hcju/isabelleps huggingface dataset:
lcm_neg1_int: fixes x :: "int"   and y :: "int" shows "lcm (- x) y = lcm x y"
LIM_equal2: fixes R :: "real"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "0 < R"   and "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"   and "g \<midarrow>a\<rightarrow> l" shows "f \<midarrow>a\<rightarrow> l"
locateTI: fixes cl :: "('test, 'atom, 'choice) cmd list"   and ii :: "nat" assumes "properL cl"   and "ii < brn (Par cl)" shows "locateT cl ii < length cl \<and> locateI cl ii < brn (cl ! locateT cl ii) \<and> ii = brnL cl (locateT cl ii) + locateI cl ii"
oalist_inv_list_of_oalist: fixes x :: "'x" shows "oalist_inv (list_of_oalist x)"
functions_of_is_surj_on: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P = card B" shows "univ (\<lambda>f. f ` A = B) (Twelvefold_Way_Entry7.functions_of P A B)"
measurable_emeasure_subprob_algebra: fixes a :: "'c set"   and A :: "'c measure" assumes "a \<in> sets A" shows "(\<lambda>M. emeasure M a) \<in> borel_measurable (subprob_algebra A)"
simps_4: fixes v :: "val" shows "\<A> (Val v) = \<lfloor>{}\<rfloor>"
Gauss_Jordan_column_k_PA_def: fixes A' :: "(('a, 'b) vec, 'b) vec \<times> nat \<times> (('a, 'c) vec, 'b) vec"   and k :: "nat" shows "Gauss_Jordan_column_k_PA A' k = (let P = fst A'; i = fst (snd A'); A = snd (snd A'); from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'a)) \<or> i = nrows A then (P, i, A) else let Gauss = Gauss_Jordan_in_ij_PA (P, A) from_nat_i from_nat_k in (fst Gauss, i + 1, snd Gauss))"
continuous_def: fixes F :: "ordinal \<Rightarrow> ordinal" shows "continuous F \<equiv> \<forall>f. F (oLimit f) = oLimit (\<lambda>n. F (f n))"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "arrow_of_spans C \<mu>"
\<psi>\<^sub>1_is_state: shows "state 2 \<psi>\<^sub>1"
INF_absorb: fixes k :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a" assumes "k \<in> I" shows "A k \<sqinter> \<Sqinter>(A ` I) = \<Sqinter>(A ` I)"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_fail return bind fail"   and "monad_catch_axioms return fail catch" shows "monad_catch return bind fail catch"
bij_betw_byWitness: fixes A :: "'e set"   and f' :: "'f \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'f"   and A' :: "'f set" assumes "\<forall>a\<in>A. f' (f a) = a"   and "\<forall>a'\<in>A'. f (f' a') = a'"   and "f ` A \<subseteq> A'"   and "f' ` A' \<subseteq> A" shows "bij_betw f A A'"
less_le: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(x <s y) = (x \<le>s y \<and> x \<noteq> y)"
itypeI: fixes c :: "'a"   and I :: "interface" shows "c ::\<^sub>i I"
CallObjThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
streamValue9: fixes d :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'a"   and j :: "nat" assumes "\<forall>j\<le>2 * d. str (t + j + Suc (i + k)) = x"   and "j \<le> d" shows "str (t + i + k + d + Suc j) = x"
x_y: shows "(f ^^ funpow_dist1 f x y) x = y"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "a \<in> set1_FGcontra x"   and "b \<in> set1_FGcontra y"   and "L1 a b" shows "L1' a b"
subterm_reduction_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set"   and t :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.subterm_reduction trm_ord sel S t = basic_superposition.trm_rep trm_ord sel (basic_superposition.trm_rep trm_ord sel (lhs t) S \<cdot> basic_superposition.trm_rep trm_ord sel (rhs t) S) S"
dom_if: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}"
less_eq_literal_def: fixes L :: "'a literal"   and M :: "'a literal" shows "(L \<le> M) = (atm_of L < atm_of M \<or> atm_of L \<le> atm_of M \<and> is_neg L \<le> is_neg M)"
