hcju/isabelleps huggingface dataset:
bezout_coefficients_fst_snd: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "plus (times (fst (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) a) (times (snd (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) b) = gcd a b"
simps_18: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> SKIP {vd}) = False"
assms_4: fixes V :: "'a set" assumes "subalgebra K V R"   and "set Us \<subseteq> V" shows "E \<subseteq> V"
hyps: fixes thesis :: "bool" assumes "\<And>ys zs. xs = ys @ x # zs \<Longrightarrow> thesis"   and "x \<in> set xs" shows "thesis"
0: shows "{\<lambda>x. 0::'a} equiintegrable_on {a..b}"
f_simps_5: fixes n :: "nat" shows "f n (SOURCE, vertex.SINK) = 0"
abs_eq: fixes xb :: "nat"   and xa :: "nat"   and x :: "nat \<times> nat \<Rightarrow> 'a" shows "Matrix.mat xb xa x = Abs_mat (xb, xa, mk_mat xb xa x)"
ntrp_add_context: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (ntrp fg)"   and "mon_ww fg (map le_rem_s w) \<inter> mon_c fg cn = {}" shows "((s, c + cn), w, s', c' + cn) \<in> trcl (ntrp fg)"
poly_mult_const_1: fixes K :: "'a list set"   and p :: "'a list list"   and a :: "'a list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p"   and "a \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}" shows "UP.poly_mult [a] p = map ((\<otimes>\<^bsub>K [X]\<^esub>) a) p"
Gamma_integral_complex': fixes z :: "complex" assumes "0 < Re z" shows "((\<lambda>t. complex_of_real t powr (z - 1) / complex_of_real (exp t)) has_integral Gamma z) {0<..}"
scale_zero_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "module scale" shows "scale (0::'b) x = (0::'c)"
0: shows "Max (sq_norm ` set fs_init) \<in> sq_norm ` set fs_init"
sconst_cycle: fixes x :: "'d" shows "pure x = cycle [x]"
boundedI: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> b \<sqinter> c"
strategyproofness: shows "(pmf (sds R23) c < pmf (sds R18) d \<or> pmf (sds R23) c + pmf (sds R23) d < pmf (sds R18) d + pmf (sds R18) c) \<or> pmf (sds R23) c = pmf (sds R18) d \<and> pmf (sds R23) c + pmf (sds R23) d = pmf (sds R18) d + pmf (sds R18) c"
norm_scaleR: fixes a :: "real"   and x :: "'b" shows "\<parallel>a *\<^sub>R x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
Lcm_atLeast1AtMost_conv_\<psi>: fixes n :: "nat" shows "real (Lcm {1..n}) = exp (primes_psi (real n))"
mono: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c llist" shows "mono_llist (\<lambda>lfilter. case x of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then LCons x (lfilter xs') else lfilter xs')"
prems_3: shows "gallery (C # (E # Es) @ [D])"
curry_conv: fixes f :: "'e \<times> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "curry f a b = f (a, b)"
fold_graph_fold: fixes B :: "'a set"   and z :: "'a" assumes "finite B"   and "B \<subseteq> A"   and "z \<in> A" shows "fold_graph f z B (Finite_Set.fold f z B)"
Set_def: fixes t :: "('b, unit) RBT.rbt" shows "Set t = {x. RBT.lookup t x = Some ()}"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x2 :: "'b" shows "(case Abrupt x2 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f2 x2"
vertical_chop_leq1: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<longrightarrow> u \<le> v"
locally_essentially_surjective: fixes a :: "'c"   and b :: "'c"   and g :: "'d" assumes "C.obj a"   and "C.obj b"   and "\<guillemotleft>g : F.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 b\<guillemotright>"   and "D.ide g" shows "\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) g"
r_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = a \<otimes> x) = (a = \<one>)"
not_AE_zero_ennreal_E: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "\<not> (AE x in M. f x = 0)"   and "f \<in> borel_measurable M" shows "\<exists>A\<in>sets M. \<exists>e>0. 0 < emeasure M A \<and> (\<forall>x\<in>A. ennreal e \<le> f x)"
del_type_1: fixes t :: "'b bro"   and h :: "nat"   and x :: "'b" assumes "t \<in> B h" shows "del x t \<in> T h"
Eps_cong: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Eps P = Eps Q"
INJECT: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I" shows "G \<rhd> \<langle> A \<rangle> C,m,l \<langle> B \<rangle> I"
