hcju/isabelleps huggingface dataset:
map_dds_parametric: fixes A' :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and B' :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "((A' ===> A) ===> (B ===> B') ===> rel_dds A B ===> rel_dds A' B') map_dds map_dds"
inverse_split: fixes G :: "('d, 'e) cyclic_group_scheme"   and a :: "'d"   and b :: "'d" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "inv\<^bsub>G\<^esub> (a \<otimes>\<^bsub>G\<^esub> b) = inv\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> b"
B5: shows "weak_reduction_bisimulation Rel (STCal Source Target)"
sources_iso_closed: fixes a :: "'a"   and \<mu> :: "'a"   and a' :: "'a" assumes "a \<in> sources \<mu>"   and "a \<cong> a'" shows "a' \<in> sources \<mu>"
inner_rot_in_segment: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "(x - a) \<bullet> rot (b - a) = 0"   and "a \<noteq> b" shows "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))"
dim_raw_def: shows "dim \<equiv> vector_space.dim (*\<^sub>R)"
prv_subst_scnj_imp: fixes F :: "'fmla set"   and t :: "'trm"   and x :: "'var" assumes "F \<subseteq> fmla"   and "finite F"   and "t \<in> trm"   and "x \<in> var" shows "prv (imp (scnj ((\<lambda>\<phi>. subst \<phi> t x) ` F)) (subst (scnj F) t x))"
T_1: shows "m = \<langle>f T\<rangle>\<^sub>t"
qfN_prime_unique: fixes a :: "int"   and N :: "int"   and b :: "int"   and c :: "int"   and d :: "int" assumes "prime (a\<^sup>2 + N * b\<^sup>2)"   and "1 < N"   and "a\<^sup>2 + N * b\<^sup>2 = c\<^sup>2 + N * d\<^sup>2" shows "\<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>b\<bar> = \<bar>d\<bar>"
mult_cancel_right2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times a c = c) = (c = zero \<or> a = one)"
assms: fixes A :: "'a tensor" assumes "A \<in> set As" shows "dims A = ds"
is_idirprod_generate_imp_generate: fixes A :: "'a set"   and gs :: "'a set" assumes "is_idirprod A (\<lambda>g. generate G {g}) gs" shows "A = generate G gs"
convex_closure_interior: fixes S :: "'a set" assumes "convex S"   and "interior S \<noteq> {}" shows "closure (interior S) = closure S"
crypts_mono: fixes H :: "msg set"   and H' :: "msg set" assumes "H \<subseteq> H'" shows "crypts H \<subseteq> crypts H'"
equivpI: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
foldr_replicate: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and n :: "nat"   and x :: "'h" shows "foldr f (replicate n x) = f x ^^ n"
le_less: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x \<subseteq># y) = (x \<subset># y \<or> x = y)"
inducts: fixes x :: "ind"   and P :: "ind \<Rightarrow> bool" assumes "Nat x"   and "P Zero_Rep"   and "\<And>i. \<lbrakk>Nat i; P i\<rbrakk> \<Longrightarrow> P (Suc_Rep i)" shows "P x"
simps: fixes P1 :: "pol"   and M :: "mon"   and P2 :: "pol"   and n :: "nat" shows "pnsubst1 P1 M P2 n = (case ponesubst P1 M P2 of None \<Rightarrow> P1 | Some P3 \<Rightarrow> if n = 0 then P3 else pnsubst1 P3 M P2 (n - 1))"
witness_3: fixes b :: "'d" assumes "b \<in> T" shows "j (i b) = b"
monotone_on_dual: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "monotone_on X r s f" shows "monotone_on X (\<lambda>x y. r y x) (\<lambda>x y. s y x) f"
magQ_def: fixes x :: "'a['b, 'c]" shows "\<lbrakk>x\<rbrakk>\<^sub>Q = mag (fromQ x)"
