hcju/isabelleps huggingface dataset:
Aleph_increasing: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> < \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" shows "\<aleph>\<alpha> < \<aleph>\<beta>"
distinct_butlast: fixes xs :: "'d list" assumes "distinct xs" shows "distinct (butlast xs)"
rep_eq: fixes x :: "('d, 'e) mapping" shows "Mapping.lookup x = mapping.rep x"
inj_on_image_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "(f ` A = f ` B) = (A = B)"
simps_1: fixes t :: "char list \<Rightarrow> int option" shows "defs SKIP t = t"
cancel: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys = xs \<cdot> zs) = (ys = zs)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "\<And>x. hom x = (0::'c) \<Longrightarrow> x = (0::'b)" shows "zero_hom_0_axioms hom"
is_asig_def: fixes triple :: "'c signature" shows "is_asig triple \<equiv> inputs triple \<inter> outputs triple = {} \<and> outputs triple \<inter> internals triple = {} \<and> inputs triple \<inter> internals triple = {}"
simps_15: fixes x2 :: "'c" shows "set_option (Some x2) = {x2}"
upper_circle_def: fixes l :: "real"   and r :: "real" shows "upper_circle l r = {x. cmod ((x - complex_of_real l) / complex_of_real (r - l) - (1 / 2 + complex_of_real (sqrt 3) / 6 * \<i>)) < sqrt 3 / 3}"
minus_less_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) b = less (uminus b) a"
b_def': shows "b = \<phi> i\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
Cons_1: fixes j :: "nat"   and i :: "nat" assumes "\<And>A j i. \<lbrakk>A \<in> set As; j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)"   and "\<forall>a\<in>set As. square_mat a"   and "j < i"   and "i < dim_row (diag_block_mat As)" shows "diag_block_mat As $$ (i, j) = (0::'a)"
Int_Union: fixes A :: "'b set"   and B :: "'b set set" shows "A \<inter> \<Union> B = \<Union> ((\<inter>) A ` B)"
unwind_exit_def: fixes \<Delta>e :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'a \<Rightarrow> bool" shows "unwind_exit \<Delta>e \<equiv> \<forall>s vl s1 vl1. reachNT s \<and> reach s1 \<and> \<Delta>e s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> exit s (hd vl)"
circline_type_pos_card_eq0_diag: fixes H :: "circline" assumes "circline_diag H"   and "0 < circline_type H" shows "circline_set H = {}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "group_add_hom hom" shows "ab_group_add_hom hom"
dg_singleton_is_arrI: fixes j :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "j \<in>\<^sub>\<circ> Vset \<alpha>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "ZFC_in_HOL.set {\<langle>j, f\<rangle>} : ZFC_in_HOL.set {\<langle>j, a\<rangle>} \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>\<^esub> ZFC_in_HOL.set {\<langle>j, b\<rangle>}"
simps_1: fixes x11 :: "'f"   and x12 :: "'g"   and x13 :: "'h"   and y11 :: "'f"   and y12 :: "'g"   and y13 :: "'h" shows "(NewThread x11 x12 x13 = NewThread y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
rank_Gauss_Jordan_code: fixes A :: "(('d, 'e) vec, 'f) vec" shows "rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \<noteq> 0) + 1)"
prems_2: shows "BSD \<V>i Tr\<^bsub>ESi\<^esub>"
absorb1: fixes a :: "'g"   and b :: "'g" assumes "a \<le> b" shows "min a b = a"
Domainp_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (rel_option R) = pred_option (Domainp R)"
rotate_D_def: shows "rotate_D = (\<lambda>v. case v of A \<Rightarrow> C | B \<Rightarrow> A | C \<Rightarrow> B | D \<Rightarrow> D)"
coprime_poly_0: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "coprime p q" shows "poly p x \<noteq> (0::'a) \<or> poly q x \<noteq> (0::'a)"
geom_proc_spick: fixes n :: "nat"   and w :: "bool stream"   and x :: "bool" shows "geom_proc (Suc n) (spick w n x) = (if x then u else d) * geom_proc n w"
int_pow_int: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> int n = x [^]\<^bsub>G\<^esub> n"
outer_measure_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal"   and X :: "'b set" shows "outer_measure M f X = (INF A\<in>{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> \<Union> (range A)}. \<Sum>i. f (A i))"
z_hyp: shows "z_R I sign = 1 \<or> z_R I sign = 0 \<or> z_R I sign = - 1"
