hcju/isabelleps huggingface dataset:
step_t_r: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and R :: "('d \<Rightarrow> real) set"   and R' :: "('d \<Rightarrow> real) set"   and l :: "'e" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "R \<in> \<R>"   and "R' \<in> Succ \<R> R"   and "R \<subseteq> \<lbrace>inv_of A l\<rbrace>"   and "R' \<subseteq> \<lbrace>inv_of A l\<rbrace>" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l, R'\<rangle>"
currentLevelPrefixToLevel_aux: fixes i :: "nat"   and l :: "nat"   and M :: "('b \<times> bool) list" assumes "i \<le> l" shows "currentLevel (prefixToLevel_aux M l i) \<le> l - i"
mult1I: fixes M :: "'b multiset"   and a :: "'b"   and M0 :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "M = add_mset a M0"   and "N = M0 + K"   and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" shows "(N, M) \<in> mult1 r"
writerT_map_def: shows "writerT_map \<equiv> \<Lambda> w a. \<mu> uu. writerT_abs oo fmap\<cdot>(writer_map\<cdot>w\<cdot>a) oo writerT_rep"
dvd_mult_right: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times b c"
scale_cancel_left: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "(a *\<^sub>R x = a *\<^sub>R y) = (x = y \<or> a = 0)"
awalkI_apath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.apath u p v" shows "S.awalk u p v"
assms_2: shows "fo_interpretation (same_values (\<lambda>t. trm_rep t S))"
fields_def: fixes P :: "'b prog"   and C :: "String.literal" shows "TypeRel.fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
disj_ac_2: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj a b = disj b a"
UNIV_enum: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "UNIV = set enum"
prod_set_simps_1: fixes x :: "'a"   and y :: "'b" shows "Basic_BNFs.fsts (x, y) = {x}"
A_def: shows "A \<equiv> \<lambda>X. {y |y. Xd y \<in> ds \<and> y \<in> Field (Pd (Xd y)) \<and> (\<forall>x\<in>X. Xd x = Xd y \<longrightarrow> (x, y) \<in> Pd (Xd x))}"
strip_while_eq_Nil: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(strip_while P xs = []) = (\<forall>x\<in>set xs. P x)"
iso_\<phi>: shows "D.iso (\<l>\<^sub>D[g] \<cdot>\<^sub>D (\<sigma>.counit a' \<star>\<^sub>D g) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a', \<sigma>.map\<^sub>0 a', g] \<cdot>\<^sub>D (\<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.counit a) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>.map\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]))"
Square_impl: shows "\<Gamma> Square_'proc = Some local.Square_body"
assms_1: shows "convert_atom var A (xs' @ x # xs) = Some A'"
AE_integrable_fst': fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
Sigma_Union: fixes X :: "'a set set"   and B :: "'a \<Rightarrow> 'b set" shows "Sigma (\<Union> X) B = (\<Union>A\<in>X. Sigma A B)"
assms_1: shows "v \<in> fv (t \<cdot> \<delta>)"
HCond_Idempotent: shows "Idempotent \<H>"
non0: fixes n :: "nat" shows "Suc 0 \<le> \<sigma> n"
f_def: shows "f \<equiv> \<lambda>x. \<Sum>i<n. x i *\<^sub>R b i"
power_up_less: fixes x :: "real"   and xa :: "real"   and n :: "nat"   and p :: "nat" assumes "x < xa ^ n"   and "0 \<le> xa" shows "x < power_up p xa n"
assms: shows "f1 |`| S1 = f2 |`| S2"
lens_put_comm: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'g"   and u :: "'e" assumes "X \<bowtie> Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"
close_subset: fixes C :: "'b fm set set" shows "C \<subseteq> close C"
irsSp_Cons_return: fixes n'' :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname" assumes "n'' irs-ns\<rightarrow>\<^sub>d* n'"   and "n s-p\<rightarrow>\<^bsub>ret\<^esub> n''" shows "n irs-n # ns\<rightarrow>\<^sub>d* n'"
normf: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> F" shows "cmod (deriv h 0) \<le> cmod (deriv f 0)"
normalize_trick: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "p = replicate (length p - length (ring.normalize R p)) \<zero>\<^bsub>R\<^esub> @ ring.normalize R p"