strip_while_map: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "strip_while P (map f xs) = map f (strip_while (P \<circ> f) xs)"
Sup_ereal_def: fixes S :: "ereal set" shows "Sup S = (SOME x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z))"
fpairsI: fixes x :: "V"   and r :: "V"   and a :: "V"   and b :: "V" assumes "x \<in>\<^sub>\<circ> r"   and "x = [a, b]\<^sub>\<circ>" shows "x \<in>\<^sub>\<circ> fpairs r"
simps_4: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, False, MEq, t2) = formula.Neg (formula.Eq t1 t2)"
tendsto_add: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'f"   and F :: "'e filter"   and g :: "'e \<Rightarrow> 'f"   and b :: "'f" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
SystemVSubst_def: shows "SystemVSubst = \<lparr>SFunctions = Map.empty, SPredicates = \<lambda>p. if p = vid1 then Some (Geq ($f (Inl fid1) local.empty) (Const 0)) else None, SContexts = Map.empty, SPrograms = \<lambda>a. if a = vid1 then Some (EvolveODE (OProd (OSing vid1 ($f fid1 local.empty)) (OSing vid2 (Var vid1))) (TT && Geq ($f fid1 local.empty) (Const 0))) else None, SODEs = Map.empty\<rparr>"
unwinding_snd: fixes I :: "('a \<times> 'a) set"   and D :: "event option \<Rightarrow> 'a" shows "secure (ts_process {[], [Some b]}) I D = d_weakly_future_consistent (ts_process {[], [Some b]}) I D (rel_ipurge (ts_process {[], [Some b]}) I D)"
order_refl: fixes x :: "'a list" shows "x \<preceq> x"
preserve_min_P: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and C :: "'a literal multiset"   and j :: "nat"   and i :: "nat" assumes "St \<leadsto>\<^sub>w St'"   and "(C, j) \<in># wP_of_wstate St'"   and "(C, i) \<in># wP_of_wstate St"   and "\<forall>k. (C, k) \<in># wP_of_wstate St \<longrightarrow> i \<le> k" shows "(C, i) \<in># wP_of_wstate St'"
adjoint_row: fixes i :: "nat"   and A :: "'c mat" assumes "i < dim_col A" shows "row (adjoint A) i = conjugate (col A i)"
assms_1: shows "constructive_security middle ideal sim_inner \<I>_middle \<I>_inner \<I>_common bound_inner lossless_inner w1"
mult: fixes a :: "'a"   and H :: "'a set"   and b :: "'a" assumes "a \<in> generate H"   and "b \<in> generate H" shows "a \<cdot> b \<in> generate H"
isnonconstant_pnormal_iff: fixes p :: "'b poly"   and bs :: "'b list" assumes "isnonconstant p" shows "pnormal (polypoly bs p) = (Ipoly bs (head p) \<noteq> (0::'b))"
resTauRight: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "\<tau>.(<\<nu>x>P) \<leadsto>[Rel] <\<nu>x>\<tau>.P"
scalar_prod_comm: fixes v :: "'c vec"   and w :: "'c vec" assumes "dim_vec v = dim_vec w" shows "v \<bullet> w = w \<bullet> v"
Gramian_matrix_alt_def: fixes k :: "nat"   and G :: "'a vec list" assumes "k \<le> length G" shows "gs.Gramian_matrix G k = (let M = mat_of_rows n (take k G) in M * M\<^sup>T)"
llength_eq_enat_lfiniteD: fixes xs :: "'a llist"   and n :: "nat" assumes "llength xs = enat n" shows "lfinite xs"
floor_mono: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
less_sets_UN1: fixes \<A> :: "'a set set"   and B :: "'a set" shows "(\<Union> \<A> \<lless> B) = (\<forall>A\<in>\<A>. A \<lless> B)"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "I.wtE \<xi>"   and "wt T" shows "I_intT (tpOf T) (I.int \<xi> T)"