WHILE: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<forall>s t.  s \<approx> t  \<longrightarrow> evalB b s = evalB b t"   and "G \<rhd> c : Sec \<Phi>" shows "G \<rhd> While b c : Sec (PhiWhile b \<Phi>)"
hb_deliver: fixes m1 :: "'a"   and i :: "nat"   and m2 :: "'a" assumes "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
complete_sets_sandwich_fmeasurable: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A \<in> fmeasurable M"   and "C \<in> fmeasurable M"   and "A \<subseteq> B"   and "B \<subseteq> C"   and "Sigma_Algebra.measure M A = Sigma_Algebra.measure M C" shows "B \<in> fmeasurable M"
simps_2: fixes d :: "bool"   and ds :: "bool list"   and T1 :: "tree"   and T2 :: "tree" shows "path (d # ds) (Branching T1 T2) = (if d then path ds T1 else path ds T2)"
1: shows "stone_phi.rho_pair (Abs_regular (- - x), stone_phi (Abs_regular (- x)) \<squnion> up_filter (Abs_dense (x \<squnion> - x))) = Abs_dense (x \<squnion> - x)"
eq_up_to_sym: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n" assumes "\<pi> =\<^bsub>k\<^esub> \<pi>'" shows "\<pi>' =\<^bsub>k\<^esub> \<pi>"
fmlookup_ran_iff: fixes y :: "'d"   and m :: "'e \<rightharpoonup>\<^sub>f 'd" shows "(y |\<in>| fmran m) = (\<exists>x. fmlookup m x = Some y)"
zero_least1: fixes x :: "'a" shows "o \<oplus> !x = !x"
not_arr_null: shows "\<not> VxV.arr VxV.null"
stationary_distribution_def: fixes N :: "'s pmf" shows "stationary_distribution N = (N = N \<bind> K)"
apply_cltn2_right_abs: fixes M :: "((real, 3) vec, 3) vec"   and p :: "proj2" assumes "invertible M" shows "apply_cltn2 p (cltn2_abs M) = proj2_abs (proj2_rep p v* M)"
islimpt_approachable_real: fixes x :: "real"   and s :: "real set" shows "x islimpt s = (\<forall>e>0. \<exists>x'\<in>s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"
center_as_intersection: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "{x--y} \<inter> {x--z} \<inter> {y--z} = {center x y z}"
He_eq_He_THe_star: fixes e :: "'a"   and H :: "'a" assumes "bijective (e * top) \<and> bijective (e[[top]])"   and "equivalence H" shows "H * e = H * e * (top * H * e)\<^sup>\<star>"
n_valid_Rmi_alt: fixes bdd :: "bdd"   and na :: "nat" shows "bdd_node_valid bdd na = (\<exists>b. (na, b) \<in> Rmi bdd)"
CapSS: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<psi> :: "'c"   and f :: "'b \<Rightarrow> 'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap"   and "\<psi> \<in> fmla"   and "Fvars \<psi> = {Generic_Syntax.Variable var (Suc 0)}"   and "f \<in> F" shows "ssap \<psi> (enc (encF f)) = enc (Deduct_with_False_Disj_Rename.instInpP var Var subst cnj exi \<psi> 0 (Generic_Syntax.instInp var subst (encF f) (enc (encF f))))"
rPar2: fixes P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> \<tau> \<prec> P'"   and "\<And>C. Prop C \<Psi> (!P) P'" shows "Prop C \<Psi> (P \<parallel> !P) (P \<parallel> P')"
rel_interior_eq_empty: fixes S :: "'b set" assumes "convex S" shows "(rel_interior S = {}) = (S = {})"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H G" shows "u \<rightarrow>\<^sup>* v"
time_closest_pair_bf_tm: fixes ps :: "(int \<times> int) list" shows "time (closest_pair_bf_tm ps) \<le> length ps * length ps + 1"
select_convs_2: fixes processes :: "('d, 'e) local_config multiset"   and state :: "'b"   and more :: "'f" shows "global_config.state \<lparr>global_config.processes = processes, state = state, \<dots> = more\<rparr> = state"
Instantiate: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[\<^bold>\<exists>x. \<phi> x in v]" obtains x :: "'a" where "[\<phi> x in v]"
simps_2: fixes v :: "'b"   and Q :: "'b \<Rightarrow> bool" shows "ospec (Some v) Q = Q v"
l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf \<equiv> type_wf = ShadowRootClass.type_wf"
