hcju/isabelleps huggingface dataset:
vrat_mult_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x *\<^sub>\<rat> y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
CatchSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c\<^sub>2 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,R" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Catch c\<^sub>1 c\<^sub>2 Q,A"
poly_deg_zero: shows "poly_deg 0 = (0::'c)"
le_infI2: fixes b :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and a :: "'b \<times> 'c" assumes "less_eq_prod b x" shows "less_eq_prod (inf_prod a b) x"
sdiv_i_def: fixes ops :: "'b arith_ops_record"   and pp :: "'b list"   and a :: "'b" shows "sdiv_i ops pp a = strip_while ((=) (arith_ops_record.zero ops)) (map (\<lambda>c. arith_ops_record.divide ops c a) pp)"
Gromov_product_commute: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "Gromov_product_at e x y = Gromov_product_at e y x"
simps: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and stack :: "'b \<Rightarrow> 'c list"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" assumes "balance_trace step stack" shows "balance_trace.bal step stack a1 a2 a3 = (\<exists>c T c'. a1 = c \<and> a2 = T \<and> a3 = c' \<and> traces.trace step c T c' \<and> (\<forall>c'\<in>set T. stack c \<lesssim> stack c') \<and> stack c' = stack c)"
hvcon: shows "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
state_to_state_qbit: fixes n :: "nat"   and v :: "complex Matrix.mat" assumes "state n v" shows "Matrix.col v 0 \<in> state_qbit n"
part1: fixes u :: "nat" shows "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u)) \<and> \<T> k u = \<Z> k u"
models_in_range: fixes h :: "heap"   and as :: "nat set"   and P :: "assn" assumes "pHeap h as \<Turnstile> P" shows "in_range (h, as)"
uadmit_mkv_adjoint: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf, 'sz) ODE" assumes "ssafe \<sigma>"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"   and "osafe ODE" shows "mk_v (local.adjoint I \<sigma> \<nu>) ODE = mk_v (local.adjoint I \<sigma> \<omega>) ODE"
axioms: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F" shows "mojmir \<Sigma> \<delta> q\<^sub>0 w F"
that: fixes z :: "name"   and zvec :: "name list" assumes "xvec = z # zvec"   and "length zvec = length yvec" shows "thesis"
LTL_F_one_step_unfolding: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl" shows "w \<Turnstile> F \<phi> = (w \<Turnstile> \<phi> \<or> w \<Turnstile> X (F \<phi>))"
lex_pm_antisym: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "lex_pm s t"   and "lex_pm t s" shows "s = t"
factors_dividesI: fixes fs :: "'a list"   and a :: "'a"   and f :: "'a" assumes "factors G fs a"   and "f \<in> set fs"   and "set fs \<subseteq> carrier G" shows "f divides a"
assms: shows "e \<tturnstile> ts @ [t] : Ts"
cobounded_unbounded_component: fixes S :: "'a set" assumes "bounded (- S)" shows "\<exists>x. x \<in> S \<and> \<not> bounded (connected_component_set S x)"
hyps_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (applyUnitPropagate state')) (getF (applyUnitPropagate state'))"   and "InvariantWatchListsUniq (getWatchList (applyUnitPropagate state'))"   and "InvariantWatchListsCharacterization (getWatchList (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesEl (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesDiffer (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))" shows "let state'a = exhaustiveUnitPropagate (applyUnitPropagate state') in getSATFlag state'a = getSATFlag (applyUnitPropagate state')"
valid_path_offset: fixes g :: "real \<Rightarrow> 'a"   and z :: "'a" shows "valid_path (\<lambda>t. g t - z) = valid_path g"
homomorphism: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "Applicative.ap (pure f) (pure x) = pure (f x)"
distinct_3: fixes x1 :: "'a" shows "Fin x1 \<noteq> -\<infinity>"
plus_pp_def: shows "(+) \<equiv> map_fun pp.mapping_of (map_fun pp.mapping_of PP) (+)"