lseg_if_splitf2: fixes x :: "'a"   and xs :: "'a list"   and p :: "'a node ref option"   and q :: "'a node ref option" shows "lseg (x # xs) p q = (\<exists>\<^sub>App n. pp \<mapsto>\<^sub>r Node x n * lseg xs n q * \<up> (p = Some pp))"
abs_eq: fixes x :: "'c list" shows "fset_from_list x = Abs_fset (set x)"
rp_resample: fixes B :: "'b set"   and A :: "'b set"   and C :: "'b set" assumes "B \<subseteq> A \<union> C"   and "A \<inter> C = {}"   and "C \<subseteq> B"   and "finite B" shows "spmf_of_set B \<bind> (\<lambda>x. if x \<in> A then spmf_of_set C else return_spmf x) = spmf_of_set C"
eint_pos_int_times_gt: fixes c :: "eint"   and n :: "int" assumes "0 < c"   and "c \<noteq> \<infinity>"   and "1 < n" shows "c < eint n * c"
D_seq: fixes P :: "'b process"   and Q :: "'b process" shows "D (P `;` Q) = div_seq P Q"
group_add_ow_def: fixes U :: "'b set"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" shows "group_add_ow U minus plus zero uminus \<equiv> (minus_ow U minus \<and> uminus_ow U uminus) \<and> monoid_add_ow U plus zero \<and> group_add_ow_axioms U minus plus zero uminus"
star_of_real_def: fixes r :: "real" shows "of_real r = star_of (of_real r)"
entt_disjD1: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<or>\<^sub>A B \<Longrightarrow>\<^sub>t C" shows "A \<Longrightarrow>\<^sub>t C"
assms_4: fixes y :: "'al" assumes "y \<in> U"   and "\<And>i. i \<in> A \<Longrightarrow> y \<le>\<^sub>o\<^sub>w i" shows "y \<le>\<^sub>o\<^sub>w x"
that: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "s' \<approx> t'"   and "c' \<approx>wT d'" shows "thesis"
ivl_disj_un_singleton_6: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l..<u} \<union> {u} = {l..u}"
fmimage_Union: fixes m :: "('b, 'a) fmap"   and A :: "'b fset fset" shows "fmimage m (ffUnion A) = ffUnion (fmimage m |`| A)"
vars_of_type_eq_type_max_eq: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "\<forall>x\<in>\<Union> (bexp_vars ` t). mem\<^sub>1 x = mem\<^sub>2 x" shows "sifum_types.type_max ev\<^sub>B t mem\<^sub>1 = sifum_types.type_max ev\<^sub>B t mem\<^sub>2"
transfer_update: shows "rel_fun (=) (rel_fun (=) (local.rel_state (=))) update' update"
trss_modify_context_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), e, s', c') \<in> trss fg"   and "mon_e fg e \<inter> mon_c fg cn = {}" shows "\<exists>csp. c' = csp + c \<and> mon_c fg csp = {} \<and> ((s, cn), e, s', csp + cn) \<in> trss fg"
fls_eqI: fixes x :: "'a fls"   and y :: "'a fls" assumes "\<And>xa. fls_nth x xa = fls_nth y xa" shows "x = y"
case_eq_if: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and llist :: "'d llist" shows "(case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = (if lnull llist then f1 else f2 (lhd llist) (ltl llist))"
leadsETo_Union_Int: fixes S :: "'d set set"   and F :: "'d program"   and C :: "'d set"   and CC :: "'d set set"   and B :: "'d set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> (A \<inter> C) leadsTo[CC] B" shows "F \<in> (\<Union> S \<inter> C) leadsTo[CC] B"
domI: fixes m :: "'g \<Rightarrow> 'f option"   and a :: "'g"   and b :: "'f" assumes "m a = Some b" shows "a \<in> dom m"
sel: fixes conv :: "('a, 'b, 'c, 'd) converter"   and res :: "('c, 'd) resource" shows "run_resource (conv \<rhd> res) = (\<lambda>a. map_spmf (\<lambda>((b, conv'), res'). (b, conv' \<rhd> res')) (exec_gpv run_resource (run_converter conv a) res))"
