hcju/isabelleps huggingface dataset:
vsplit: fixes c :: "V \<Rightarrow> V \<Rightarrow> 'a"   and a :: "V"   and b :: "V" shows "vsplit c \<langle>a, b\<rangle> = c a b"
double_negation_dist_implies: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<squnion> y) = - - - x \<squnion> - - y"
not_le: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<le> y) = (y < x)"
test_iso_eq: fixes p :: "'a"   and q :: "'a" shows "(p \<le> q) = (\<iota> p \<le> \<iota> q)"
bet2_le2__le: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and P :: "'a"   and b :: "'a"   and A :: "'a"   and Q :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet a P b"   and "Bet A Q B"   and "Tarski_neutral_dimensionless.Le Bet Cong P a Q A"   and "Tarski_neutral_dimensionless.Le Bet Cong P b Q B" shows "Tarski_neutral_dimensionless.Le Bet Cong a b A B"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c p. P p \<Longrightarrow> P (CN c 0 p)"   and "\<And>v. P (v)\<^sub>p"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)"   and "\<And>v vc vb. P (CN v (Suc vc) vb)" shows "P a0"
l2p_aux2: fixes list :: "('a, 'b) Combinators list"   and a :: "('a, 'b) Combinators" assumes "list = []" shows "list2FWpolicy (a # list) = a"
intros_2: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and p :: "value list"   and t :: "(String.literal \<times> value list \<times> value list) list" assumes "\<exists>(s', T)|\<in>|possible_steps e s r l i. evaluate_outputs T i r = map Some p \<and> accepts_trace e s' (evaluate_updates T i r) t" shows "accepts_trace e s r ((l, i, p) # t)"
foldD_not_depend: fixes B :: "'h set"   and D :: "'i set"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'i"   and E :: "'i set"   and e :: "'i"   and A :: "'h set" assumes "LCD B D f"   and "LCD B E f"   and "D \<subseteq> E"   and "e \<in> D"   and "A \<subseteq> B"   and "finite B" shows "foldD D f e A = foldD E f e A"
DSourcesA41_L1: shows "DSources level1 sA41 = {sA31, sA32}"
keys_monom_multE: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and v :: "'b"   and c :: "'e"   and t :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'e" assumes "term_powerprod pair_of_term term_of_pair"   and "v \<in> keys (term_powerprod.monom_mult pair_of_term term_of_pair c t p)" obtains u :: "'b" where "u \<in> keys p" and "v = term_powerprod.splus pair_of_term term_of_pair t u"
z_zero: fixes p :: "proj2" assumes "proj2_rep p $ 3 = 0" shows "proj2_incident p z_zero"
uminus_aform_def: fixes X :: "'a \<times> 'b pdevs" shows "uminus_aform X = (- fst X, uminus_pdevs (snd X))"
preserves_inv: fixes f :: "'c \<times> 'c" assumes "C'.VV.iso f" shows "fst (C'.VV.inv f) \<star>\<^sub>C\<^sup>o\<^sup>p snd (C'.VV.inv f) = C.inv (fst f \<star>\<^sub>C\<^sup>o\<^sup>p snd f)"
rtranclp_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "(equivclp r)\<^sup>*\<^sup>* = equivclp r"
obs_neq_contradicts_term: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and n :: "nat"   and \<sigma>' :: "'b \<Rightarrow> 'c"   and n' :: "nat" assumes "IFC prob"   and "IFC_def.path prob \<sigma> n = IFC_def.return prob"   and "IFC_def.path prob \<sigma>' n' = IFC_def.return prob"   and "IFC_def.obs prob \<sigma> \<noteq> IFC_def.obs prob \<sigma>'" shows "\<exists>k k'. IFC_def.contradicts prob (\<sigma>', k') (\<sigma>, k) \<and> IFC_def.path prob \<sigma> k \<in> dom (IFC_def.att prob) \<or> IFC_def.contradicts prob (\<sigma>, k) (\<sigma>', k') \<and> IFC_def.path prob \<sigma>' k' \<in> dom (IFC_def.att prob)"
G_def: shows "G \<equiv> \<lambda>l m M. SOME (n, N, j). \<Phi> n N \<and> \<Psi> (Suc l) m n M N j"
l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
distinct_1: fixes x21 :: "('b btree \<times> 'b) list"   and x22 :: "'b btree" shows "Leaf \<noteq> Node x21 x22"
in_set_selsort_eq: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set (selsort xs)) = (x \<in> set xs)"