fps_ln_deriv: fixes c :: "'a" shows "fps_deriv (fps_ln c) = fps_const ((1::'a) / c) * inverse (1 + fps_X)"
cy0: shows "\<forall>i<length Vs. (0::'a) \<le> cy i"
mat_group: fixes nr :: "nat"   and nc :: "nat"   and b :: "'d" shows "Group.group (mat_monoid nr nc b)"
strong_impl_weak_reduction_bisimulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))" shows "weak_reduction_bisimulation Rel Cal"
endofunctor_axioms: shows "endofunctor (\<cdot>\<^sub>B) trg\<^sub>B"
ffold_empty: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "ffold f z {||} = z"
neg_bounds_compare_8: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c >\<^sub>l\<^sub>b b" shows "c \<le>\<^sub>l\<^sub>b b"
Upsilon1_subsetof_E1: shows "\<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"
finite_has_minimal: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq b m \<longrightarrow> m = b"
max_li_is_gen: fixes A :: "'c set" assumes "maximal A (\<lambda>S. S \<subseteq> carrier V \<and> lin_indpt S)" shows "gen_set A"
atLeastLessThanSuc: fixes m :: "nat"   and n :: "nat" shows "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
elims: fixes q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and x :: "'b ltl"   and y :: "'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)" assumes "ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M x = y" obtains \<phi> :: "'b ltl" where "x = \<phi>" and "y = (q\<^sub>0 \<phi>, \<iota>\<^sub>\<times> (\<^bold>G \<phi>) (semi_mojmir_def.initial \<circ> q\<^sub>0\<^sub>M \<circ> theG))"
drf_weak: fixes \<E> :: "('a \<times> ('b, 'a) obs_event action) llist set"   and P :: "'c prog"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "drf \<E> P"   and "correctly_synchronized P \<E>"   and "weakly_legal_execution P \<E> (E, ws)" shows "sequentially_consistent P (E, ws)"
conjugate_vec_def: fixes v :: "'b vec" shows "conjugate v = vec (dim_vec v) (\<lambda>i. conjugate (v $ i))"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a \<le> b"
Infinitesimal_of_hypreal_iff: fixes x :: "real star" shows "(of_hypreal x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
dg_prod_2_op_dg_dg_Obj: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "(op_dg \<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr> = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
map_mono_prefix: fixes xs :: "'e list"   and ys :: "'e list"   and f :: "'e \<Rightarrow> 'f" assumes "prefix xs ys" shows "prefix (map f xs) (map f ys)"
linearly_independent_over_int_def: shows "linearly_independent_over_int = (\<lambda>s. \<not> module.dependent (\<lambda>r. (*) (of_int r)) s)"
run_owner_ii_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_owner_ii n s))) \<notin> used s"
solutions_less_iff_pell_valuation_less: fixes a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "solution (a, b)"   and "solution (x, y)" shows "(a < x \<and> b < y) = (pell_valuation (case (a, b) of (x, y) \<Rightarrow> (int x, int y)) < pell_valuation (case (x, y) of (x, y) \<Rightarrow> (int x, int y)))"
gcd_coprime_exists: fixes a :: "'a"   and b :: "'a" assumes "gcd a b \<noteq> (0::'a)" shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
intro: fixes T :: "'b \<Rightarrow> 'b"   and M :: "'b measure" assumes "T \<in> quasi_measure_preserving M M" shows "qmpt_axioms M T"
smaller_sys_are_good_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "0 < length qs2"   and "set (find_consistent_signs_at_roots_R p qs1) = set (characterize_consistent_signs_at_roots p qs1)"   and "set (find_consistent_signs_at_roots_R p qs2) = set (characterize_consistent_signs_at_roots p qs2)" shows "set (snd (snd (reduce_system_R p (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))) = set (characterize_consistent_signs_at_roots p (qs1 @ qs2))"