nat2Nat_Suc: fixes n :: "nat" shows "nat2Nat (Suc n) = SucNat (nat2Nat n)"
inv_id_lens: shows "inv\<^sub>L \<Sigma> = \<Sigma>"
asIMOD_igFresh: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom"   and ys :: "'varSort"   and y :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> 'a" shows "igFresh (asIMOD SEM) ys y X = (\<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> X val = X val')"
simps: fixes a1 :: "'b list"   and a2 :: "'b list" shows "(a1 <~~> a2) = (a1 = [] \<and> a2 = [] \<or> (\<exists>y x l. a1 = y # x # l \<and> a2 = x # y # l) \<or> (\<exists>xs ys z. a1 = z # xs \<and> a2 = z # ys \<and> xs <~~> ys) \<or> (\<exists>xs ys zs. a1 = xs \<and> a2 = zs \<and> xs <~~> ys \<and> ys <~~> zs))"
semiattractive_axioms: shows "semiattractive X (\<sqsubseteq>)"
finite_iff_ordLess_natLeq: fixes A :: "'b set" shows "finite A = ordLess2 (card_of A) natLeq"
1: shows "(\<sqinter>xa\<in>range left \<union> range right \<rightarrow>  P) \<le> (\<sqinter>xa\<in>range right \<rightarrow>  P)"
insert_select: fixes f :: "'b \<Rightarrow> 'a option"   and l :: "'b"   and t :: "'a" shows "the ((f(l \<mapsto> t)) l) = t"
mk_coeff_carrier: fixes c :: "nat \<Rightarrow> 'a"   and vs :: "'f list"   and w :: "'f" assumes "c \<in> {0..<length vs} \<rightarrow> carrier R" shows "mk_coeff vs c w \<in> carrier R"
simplicial_chain_simplicial_subdivision: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "simplicial_chain p S (simplicial_subdivision p c)"
lmap_lappend_dist: fixes f :: "'c \<rightarrow> 'b"   and xs :: "'c llist"   and ys :: "'c llist" shows "lmap\<cdot>f\<cdot>(xs :++ ys) = lmap\<cdot>f\<cdot>xs :++ lmap\<cdot>f\<cdot>ys"
simps_1: fixes P :: "('a, 'b, 'c) psi" shows "\<lparr>\<nu>*[]\<rparr>P = P"
cap_def: shows "cap \<equiv> sum c (outgoing' (cf.reachableNodes s))"
nscal_0: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" shows " 0 \<times>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
minus_add_distrib: fixes a :: "'c"   and b :: "'c" shows "- (a + b) = - a + - b"
insertion_fun_def: fixes f :: "nat \<Rightarrow> 'b"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'b" shows "insertion_fun f p = (\<Sum>m. p m * (\<Prod>v. f v ^ lookup m v))"
pr_sign_key_used: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and X :: "agent" assumes "(evs, S, A, U) \<in> protocol" shows "Key (priSK X) \<in> U"
finite_list: fixes A :: "'b set" assumes "finite A" shows "\<exists>xs. set xs = A"
negative_len_shortest: fixes xs :: "nat list"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat" assumes "length xs = n"   and "linordered_monoid.len m i i xs \<prec> Le (0::'c)" shows "\<exists>j ys. distinct (j # ys) \<and> linordered_monoid.len m j j ys \<prec> Le (0::'c) \<and> j \<in> set (i # xs) \<and> set ys \<subseteq> set xs"
rdomp_pure: fixes R :: "('b \<times> 'a) set"   and a :: "'a" shows "rdomp (pure R) a = (a \<in> Range R)"
extends_not_cons: fixes x :: "'a"   and S :: "'a list" shows "\<not> x # S \<lesssim> S"
Sup_subset_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "\<Squnion> A \<le> \<Squnion> B"
diff_plus_sym_hmset: fixes c :: "hmultiset"   and b :: "hmultiset" shows "c - b + b = b - c + c"
prod_cases5: fixes y :: "'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f" obtains a :: "'b" and b :: "'c" and c :: "'d" and d :: "'e" and e :: "'f" where "y = (a, b, c, d, e)"
invertible_def: fixes u :: "'a set" assumes "u \<in> Partition" shows "quotient.invertible u = (\<exists>v\<in>Partition. u [\<cdot>] v = Class \<one> \<and> v [\<cdot>] u = Class \<one>)"
permute_profile_map_relation: fixes \<sigma> :: "'e \<Rightarrow> 'd"   and R :: "'c \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool" shows "permute_profile \<sigma> R = (\<lambda>i. map_relation (inv \<sigma>) (R i))"
lappend_lnull1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lnull xs" shows "lappend xs ys = ys"