reduced_homology_group_nsphere_2: fixes p :: "int"   and n :: "nat" assumes "p \<noteq> int n" shows "trivial_group (reduced_homology_group p (nsphere n))"
orientable_11_implies_12: fixes uminus :: "'b \<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 one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "inf x (conv x) = one \<and> sup x (conv x) = top" shows "inf (inf x (uminus one)) (conv (inf x (uminus one))) = bot \<and> sup (inf x (uminus one)) (conv (inf x (uminus one))) = uminus one"
s4: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "rel_ad x O rel_ad (rel_ad x O y) = rel_ad x O rel_ad y"
assms_1: shows "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'"
countable_Int1: fixes A :: "'f set"   and B :: "'f set" assumes "countable A" shows "countable (A \<inter> B)"
REFC: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma>' :: "'\<sigma>a"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "\<Phi> it \<sigma>"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R" shows "c \<sigma> = c' \<sigma>'"
replicate_length_same: fixes xs :: "'b list"   and x :: "'b" assumes "\<forall>y\<in>set xs. y = x" shows "replicate (length xs) x = xs"
strict_suffix_set_subset: fixes xs :: "'c list"   and ys :: "'c list" assumes "strict_suffix xs ys" shows "set xs \<subseteq> set ys"
diff_Union_pairwise_disjoint: fixes \<A> :: "'a set set"   and \<B> :: "'a set set" assumes "disjoint \<A>"   and "\<B> \<subseteq> \<A>" shows "\<Union> \<A> - \<Union> \<B> = \<Union> (\<A> - \<B>)"
indmhom_someTr2: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and H :: "'a set"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "submodule R M H"   and "m \<in> carrier M"   and "H \<subseteq> ker\<^bsub>M,N\<^esub> f" shows "f (SOME xa. xa \<in> m \<uplus>\<^bsub>M\<^esub> H) = f m"
dWWPk: shows "WS (W P k) \<in> carrier_mat d d"
tau_steps_eqvt': fixes p :: "perm"   and x :: "('idx, 'pred, 'act) formula" shows "p \<bullet> tau_steps x = tau_steps (p \<bullet> x)"
a: shows "set p \<subseteq> Xs \<times> Ys"
simps_6: fixes x21 :: "qtname"   and x22 :: "qtname"   and x23 :: "bool"   and x24 :: "expr"   and x25 :: "vname"   and x1 :: "lname" shows "{x21,x22,x23}x24..x25 \<noteq> LVar x1"
const_Lin_Prof: fixes L :: "(alt \<times> alt) set" assumes "L \<in> Lin" shows "(\<lambda>p. L) \<in> Prof"
injectivity: fixes x :: "'b"   and d :: "nat"   and y :: "'b" assumes "monom x d = monom y d" shows "x = y"
simps: fixes p :: "linear_poly"   and v :: "nat" shows "linear_poly_to_eq p v = (v, p)"
nxt_mono: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "nxt \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "nxt \<psi> xs"
reduction_in_concls_of: fixes C\<mu> :: "'a literal multiset"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and L' :: "'a literal"   and CC :: "'a literal multiset set"   and L :: "'a literal"   and \<sigma> :: "'s" assumes "C\<mu> \<in> grounding_of_cls C"   and "D + {#L'#} \<in> CC"   and "- L = L' \<cdot>l \<sigma>"   and "D \<cdot> \<sigma> \<subseteq># C" shows "C\<mu> \<in> concls_of (sr_ext.inferences_from (grounding_of_clss (CC \<union> {C + {#L#}})))"
G_reach_2: shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
noFault_startn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t \<noteq> Fault f" shows "s \<noteq> Fault f"
rel_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a multiset"   and b :: "'b multiset" shows "rel_mset R\<inverse>\<inverse> a b = rel_mset R b a"
isAbr_def: fixes S :: "('a, 'b) xstate" shows "isAbr S = (\<exists>s. S = Abrupt s)"
base_in: fixes d :: "nat"   and dm :: "nat"   and ds :: "nat set"   and p :: "(nat \<times> int) list" assumes "d < dm"   and "d \<in> ds"   and "p \<in> sparsegrid' dm" shows "base ds p ! d = start dm ! d"
i_take_o: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'e" shows "(f \<circ> g) \<Down> n = map f (g \<Down> n)"
adj_not_same: fixes a :: "'a" assumes "(a, a) \<in> parcs G" shows "False"
\<MM>: fixes x :: "real" shows "\<MM> x = ln x + R\<^sub>\<MM> x"
