hcju/isabelleps huggingface dataset:
close_eq_predAtm: fixes s :: "predicate \<times> 'a list \<Rightarrow> bool"   and p :: "predicate"   and xs :: "'a list" shows "close_eq s (predAtm p xs) = s (p, xs)"
of_nat_id: fixes n :: "nat" shows "of_nat n = n"
rqrm_propD: fixes \<delta> :: "('a, 'b) ta_rule set"   and rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and q :: "'a" assumes "rqrm_prop \<delta> rqrm" shows "ls.\<alpha> (rqrm_lookup rqrm q) = {r \<in> \<delta>. q \<in> set (rhsq r)}"
completion_upper: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets (completion M)" shows "\<exists>A'\<in>sets M. A \<subseteq> A' \<and> emeasure (completion M) A = emeasure M A'"
diff_eq: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
match_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and h :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C h g)" shows "partial_magma.arr C (C (C h g) f)"
mk_op_rl1: fixes mf :: "'e \<Rightarrow> 'f nres"   and P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x. mf x \<equiv> ASSERT (P x) \<bind> (\<lambda>_. RETURN (f x))" shows "(mf, RETURN \<circ> f) \<in> [P]\<^sub>f Id \<rightarrow> \<langle>Id\<rangle>nres_rel"
extendContain: fixes r :: "'b sequent list \<times> 'b sequent"   and ps :: "'b sequent list"   and c :: "'b sequent"   and Ps :: "'b sequent list"   and C :: "'b sequent"   and S :: "'b sequent"   and p :: "'b sequent" assumes "r = (ps, c)"   and "(Ps, C) = extendRule S r"   and "p \<in> set ps" shows "extend S p \<in> set Ps"
funion_upper1: fixes A :: "'c fset"   and B :: "'c fset" shows "A |\<subseteq>| A |\<union>| B"
ExFalso: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<not> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
drop_Cons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
mod_ring_locale_def: fixes p :: "int" shows "mod_ring_locale TYPE('a) p \<equiv> p = int CARD('a)"
hyps: fixes \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" assumes "weak_sharing_consistent \<O> sb"   and "a \<notin> \<O> \<union> all_acquired sb"   and "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only \<S>"
true_cp: shows "\<forall>i<length CAs. \<forall>\<sigma>. is_ground_subst \<sigma> \<longrightarrow> I \<Turnstile> CAs ! i \<cdot> P ! i \<cdot> \<sigma>"
diff_divide_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 b :: "'b"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
in_hhomI: fixes \<mu> :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "src \<mu> = a"   and "trg \<mu> = b" shows "in_hhom \<mu> a b"
polynomial_of_mset_Cons: fixes x :: "char list multiset \<times> int"   and ys :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset (add_mset x ys) = Const (snd x) * poly_of_vars (fst x) + polynomial_of_mset ys"
zero_cong: shows "\<zero>\<^bsub>R\<^esub> = 0"
ereal_diff_eq_MInfty_iff: fixes x :: "ereal"   and y :: "ereal" shows "(x - y = - \<infinity>) = (x = - \<infinity> \<and> y \<noteq> - \<infinity> \<or> y = \<infinity> \<and> \<bar>x\<bar> \<noteq> \<infinity>)"
select_convs_1: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "sp_impl_subj_subj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = sp_impl_subj_subj"
sets: fixes i :: "'b"   and A :: "'c set" assumes "i \<in> I"   and "A \<in> sets (M i)" shows "{\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
gamma2: shows "k\<^sup>2 + l\<^sup>2 = \<gamma>\<^sup>2"
poly_add_coeff: fixes p1 :: "'c list"   and p2 :: "'c list" assumes "set p1 \<subseteq> carrier S"   and "set p2 \<subseteq> carrier S" shows "S.coeff (S.poly_add p1 p2) = (\<lambda>i. S.coeff p1 i \<oplus>\<^bsub>S\<^esub> S.coeff p2 i)"
bind_pause: fixes out :: "'o"   and c :: "'i \<Rightarrow> 'm"   and f :: "'a option \<Rightarrow> 'm" shows "bind (pause out c) f = pause out (\<lambda>i. bind (c i) f)"
IH: assumes "(x + y) # xs \<noteq> []"   and "last ((x + y) # xs) \<noteq> (0::'a)"   and "sum_list ((x + y) # xs) = (0::'a)" shows "sign_changes (psums ((x + y) # xs)) < sign_changes ((x + y) # xs) \<and> odd (sign_changes ((x + y) # xs) - sign_changes (psums ((x + y) # xs)))"
pgallery_map: fixes Cs :: "'a set list" assumes "domain.pgallery Cs" shows "codomain.pgallery (f \<Turnstile> Cs)"
