hcju/isabelleps huggingface dataset:
intro: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" assumes "ccw_system123 ccw S"   and "ccw_system1235_axioms ccw S" shows "ccw_system1235 ccw S"
equiv_subst_right: fixes cmp :: "'b comparator"   and z :: "'b"   and y :: "'b"   and x :: "'b"   and comp :: "comp" assumes "compare cmp z y = Equiv" shows "(compare cmp x z = comp) = (compare cmp x y = comp)"
n: shows "n \<in> {idx i..<idx (Suc i)}"
bernoulli'_fps_aux: shows "(fps_exp (1::'a) - 1) * Abs_fps (\<lambda>n. of_real (bernoulli' n) / fact n) = fps_exp (1::'a) * fps_X"
\<theta>_eq_0: fixes x :: "real" assumes "x < 2" shows "\<theta> x = 0"
cone_\<chi>: shows "D.mkCone p0 p1 \<in> D.cones (C.dom p0)"
vars_of_cnf_def: fixes cnf :: "lit uprod set" shows "vars_of_cnf cnf \<equiv> \<Union>cl\<in>cnf. var ` set_uprod cl"
C: shows "n \<notin> bad_shakey \<inter> (bad_id_password \<union> bad_id_pubkey)"
godel_code'_butlast_last_id': fixes ys :: "nat list"   and y :: "nat"   and j :: "nat" shows "godel_code' (ys @ [y]) (Suc j) = godel_code' ys (Suc j) * Pi (Suc (length ys + j)) ^ y"
intros_1: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
state_not_singleton_def: shows "state_not_singleton = (\<exists>s t. s \<noteq> t)"
sum_integral_diff_series_antimono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "sum_integral_diff_series n \<le> sum_integral_diff_series m"
dist_pos_lt: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open"   and "x \<noteq> y" shows "0 < dist x y"
elims: fixes x :: "trm option"   and xa :: "trm option"   and y :: "trm option" assumes "Pluso x xa = y" obtains \<theta> :: "trm" and \<eta> :: "trm" where "x = Aterm \<theta>" and "xa = Aterm \<eta>__" and "y = Aterm (Plus \<theta> \<eta>__)"   | \<eta> :: "trm option" where "x = undeft" and "xa = \<eta>" and "y = undeft"   | v :: "trm" where "x = Aterm v" and "xa = undeft" and "y = undeft"
root_cancel2: fixes n :: "nat" assumes "0 < n" shows "FFT.root (2 * n) ^ n = - 1"
ask_option_def: fixes f :: "'r \<Rightarrow> ('a, 'm) optionT" shows "local.ask_option f = OptionT (ask (\<lambda>r. run_option (f r)))"
simps_5: fixes bs :: "int list"   and a :: "Cooper.num"   and b :: "Cooper.num" shows "Inum bs (Add a b) = Inum bs a + Inum bs b"
Rep_resumption_Branch: fixes c :: "'f"   and rs :: "'i \<Rightarrow> ('h, 'g, 'f, 'i) resumption" shows "Rep_resumption (Branch c rs) = BRANCH c (\<lambda>d. Rep_resumption (rs d))"
are_operator_effects_consistent_def: fixes op :: "('a, 'b) sas_plus_operator"   and op' :: "('a, 'b) sas_plus_operator" shows "are_operator_effects_consistent op op' \<equiv> let effect = effect_of op; effect' = effect_of op' in list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') effect') effect"
icount_correct: fixes M :: "nat" assumes "2 < M" shows "icount M \<le> SPEC (\<lambda>r. r = lcount M)"
count_partitions_iff: fixes N :: "nat multiset"   and n :: "nat" shows "(count N partitions n) = number_partition n N"
take_Suc_Cons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "take (Suc n) (x # xs) = x # take n xs"
integer_of_uint8_def: shows "integer_of_uint8 = integer_of_int \<circ> int_of_uint8"
fact_ge_power: fixes c :: "nat"   and n0 :: "nat"   and n :: "nat" assumes "c ^ n0 \<le> fact n0"   and "c \<le> n0 + 1"   and "n0 \<le> n" shows "c ^ n \<le> fact n"
Log_def: shows "Log = nat \<lceil>log (real_of_rat (4 * \<alpha> / (4 + \<alpha>))) (real N)\<rceil>"
match_def: fixes t :: "('e, 'd) Term.term"   and l :: "('e, 'd) Term.term" shows "match t l = match_list Var [(l, t)]"
runProc_def: fixes name :: "String.literal"   and args :: "expr list"   and prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState" shows "runProc name args prog g p = (if max_procs \<le> length (procs g) then abort STR ''Too many processes'' (\<lambda>_. (g, p)) else let pid = length (procs g) + 1 in case lm.lookup name (proc_data prog) of None \<Rightarrow> abortv STR ''No such process: '' name (\<lambda>_. (g, p)) | Some proc_idx \<Rightarrow> let (g', proc) = mkProc g p name args (processes prog !! proc_idx) pid in (g'\<lparr>procs := procs g @ [proc]\<rparr>, p))"