funpow_increasing: fixes m :: "nat"   and n :: "nat"   and f :: "'c \<Rightarrow> 'c" assumes "m \<le> n"   and "mono f" shows "(f ^^ n) top \<le> (f ^^ m) top"
assms_3: shows "eval (Pr n f g) (Suc x # xs) \<down>"
rotate_pow_comm: fixes n :: "nat"   and u :: "'b list"   and k :: "nat" shows "rotate n (u \<^sup>@ k) = rotate n u \<^sup>@ k"
distinct_126: fixes x6 :: "Syntax.term"   and x5 :: "Syntax.term" shows "Inj2 x6 \<noteq> Inj1 x5"
am6_lem: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) \<cdot> x \<cdot> ad y = ad (x \<cdot> y) \<cdot> x"
assms_3: shows "case_sum f1 g1 \<in> rst.FINFUNC"
length_rotate: fixes n :: "nat"   and xs :: "'g list" shows "length (rotate n xs) = length xs"
trace_rest_conv_trace_rest_aux: fixes rest :: "('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme"   and p :: "'f spmf"   and tr :: "('a \<times> 'b \<times> 'c list + 'd \<times> 'e \<times> 'c list) list" shows "trace_rest' rest p tr = (\<lambda>ia. trace_rest_aux rest p tr \<bind> (\<lambda>s. map_spmf fst (rfunc_adv rest s ia)), \<lambda>iu. trace_rest_aux rest p tr \<bind> (\<lambda>s. map_spmf fst (rfunc_usr rest s iu)))"
order_le_imp_less_or_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x < y \<or> x = y"
bilinear_rneg: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'b" assumes "bilinear h" shows "h x (- y) = - h x y"
ee_refl: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list" assumes "Group.monoid G"   and "set as \<subseteq> carrier G" shows "essentially_equal G as as"
simps_2: fixes literal :: "Literal"   and clause :: "Literal list"   and w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list" shows "getNonWatchedUnfalsifiedLiteral (literal # clause) w1 w2 M = (if literal \<noteq> w1 \<and> literal \<noteq> w2 \<and> \<not> literalFalse literal (elements M) then Some literal else getNonWatchedUnfalsifiedLiteral clause w1 w2 M)"
tlset2_def: shows "tlset2 \<equiv> (\<lambda>x. \<Inter>mx\<in>{mx. eq_tllist (map_prod (lmap Inr) Inr x) mx}. \<Union> (Basic_BNFs.setr ` Basic_BNFs.snds mx)) \<circ> rep_tllist"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "pred_mset P = (\<lambda>x. Multiset.Ball x P)"
rtrancl_subset_rtrancl: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s\<^sup>*" shows "r\<^sup>* \<subseteq> s\<^sup>*"
freshEnv_def: fixes xs :: "'d"   and x :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm set option" shows "freshEnv xs x rho \<equiv> rho xs x = None \<and> (\<forall>ys. liftAll (fresh xs x) (rho ys))"
image_mset_single: fixes f :: "'l \<Rightarrow> 'k"   and x :: "'l" shows "image_mset f {#x#} = {#f x#}"
ae: shows "AE x in MX. RN_deriv MX (density MX (\<lambda>x. ennreal (f x))) x = ennreal (f x)"
step_def: fixes astep :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool" shows "asystem.step astep \<equiv> {(c, c'). \<exists>a. astep c a c'}"
scf_term_subst: fixes scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "scf_term scf (t \<cdot> \<sigma>) = scf_term scf t \<cdot> (\<lambda>x. scf_term scf (\<sigma> x))"
equivalence_of_categories_axioms: shows "equivalence_of_categories (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F \<eta> \<epsilon>"
let1: fixes l :: "nat"   and binds :: "((nat \<times> char list) \<times> lambda) list"   and c' :: "call"   and x :: "lambda" assumes "call.Let l binds c' \<in> calls x" shows "l \<in> labels x"
f_non_negative: fixes e :: "nat \<times> nat" shows "0 \<le> max_flow e"
finT_simp: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<star>" shows "r \<in> UNIV\<^sup>\<star>"
divide_leq1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "rat"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq (scaleRat c a) b"   and "c < 0" shows "less_eq (scaleRat (1 / c) b) a"
flip_altsublist_adjacent_refl: fixes ss :: "'w list" assumes "ss \<noteq> []"   and "ss \<in> lists S" shows "flip_altsublist_adjacent ss ss"