fmdom'_drop_set: fixes A :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdom' (fmdrop_set A m) = fmdom' m - A"
iT_Mult_Max: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "Max (I \<otimes> k) = Max I * k"
Rep_List_in_sexp: fixes x :: "'b SList.list" shows "Rep_List x \<in> sexp"
has_vector_derivative_continuous: fixes f :: "real \<Rightarrow> 'a"   and D :: "'a"   and x :: "real"   and s :: "real set" assumes "(f has_vector_derivative D) (at x within s)" shows "continuous (at x within s) f"
\<Phi>_n_def: fixes x :: "real" shows "\<Phi>_n x = x ^ 3 / 3 * 2 ^ (l' + 1) * 2 ^ (l + 1) + x * (real_of_int i - 1) * (real_of_int i' - 1) - x\<^sup>2 / 2 * ((real_of_int i' - 1) * 2 ^ (l + 1) + (real_of_int i - 1) * 2 ^ (l' + 1))"
componentsE: fixes S :: "'c set"   and U :: "'c set" assumes "S \<in> components U" obtains x :: "'c" where "x \<in> U" and "S = connected_component_set U x"
td_ext_def': fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and norm :: "'b \<Rightarrow> 'b" shows "td_ext Rep Abs A norm \<equiv> ((\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)) \<and> (\<forall>x. Rep (Abs x) = norm x)"
reduction_step2: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "\<exists>A1. \<bar>spmf (R1_14_interm1 M (c0, c1) \<bind> D) True - spmf (R1_14_interm2 M (c0, c1) \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar>"
lprefix_code_1: fixes ys :: "'c llist" shows "LNil \<sqsubseteq> ys = True"
subdef: shows "usubstappt \<sigma> {RVar x, DVar x} \<theta> \<noteq> undeft"
filterlim_at_bot_linear_iff: fixes c :: "'a"   and f :: "'a \<Rightarrow> 'b"   and b :: "'a"   and F2 :: "'b filter" assumes "c \<noteq> (0::'a)" shows "(LIM x at_bot. f (x * c + b) :> F2) = (if (0::'a) < c then filterlim f F2 at_bot else filterlim f F2 at_top)"
nofail_simps_3: fixes x :: "'e" shows "nofail (RETURN x) = True"
init: fixes s :: "'a" assumes "s \<in> init A" shows "P s"
assms_3: shows "agree (fmrestrict_set vs (snd a)) (fmrestrict_set vs s)"
base: shows "b \<in> {..<n} \<rightarrow> {..<p}"
Gcd_in_group_closure: fixes S :: "int set" shows "Gcd S \<in> group_closure S"
act_inv_act: fixes g :: "'a"   and b :: "'b" assumes "g \<in> carrier G" shows "b <o inv g <o g = b"
simps: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "sfirst P \<omega> = (if P \<omega> then 0 else eSuc (sfirst P (stl \<omega>)))"
list_constr_append: fixes a :: "nat list"   and n :: "nat"   and b :: "nat list" shows "list_constr a n \<and> list_constr b n \<longrightarrow> list_constr (a @ b) n"
holX: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> X" shows "h holomorphic_on S"
lowner_lub_trace: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and x :: "complex" assumes "matrix_seq dim f"   and "\<forall>n. trace (f n) \<le> x" shows "trace (matrix_seq.lowner_lub f) \<le> x"
bv_sadd_length: fixes w1 :: "bit list"   and w2 :: "bit list" shows "length (bv_sadd w1 w2) \<le> Suc (max (length w1) (length w2))"
rec_max1_def: fixes f :: "recf" shows "rec_max1 f = PR Z (CN rec_ifz [CN f [CN S [recf.Id 3 0], recf.Id 3 2], CN S [recf.Id 3 0], recf.Id 3 1])"