tail_in_verts: fixes e :: "'b" assumes "e \<in> pre_digraph.arcs G" shows "tail G e \<in> pre_digraph.verts G"
a3: shows "x:N \<in> \<parallel>(B)\<parallel>"
vars_chop: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "vars (chop t) \<union> vars_hd (head t) = vars t"
seqI': fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and c :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] c\<guillemotright>" shows "Cop_S.seq g f"
rep_eq: shows "(*\<^sub>V) 0 = (\<lambda>x. 0::'b)"
happ_expansion: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" shows "F = \<lbrace>XY \<^bold>\<in> B * C. hsnd XY = happ F (hfst XY)\<rbrace>"
square_free_factorizationD_5: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "distinct bs"
disc_2: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "is_Fun (Fun x21 x22)"
essentially_surjective_functor_axioms: shows "essentially_surjective_functor A B F"
consistency_def: fixes f :: "'d set \<Rightarrow> 'd set" shows "consistency f = (\<forall>B C. C \<subseteq> B \<longrightarrow> f B \<subseteq> C \<longrightarrow> f B = f C)"
infsetsum_subset_real: fixes f :: "'c \<Rightarrow> real"   and B :: "'c set"   and A :: "'c set" assumes "f abs_summable_on B"   and "A \<subseteq> B"   and "\<And>x. x \<notin> A \<Longrightarrow> 0 \<le> f x" shows "infsetsum f A \<le> infsetsum f B"
zip_Cons_Cons: fixes x :: "'k"   and xs :: "'k list"   and y :: "'l"   and ys :: "'l list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
return_partition_basics_3: fixes A :: "'a set" assumes "A \<in> sets M" shows "(\<Union>n. return_partition A (n + 1)) = recurrent_subset A"
dependent_insertD: fixes a :: "'d"   and S :: "'d set" assumes "a \<notin> cspan S"   and "cdependent (insert a S)" shows "cdependent S"
simps: fixes x :: "xcpt" shows "the_Xcpt (Xcpt x) = x"
coeff_uminus: fixes p :: "'a mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (- p) x = - MPoly_Type.coeff p x"
RedLAss: fixes V :: "char list"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>V:=Val v,(h, l)\<rangle> \<rightarrow> \<langle>unit,(h, l(V \<mapsto> v))\<rangle>"
ee_3: shows "dist (tail G e) \<noteq> \<infinity>"
canceling_def: fixes a :: "bool \<times> 'd"   and b :: "bool \<times> 'd" shows "canceling a b = (snd a = snd b \<and> fst a \<noteq> fst b)"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K" shows "x \<oplus>\<^bsub>K\<^esub> y \<in> carrier K"
mr_not_Conc: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "singleCombinators p" shows "applied_rule_rev Cp x p \<noteq> \<lfloor>a \<oplus> b\<rfloor>"
basis_reduction_add_rows_loop_1: fixes i :: "nat"   and fs :: "int vec list"   and j :: "nat"   and fs' :: "int vec list" assumes "LLL_invariant True i fs"   and "\<mu>_small_row i fs j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_invariant False i fs'"
enc_and_TRel_impl_indRelRTPO_weakly_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>S x. (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<and> (\<forall>S x. (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x))"   and "rel_weakly_preserves_barbs TRel TWB"   and "rel_weakly_reflects_barbs TRel TWB" shows "rel_weakly_respects_barbs (indRelRTPO TRel) (STCalWB SWB TWB)"
normalize_mult_unit_left: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a one" shows "normalize (times a b) = normalize b"
find_exponent_def: fixes p :: "int"   and bnd :: "int" shows "find_exponent p bnd = ??.Berlekamp_Zassenhaus.find_exponent_main p p 1 bnd"
mono_Max_commute: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "finite A"   and "A \<noteq> {}" shows "f (linorder.Max less_eq A) = Max (f ` A)"
discounted_mult_times_rfr: fixes r :: "real"   and X :: "'b \<Rightarrow> real"   and n :: "nat"   and w :: "'b" assumes "- 1 < r" shows "discounted_value r (\<lambda>m w. (1 + r) * X w) (Suc n) w = discounted_value r (\<lambda>m. X) n w"
le_numeral_extra_3: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
length_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>xs. \<forall>ys. \<^bold>|ys\<^bold>| < \<^bold>|xs\<^bold>| \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