seqE: fixes g :: "'b"   and f :: "'b" assumes "C'.seq g f" obtains "C'.arr f" and "C'.arr g" and "C'.dom g = C'.cod f"
eq_impD: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i"   and s :: "'g"   and s' :: "'g" assumes "eq_imp f g"   and "\<forall>x. f x s = f x s'" shows "g s = g s'"
ESem_fresh_cong_below_subset: fixes e :: "exp"   and S :: "Vars.var set"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and \<rho>' :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "fv e \<subseteq> S"   and "\<rho> f|` S \<sqsubseteq> \<rho>' f|` S" shows "\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> \<sqsubseteq> \<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
a: shows "steps0 (Suc 0, l, r) (findnth n) stp = (Suc (2 * n), l', r') \<and> inv_locate_a (as, lm) (n, l', r') ires"
Length_CONS: fixes x :: "'x"   and \<AA> :: "'i" shows "Length (CONS x \<AA>) = Length \<AA> + 1"
disjoint_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
simps_4: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and a :: "'e"   and \<phi> :: "('e, 'f) Kform" shows "M, w \<Turnstile> \<^bold>K\<^sub>a \<phi> = (\<forall>w'\<in>relations M a `` {w}. M, w' \<Turnstile> \<phi>)"
subspace_span: fixes S :: "'b set" shows "local.subspace (local.span S)"
null_eqI: fixes n :: "'a" assumes "\<And>f. R.comp n f = n \<and> R.comp f n = n" shows "n = R.null"
simps_5: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "aval (Times a1 a2) s = value_times (aval a1 s) (aval a2 s)"
Cantor_2: shows "list.set ms \<subseteq> {0<..}"
perm_remove: fixes x :: "'a"   and ys :: "'a list" assumes "x \<in> set ys" shows "ys <~~> x # remove1 x ys"
T_del_def: fixes x :: "'a"   and t :: "'a tree" shows "T_del x t = time (del_tm x t)"
shared_weight_net_insert_weights: fixes m :: "(nat \<times> nat) convnet"   and w :: "nat \<Rightarrow> real" assumes "balanced_net m" shows "shared_weight_net (insert_weights True m w)"
extend_state_oracle_plus_oracle: fixes oracle1 :: "'b \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'b) spmf"   and oracle2 :: "'b \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'b) spmf" shows "\<dagger>(oracle1 \<oplus>\<^sub>O oracle2) = \<dagger>oracle1 \<oplus>\<^sub>O \<dagger>oracle2"
bezout_coefficients_def: fixes a :: "'b"   and b :: "'b" shows "bezout_coefficients a b = fst (euclid_ext_aux (1::'b) (0::'b) (0::'b) (1::'b) a b)"
split_length: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list" assumes "split ts x = (ls, rs)" shows "length ls + length rs = length ts"
cond_L6: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  (Q \<triangleleft> b \<triangleright>  R)) = (P \<triangleleft> b \<triangleright>  R)"
pp_of_map_component: fixes f :: "nat \<Rightarrow> nat"   and v :: "'t" shows "pp_of_term (map_component f v) = pp_of_term v"
dom_distrib: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "ad (ad x) + ad (ad y) \<cdot> ad (ad z) = (ad (ad x) + ad (ad y)) \<cdot> (ad (ad x) + ad (ad z))"
top_mult_annil: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and strong_iteration :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.dra plus times one zero less_eq less star strong_iteration" shows "times (strong_iteration one) x = strong_iteration one"
prec_after_EV_WAIT_step: fixes s :: "rstate_t"   and aseq :: "int_point_t list"   and n :: "nat" assumes "rprecondition s (rcurrent s) (aseq ! n)"   and "Suc n < length aseq"   and "is_SK_EV_WAIT aseq"   and "\<not> raborting s (rcurrent s) (aseq ! n)"   and "\<not> rwaiting s (rcurrent s) (aseq ! n)" shows "rprecondition (rstep s (aseq ! n)) (rcurrent s) (aseq ! Suc n)"
dims_tensor_prod: fixes A :: "'b tensor"   and B :: "'b tensor" shows "dims (A \<otimes> B) = dims A @ dims B"
simps_2: fixes x :: "'b"   and k :: "'b"   and v :: "'a"   and l :: "('b \<times> 'a) list" shows "ls_lookup x ((k, v) # l) = (if x = k then Some v else ls_lookup x l)"
h1: shows "\<forall>m. ks = kKS m \<longrightarrow> \<not> ine P (kE m) \<and> \<not> ine Q (kE m)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) \<chi>'.A.map D (cones_map f \<chi>)"