ivl_disj_int_one_4: fixes l :: "'b"   and u :: "'b" shows "{..<l} \<inter> {l..u} = {}"
1: shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
Times_PlusL: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times (Plus r s) t = Plus (Times r t) (Times s t)"
imp_cnj_compat_eqv: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (eqv \<phi>1 \<phi>2) (imp (eqv \<chi>1 \<chi>2) (eqv (cnj \<phi>1 \<chi>1) (cnj \<phi>2 \<chi>2))))"
blinfun_bij1: fixes f :: "'a \<Rightarrow>\<^sub>L 'a"   and g :: "'a \<Rightarrow>\<^sub>L 'a" assumes "f o\<^sub>L g = id_blinfun" shows "bij (blinfun_apply f)"
valid_player0_set: shows "V0\<^bsub>G'\<^esub> \<subseteq> V\<^bsub>G'\<^esub>"
distinct_3: fixes x1 :: "'a"   and x31 :: "nat"   and x32 :: "'a" shows "broadcast x1 \<noteq> unicast x31 x32"
is_zero_def: fixes a :: "('c, 'd) IEEE.float" shows "is_zero a = (IEEE.exponent a = 0 \<and> fraction a = 0)"
leg0_composite_2: shows "HH\<mu>\<nu>H\<pi>\<rho>.dom.leg0 = \<pi>\<rho>.leg0 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0"
invh_paint: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and c :: "tcolor" assumes "invh t" shows "invh (paint c t)"
StrongEq_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "X \<triangleq> Y \<equiv> \<lambda>\<tau>. \<lfloor>\<lfloor>X \<tau> = Y \<tau>\<rfloor>\<rfloor>"
val_equiv_eq_pdiv: fixes P :: "('a \<Rightarrow> ant) set"   and P' :: "('a \<Rightarrow> ant) set"   and v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "P \<in> Pds"   and "P' \<in> Pds"   and "valuation K v"   and "valuation K v'"   and "v_equiv K v v'"   and "v \<in> P"   and "v' \<in> P'" shows "P = P'"
eq_Nil_null: fixes xs :: "'g list" shows "(xs = []) = List.null xs"
while_option_def: fixes b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "while_option b c s = (if \<exists>k. \<not> b ((c ^^ k) s) then Some ((c ^^ (LEAST k. \<not> b ((c ^^ k) s))) s) else None)"
card_Suc_Diff1: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "Suc (card (A - {x})) = card A"
assms: fixes x :: "'a" assumes "x \<in> keys t" shows "poly_deg (f x) \<le> 1"
inject_6: fixes x7 :: "'b ptrm"   and y7 :: "'b ptrm" shows "(PSnd x7 = PSnd y7) = (x7 = y7)"
list_all_iff: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b list" shows "list_all P x = Ball (set x) P"
i_join_i_expand_iMODb: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "f \<odot> k \<Join>\<^sub> [ n * k, mod k, d ] = f \<Join>\<^sub> [n\<dots>,d]"
emeasure_Diff: fixes M :: "'c measure"   and B :: "'c set"   and A :: "'c set" assumes "emeasure M B \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "emeasure M (A - B) = emeasure M A - emeasure M B"
s_sub_all_clauses: fixes S :: "'b literal set set" shows "S \<subseteq> all_clauses S"
\<R>\<^sub>1_weak_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "weak_bisim (\<R>\<^sub>1 \<Gamma>') (\<R> \<Gamma>')"
assm: shows "R_Chop(r,r1,r2) \<and> R_Chop(r2,r3,r4)"
\<alpha>\<beta>_refl: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" shows "s \<approx> s"
case_prod_Pair_iden: fixes p :: "'a \<times> 'c" shows "(case p of (x, y) \<Rightarrow> (x, y)) = p"
inv_safe_safe: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in"   and "\<And>loc. c_work c loc = {#}\<^sub>z"   and "dataflow_topology.inv_safe summary results_in c" shows "dataflow_topology.safe summary results_in c"
split_fst: fixes R :: "'a \<Rightarrow> bool"   and p :: "'a \<times> 'b" shows "R (fst p) = (\<forall>x y. p = (x, y) \<longrightarrow> R x)"
2: shows "z (subsets_smash n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j) = z (subs1 ! (i div length subs2) @ map ((+) n1) (subs2 ! (i mod length subs2))) (signs1 ! (j div length signs2) @ signs2 ! (j mod length signs2))"
RA: fixes \<phi> :: "\<o>" assumes "[\<phi> in dw]" shows "[\<^bold>\<A>\<phi> in dw]"
deg_sect_def: fixes X :: "'b set"   and d :: "nat" shows "deg_sect X d = .[X] \<inter> {t. deg_pm t = d}"
space_def: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "space ts v c \<equiv> Abs_real_int (pos ts c, pos ts c + sensors (own v) ts c)"
pf': fixes k :: "nat" assumes "k < n" shows "positive (f' k)"