assms_2: shows "negchecks_model \<I> D X F F'"
assms_2: assumes "A \<subseteq>\<^sub>\<circ> B" shows "P"
mirror_\<alpha>_\<beta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and i :: "int" assumes "iszlfm p (a # bs)" shows "Inum (real_of_int i # bs) ` set (\<alpha> p) = Inum (real_of_int i # bs) ` set (\<beta> (mirror p))"
derivative_component_fun_component_at_within: fixes \<gamma> :: "real \<Rightarrow> 'a"   and i :: "'a" assumes "\<forall>x\<in>{0..1}. \<gamma> differentiable at x" shows "\<forall>x\<in>{0..1}. vector_derivative \<gamma> (at x within {0..1}) \<bullet> i = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
poly_cpoly_of_real: fixes pR :: "real poly"   and pI :: "real poly"   and x :: "real" shows "poly (cpoly_of pR pI) (complex_of_real x) = Complex (poly pR x) (poly pI x)"
inject_5: fixes x5 :: "'c val"   and y5 :: "'c val" shows "(Val x5 = Val y5) = (x5 = y5)"
phi_def: fixes g :: "'g"   and v :: "'val" shows "uninst_code.ssa.phi phis g v = Mapping.lookup (phis g) (uninst_code.ssa.defNode phis g v, v)"
assms: fixes l :: "'a"   and l' :: "'a" assumes "l \<in> labels \<Gamma> p"   and "l' \<in> labels \<Gamma> p'" shows "P ((\<xi>, l), a, \<xi>', l')"
takeWhile_map: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
cong_3421: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong C D B A"
repeatable_ccTTree_ccSquare: fixes S :: "Vars.var set"   and S' :: "Vars.var set" assumes "S \<subseteq> S'" shows "repeatable (ccTTree S (S'\<^sup>2))"
ipurge_tr_rev_t_def: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_rev_t I D u xs \<equiv> ipurge_tr_rev_t_out (ipurge_tr_rev_t_aux (ipurge_tr_rev_t_in I D u xs))"
abs_eq: fixes xa :: "'a \<Rightarrow> 'b option"   and x :: "'b set" shows "Abs_pfun xa \<rhd>\<^sub>p x = Abs_pfun (xa\<upharpoonleft>\<^bsub>x\<^esub>)"
permutes_inverses_1: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" assumes "p permutes S" shows "p (inv p x) = x"
ord_option_simps_4: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c" shows "ord_option ord (Some x) None = False"
sr_1: shows "(s \<circ> r) n \<in> \<omega>_limit_set x"
g_supp_le_V: fixes i :: "nat" assumes "i \<in> N" shows "support_on carrier (g i) \<subseteq> V i"
pos_add_strict: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less b c" shows "less b (plus a c)"
poly_deg_times_le: fixes p :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd" shows "poly_deg (p * q) \<le> poly_deg p + poly_deg q"
bullet_app_redR: fixes u :: "CL"   and u' :: "CL"   and t :: "CL" assumes "(u, u') \<in> red" shows "(bullet_app t u, bullet_app t u') \<in> red\<^sup>*"
assms_1: shows "finite {a. f a \<noteq> (0::'b)}"
prv_cnj_com_imp3: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla" shows "prv (imp (cnj \<phi>1 (cnj \<phi>2 \<phi>3)) (cnj \<phi>2 (cnj \<phi>1 \<phi>3)))"
F: shows "(h, as) \<Turnstile> P1 a ap * P2 b bp * F \<and>\<^sub>A P1 a' ap * P2 b' bp * F'"
legs_form_cospan_1: shows "Maps.cospan SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "C.ide (F a)"
in_idealE_Polys_finite: fixes B :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite B"   and "B \<subseteq> P[X]"   and "p \<in> P[X]"   and "p \<in> ideal B" obtains q :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "\<And>b. q b \<in> P[X]" and "p = (\<Sum>b\<in>B. q b * b)"
partition_set: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and yes :: "'b list"   and no :: "'b list" assumes "List.partition P xs = (yes, no)" shows "set yes \<union> set no = set xs"
comp_left_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times bot x = bot"
eq_Apps_conv_10: fixes f :: "expr"   and ts :: "expr list"   and T :: "Unary_PCF.type"   and t :: "expr" shows "(f \<bullet> ts = \<Lambda>\<langle>T\<rangle> t) = (\<Lambda>\<langle>T\<rangle> t = f \<and> ts = [])"