p2_decomp: shows "p2 = (y, z) # p2'"
prefeq_reflection: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "|~ x = y" shows "x \<equiv> y"
is_mgu_subst_set_subst: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" assumes "x \<notin> vars_term t"   and "is_mgu \<sigma> (subst_set (subst x t) E)" shows "is_mgu (subst x t \<circ>\<^sub>s \<sigma>) (insert (Var x, t) E)"
antisym_def: fixes r :: "('a \<times> 'a) set" shows "antisym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
strictly_generalizes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.strictly_generalizes subst_atm C D = strict (substitution_ops.generalizes subst_atm) C D"
insert_in_apply_set: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and a :: "'a"   and j :: "nat"   and i' :: "'a"   and e' :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and s :: "('a \<times> 'b \<times> bool) list" assumes "es @ [Deliver (i, Insert e (Some a))] prefix of j"   and "Deliver (i', Insert e' n) \<in> set es"   and "apply_operations es = Some s" shows "fst e' \<in> element_ids s"
fun_mono: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and D :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "C \<le> A"   and "B \<le> D" shows "rel_fun A B \<le> rel_fun C D"
length_sorted_list_of_set: fixes A :: "complex set" shows "length (complex_lex.sorted_list_of_set A) = card A"
measure_lborel_box: fixes l :: "'b"   and u :: "'b" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "Sigma_Algebra.measure lborel (box l u) = prod ((\<bullet>) (u - l)) Basis"
induces_elementary_monoidal_category: shows "elementary_monoidal_category (\<cdot>) C.tensor C.unity C.lunit C.runit C.assoc"
module_hom_eq_linear: shows "module_hom = Vector_Spaces.linear"
finite_measure_finite_Union: fixes S :: "'b set"   and A :: "'b \<Rightarrow> bool stream set" assumes "finite S"   and "A ` S \<subseteq> events"   and "disjoint_family_on A S" shows "prob (\<Union> (A ` S)) = (\<Sum>i\<in>S. prob (A i))"
prod_topology_subtopology_1: fixes X :: "'b topology"   and S :: "'b set"   and Y :: "'c topology" shows "prod_topology (subtopology X S) Y = subtopology (prod_topology X Y) (S \<times> topspace Y)"
split: fixes P :: "'k \<Rightarrow> bool"   and f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'k"   and option :: "'l option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
closed_of_nat_image: fixes A :: "nat set" shows "closed (of_nat ` A)"
cball_eq_empty: fixes x :: "'b"   and e :: "real" shows "(cball x e = {}) = (e < 0)"
inj_on_span_iff_independent_image: fixes B :: "'b set" assumes "independent (f ` B)" shows "inj_on f (span B) = inj_on f B"
indep_vars_PiM_components: fixes A :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" assumes "\<And>i. i \<in> A \<Longrightarrow> prob_space (M i)" shows "prob_space.indep_vars (Pi\<^sub>M A M) M (\<lambda>i f. f i) A"
5: shows "(x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * z\<^sup>\<star> \<le> (x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * (z * Rf y (x\<^sup>\<star>))\<^sup>\<star>"
simps_13: fixes var :: "nat"   and i :: "nat"   and a :: "atom fm" shows "freeIn var (ExN i a) = freeIn (var + i) a"
isoI: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" shows "local.iso f"
h_in_hs: fixes i :: "nat" assumes "i < k" shows "hs ! i \<in> set hs"
boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "max b c \<le> a" obtains "b \<le> a" and "c \<le> a"
af\<^sub>G_decompose_1: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and w :: "'a set list" shows "af\<^sub>G (\<phi> and \<psi>) w = af\<^sub>G \<phi> w and af\<^sub>G \<psi> w"
assms_1: fixes a :: "name"   and y :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a<\<nu>y> \<prec> Q'"   and "y \<sharp> P" shows "\<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel"
fmrestrict_set_null: fixes m :: "('d, 'e) fmap" shows "fmrestrict_set {} m = fmempty"
complete_lattice_ow_def: fixes U :: "'a set"   and Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" shows "complete_lattice_ow U Inf Sup inf le ls sup bot top \<equiv> (lattice_ow U inf le ls sup \<and> Inf_ow U Inf \<and> Sup_ow U Sup) \<and> bot_ow U bot \<and> top_ow U top \<and> complete_lattice_ow_axioms U Inf Sup le bot top"