cring_simprules_4: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus>\<^bsub>R\<^esub> y \<in> carrier R"
is_acc_run_def: fixes r :: "nat \<Rightarrow> 'v" shows "G'.is_acc_run r \<equiv> G'.is_run r \<and> G'.is_acc r"
fls_base_factor_shift: fixes n :: "int"   and f :: "'a fls" shows "fls_base_factor (fls_shift n f) = fls_base_factor f"
valid_combine_if_valid: fixes a :: "'a tree"   and z :: "'a tree" assumes "0 < l a"   and "local.valid a"   and "local.valid z"   and "a \<noteq> Leaf"   and "z \<noteq> Leaf"   and "l z = Suc (r a)" shows "local.valid (combine a z)"
mod_diff_left_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 uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus (modulo a c) b) c = modulo (minus a b) c"
mdpt_val: shows "a * (- b / (2 * a))\<^sup>2 + b * (- b / (2 * a)) + c < 0"
smallo_iff: shows "((\<lambda>_. 1) \<in> o(\<lambda>x. \<Prod>g\<leftarrow>group_primfuns fs. (eval_primfun' \<circ> fst) g x powr snd g)) = pos_list (map snd (group_primfuns fs))"
int: shows "dom (map_of (map (\<lambda>(F, b, T). ((F, C), b, T)) fs)) \<inter> dom (map_of FDTs') = {}"
represents_nth_root_pos_real: fixes p :: "int poly"   and x :: "real" assumes "p represents x"   and "0 < x" shows "local.poly_nth_root p represents root n x"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
ff: shows "f \<langle>encF f\<rangle> = \<langle>instInpP \<psi> 0 (\<tau>J \<psi>)\<rangle>"
indicator_eq_0_iff: fixes A :: "'c set"   and x :: "'c" shows "(indicator A x = (0::'b)) = (x \<notin> A)"
fmrel_on_fset_fmdom: fixes ym :: "'d \<rightharpoonup>\<^sub>f 'f"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e"   and k :: "'d" assumes "subtuple f xm ym"   and "k |\<in>| fmdom ym" shows "k |\<in>| fmdom xm"
wlsInp_castOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wlsInp (fst3 (castOp s X)) (snd3 (castOp s X))"
prod_dvd_prod_subset2: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.comm_semiring_1 times one plus zero"   and "finite B"   and "A \<subseteq> B"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times (f a) (g a)" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g B)"
removing_winning_region_induces_no_deadends: fixes v :: "'a"   and p :: "Player" assumes "v \<in> V\<^bsub>G'\<^esub> - G'.winning_region p"   and "\<not> G'.deadend v" shows "\<exists>w\<in>V\<^bsub>G'\<^esub> - G'.winning_region p. v \<rightarrow>\<^bsub>G'\<^esub> w"
similar_mat_witD_7: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "Q \<in> carrier_mat n n"
not_less_iff_gr_or_eq: fixes x :: "'a"   and y :: "'a" shows "(\<not> lt x y) = (lt y x \<or> x = y)"
g2_scale_i_contin: shows "continuous_on (cbox a b) (\<lambda>x. (g2 x, 0::'a))"
adv: assumes "exception_\<I> (\<I>_real1 \<oplus>\<^sub>\<I> \<I>_common1) \<turnstile>g absorb \<A> (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2)))) \<surd>" shows "advantage (absorb \<A> (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) (obsf_resource (sim1 |\<^sub>= 1\<^sub>C \<rhd> ideal1)) (obsf_resource real1) \<le> adv1"
poly_rel_div: shows "(poly_rel ===> poly_rel ===> poly_rel) (div_field_poly_i ops) (div)"
continuous_on_iff: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
bilateral_substitutesD: fixes f :: "'a set \<Rightarrow> 'a set"   and a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "bilateral_substitutes_on UNIV f"   and "Xd a \<notin> Xd ` B"   and "Xd b \<notin> Xd ` B"   and "b \<notin> f (insert b B)" shows "b \<notin> f (insert a (insert b B))"
nlea__lta: 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"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "Tarski_neutral_dimensionless.LtA Bet Cong D E F A B C"
d: shows "Geq x d \<in> snd ` insert ia as"
deg_v_pj: shows "degree (v mod p_j) = 0"
Smith_2x2: shows "\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2 A)"
partial_summation_strong: shows "((\<lambda>t. A t * f' t) has_integral A x * f x - A y * f y - (\<Sum>n\<in>real -` {y<..x}. a n * f (real n))) {y..x}"
assms: shows "- pi < \<alpha> \<and> \<alpha> \<le> pi"
PO_s0g_dom: shows "reach s0g \<subseteq> s0g_dom"
lm082: fixes X :: "'c set"   and Y :: "'c set"   and P :: "('c \<times> 'd) set" assumes "X \<inter> Y = {}" shows "Outside' Y P `` X = P `` X"