finsum_closed: fixes f :: "'e \<Rightarrow> 'a"   and A :: "'e set" assumes "f \<in> A \<rightarrow> carrier R" shows "finsum R f A \<in> carrier R"
convex_lincomb_empty: fixes n :: "nat"   and c :: "'b vec \<Rightarrow> 'b"   and x :: "'b vec" shows "\<not> gram_schmidt.convex_lincomb n c {} x"
g1_g2_continuous_2: shows "continuous_on (cbox a b) g2"
eq_shift: fixes w :: "'b stream"   and u :: "'b list"   and v :: "'b stream" shows "(w = u @- v) = (stake (length u) w = u \<and> sdrop (length u) w = v)"
cod_MkArr: fixes A :: "'a"   and B :: "'a"   and F :: "'a set" assumes "Maps.arr (Maps.MkArr A B F)" shows "Maps.cod (Maps.MkArr A B F) = Maps.MkIde B"
t_non_zero: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_packing_design point_set block_collection u_block_size grouping min_index" shows "1 \<le> grouping"
map_id: fixes t :: "('a, 'b, 'c) expr" shows "\<R>\<^sub>E id id t = t"
list_all2_swap: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P xs ys = list_all2 (\<lambda>x y. P y x) ys xs"
pointermap_p_valid_inv: fixes p :: "nat"   and m :: "'a pointermap"   and a :: "'a"   and x :: "nat"   and u :: "'a pointermap" assumes "pointermap_p_valid p m"   and "pointermap_getmk a m = (x, u)" shows "pointermap_p_valid p u"
wellformed1_alternative_sorted: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (FWNormalisationCore.sort p l)"
cong_add_rcancel_0_nat: fixes x :: "nat"   and a :: "nat"   and n :: "nat" shows "[x + a = a] (mod n) = [x = 0] (mod n)"
power2_norm_eq_inner: fixes x :: "'a" shows "(norm x)\<^sup>2 = x \<bullet> x"
trees_of_size_disjoint: fixes m :: "nat"   and n :: "nat" assumes "m \<noteq> n" shows "trees_of_size m \<inter> trees_of_size n = {}"
Times_Int_Times: fixes A :: "'b set"   and B :: "'c set"   and C :: "'b set"   and D :: "'c set" shows "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
y'_def: shows "y' \<equiv> snd (normalize_quot (x, y))"
is_of_type_const_conv: fixes x :: "object"   and T :: "PDDL_STRIPS_Semantics.type" shows "is_of_type (ty_term Q objT) (term.CONST x) T = is_of_type objT x T"
IH: fixes ms :: "'node list" assumes "var g v \<in> uses g m'"   and "g \<turnstile> m-ms\<rightarrow>m'"   and "\<forall>x\<in>set (tl ms). var g v \<notin> defs g x" shows "v \<in> braun_ssa.allDefs g (defNode v)"
K: shows "\<forall>a b. norm b \<le> norm b * K"
assms_2: shows "V \<subseteq> U\<^sub>M\<^sub>_\<^sub>1"
f_inv_on_f: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "y \<in> f ` A" shows "f (inv_on f A y) = y"
zeta_Re_gt_1_nonzero: fixes s :: "complex" assumes "1 < Re s" shows "zeta s \<noteq> 0"
dist_triangle2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "dist x y \<le> dist x z + dist y z"
index_simps_2: fixes n :: "nat"   and l :: "'a list" assumes "n < length l" shows "index l n = Some (l ! n)"
quicksort_post_set: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and lo :: "nat"   and hi :: "nat"   and xs :: "'d list"   and xs' :: "'d list" assumes "quicksort_post R h lo hi xs xs'"   and "lo \<le> hi"   and "hi < length xs" shows "set (WB_Sort.sublist xs' lo hi) = set (WB_Sort.sublist xs lo hi)"
uncountable_minus_countable: fixes A :: "'b set"   and B :: "'b set" assumes "uncountable A"   and "countable B" shows "uncountable (A - B)"
steps_c\<^sub>1: shows "\<Gamma> \<turnstile> (c\<^sub>1, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
ccpo_lfp_simp: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "ccpo.fixp Sup (\<le>) f = lfp f"
parts_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "parts G \<subseteq> parts H"