eq_2: shows "typ_of (Abs \<tau> f) = Some (\<tau> \<rightarrow> \<tau>')"
inference_computation: fixes N :: "'a literal multiset set"   and Q :: "'a literal multiset set"   and C :: "'a literal multiset"   and P :: "'a literal multiset set" assumes "N = concls_of (ord_FO_resolution.inferences_between Q C)" shows "({}, P \<union> {C}, Q) \<leadsto> (N, P, Q \<union> {C})"
S_quadratic: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and k :: "real" assumes "p \<noteq> q"   and "r = proj2_abs (k *\<^sub>R proj2_rep p + proj2_rep q)" shows "(r \<in> S) = (proj2_rep p \<bullet> (M *v proj2_rep p) * k\<^sup>2 + proj2_rep p \<bullet> (M *v proj2_rep q) * 2 * k + proj2_rep q \<bullet> (M *v proj2_rep q) = 0)"
rvE: fixes V :: "'var"   and S :: "'node SDG_node set"   and n :: "'node SDG_node" assumes "V \<in> rv S n" obtains as :: "'edge list" and n' :: "'node SDG_node" where "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'" and "n' \<in> HRB_slice S" and "V \<in> Use\<^bsub>SDG\<^esub> n'" and "\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''"
card_of_mono2: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r \<le>o r'" shows "|Field r| \<le>o |Field r'|"
le_infI1: fixes a :: "'b multiset"   and x :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># x" shows "a \<inter># b \<subseteq># x"
mult_nonneg_nonpos2: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "b \<le> (0::'a)" shows "b * a \<le> (0::'a)"
l1_inv3I: fixes x :: "l1_state" assumes "\<And>A B Rb Ra gny. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progressS (progress x Rb) {xny, xgnx, xgny, xsk}; guessed_frame Rb xgny = Some gny; guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; progress x Ra = Some {xnx, xgnx}; can_signal x A B\<rbrakk> \<Longrightarrow> signalsInit x (Commit A B (Exp gny (NonceF (Ra $ nx)))) < signalsInit x (Running A B (Exp gny (NonceF (Ra $ nx))))" shows "x \<in> l1_inv3"
sub_4: shows "\<Gamma>\<^sub>1' \<sqsubseteq>\<^sub>c \<Gamma>'"
cut_greater_empty: fixes t :: "'a" shows "{} \<down>> t = {}"
intro: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "aGroup M"   and "Module_axioms M R" shows "R module M"
discount_factor_real_cond_exp: fixes X :: "'a \<Rightarrow> real"   and G :: "'a measure"   and r :: "real"   and n :: "nat" assumes "integrable M X"   and "subalgebra M G"   and "- 1 < r" shows "AEeq M (real_cond_exp M G (\<lambda>x. discount_factor r n x * X x)) (\<lambda>w. discount_factor r n w * real_cond_exp M G X w)"
idem: fixes f :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and a :: "'a::type" assumes "semilattice (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type)" shows "(f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) a = a"
InterpFnE: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and f :: "'b"   and e :: "('a, 'b) Expression"   and C :: "'a"   and i :: "('c, 'd) IType" assumes "L\<lbrakk>Vx : A \<turnstile> f E@ e : C\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" obtains B :: "'a" and g :: "'d" where "i = IMor (g ;;\<^bsub>iC\<^bsub>I\<^esub>\<^esub> Fn\<lbrakk>f\<rbrakk>\<^bsub>I\<^esub>)" and "Sig iS\<^bsub>I\<^esub> \<triangleright> Vx : A \<turnstile> e : B" and "f \<in> Sig iS\<^bsub>I\<^esub> : B \<rightarrow> C" and "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g"
prems: shows "wfrestrictedvars\<^sub>s\<^sub>t (x # S) \<subseteq> V"
id2: shows "poly_coeff_sum (mc # p) * x ^ poly_degree (mc # p) = x ^ max (monom_degree m) (poly_degree p) * max (0::'a) c + poly_coeff_sum p * x ^ max (monom_degree m) (poly_degree p)"
minus_mod_self1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus b a) b = modulo (uminus a) b"
suba2: shows "{a--flow0 x t2} \<subseteq> {a--b}"
fold_end_sum_chain_gf: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and as :: "'a permutation list"   and s :: "'a permutation"   and bs :: "'a permutation list" assumes "(f, g) \<in> foldpairs"   and "as \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "bs \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "sum_list as `\<rightarrow> C0 \<in> g \<turnstile> \<C>"   and "sum_list (as @ [s]) `\<rightarrow> C0 \<in> f \<turnstile> \<C>" shows "induced_automorph f g ` sum_list (as @ [s] @ bs) `\<rightarrow> C0 = sum_list (as @ bs) `\<rightarrow> C0"