take_Suc: fixes xs :: "'c list"   and n :: "nat" assumes "xs \<noteq> []" shows "take (Suc n) xs = hd xs # take n (tl xs)"
strict_monoidal_extension_to_free_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd"   and F :: "'b local.term set \<Rightarrow> 'd" shows "strict_monoidal_extension_to_free_monoidal_category C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V F \<equiv> (category C \<and> monoidal_language C) \<and> free_monoidal_category C \<and> strict_monoidal_extension C (free_monoidal_category.comp C) (elementary_monoidal_category.T (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C)) (elementary_monoidal_category.\<alpha> (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C)) (elementary_monoidal_category.\<iota> (free_monoidal_category.unity\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C)) D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D (free_monoidal_category.inclusion_of_generators C) V F"
OAlist_tc_filter_def: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_filter P xs = OAlist_tc (filter P (list_of_oalist_tc xs))"
mpt_axioms_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "mpt_axioms M T \<equiv> T \<in> measure_preserving M M"
maximal_def: fixes f :: "'c \<Rightarrow> 'a"   and S :: "'c set" shows "maximal f S = {x \<in> S. \<forall>y\<in>S. f y \<le> f x}"
fin_cut_same_singleton: fixes x :: "'b"   and xs :: "'b list" shows "fin_cut_same x (xs @ [x]) = fin_cut_same x xs"
block_complement_inv: fixes bl :: "'a set"   and bl2 :: "'a set" assumes "bl \<in># \<B>"   and "source.block_complement bl = bl2" shows "source.block_complement bl2 = bl"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<rightarrow>\<^sub>\<beta> x2"   and "\<And>x t s y. \<lbrakk>x \<sharp> y; x \<sharp> t\<rbrakk> \<Longrightarrow> P y ((Lam [x].s) \<degree> t) (s[x::=t])"   and "\<And>s t u y. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; \<And>z. P z s t\<rbrakk> \<Longrightarrow> P y (s \<degree> u) (t \<degree> u)"   and "\<And>s t u y. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; \<And>z. P z s t\<rbrakk> \<Longrightarrow> P y (u \<degree> s) (u \<degree> t)"   and "\<And>s t x y. \<lbrakk>x \<sharp> y; s \<rightarrow>\<^sub>\<beta> t; \<And>z. P z s t\<rbrakk> \<Longrightarrow> P y (Lam [x].s) (Lam [x].t)" shows "P z x1 x2"
D_2: shows "AE x in M. 0 \<le> D x"
branch_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {bicc_type BE, bicc_type BNE, bicc_type BGU, bicc_type BLE, bicc_type BL, bicc_type BGE, bicc_type BNEG, bicc_type BG, bicc_type BCS, bicc_type BLEU, bicc_type BCC, bicc_type BA, bicc_type BN}" shows "snd (branch_instr i s) = False"
WTrt_elim_cases_8: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'c expr"   and U :: "ty"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (e instanceof U) T" obtains T :: "ty" where "Ta__ = Boolean" and "J_heap_base.WTrt typeof_addr P h E e T" and "is_type P U"
decisive6_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} y x" shows "decisive swf A Is {j} y z"
assms_5: shows "\<forall>x y z. fst (g x y) \<longrightarrow> fst (g y z) \<longrightarrow> fst (g x z)"
repeatable_both_nxt: fixes t :: "'b ttree"   and t' :: "'b ttree"   and x :: "'b" assumes "repeatable t"   and "possible t' x"   and "t' \<otimes>\<otimes> t = t'" shows "nxt t' x \<otimes>\<otimes> t = nxt t' x"
good_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" shows "good P f = (\<exists>i j. i < j \<and> P (f i) (f j))"
mirror_elem_imirror: fixes I :: "nat set"   and t :: "nat" assumes "finite I" shows "mirror_elem t (imirror I) = mirror_elem t I"
mult_numeral_1_right: fixes a :: "'c" shows "a * Numeral1 = a"
wf0_decr0: fixes k :: "'k"   and idx :: "'n"   and a :: "'a"   and l :: "nat" assumes "wf0 (SUC k idx) a"   and "LEQ k l idx"   and "\<not> find0 k l a" shows "wf0 idx (decr0 k l a)"
le_list_Cons2_iff: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(x # xs \<preceq> y # ys) = (if x = y then xs \<preceq> ys else x # xs \<preceq> ys)"
HInter_def: fixes A :: "hf" shows "\<Sqinter>A = \<lbrace>x \<^bold>\<in> \<Squnion>A. \<forall>y. y \<^bold>\<in> A \<longrightarrow> x \<^bold>\<in> y\<rbrace>"