pdevs_val_partition: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and xs :: "'a list"   and p :: "'a \<Rightarrow> bool" assumes "e \<in> UNIV \<rightarrow> I" obtains f :: "nat \<Rightarrow> real" and g :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list xs) = pdevs_val f (pdevs_of_list (filter p xs)) + pdevs_val g (pdevs_of_list (filter (Not \<circ> p) xs))" and "f \<in> UNIV \<rightarrow> I" and "g \<in> UNIV \<rightarrow> I"
effective_matrix_Tensor_elements: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and i :: "nat"   and M1 :: "'a list list"   and M2 :: "'a list list"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "i < mult.row_length M1 * mult.row_length M2"   and "j < length M1 * length M2"   and "Matrix_Legacy.mat (mult.row_length M1) (length M1) M1"   and "Matrix_Legacy.mat (mult.row_length M2) (length M2) M2" shows "mult.Tensor f M1 M2 ! j ! i = f (M1 ! (j div length M2) ! (i div mult.row_length M2)) (M2 ! (j mod length M2) ! (i mod mult.row_length M2))"
zero: fixes f :: "'a \<Rightarrow> 'b" assumes "additive f" shows "f (0::'a) = (0::'b)"
assms: shows "x \<in> set (vwalk_arcs p) \<or> x \<in> set (vwalk_arcs q)"
AE_measure_pmf: fixes M :: "'b pmf" shows "AE x in measure_pmf M. x \<in> set_pmf M"
beh_obs_upd: fixes x :: "'e \<Rightarrow> 'f"   and S :: "('e, 'f, 'g) spec_scheme" shows "beh (S\<lparr>obs := x\<rparr>) = beh S"
adjacent_Cons_r: fixes a :: "nat"   and t :: "'c tree"   and x :: "'b tree"   and xs :: "'b tree list" assumes "adjacent (a, r t) (x # xs)" shows "adjacent (a, r x) (x # xs)"
sig_trd_aux_domI: fixes args0 :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)" assumes "fst args0 \<in> keys (rep_list (snd args0))" shows "sig_trd_aux_dom args0"
hyps_2: shows "ix p d < (ix b d + 1) * 2 ^ (lv p d - lv b d) \<and> (ix b d - 1) * 2 ^ (lv p d - lv b d) < ix p d"
image_subset_iff_funcset: fixes F :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and B :: "'e set" shows "(F ` A \<subseteq> B) = (F \<in> A \<rightarrow> B)"
w_simps_4: shows "trg THfgh_HHfgh.chine = src (tab\<^sub>0 ((f \<star> g) \<star> h))"
stmt_step_load_tmps_range: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)" shows "\<forall>i\<in>load_tmps is. t \<le> i \<and> i < t'"
ratpolyE: fixes p :: "'a poly" assumes "\<forall>i. poly.coeff p i \<in> \<rat>" obtains q :: "rat poly" where "p = map_poly of_rat q"
L'_def: fixes F' :: "'a filter"   and g' :: "'a \<Rightarrow> 'b" shows "O[F'](g') = {f. \<exists>c>0. \<forall>\<^sub>F x in F'. norm (f x) \<le> c * norm (g' x)}"
simps_1: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat" shows "denote_while_n_iter M0 M1 DS 0 \<rho> = \<rho>"
horizontal_homs_axioms: shows "horizontal_homs (\<bullet>) src trg"
simps_3: fixes x :: "'i"   and f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "sum_bot.Right x \<bind> f = f x"
FW_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "dbm_int M n" shows "dbm_int (FW M n) n"
two_list_induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and S :: "'b list"   and T :: "'b list" assumes "P [] []"   and "\<And>a S T. P S T \<Longrightarrow> P (a # S) T &&& P S (a # T)" shows "P S T"
mono_Un: fixes A :: "'a \<Rightarrow> 'b set"   and n :: "'a" assumes "mono A" shows "\<Union> (A ` {..n}) = A n"
of_nat_rat: fixes k :: "nat" shows "rat_of_nat k = Fract (int k) 1"
unital_quantale_module_axioms: shows "unital_quantale_module \<alpha>"
harry_meas_eq_0_iff: fixes xs :: "bool list" shows "(harry_meas xs = 0) = (True \<notin> set xs)"
epm_10: fixes e :: "'b"   and v :: "'b"   and w :: "'b" assumes "e \<le> v \<sqinter> - id v"   and "bounded_distrib_lattice_class.comp_inf.vector v" shows "id (stone_algebra_class.comp_inf.prim_P w v e) \<sqinter> e = bot"
awalk_verts_non_Nil: fixes u :: "'a"   and p :: "'b list" shows "awalk_verts u p \<noteq> []"
assert_bound_nolhs_tableau_id: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s" shows "\<T> (assert_bound a s) = \<T> s"