uminus_set_def: fixes A :: "'m set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
arr_cod_iff_arr: fixes f :: "'a" shows "L.arr (L.cod f) = L.arr f"
distinct_3: fixes x31 :: "cmd"   and x32 :: "cmd" shows "Skip \<noteq> x31;; x32"
elims: fixes x :: "'b tree"   and y :: "'b multiset" assumes "mset_tree x = y" obtains "x = \<langle>\<rangle>" and "y = {#}"   | l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = \<langle>l, a, r\<rangle>" and "y = {#a#} + mset_tree l + mset_tree r"
C1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"
Fvars_LLq: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "Fvars (LLq t1 t2) = FvarsT t1 \<union> FvarsT t2"
exhaust: fixes y :: "opubx" obtains "y = OpAddNumUbx"
disjunctive_diff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> '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 "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "\<And>n. bit b n \<Longrightarrow> bit a n" shows "minus a b = and a (not b)"
sem_5bd4: fixes X :: "w \<Rightarrow> bool"   and Y :: "w \<Rightarrow> bool"   and Z :: "w \<Rightarrow> bool" shows "ob X Y \<and> X \<sqsubseteq> Z \<longrightarrow> ob Z ((\<sim>X) \<squnion> X \<sqinter> Y)"
mono_neutral_left: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'b)" shows "sum g S = sum g T"
fps_conv_radius_norm: fixes f :: "'a fps" shows "fps_conv_radius (Abs_fps (\<lambda>n. norm (fps_nth f n))) = fps_conv_radius f"
fbox_add2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x + y] z =  |x] z \<cdot>  |y] z"
simps_7: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "Formula.future_bounded (formula.Or \<phi> \<psi>) = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi>)"
AE_density: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "f \<in> borel_measurable M" shows "almost_everywhere (density M f) P = (AE x in M. 0 < f x \<longrightarrow> P x)"
trivial_subset: fixes Y :: "'c set"   and X :: "'c set" assumes "trivial Y"   and "X \<subseteq> Y" shows "trivial X"
P_def: shows "P \<equiv> distr M S X \<Otimes>\<^sub>M distr M T Y"
cMatch: assumes "P \<longmapsto> Rs"   and "a \<noteq> b" shows "F a b"
Case_def: fixes c :: "('b, 'c) node set \<Rightarrow> 'a"   and d :: "('b, 'c) node set \<Rightarrow> 'a"   and M :: "('b, 'c) node set" shows "Case c d M \<equiv> THE u. (\<exists>x. M = In0 x \<and> u = c x) \<or> (\<exists>y. M = In1 y \<and> u = d y)"
A: shows "((s, c), e, s', c') \<in> trcl (ntrp fg)"
butlast_split: fixes n :: "nat"   and q :: "'b list" assumes "n < length q" shows "butlast q = take n q @ drop n (butlast q)"
alg_dvd_diff: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x alg_dvd y"   and "x alg_dvd z" shows "x alg_dvd y - z"
basis_indep_card: fixes X :: "'a set"   and B :: "'a set" assumes "indep_in \<E> X"   and "\<E>.basis B" shows "card X \<le> card B"
R1: fixes p :: "nat"   and a :: "nat"   and x :: "nat"   and xs :: "pratt list" shows "valid_cert (Triple p a x # xs) = (1 < p \<and> 0 < x \<and> valid_cert xs \<and> (x = 1 \<or> (\<exists>q y. x = q * y \<and> Prime q \<in> set xs \<and> Triple p a y \<in> set xs \<and> [a ^ ((p - 1) div q) \<noteq> 1] (mod p))))"
tau_diverg: fixes x :: "nat"   and i :: "nat" assumes "\<not> determined i x" shows "\<tau> i x \<up>"
finsum_cong: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier R) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum R f A = finsum R g B"
S_Min_Lt: shows "S_Min_Lt = {u c' - d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c') (v c) = Lt d} \<union> {- d |d. M 0 (v c) = Lt d}"
d_refl: fixes C :: "'a set"   and Is :: "'a set"   and scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b" assumes "C \<subseteq> Is"   and "C \<noteq> {}" shows "decisive scf A Is C x x"
starset_subset: fixes A :: "'b set"   and B :: "'b set" shows "(*s* A \<subseteq> *s* B) = (A \<subseteq> B)"
igAbsIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