defined_and_I: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> y" shows "\<tau> \<Turnstile> \<delta> (x and y)"
assms_2: shows "\<A> \<Turnstile> \<Phi> \<Pi> t"
preserves_cod: fixes f :: "'b" assumes "D.arr f" shows "D.cod (trg\<^sub>D f) = trg\<^sub>D (D.cod f)"
word_less_sub_le: fixes n :: "nat"   and x :: "'b word" assumes "n < LENGTH('b)" shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)"
eq1_1: shows "y1 \<bullet> y2 = x \<bullet> y2"
linear_homeomorphic_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "inj f" shows "S homeomorphic f ` S"
inv_partD_1: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m_i :: "nat"   and m_j :: "nat"   and i :: "nat"   and j :: "nat" assumes "local.inv_part p A m_i m_j"   and "i < n"   and "j < n"   and "j < m_j" shows "p A i j"
simps_3: fixes t :: "nat"   and a :: "expr"   and c\<^sub>e :: "expr"   and s\<^sub>e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set" shows "valid_sops_stmt t (CAS a c\<^sub>e s\<^sub>e A L R W) = (valid_sops_expr t a \<and> valid_sops_expr t c\<^sub>e \<and> valid_sops_expr t s\<^sub>e)"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "ring R"   and "noetherian_ring_axioms R" shows "noetherian_ring R"
rel_interior_subset_closure: fixes S :: "'a set" shows "rel_interior S \<subseteq> closure S"
nat_cases_2Suc: fixes n :: "nat" obtains "n = 0"   | "n = 1"   | m :: "nat" where "n = Suc (Suc m)"
Rep_uexpr_inject: fixes x :: "('c, 'b) uexpr"   and y :: "('c, 'b) uexpr" shows "(\<lbrakk>x\<rbrakk>\<^sub>e = \<lbrakk>y\<rbrakk>\<^sub>e) = (x = y)"
equivalence_functor_G: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>'. trg \<mu> \<star> \<mu>' \<star> e)"
B: shows "(evsFC5, S, A, U) \<in> protocol"
tensor_P_left_right_partial1: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat"   and m4 :: "'a mat" assumes "m1 \<in> carrier_mat N N"   and "m2 \<in> carrier_mat N N"   and "m3 \<in> carrier_mat K K"   and "m4 \<in> carrier_mat N N" shows "tensor_P m1 (1\<^sub>m K) * tensor_P m2 m3 * tensor_P m4 (1\<^sub>m K) = tensor_P (m1 * m2 * m4) m3"
finite: fixes m :: "'m1" assumes "m1.invar m" shows "finite (dom (m1.\<alpha> m))"
partitioning1_disjoint_list: fixes ts :: "'a set list"   and ss :: "'a set list" assumes "{} \<notin> set ts"   and "\<Union> (set ss) \<subseteq> \<Union> (set ts)"   and "disjoint_list ts" shows "disjoint_list (partitioning1 ss ts)"
create_gba_from_nodes__invar: shows "gba (create_gba_from_nodes \<phi> qs)"
DERIV_tan: fixes x :: "'a" assumes "cos x \<noteq> (0::'a)" shows "(tan has_field_derivative inverse ((cos x)\<^sup>2)) (at x)"
of_nat_simps_2: shows "of_nat 1 = (1::'a)"
AboveS_Pd_Xd: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> AboveS (Pd d) X" shows "Xd x = d"
triv_admI: fixes P :: "'b \<Rightarrow> bool" assumes "\<forall>x. P x" shows "adm P"
s_of_v_is_not_None: shows "s v \<noteq> None"
has_ticket_in_range: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and tk :: "nat" assumes "A.reachable (c, n, ts)"   and "t < N"   and "has_ticket (ts t) tk" shows "c \<le> tk \<and> tk < c + N"
Field_singleton: fixes z :: "'d" shows "Field {(z, z)} = {z}"
inductive_step: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and subsets1 :: "nat list list"   and subsets2 :: "nat list list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "0 < length qs2"   and "\<forall>q. List.member qs1 q \<longrightarrow> coprime p q"   and "\<forall>q. List.member qs2 q \<longrightarrow> coprime p q"   and "well_def_signs (length qs1) signs1"   and "well_def_signs (length qs2) signs2"   and "distinct signs1"   and "distinct signs2"   and "set (characterize_consistent_signs_at_roots_copr p qs1) \<subseteq> set signs1"   and "set (characterize_consistent_signs_at_roots_copr p qs2) \<subseteq> set signs2"   and "all_list_constr subsets1 (length qs1)"   and "all_list_constr subsets2 (length qs2)"   and "invertible_mat (M_mat signs1 subsets1)"   and "invertible_mat (M_mat signs2 subsets2)" shows "satisfy_equation p (qs1 @ qs2) (subsets_smash (length qs1) subsets1 subsets2) (signs_smash signs1 signs2) \<and> invertible_mat (M_mat (signs_smash signs1 signs2) (subsets_smash (length qs1) subsets1 subsets2))"
