_plan s2 as"
path_connected_space_path_component_set: fixes X :: "'b topology" shows "path_connected_space X = (\<forall>x\<in>topspace X. path_component_of_set X x = topspace X)"
LeftDerivation_Derive_take_LeftDerives1: fixes N :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and \<beta> :: "symbol list" assumes "N \<noteq> 0"   and "N \<le> length D"   and "LeftDerivation a D b"   and "\<alpha> = Derive a (take (N - 1) D)"   and "\<beta> = Derive a (take N D)" shows "LeftDerives1 \<alpha> (fst (D ! (N - 1))) (snd (D ! (N - 1))) \<beta>"
filter_ex1: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<exists>!x. x \<in> set xs \<and> P x"   and "P x"   and "x \<in> set xs"   and "distinct xs" shows "filter P xs = [x]"
T_composition_borel_2: fixes f :: "'a \<Rightarrow> 'b"   and k :: "nat" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f ((T ^^ k) x)) \<in> borel_measurable M"
sorted_cons: fixes x :: "'a"   and xs :: "'a list" assumes "Sorted_Less.sorted (x # xs)" shows "Sorted_Less.sorted xs"
A: fixes u :: "nat \<Rightarrow> 'a" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi" shows "ereal (dist x y) - extended_Gromov_product_at y (to_Gromov_completion x) xi - ereal (deltaG TYPE('a)) \<le> liminf (\<lambda>n. ereal (Gromov_product_at x y (u n)))"
l_delete_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_delete_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs"
STCal_divergent_2: fixes T :: "'b"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus" shows "TargetTerm T \<longmapsto>STCal Source Target\<omega> = T \<longmapsto>Target\<omega>"
tranclp_on_def: shows "tranclp_on \<equiv> \<lambda>U r. lfp (\<lambda>p x1 x2. (\<exists>a b. x1 = a \<and> x2 = b \<and> U a \<and> U b \<and> r a b) \<or> (\<exists>a b c. x1 = a \<and> x2 = c \<and> U a \<and> U b \<and> U c \<and> p a b \<and> r b c))"
l_one: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
any_closest_point_dot: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and a :: "'a" assumes "convex S"   and "closed S"   and "x \<in> S"   and "y \<in> S"   and "\<forall>z\<in>S. dist a x \<le> dist a z" shows "(a - x) \<bullet> (y - x) \<le> 0"
card_le_Suc0_iff_eq: fixes A :: "'c set" assumes "finite A" shows "(card A \<le> Suc 0) = (\<forall>a1\<in>A. \<forall>a2\<in>A. a1 = a2)"
inext_nth_mono2_infin: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "infinite I" shows "(I \<rightarrow> a < I \<rightarrow> b) = (a < b)"
prems: shows "KD_Tree.size_kdt kdt = 2 ^ min_height kdt"
simps_5: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula"   and G :: "'b formula" shows "\<A> \<Turnstile> F \<^bold>\<or> G = (\<A> \<Turnstile> F \<or> \<A> \<Turnstile> G)"
bi_unique_mod_ring_rel_3: shows "right_unique mod_ring_rel"
prime_factorization_nat_3: fixes n :: "nat" assumes "n = 0" shows "prime_factorization_nat n = []"
mem_notdisjoint: fixes x :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" assumes "x mem l1"   and "x mem l2" shows "\<not> disjoint l1 l2"
top_weak_eq: fixes t :: "'a" assumes "t \<in> carrier L"   and "\<And>x. x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> t" shows "t .= \<top>"
arr_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.arr C (partial_magma.cod C f)"
ode_sem_equiv_def: fixes \<alpha> :: "('d, 'e, 'f) hp"   and I :: "('d, 'e, 'f) interp" shows "ode_sem_equiv \<alpha> I = (\<forall>ODE \<phi>. osafe ODE \<longrightarrow> fsafe \<phi> \<longrightarrow> \<alpha> = EvolveODE ODE \<phi> \<longrightarrow> {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> VSagree (sol 0) (fst \<nu>) {x |x. Inl x \<in> FVP (EvolveODE ODE \<phi>)}} = {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> sol 0 = fst \<nu>})"
l_get_owner_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_owner_document_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.known_ptrs DocumentClass.type_wf heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs get_owner_document"