seq_char: fixes g :: "'a"   and f :: "'a" shows "seq g f = (arr f \<and> arr g \<and> C.seq g f)"
getEnv_updEnv2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort" shows "(rho [x \<leftarrow> X]_xs) xs x = Some X"
PS\<^sub>L: shows "sparse (dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) PS\<^sub>L"
hyps: shows "coeffs (Poly as) = strip_while ((=) (0::'a)) as"
true_clss_singleton: fixes I :: "'a set"   and C :: "'a literal multiset" shows "(I \<Turnstile>s {C}) = (I \<Turnstile> C)"
arc_to_block_is_inj_in_arcs: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" assumes "graph G" shows "inj_on (graph.arc_to_block G) (in_arcs G v)"
take_Cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
cscalar_prod_conjugate_transpose: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and x :: "'b vec"   and y :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "x \<in> carrier_vec nr"   and "y \<in> carrier_vec nc" shows "x \<bullet>c (A *\<^sub>v y) = (A\<^sup>H *\<^sub>v x) \<bullet>c y"
filterlim_at_split: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter"   and x :: "'b" shows "filterlim f F (at x) = (filterlim f F (at_left x) \<and> filterlim f F (at_right x))"
cc_nr: shows "\<forall>C\<in>CC. \<forall>DD\<subseteq>N. DD \<Turnstile> {C} \<longrightarrow> (\<exists>D\<in>DD. \<not> D < C)"
minus_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A - B = Collect ((\<lambda>x. x \<in> A) - (\<lambda>x. x \<in> B))"
zero_multiset_def: shows "{#} \<equiv> Abs_multiset (\<lambda>a. 0)"
mbisim_simulation_silent1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<tau>mred1 s1 s1'"   and "s1 \<approx>m s2" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> s1' \<approx>m s2'"
locally_full: fixes f :: "'c"   and f' :: "'c"   and \<nu> :: "'d" assumes "C.ide f"   and "C.ide f'"   and "src\<^sub>C f = src\<^sub>C f'"   and "trg\<^sub>C f = trg\<^sub>C f'"   and "\<guillemotleft>\<nu> : F f \<Rightarrow>\<^sub>D F f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> F \<mu> = \<nu>"
apfst_eq_conv: fixes f :: "'f \<Rightarrow> 'd"   and x :: "'f \<times> 'e"   and g :: "'f \<Rightarrow> 'd" shows "(apfst f x = apfst g x) = (f (fst x) = g (fst x))"
mono_Scan: fixes T :: "(symbol \<times> character list) set"   and k :: "nat" shows "mono (Scan T k)"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<longrightarrow>\<^sub>1 x2"   and "\<And>x y. P y (Var x) (Var x)"   and "\<And>t1 t2 s1 s2 y. \<lbrakk> t1 \<longrightarrow>\<^sub>1 t2; \<And>z. P z t1 t2;  s1 \<longrightarrow>\<^sub>1 s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App t1 s1) (App t2 s2)"   and "\<And>t1 t2 x y. \<lbrakk>x \<sharp> y;  t1 \<longrightarrow>\<^sub>1 t2; \<And>z. P z t1 t2\<rbrakk> \<Longrightarrow> P y (Lam [x].t1) (Lam [x].t2)"   and "\<And>x s1 s2 t1 t2 y. \<lbrakk>x \<sharp> y; x \<sharp> (s1, s2);  t1 \<longrightarrow>\<^sub>1 t2; \<And>z. P z t1 t2;  s1 \<longrightarrow>\<^sub>1 s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App (Lam [x].t1) s1) (t2[x::=s2])" shows "P z x1 x2"
collapse: fixes list :: "'c list" assumes "list \<noteq> []" shows "hd list # tl list = list"
positively_invariant_le_domain: fixes D :: "'a set"   and g :: "'a \<Rightarrow> real"   and V :: "'a \<Rightarrow> real"   and V' :: "'a \<Rightarrow> 'a \<Rightarrow> real" assumes "positively_invariant D"   and "continuous_on D g"   and "\<And>x. (V has_derivative V' x) (at x)"   and "\<And>s. s \<in> D \<Longrightarrow> V' s (f s) \<le> g s * V s" shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
simps: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set" shows "separating_gen G A B S = (\<forall>x xa xb. x \<in> A \<longrightarrow> xa \<in> B \<longrightarrow> path G x xb xa \<longrightarrow> (\<exists>z\<in>set xb. z \<in> S) \<or> x \<in> S)"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
cod_null: shows "VoV.cod VoV.null = VoV.null"