visible_method_exists: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>(b, Ts, T, m)\<rfloor>"
powr_minus_divide: fixes x :: "'c"   and a :: "'c" shows "x powr - a = (1::'c) / x powr a"
assms_2: shows "pd_kleene_allegory_class.acyclic f"
intros_2: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
AssertAllState''Induct: fixes t :: "(nat \<times> linear_poly) list"   and P :: "('i \<times> 'a atom) set \<Rightarrow> ('i, 'a) state \<Rightarrow> bool"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t"   and "P {} (init t)"   and "\<And>as bs t. \<lbrakk>as \<subseteq> bs; P as t\<rbrakk> \<Longrightarrow> P bs t"   and "\<And>s a ats. \<lbrakk>\<not> \<U> s; \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s; \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<triangle> (\<T> s); \<nabla> s; \<diamond> s; P (set ats) s; index_valid (set ats) s\<rbrakk> \<Longrightarrow> P (insert a (set ats)) (assert_bound a s)" shows "P (set ats) (assert_bound_loop ats (init t))"
coclop_coclosure_set: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "range f = Fix f"
IH: fixes f :: "'c \<Rightarrow> 'd" assumes "higher_differentiable_on S f n"   and "(0::'d) \<notin> f ` S"   and "open S" shows "higher_differentiable_on S (\<lambda>x. norm (f x)) n"
ik_append: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t (S @ S') = ik\<^sub>s\<^sub>t S \<union> ik\<^sub>s\<^sub>t S'"
norm_signed_result: fixes w :: "bit list" shows "norm_signed w = [] \<or> norm_signed w = [\<one>] \<or> bv_msb (norm_signed w) \<noteq> bv_msb (tl (norm_signed w))"
integral_restrict_UNIV: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue" shows "LINT x|lebesgue. (if x \<in> S then f x else (0::'d)) = integral\<^sup>L (lebesgue_on S) f"
rep_eq: fixes x :: "'d ccsubspace"   and xa :: "'d ccsubspace" shows "space_as_set (x \<sqinter> xa) = space_as_set x \<inter> space_as_set xa"
converse_reachable_cases: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" obtains "u = v" and "u \<in> verts G"   | w :: "'c" where "u \<rightarrow>\<^bsub>G\<^esub> w" and "w \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
finite_measure_Union': fixes A :: "real set"   and B :: "real set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M (B - A)"
inv_eq_1_iff: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "(inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>) = (x = \<one>\<^bsub>G\<^esub>)"
simps: fixes a1 :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and a2 :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "Eq_RESUMPTION a1 a2 = ((\<exists>a. a1 = TERMINAL a \<and> a2 = TERMINAL a) \<or> (\<exists>r r' b. a1 = LINEAR b r \<and> a2 = LINEAR b r' \<and> Eq_RESUMPTION r r') \<or> (\<exists>rs rs' c. a1 = BRANCH c rs \<and> a2 = BRANCH c rs' \<and> (\<forall>x. Eq_RESUMPTION (rs x) (rs' x))))"
simps: fixes I :: "'b set"   and v :: "nat \<Rightarrow> real"   and cs :: "('b \<times> constraint) set" shows "(I, v) \<Turnstile>\<^sub>r\<^sub>i\<^sub>c\<^sub>s cs = v \<Turnstile>\<^sub>r\<^sub>c\<^sub>s (snd ` (cs \<inter> I \<times> UNIV))"
ce_finite: fixes A :: "nat set" assumes "finite A" shows "A \<in> ce_sets"
emeasure_Un: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
pfsubset_finsert_iff: fixes A :: "'c fset"   and x :: "'c"   and B :: "'c fset" shows "(A |\<subset>| finsert x B) = (if x |\<in>| B then A |\<subset>| B else if x |\<in>| A then A |-| {|x|} |\<subset>| B else A |\<subseteq>| B)"
Int_Un_eq_3: fixes S :: "'c set"   and T :: "'c set" shows "S \<union> S \<inter> T = S"
norm_le_zero_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x \<le> 0) = (x = zero)"