U_def: shows "U = (\<lambda>x. {n. \<exists>l\<in>{n - s<..n}. u n x - u (n - l) x \<le> - d * real l})"
STAR_exp_add: fixes x :: "'a star"   and y :: "'a star" shows "(*f* exp) (x + y) = (*f* exp) x * (*f* exp) y"
det_def: shows "det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>i = 0..<n. A $$ (i, p i)))"
exhaust: fixes y :: "fmla" obtains x11 :: "trm" and x12 :: "trm" where "y = x11 EQ x12"   | x21 :: "fmla" and x22 :: "fmla" where "y = x21 OR x22"   | x3 :: "fmla" where "y = Robinson_Arithmetic.neg x3"   | x41 :: "name" and x42 :: "fmla" where "y = exi x41 x42"
nAct_not_active_same: fixes n :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "enat n \<le> n'"   and "n' - 1 < llength t"   and "\<nexists>k. enat n \<le> enat k \<and> enat k < n' \<and> active c (lnth t k)" shows "dynamic_component.nAct tCMP active c n' t = dynamic_component.nAct tCMP active c (enat n) t"
mHom_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme" shows "mHom R M N = {f \<in> aHom M N. \<forall>a\<in>carrier R. \<forall>m\<in>carrier M. f (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m}"
pinduct: fixes a0 :: "'x \<Rightarrow>\<^sub>0 nat"   and a1 :: "'x set"   and a2 :: "('x \<Rightarrow>\<^sub>0 nat) set"   and P :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'x set \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) set \<Rightarrow> bool" assumes "split_dom TYPE('b) (a0, a1, a2)"   and "\<And>t U S. \<lbrakk>split_dom TYPE('b) (t, U, S); \<And>x. \<lbrakk>0 \<notin> S; S \<inter> .[U] \<noteq> {}; x = (SOME x'. x' \<in> U - max_subset U (\<lambda>V. S \<inter> .[V] = {}))\<rbrakk> \<Longrightarrow> P t (U - {x}) S; \<And>x xa xb y. \<lbrakk>0 \<notin> S; S \<inter> .[U] \<noteq> {}; x = (SOME x'. x' \<in> U - max_subset U (\<lambda>V. S \<inter> .[V] = {})); xa = split t (U - {x}) S; (xb, y) = xa\<rbrakk> \<Longrightarrow> P (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S)\<rbrakk> \<Longrightarrow> P t U S" shows "P a0 a1 a2"
xor_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> y = y \<oplus> x"
basic_monos_6: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
select_convs_3: fixes CatDom :: "('c, 'a, 'e) Category_scheme"   and CatCod :: "('d, 'b, 'f) Category_scheme"   and MapM :: "'a \<Rightarrow> 'b"   and more :: "'g" shows "(##) \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = MapM"
expand_new_name_def: shows "expand_new_name \<equiv> Suc"
Node_Push_I: fixes p :: "(nat \<Rightarrow> 'f + nat) \<times> ('g + nat)"   and i :: "'f + nat" assumes "p \<in> Old_Datatype.Node" shows "apfst (Old_Datatype.Push i) p \<in> Old_Datatype.Node"
diff_add_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
null_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "product_category C1 C2" shows "partial_magma.null (product_category.comp C1 C2) = (partial_magma.null C1, partial_magma.null C2)"
in_hom_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "partial_magma C" shows "partial_magma.in_hom C f a b \<equiv> a \<in> partial_magma.domains C f \<and> b \<in> partial_magma.codomains C f"
pr_algo_rel_alt: shows "pr_algo_rel = {((push_effect f e, l), f, l) |f e l. Height_Bounded_Labeling c s t f l \<and> push_precond f l e} \<union> {((f, relabel_effect f l u), f, l) |f u l. Height_Bounded_Labeling c s t f l \<and> relabel_precond f l u}"
simps: fixes args :: "Monitor.args"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping" shows "result_mmsaux args (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = Mapping.keys tuple_in"
qAbs_preserves_alpha2: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGoodAbs (qAbs xs x X) \<or> qGoodAbs (qAbs xs x X')" shows "qAbs xs x X $= qAbs xs x X'"