ole_def: fixes D :: "('h, 'i) Order_scheme"   and a :: "'h"   and b :: "'h" shows "a \<preceq>\<^bsub>D\<^esub> b = ((a, b) \<in> rel D)"
lcl_conv: fixes h :: "'c"   and l :: "'b" shows "lcl (h, l) = l"
Suffixes_compower: fixes A :: "'a list set"   and n :: "nat" assumes "A \<noteq> {}" shows "Suffixes (A ^^ n) = insert [] (Suffixes A @@ \<Union> ((^^) A ` {..<n}))"
unfree_tm_def: fixes t :: "('c, 'd) tm" shows "unfree t = (case t of Hd (hd.Var a) \<Rightarrow> Some (to_name a) | Hd (Sym x) \<Rightarrow> Map.empty x | tm.App tm1 x \<Rightarrow> Map.empty x)"
conj_assoc: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> bool" shows "((P \<and> Q) \<and> S) = (P \<and> Q \<and> S)"
wf_int_ge_less_than: fixes d :: "int" shows "wf (int_ge_less_than d)"
sel_7: fixes x2 :: "('d, 'c) Term.term" shows "bvars\<^sub>s\<^sub>t\<^sub>p (receive\<langle>x2\<rangle>\<^sub>s\<^sub>t) = []"
lift_state_oracle_extend_state_oracle: fixes G :: "('c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf) \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'e) spmf"   and F :: "('c \<Rightarrow> 'd \<Rightarrow> (('a \<times> 'b) \<times> 'c) spmf) \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> (('a \<times> 'b) \<times> 'e) spmf"   and "oracle" :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf" assumes "\<And>B. Transfer.Rel (rel_fun (rel_fun (=) (rel_fun (=) (rel_spmf (rel_prod B (=))))) (rel_fun (=) (rel_fun (=) (rel_spmf (rel_prod B (=)))))) G F" shows "lift_state_oracle F \<dagger>oracle = \<dagger>(G oracle)"
unstream_maps_trans_Some: fixes f :: "'h \<Rightarrow> ('e, 'g) generator \<times> 'g"   and g :: "('h, 'f) generator"   and s :: "'f"   and g'' :: "('e, 'g) generator"   and s'' :: "'g" shows "unstream (maps_trans f g) (s, Some (g'', s'')) = unstream g'' s'' @ unstream (maps_trans f g) (s, None)"
less_ge_neq_conv: fixes n :: "'a"   and a :: "'a" shows "(n < a) = (\<forall>x\<ge>a. n \<noteq> x)"
pointermap_insert_def: fixes a :: "'a"   and m :: "('a, 'b) pointermap_scheme" shows "pointermap_insert a m \<equiv> \<lparr>entries = entries m @ [a], getentry = getentry m(a \<mapsto> length (entries m))\<rparr>"
LINEAR_not_BRANCH: fixes b :: "'g"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and c :: "'f"   and rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "LINEAR b r \<noteq> BRANCH c rs"
add_zero_left: fixes x :: "'a" assumes "x \<in> E" shows "(0::'a) + x = x"
insertI1: fixes a :: "'a"   and B :: "'a set" shows "a \<in> set_ins a B"
refine: fixes succi :: "'b \<Rightarrow> 'b list"   and E :: "('b \<times> 'b) set"   and src :: "'b"   and tgt :: "'b" assumes "(succi, E) \<in> \<langle>Id\<rangle>succg_rel" shows "(RETURN (dfs_impl2 succi src tgt), dfs E src tgt) \<in> \<langle>bool_rel\<rangle>nres_rel"
transform_simple_fw_upper_2: fixes rs :: "'c common_primitive rule list" assumes "simple_ruleset rs" shows "_Collect p ((common_matcher, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New) \<subseteq> _Collect p (simple_fw (to_simple_firewall (upper_closure (optimize_matches abstract_for_simple_firewall (upper_closure (packet_assume_new rs))))) p = Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New)"
simps_14: fixes x1 :: "'c" shows "WokenUp \<noteq> Suspend x1"
bin_sc_eq_2: fixes n :: "nat" shows "bin_sc n True = set_bit n"
bounded_def: fixes step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat" shows "bounded step n = (\<forall>p<n. \<forall>\<tau>. \<forall>(q, \<tau>')\<in>set (step p \<tau>). q < n)"
code: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "sscan f xs a = f (shd xs) a ## sscan f (stl xs) (f (shd xs) a)"
k: shows "k \<in> Units (R\<lparr>carrier := K\<rparr>)"
arctan_float_interval: fixes x :: "float interval"   and p :: "nat" shows "arctan ` set_of (real_interval x) \<subseteq> set_of (real_interval (arctan_float_interval p x))"
assms_1: shows "abc = [a, b, c]\<^sub>\<circ>"
seqr_or_distl: fixes P :: "('b, 'd) urel"   and Q :: "('b, 'd) urel"   and R :: "('d, 'c) urel" shows "(P \<or> Q) ;; R = (P ;; R \<or> Q ;; R)"