pivot_tableau_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_eq :: "nat \<times> linear_poly \<Rightarrow> nat \<Rightarrow> nat \<times> linear_poly"   and subst_var :: "nat \<Rightarrow> linear_poly \<Rightarrow> linear_poly \<Rightarrow> linear_poly"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "Pivot' TYPE('a) eq_idx_for_lvar pivot_eq subst_var" shows "Pivot'.pivot_tableau eq_idx_for_lvar pivot_eq subst_var x\<^sub>i x\<^sub>j t \<equiv> let eqa = EqForLVar.eq_for_lvar eq_idx_for_lvar t x\<^sub>i; eq' = pivot_eq eqa x\<^sub>j in map (\<lambda>e. if lhs e = lhs eqa then eq' else SubstVar.subst_var_eq subst_var x\<^sub>j (rhs eq') e) t"
IH: assumes "\<forall>a. ListMem a as \<longrightarrow> agree (fst a) s \<and> agree (snd a) s"   and "as \<in> valid_plans PROB" shows "as \<in> valid_plans (snapshot PROB s)"
filter_syzygy_basis_alt: fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "filter_syzygy_basis n bs = filter (\<lambda>b. proj_poly_syz n b = 0) bs"
product_run_embed_limit_finiteness: fixes \<iota>\<^sub>m :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and q\<^sub>0 :: "'b"   and \<delta>\<^sub>m :: "'c \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> 'b"   and w :: "nat \<Rightarrow> 'd"   and S :: "('b \<times> 'd \<times> 'b) set" assumes "\<iota>\<^sub>m k = Some q\<^sub>0"   and "finite (range (run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w))" shows "(limit (run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w) \<inter> \<Union> (\<upharpoonleft>\<^sub>k ` S) = {}) = (limit (run\<^sub>t (\<delta>\<^sub>m k) q\<^sub>0 w) \<inter> S = {})"
contra: fixes d :: "nat" assumes "xs \<noteq> []"   and "list_all (\<lambda>x. x \<noteq> 0) xs"   and "list_all ((=) 0) (take (Suc d) xs)" shows "False"
assms_4: shows "(y + x * x') mod q = (y + x * y') mod q"
VcompNml_Cod_Nml: fixes t :: "'c local.term" assumes "Nml t" shows "Cod t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
ts_ipurge_unwinding: fixes T :: "'a list set"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" assumes "trace_set T" shows "secure (ts_process T) I D = d_weakly_future_consistent (ts_process T) I D (rel_ipurge (ts_process T) I D)"
dist_minus: fixes x :: "'b"   and y :: "'b" shows "dist (- x) (- y) = dist x y"
tm_dghm_ArrMap_in_Vset: shows "\<GG>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
reflp_eq: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "reflp R = ((=) \<le> R)"
isomorphism_expand: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" shows "(f \<circ> g = id \<and> g \<circ> f = id) = ((\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x))"
geodesic_segment_param_in_segment: fixes G :: "'b set"   and x :: "'b"   and t :: "real" assumes "G \<noteq> {}" shows "geodesic_segment_param G x t \<in> G"
lm037: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P outside X \<subseteq> P || (Domain P - X)"
mod_mult_self2_is_0: fixes a :: "'a"   and b :: "'a" shows "a * b mod b = (0::'a)"
absolutely_integrable_on_subcbox: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a" assumes "f absolutely_integrable_on S"   and "cbox a b \<subseteq> S" shows "f absolutely_integrable_on cbox a b"
induct: fixes \<Gamma> :: "'b formula set"   and x :: "'b formula"   and P :: "'b formula \<Rightarrow> bool" assumes "\<Gamma> \<turnstile>\<^sub>H x"   and "\<And>F. F \<in> \<Gamma> \<Longrightarrow> P F"   and "\<And>F G. \<lbrakk>\<Gamma> \<turnstile>\<^sub>H F; P F; \<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G; P (F \<^bold>\<rightarrow> G)\<rbrakk> \<Longrightarrow> P G" shows "P x"
simple_syntactic_and_axioms: shows "simple_syntactic_and pre_strong_term_class.wellformed"
tendsto_ine\<^sub>N_iff_limitin: fixes N :: "'b quasinorm"   and u :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "defect N = 1" shows "tendsto_ine\<^sub>N N u x = limitin (topology\<^sub>N N) u x sequentially"
comp_ipresIGSwap: fixes h :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'w, 'ae, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ab, 'ai, 'ad, 'x, 'aj, 'ak) model_scheme"   and h' :: "'x \<Rightarrow> 'al"   and MOD'' :: "('am, 'an, 'aa, 'ao, 'ap, 'ad, 'al, 'aq, 'ar) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGSwap h MOD MOD'"   and "ipresIGSwap h' MOD' MOD''" shows "ipresIGSwap (h' \<circ> h) MOD MOD''"