elims: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "'a list" assumes "remdups_on x xa = y" obtains uu_ :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = uu_" and "xa = []" and "y = []"   | cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and xs :: "'a list" where "x = cmp" and "xa = xb # xs" and "y = (if Bex (set xs) (cmp xb) then remdups_on cmp xs else xb # remdups_on cmp xs)"
rbl_word_not: fixes x :: "'a word" shows "rev (to_bl (NOT x)) = map Not (rev (to_bl x))"
WTrt_binop_simps_3: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>LessThan\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Boolean)"
vars_empty_iff: fixes p :: "'b mpoly" shows "(vars p = {}) = (p = Const (Symmetric_Polynomials.lead_coeff p))"
assms_8: shows "set us \<inter> \<Union> (FvarsT ` set ts) = {}"
infinite_Un: fixes S :: "'x set"   and T :: "'x set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
UN_ball_bex_simps_2: fixes B :: "'e \<Rightarrow> 'd set"   and A :: "'e set"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> (B ` A). P x) = (\<forall>a\<in>A. \<forall>x\<in>B a. P x)"
asIMOD_igFreshIGAbs2: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igFreshIGAbs2 (asIMOD SEM)"
nb\<^sub>1: fixes C :: "sat_plan_variable literal set" assumes "C \<in> cnf (encode_interfering_operator_exclusion \<Pi> t)" shows "clause_semantics \<A> C"
top_clock_monotone: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_top ck env s tp (s', r)"   and "ck = True" shows "clock s' \<le> clock s"
preL: fixes b :: "'a" shows "pre_adjust \<langle>delete x l, (b, lv), r\<rangle>"
Nat_atleast_NullD: fixes s :: "state"   and n :: "nat" assumes "s:Null \<ge> n" shows "False"
rule_charn3: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and po :: "int" assumes "wellformed_policy1 p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3 p"   and "applied_rule_rev C x p = \<lfloor>DenyAllFromTo c d\<rfloor>"   and "AllowPortFromTo a b po \<in> set p" shows "x \<notin> dom (C (AllowPortFromTo a b po))"
insert_eq_iff: fixes a :: "'b"   and A :: "'b set"   and b :: "'b"   and B :: "'b set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
alpha_tAct: fixes \<alpha>1 :: "'g"   and t1 :: "('e, 'f, 'g) Tree"   and \<alpha>2 :: "'g"   and t2 :: "('e, 'f, 'g) Tree" shows "(tAct \<alpha>1 t1 =\<^sub>\<alpha> tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
assms_1: shows "Insert m' \<in> M \<union> {Insert m}"
Ri_spec: fixes x :: "'c"   and y :: "'d" shows "x \<le> Ri x y y"
sublinear_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "sublinear t = (\<forall>a b c P Q s. sound P \<and> sound Q \<and> 0 \<le> a \<and> 0 \<le> b \<and> 0 \<le> c \<longrightarrow> a * t P s + b * t Q s \<ominus> c \<le> t (\<lambda>s'. a * P s' + b * Q s' \<ominus> c) s)"
fg_freeTr: fixes M :: "('e, 'a, 'f) Module_scheme"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'a \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'e"   and fa :: "nat \<Rightarrow> 'e"   and n :: "nat"   and sa :: "nat \<Rightarrow> 'a" assumes "R module M"   and "free_generator R M A"   and "R module fgmodule R A z i f s"   and "free_generator R (fgmodule R A z i f s) A"   and "g \<in> mHom R M (fgmodule R A z i f s)"   and "\<forall>x\<in>A. g x = x"   and "fa \<in> {j. j \<le> n} \<rightarrow> A"   and "sa \<in> {j. j \<le> n} \<rightarrow> carrier R" shows "l_comb R (fgmodule R A z i f s) n sa (cmp g fa) = l_comb R (fgmodule R A z i f s) n sa fa"