eval_ring_hom: fixes K :: "'a set"   and a :: "'a" assumes "subring K R"   and "a \<in> carrier R" shows "ring_hom_ring (K [X]) R (\<lambda>p. local.eval p a)"
eqSnd: shows "(\<chi>y. if vid1 = y then snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ vid1 else snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ y) = snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))"
linepath_trivial: fixes a :: "'c"   and x :: "real" shows "linepath a a x = a"
linear_Poly_list_of_vec': fixes R :: "('a mod_ring, 'b) ring_scheme" assumes "0 < degree u" shows "Poly \<circ> list_of_vec \<in> module_hom R local.V (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
simps: fixes t :: "'a tree"   and n :: "nat"   and dl :: "nat" shows "\<Phi>\<^sub>d (t, n, dl) = \<Phi> t + 4 * real dl / cd"
r_consist_phi: fixes e :: "nat"   and i :: "nat" assumes "\<forall>k<e_length e. \<phi> i k \<down>" shows "eval r_consist_phi [i, e] \<down>= (if \<forall>k<e_length e. \<phi> i k \<down>= e_nth e k then 0 else 1)"
mc: fixes A :: "'a set"   and \<gamma> :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set set"   and \<phi> :: "'a" shows "A \<div>\<^bsub>\<gamma>\<^esub> \<phi> \<equiv> \<Inter> (\<gamma> A \<phi>)"
h1: shows "(nat p - 1) div 2 = nat ((p - 1) div 2)"
init': fixes i :: "nat"   and w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list" assumes "i + 1 \<le> length w" shows "set (init' G w 0 (i, 1)) = CYK G w i 1"
mat_of_cols_carrier_2: fixes n :: "nat"   and vs :: "'c vec list" shows "dim_row (mat_of_cols n vs) = n"
absorb2: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "min a b = b"
finite_llast_deadend: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0"   and "lfinite P" shows "\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. llast P \<rightarrow>\<^bsub>G\<^esub> w)"
mem_subring_mem_ring: fixes S :: "('a, 'g) Ring_scheme"   and x :: "'a" assumes "Subring R S"   and "x \<in> carrier S" shows "x \<in> carrier R"
A_1: fixes d' :: "'b" assumes "u(c := d') \<turnstile>\<^bsub>v,n\<^esub> M" shows "thesis"
assms: fixes x :: "'a" assumes "x \<in> lset xs" shows "f x \<le> g x"
atLeastLessThan_less_eq3: fixes m :: "nat"   and n :: "nat" shows "(\<exists>f. inj_on f {0..<m} \<and> f ` {0..<m} \<subseteq> {0..<n}) = (m \<le> n)"
simps: fixes ant :: "form multiset"   and suc :: "form multiset" shows "NominalSequents.mset ( ant \<Rightarrow>* suc) = ant + suc"
run_one_step_basic_loop_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x61 :: "tf"   and x62 :: "b_e list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Loop x61 x62) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
NSLIMSEQ_cos_one: shows "(\<lambda>n. cos (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S 1"
45: fixes x :: "'a" shows "- bot \<squnion> - - x = - bot"
add: fixes u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal" assumes "P u"   and "P v" shows "P (\<lambda>x. v x + u x)"
ts_process_d: fixes T :: "'c list set" assumes "trace_set T" shows "deterministic (ts_process T)"
NEG_INF_def: shows "NEG_INF = - (2 ^ 31 - 1)"
strict_trans1: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" assumes "b \<subseteq># a"   and "c \<subset># b" shows "c \<subset># a"
iteratei_def: fixes al :: "('c, 'd) assoc_list"   and c :: "'b \<Rightarrow> bool"   and f :: "'c \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'b" shows "Assoc_List.iteratei al c f = foldli (assoc_list.impl_of al) c f"
exhaust: fixes y :: "hterm" obtains x1 :: "char list" and x2 :: "hterm list" where "y = HFun x1 x2"
assms: shows "lt q \<prec>\<^sub>t lt (p + q)"
IN_j_le_weight: fixes x :: "'v" shows "d_IN j \<langle>x\<rangle> \<le> weight \<Gamma> x"
freshAbs_vsubstAbs_E2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "freshAbs zs z (A $[y1 // y]_ys)" shows "freshAbs ys y A \<or> zs \<noteq> ys \<or> z \<noteq> y1"
Greatest_ex_le_nat: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>k. P k \<and> (\<forall>k'. P k' \<longrightarrow> k' \<le> k)" shows "\<not> (\<exists>n'>Greatest P. P n')"