size_4: fixes x21 :: "'k"   and x22 :: "'k list" shows "length (x21 # x22) = length x22 + Suc 0"
assms_4: fixes x :: "'a"   and y :: "'a" assumes "\<not> P x y" shows "Q x y"
ring_closed_def: fixes A :: "'a set" shows "ring_closed A \<equiv> ((0::'a) \<in> A \<and> (1::'a) \<in> A) \<and> (\<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x + y \<in> A) \<and> (\<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x * y \<in> A) \<and> (\<forall>x. x \<in> A \<longrightarrow> - x \<in> A)"
loc_imps_fw_M_in_implications': fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and loc1 :: "'a"   and loc2 :: "'a"   and M :: "'c zmultiset"   and xs :: "('a \<times> 'b \<times> 'a) list"   and t :: "'c" assumes "dataflow_topology summary results_in"   and "dataflow_topology.loc_imps_fw summary results_in c loc1 loc2 M xs"   and "dataflow_topology.inv_imps_work_sum summary results_in c"   and "inv_implications_nonneg c"   and "dataflow_topology.worklists_vacant_to summary results_in c t"   and "0 < zcount M t" shows "\<exists>s\<le>t. s \<in>\<^sub>A frontier (c_imp c loc2)"
less: fixes xs :: "'a list" assumes "length xs < length xsa__" shows "orbits_list f xs = orbits_list_impl f xs"
inext_mono: fixes n :: "nat"   and I :: "nat set" shows "n \<le> inext n I"
wf_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "wf_digraph G \<equiv> (\<forall>e. e \<in> arcs G \<longrightarrow> tail G e \<in> verts G) \<and> (\<forall>e. e \<in> arcs G \<longrightarrow> head G e \<in> verts G)"
smc_Dom_vempty_if_Arr_vempty: assumes "\<AA>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<AA>\<lparr>Dom\<rparr> = []\<^sub>\<circ>"
dvd_Lcm: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times a (Lcm A)"
LID\<^sub>SI_1: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and l :: "'e"   and v :: "('d, 'e, 'f) val" assumes "\<sigma> l = Some v" shows "l \<in> LID\<^sub>S \<sigma>"
Sup_sigma: fixes M :: "'b set set set"   and \<Omega> :: "'b set" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>" shows "Sup (sigma \<Omega> ` M) = sigma \<Omega> (\<Union> M)"
streamValue11: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> 'a"   and t :: "nat"   and x :: "'a"   and j :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = x"   and "j \<le> Suc d" shows "lose (t + 2 + k + j) = x"
e: shows "eval (Cn 2 r_phi [recf.Id 2 0, Cn 2 r_take [Cn 2 r_delay [recf.Id 2 0, recf.Id 2 1], recf.Id 2 1]]) [i, x] = \<phi> i (e_take (the (delay i x)) x)"
is_sig_red_monom_mult: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and s :: "'a" assumes "is_sig_red sing_reg top_tail F p"   and "c \<noteq> (0::'b)" shows "is_sig_red sing_reg top_tail F (monom_mult c s p)"
kleisli_state_def: fixes g :: "'e \<Rightarrow> ('c, 'd) state"   and f :: "'f \<Rightarrow> ('c, 'e) state"   and a :: "'f" shows "(g \<bullet> f) a = f a \<bind> g"
0: shows "bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
pm_of_poly_times: fixes x :: "'a"   and p :: "'b poly"   and q :: "'b poly" shows "pm_of_poly x (p * q) = pm_of_poly x p * pm_of_poly x q"
exi_ren_subst_fresh: fixes y :: "name"   and \<phi> :: "fmla"   and x :: "name" assumes "atom y \<sharp> \<phi>" shows "exi x \<phi> = exi y (\<phi>(x::=Var y))"
elims: fixes x :: "char list"   and y :: "char list set" assumes "internal_iface_name_to_set x = y" obtains i :: "char list" where "x = i" and "y = (if \<not> iface_name_is_wildcard i then {i} else {butlast i @ cs |cs. True})"
len_upd_until: shows "length done + (len + 1) = length (update_until args rel1 rel2 nt auxlist)"
distr_marginal1: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e"   and Pxy :: "'d \<times> 'e \<Rightarrow> ennreal" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "distributed M S X (\<lambda>x. \<integral>\<^sup>+ z. Pxy (x, z) \<partial>T)"
assms_2: fixes c :: "'a set" assumes "c \<in> components S" shows "continuous_on c f"
circ_simulate: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z * x \<le> y * z \<longrightarrow> z * x\<^sup>\<star> \<le> y\<^sup>\<star> * z"