Fubini_integrable: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)"   and "integrable lborel (\<lambda>x. LBINT y. norm (f (x, y)))"   and "AE x in lborel. integrable lborel (\<lambda>y. f (x, y))" shows "integrable (lborel \<Otimes>\<^sub>M lborel) f"
LstSeq_def: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "LstSeq s k y \<equiv> Seq s (succ k) \<and> Ord k \<and> \<langle>k, y\<rangle> \<^bold>\<in> s \<and> OrdDom s"
dgrad_p_setI: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and m :: "nat" assumes "\<And>v. v \<in> keys p \<Longrightarrow> d (local.punit.pp_of_term v) \<le> m" shows "p \<in> punit.dgrad_p_set d m"
exu_rename: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y \<notin> Fvars \<phi>" shows "exu x \<phi> = exu y (subst \<phi> (Var y) x)"
lim_sequence: shows "lim_sequence \<in> M \<rightarrow>\<^sub>M prob_algebra (Pi\<^sub>M UNIV (\<lambda>i. M))"
simps_3: fixes mem :: "Eg1.var \<Rightarrow> nat"   and x :: "Eg1.var"   and c :: "nat" shows "ev\<^sub>B mem (bexp.Eq x c) = (mem x = c)"
distinct_1: fixes x1 :: "'a"   and x21 :: "'a bdd"   and x22 :: "'a bdd" shows "Leaf x1 \<noteq> Branch x21 x22"
trivial_limit_eventually: fixes net :: "'c filter"   and P :: "'c \<Rightarrow> bool" assumes "net = bot" shows "eventually P net"
fps_ln_fps_exp_inv: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "fps_ln a = fps_inv (fps_exp a - 1)"
index_mat_swaprows_2: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_row (swaprows k l A) = dim_row A"
ListSum_conv_sum: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "distinct xs" shows "ListSum xs f = sum f (set xs)"
Exp_Gen_inj: fixes X :: "msg"   and Y :: "msg" assumes "Exp Gen X = Exp Gen Y" shows "X = Y"
Min_eqI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq x y"   and "x \<in> A" shows "linorder.Min less_eq A = x"
borel_measurable_if_D: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "(\<lambda>x. if x \<in> S then f x else (0::'d)) \<in> borel_measurable lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)"
M_consts_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> M\<^sub>n false\<^sub>n"
fr: shows "finite {v. r v \<noteq> (0::'a)}"
FM_P_module: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "R module P"
ground_ord_resolve_imp_ord_resolve: fixes DA :: "'a literal multiset"   and CAs :: "'a literal multiset list"   and S_G :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset" assumes "is_ground_cls DA"   and "is_ground_cls_list CAs"   and "ground_resolution_with_selection S_G"   and "ground_resolution_with_selection.ord_resolve S_G CAs DA AAs As E" shows "\<exists>\<sigma>. ord_resolve S_G CAs DA AAs As \<sigma> E"
inf_chain_def: fixes f :: "nat \<Rightarrow> 'a" shows "local.inf_chain f = (\<forall>i. p (f i) (f (Suc i)))"
division_segment_1: 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 division_segment :: "'c \<Rightarrow> 'c" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment one = one"
go_is_done_impl_def: fixes v :: "'q"   and s :: "('q set \<times> 'q set) option \<times> ('q \<Rightarrow> node_state option)" shows "go_is_done_impl v s \<equiv> is_done_oimpl v (snd s)"
pickAnyE_def: shows "pickAnyE \<equiv> pickE [] []"
edge_to_Snoc: fixes c :: "'form"   and "is" :: "nat list"   and i :: "nat" shows "edge_to c (is @ [i]) = ((c, 0 # is), in_port_at (c, 0 # is) i)"
assms_5: shows "\<forall>n. AE w in M. real_cond_exp M (F n) (X (Suc n)) w = X n w"
length_sort: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "length (sort_key f xs) = length xs"
sum_integral_diff_series_def: fixes n :: "nat" shows "sum_integral_diff_series n = (\<Sum>k\<le>n. f (real k)) - integral {0..real n} f"
add_ac_1: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a + b + c = a + (b + c)"
relcompp_apply: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and c :: "'e" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"