component_is_connected: fixes e :: "'a"   and v :: "'a" shows "choose_component e v * choose_component e v\<^sup>T \<le> e"
poly_prime_eq_zero: fixes a :: "'a" shows "poly [a, 1::'a] \<noteq> poly []"
in_carrier2: shows "(\<^bold>g [^] x1) [^] c \<in> carrier \<G>"
good_item_simps_10: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "goodAbs (Abs xsa xa X)"
div: shows "pdivmod_monic (T * U) D = (Q, R)"
prems_5: shows "\<forall>n<length Ps - 1. deselect (Ps ! n) o\<^sub>0 (Ps ! Suc n)"
x: shows "x = Quad (a1, b1, c1, d1)"
w_def: shows "x1 < w \<and> w < x2 \<and> poly (pderiv (prod_list_var qs)) w = 0"
assms_1: shows "(f \<longlongrightarrow> l) at_top"
card_stablizer_sum: fixes orb :: "'c set" assumes "finite (carrier G)"   and "orb \<in> orbits G E \<phi>" shows "(\<Sum>x\<in>orb. card (stabilizer G \<phi> x)) = order G"
emeasure_space_1: fixes s :: "'a cfg" shows "emeasure (MC.T s) (space (MC.T s)) = 1"
CatExpCod: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme" assumes "f \<in> mor\<^bsub>CatExp A B\<^esub>" shows "cod\<^bsub>CatExp A B\<^esub> f = NTCod f"
comp_ipresIGAbs: fixes h :: "'l \<Rightarrow> 'm"   and MOD :: "('n, 'o, 'varSort, 'sort, 'p, 'q, 'l, 'r, 's) model_scheme"   and MOD' :: "('t, 'u, 'varSort, 'sort, 'v, 'q, 'm, 'w, 'x) model_scheme"   and hA :: "'r \<Rightarrow> 'w"   and h' :: "'m \<Rightarrow> 'y"   and hA' :: "'w \<Rightarrow> 'z"   and MOD'' :: "('aa, 'ab, 'varSort, 'ac, 'ad, 'q, 'y, 'z, 'ae) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGAbs h hA MOD MOD'"   and "ipresIGAbs h' hA' MOD' MOD''" shows "ipresIGAbs (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
I\<^sub>2_def: shows "I\<^sub>2 \<equiv> {(None, Some a)}\<^sup>="
assms_1: shows "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"
intros: fixes s :: "('g, 'h) locks \<times> (('h \<Rightarrow> ('x \<times> 'g \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'j wait_set_status option) \<times> 'h set"   and t :: "'h"   and ta :: "('g, 'h, 'k, 'i, 'j, 'l) thread_action" assumes "lock_ok_las' (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"   and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "cond_action_oks' s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"   and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>"   and "interrupt_actions_ok' (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>" shows "actions_ok' s t ta"
index_set_def: shows "index_set = (\<lambda>A. \<forall>n m. n \<in> A \<and> nat_to_ce_set n = nat_to_ce_set m \<longrightarrow> m \<in> A)"
bounds_consistent_leq_ub: fixes s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and c :: "'b" assumes "\<diamond> s"   and "\<B>\<^sub>l s x\<^sub>i = Some c" shows "c \<le>\<^sub>u\<^sub>b \<B>\<^sub>u s x\<^sub>i"
used_evs_rev: fixes evs :: "event list" shows "used evs = used (rev evs)"
chamber_nempty: fixes x :: "'a set" assumes "X \<noteq> {{}}"   and "chamber x" shows "x \<noteq> {}"
iIN_iTILL_subset_same: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<subseteq> [\<dots>n + d]"
g_1: fixes x :: "'a" assumes "x \<in> U" shows "g x = f x"
minit_safe_minit: fixes \<phi> :: "Formula.formula" assumes "mmonitorable \<phi>" shows "minit_safe \<phi> = minit \<phi>"
fold_cons_def: fixes g :: "('f, 'g) generator"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and s :: "'g" shows "fold_cons g f z s = fold f (unstream g s) z"
connected_uncountable: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "connected S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "uncountable S"
SUP_least: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. s2p A i \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (\<P> f A)) u"
exhaustive_above_fusion_code: fixes g :: "('a, 'd) generator"   and y :: "'a"   and s :: "'d" shows "exhaustive_above_fusion g y s = (if list.has_next g s then let (x, s') = list.next g s in \<not> proper_interval (Some y) (Some x) \<and> exhaustive_above_fusion g x s' else \<not> proper_interval (Some y) None)"