pgwt_arity: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and t :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "intruder_model.public_ground_wf_term arity public t"   and "Fun f T \<sqsubseteq> t" shows "arity f = length T"
BrnFT_Int_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "II \<in> UNpart1 cl dl P" shows "BrnFT cl dl \<inter> II = {}"
st_add: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "st (x + y) = st x + st y"
leg1_is_map: shows "is_left_adjoint t\<^sub>0u\<^sub>1.p\<^sub>1"
floor_less_one: fixes x :: "'a" shows "(\<lfloor>x\<rfloor> < 1) = (x < (1::'a))"
simps_2: fixes A :: "(('a, 'b) vec, 'c) vec"   and n :: "nat"   and i :: "'c"   and j :: "'b"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "bezout_iterate A (Suc n) i j bezout = (if Suc n \<le> mod_type_class.to_nat i then A else bezout_iterate (bezout_matrix A i (mod_type_class.from_nat (Suc n)) j bezout ** A) n i j bezout)"
const_type_inv_wf: fixes c :: "('fun, 'atom, 'sets) prot_fun"   and X :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term list"   and a :: "'atom prot_atom" assumes "\<Gamma> (Fun c X) = Var a"   and "wf\<^sub>t\<^sub>r\<^sub>m (Fun c X)" shows "X = []"
analytic_continuation_open: fixes s :: "complex set"   and s' :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "open s"   and "open s'"   and "s \<noteq> {}"   and "connected s'"   and "s \<subseteq> s'"   and "f holomorphic_on s'"   and "g holomorphic_on s'"   and "\<And>z. z \<in> s \<Longrightarrow> f z = g z"   and "z \<in> s'" shows "f z = g z"
wi_hom_neg: fixes a :: "int" shows "- word_of_int a = word_of_int (- a)"
oghoare_Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and f :: "'e \<Rightarrow> 'e"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Basic f) Q, A" shows "\<exists>P'. P = AnnExpr P' \<and> P' \<subseteq> {x. f x \<in> Q}"
cong_sym_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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = unique_euclidean_semiring.cong modulo c b a"
lnths_def: fixes xs :: "'c llist"   and A :: "nat set" shows "lnths xs A = lmap fst (lfilter (\<lambda>(x, y). y \<in> A) (lzip xs (iterates Suc 0)))"
euclidean_representation: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and x :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner x b) b) Basis = x"
ip: shows "is_path (\<pi> @\<^bsup>k\<^esup> \<pi>l)"
assms_2: shows "sum_list (map snd \<mu>) = 1"
rbt_lookup_rbt_delete: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and k :: "'b" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_delete less k t) = ord.rbt_lookup less t |` (- {k})"
lambda_predicates_3_1: fixes F :: "\<Pi>\<^sub>1" shows "[[(\<^bold>\<lambda>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>= F]]"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ThinishChamberComplex X"   and "ChamberComplexFolding X f" shows "ThinishChamberComplexFolding X f"
of_int_less_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w < z)"
vbrelation_axioms: shows "vbrelation (\<FF>\<lparr>ArrMap\<rparr>)"
assms_2: shows "theta \<subseteq> RetrT (theta \<union> BisT)"
pwd_owner_ii_def: fixes n :: "nat" shows "pwd_owner_ii n \<equiv> SOME x. pred_owner_ii n (pwd_spy_i n) x"
delete_max_induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>n l h. P (MKT n l ET h)"   and "\<And>n l v va vb vc h. P (MKT v va vb vc) \<Longrightarrow> P (MKT n l (MKT v va vb vc) h)"   and "P ET" shows "P a0"
mult_le_less_imp_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq a b"   and "less c d"   and "less zero a"   and "less_eq zero c" shows "less (times a c) (times b d)"
arrI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "Dop.in_hom f a b" shows "Dop.arr f"
take_bit_add: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit n (plus (take_bit n a) (take_bit n b)) = take_bit n (plus a b)"
cases: fixes x :: "'b list \<times> 'b list" obtains a :: "'b" and fs :: "'b list" and rs :: "'b list" where "x = (a # fs, rs)"   | va :: "'b list" where "x = ([], va)"