Keys_mono: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set"   and B :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "A \<subseteq> B" shows "Keys A \<subseteq> Keys B"
convergent_powser_def: fixes cs :: "real msllist" shows "convergent_powser cs = (\<exists>R>0. \<forall>x. \<bar>x\<bar> < R \<longrightarrow> summable (\<lambda>n. lcoeff cs n * x ^ n))"
elims: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree"   and xa :: "'b"   and xb :: "('b \<times> nat) tree"   and y :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.balanceR \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 x xa xb = y" obtains l :: "('b \<times> nat) tree" and a :: "'b" and r :: "('b \<times> nat) tree" where "x = l" and "xa = a" and "xb = r" and "y = (if WBT.balanced1 \<Delta>1 \<Delta>2 r l then N l a r else WBT.rotateR \<Gamma>1 \<Gamma>2 l a r)"
holomorphic_lower_bound_difference: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex"   and \<phi> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "\<xi> \<in> S"   and "\<phi> \<in> S"   and "f \<phi> \<noteq> f \<xi>" obtains k :: "real" and n :: "nat" and r :: "real" where "0 < k" and "0 < r" and "ball \<xi> r \<subseteq> S" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * cmod (w - \<xi>) ^ n \<le> cmod (f w - f \<xi>)"
Approzimation1_5P2: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and i :: "nat"   and j :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>l\<in>{h. h \<le> Suc n}. n_val K (vv l) = vv l"   and "i \<le> Suc n"   and "j \<le> Suc n" shows "vv i (App_base K vv (Suc n) j) = \<delta>\<^bsub>i j\<^esub>"
fmlookup_inverse: fixes x :: "('c, 'd) fmap" shows "Abs_fmap (fmlookup x) = x"
card_PiE: fixes S :: "'e set"   and T :: "'e \<Rightarrow> 'f set" assumes "finite S" shows "card (Pi\<^sub>E S T) = (\<Prod>i\<in>S. card (T i))"
unique_factorization_m_cong: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset"   and g :: "int poly" assumes "unique_factorization_m f cfs"   and "f =m g" shows "unique_factorization_m g cfs"
intro: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "\<And>x y. invert_order (comp x y) = comp y x"   and "\<And>x y. comp x y = order.Eq \<Longrightarrow> x = y"   and "\<And>x y z. \<lbrakk>comp x y = order.Lt; comp y z = order.Lt\<rbrakk> \<Longrightarrow> comp x z = order.Lt" shows "comparator comp"
not_in_segment1: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "x \<notin> segment f x y"
simps_2: fixes op :: "'a strips_operator"   and ops :: "'a strips_operator list" shows "are_all_operators_non_interfering (op # ops) = (list_all (\<lambda>op'. \<not> are_operators_interfering op op') ops \<and> are_all_operators_non_interfering ops)"
ground_fresh_2: fixes A :: "fmla"   and i :: "'a" assumes "ground_fmla A" shows "atom i \<sharp> A"
upd_None_map_le: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c" shows "f(x := None) \<subseteq>\<^sub>m f"
hyps: assumes "is_tangle_diagram ws1 \<and> is_tangle_diagram ws2 \<and> domain_wall ws2 = codomain_wall ws1" shows "kauff_mat (ws1 \<circ> ws2) = rat_poly.matrix_mult (kauff_mat ws1) (kauff_mat ws2)"
rel_intros_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and y21 :: "'b"   and x22 :: "'a list"   and y22 :: "'b list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
is_ext_def: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" shows "is_ext y' g f \<equiv> \<guillemotleft>f : Fo y \<rightarrow>\<^sub>C y'\<guillemotright> \<and> g = G f \<cdot>\<^sub>D \<eta>o y"
less_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and s' :: "'e" shows "pre_RDR.less \<delta> s s' \<equiv> pre_RDR.less_eq \<delta> s s' \<and> s \<noteq> s'"
of_int_code_2: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"