row_mat: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "i < nr" shows "row (mat nr nc f) i = vec nc (\<lambda>j. f (i, j))"
comp_assoc_assoc'_1: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "\<a>\<^sub>D[a, b, c] \<cdot>\<^sub>D D.assoc' a b c = a \<otimes>\<^sub>D b \<otimes>\<^sub>D c"
h_step_def: shows "h_step = sum steps {r1..<r2}"
Inf_eq_Sup: fixes A :: "'a set" shows "Inf A = Sup {b. \<forall>a\<in>A. b \<le> a}"
Cn_inter: fixes K :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "K = Cn A \<inter> Cn B" shows "K = Cn K"
tinc_b_def: shows "tinc_b \<equiv> [(W1, 1), (R, 2), (W1, 3), (R, 2), (W1, 3), (R, 4), (L, 7), (W0, 5), (R, 6), (W0, 5), (W1, 3), (R, 6), (L, 8), (L, 7), (R, 9), (L, 7), (R, 10), (W0, 9)]"
fv_subst_def: fixes s :: "(nat \<times> nat list \<times> form) list" shows "fv_subst s = \<Union> (fv_schema ` ran (map_of s))"
simps_1: fixes x :: "'v" shows "arity_hd (Var x) = arity_var x"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('a, 'b) lit" assumes "F.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('a, 'b) atm" where "x = \<xi>'__" and "xa = Pos a__" and "F.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> F.satA \<xi> a"
dest_def: fixes i :: "nat"   and q :: "'a" shows "dest i q \<equiv> \<exists>p. channel i = Some (p, q)"
pair_wf_digraph_axioms: shows "pair_wf_digraph (list_digraph G)"
e_cdu: shows "somegcd G (c \<otimes> a) (c \<otimes> b) = c \<otimes> somegcd G a b \<otimes> u"
not_eq_order_implies_strict: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" assumes "a \<noteq> b"   and "b \<sqsubseteq> a" shows "b \<sqsubset> a"
exhaust_sel: fixes "term" :: "('a, 'b) Term.term" obtains "term = Var (the_Var term)"   | "term = Fun (the_Fun term) (args term)"
dvd_div_mult2_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times (times b c) a" shows "divide a (times b c) = divide (divide a b) c"
ord_option_eq_simps_2: fixes x :: "'b"   and y :: "'b option" shows "ord_option (=) (Some x) y = (y = Some x)"
fset_of_list_simps_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)"
iint_less: fixes a :: "'a idx_interval"   and b :: "'a idx_interval" shows "(a < b) = (idx_interval.int a < idx_interval.int b \<or> idx_interval.int a = idx_interval.int b \<and> idx a < idx b)"
flist_props_2: fixes xs :: "'a fset" shows "distinct (flist xs)"
factorization_lattice_as_sylvester: fixes d :: "nat"   and j :: "nat"   and p :: "'a poly"   and m :: "'a" assumes "d \<le> j"   and "degree p = d" shows "mat_of_rows j (factorization_lattice p (j - d) m) = sylvester_mat_sub d (j - d) p [:m:]"
add_assoc': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y + z = x + (y + z)"
iface_methd: shows "DeclConcepts.imethds G I sig \<noteq> {}"
pag: shows "path_image \<gamma> \<subseteq> {w. b \<le> a \<bullet> w}"
showsp_unit_def: fixes p :: "nat"   and x :: "unit" shows "showsp_unit p x = shows_string ''()''"
simps_14: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "fv (e\<^sub>1;; e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
r_join: shows "revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3'"
a: shows "[] \<turnstile> t : T1 \<rightarrow> T2"
suminf_fds_zeta_aux: fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 = (0::'a)" shows "(\<Sum>n. fds_nth fds_zeta n / f n) = (\<Sum>n. (1::'a) / f n)"
remove: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<in> A" shows "aggregation.sum_0 g A = g x + aggregation.sum_0 g (A - {x})"
ring_simprules_25: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
prems: shows "gf \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
simps_15: fixes f1 :: "'i \<Rightarrow> 'g"   and f2 :: "'j \<Rightarrow> 'h"   and x21 :: "'i"   and x22 :: "('i, 'j) tllist" shows "tmap f1 f2 (TCons x21 x22) = TCons (f1 x21) (tmap f1 f2 x22)"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto>[Rel] Q"   and "\<And>P Q x. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto>[Rel'] <\<nu>x>Q"