farkas_coefficients: fixes cs :: "constraint set" assumes "finite cs" shows "(\<exists>C. farkas_coefficients cs C) = (\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s cs)"
ground_prove_IN: fixes t :: "tm"   and u :: "tm"   and e :: "(name, hf) finfun" assumes "ground t"   and "ground u"   and "\<lbrakk>t\<rbrakk>e \<^bold>\<in> \<lbrakk>u\<rbrakk>e" shows "{} \<turnstile> t IN u"
1: shows "set_integrable lborel (einterval (ereal a) \<infinity>) (\<lambda>x. (b + x) powr c)"
simps: fixes M1 :: "('e, 'f, 'g) FSM"   and TS :: "'e list set"   and \<Omega> :: "('e, 'f) ATC set"   and M2 :: "('e, 'f, 'h) FSM" shows "M1 \<preceq>\<lbrakk>TS.\<Omega>\<rbrakk> M2 = (\<forall>iseq\<in>TS. atc_io_reduction_on M1 M2 iseq \<Omega>)"
fold_congs_6: fixes r :: "'a state_scheme"   and r' :: "'a state_scheme"   and v' :: "msg"   and f :: "msg \<Rightarrow> msg"   and f' :: "msg \<Rightarrow> msg" assumes "r = r'"   and "msg r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "msg_update f r = msg_update f' r'"
sqrt_rat_floor_def: fixes x :: "rat" shows "sqrt_rat_floor x = root_rat_floor 2 x"
simps_7: fixes Texp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree"   and as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" shows "TTreeAnalysis.Fstack Texp as (Upd v # S) = TTreeAnalysis.Fstack Texp as S"
fps_nth_inverse: fixes x :: "'a fps" shows "Abs_fps (fps_nth x) = x"
next_recvupd'_def: fixes c0 :: "('c, 'd, 'e) configuration_scheme"   and c1 :: "('c, 'd, 'e) configuration_scheme"   and p :: "'c"   and q :: "'c" shows "next_recvupd' c0 c1 p q = (c_msg c0 p q \<noteq> [] \<and> c1 = c0\<lparr>c_msg := (c_msg c0)(p := (c_msg c0 p)(q := tl (c_msg c0 p q))), c_glob := (c_glob c0)(q := c_glob c0 q + hd (c_msg c0 p q))\<rparr>)"
is_derivation_implies_admissible: fixes p :: "(symbol \<times> character list) list"   and \<delta> :: "symbol list" assumes "is_derivation (terminals p @ \<delta>)"   and "is_word (terminals p)" shows "admissible p"
LeftDerivationIntro_cut_appendix: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list"   and \<delta>' :: "symbol list"   and n :: "nat" assumes "LeftDerivationIntro (\<alpha> @ \<delta>) i r ix D j (\<beta> @ \<delta>')"   and "LeftDerivation \<alpha> ((i, r) # take n D) \<beta>"   and "n \<le> length D"   and "derivation_ge (drop n D) (length \<beta>)"   and "i < length \<alpha>"   and "j < length \<beta>" shows "LeftDerivationIntro \<alpha> i r ix (take n D) j \<beta>"
power_minus_Bit0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus x) (numeral (num.Bit0 k)) = power.power one times x (numeral (num.Bit0 k))"
assms_3: shows "ptr |\<in>| object_ptr_kinds h"
lim_imp_Liminf: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
map_simp: fixes f :: "'c" assumes "J'.arr f" shows "J'.map f = f"
unique_factorization_m_cong: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset"   and g :: "int poly" assumes "q.unique_factorization_m f cfs"   and "q.eq_m f g" shows "q.unique_factorization_m g cfs"
assms_1: shows "compiler step1 step2 final1 final2 load1 load2 order1 match1 compile1"
codomain_unique: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'b"   and b' :: "'b" assumes "category C"   and "b \<in> partial_magma.codomains C f"   and "b' \<in> partial_magma.codomains C f" shows "b = b'"