fps_uminus_def: shows "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))"
distinct_1: fixes x11 :: "char list"   and x12 :: "spec list option"   and x13 :: "dec list"   and x2 :: "dec" shows "Tmod x11 x12 x13 \<noteq> Tdec x2"
z: shows "map_of (l ! bounded_hashcode_nat (length l) k) k = Some z"
nat_int_add: fixes a :: "nat"   and b :: "nat" shows "nat (int a + int b) = a + b"
unique_factorization_m_smult: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and di :: "int"   and d :: "int" assumes "p.unique_factorization_m f (c, fs)"   and "p.M (di * d) = 1" shows "p.unique_factorization_m (Polynomial.smult d f) (c * d, fs)"
countable_card_le_natLeq: fixes A :: "'f set" shows "countable A = (|A| \<le>o natLeq)"
exposed_face_of_Int: fixes T :: "'b set"   and S :: "'b set"   and u :: "'b set" assumes "T exposed_face_of S"   and "u exposed_face_of S" shows "T \<inter> u exposed_face_of S"
mset_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, a, r\<rangle>" shows "mset_tree (sift_down l a r) = {#a#} + (mset_tree l + mset_tree r)"
simps_4: fixes t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "trms_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,t'\<rangle>) = [t, t']"
mmulti_join_link: fixes A_pos :: "nat set list"   and n :: "nat"   and A_neg :: "nat set list"   and L :: "'b option list set list" assumes "A_pos \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L" shows "mmulti_join n A_pos A_neg L = mmulti_join' A_pos A_neg L"
may_acquire_all_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and ln :: "'c \<Rightarrow>f nat" shows "may_acquire_all ls t ln \<equiv> \<forall>l. 0 < ln $ l \<longrightarrow> may_lock (ls $ l) t"
vinsert_vempty: fixes a :: "V" shows "vinsert a 0 = set {a}"
costBIT_3x: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a" assumes "x \<noteq> y"   and "x \<in> {x0, y0}"   and "y \<in> {x0, y0}" shows "E (type3 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s x \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) x a))))) = 3 / 4"
Suc_llength: fixes n :: "nat"   and xs :: "'b llist" assumes "enat (Suc n) < llength xs" shows "enat n < llength xs"
hta_ensure_idx_sf_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_ensure_idx_sf H \<equiv> case hta_idx_sf H of None \<Rightarrow> H\<lparr>hta_idx_sf := Some (build_rule_index_sf (hta_\<delta> H))\<rparr> | Some x \<Rightarrow> H"
degree_prod_monom: fixes n :: "nat" shows "degree (\<Prod>i = 0..<n. monom (1::'b) 1) = n"
map_id: fixes t :: "('b, 'c) result" shows "map_result id id t = t"
well_base_bound: fixes M :: "nat multiset"   and n :: "nat" assumes "local.well_base M"   and "\<forall>m\<in>#M. m < n" shows "\<Sum>\<^sub># (image_mset ((^) base) M) < base ^ n"
assms_1: shows "orient_lit_inst L t s polarity \<sigma>"
\<delta>_statesE: fixes q :: "'a"   and \<Delta> :: "('a, 'b) ta_rule set" assumes "q \<in> \<delta>_states \<Delta>" obtains f :: "'b" and qs :: "'a list" where "q \<rightarrow> f__ qs__ \<in> \<Delta>"   | ql :: "'a" and f :: "'b" and qs :: "'a list" where "ql \<rightarrow> f qs \<in> \<Delta>" and "q \<in> set qs"
le_imp_inverse_le_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less b zero" shows "less_eq (inverse b) (inverse a)"
HEndPhase1_dblock_allInput: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv1 s"   and "Inv2a s" shows "inp (dblock s' p) \<in> allInput s'"
sep_add_left_commute: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a ## b"   and "b ## c"   and "a ## c" shows "b + (a + c) = a + (b + c)"
less_length_mono: fixes s :: "'c list"   and t :: "'c list" assumes "s < t" shows "length s < length t"
rep_eq: fixes x :: "'b fset set" shows "fset (Sup x) = (if finite (\<Union> (fset ` x)) then \<Union> (fset ` x) else {})"
assms_6: shows "e \<star> f \<cong> e \<star> f'"
iD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "iD rt \<equiv> {dip. flag rt dip = Some Aodv_Basic.inv}"
r: fixes loc1 :: "'loc"   and u :: "'t"   and s :: "'sum"   and loc2 :: "'loc" assumes "0 < zcount (c_pts c1 loc1) u"   and "s \<in>\<^sub>A path_summary loc1 loc2"   and "n \<noteq> 0"   and "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t"   and "\<forall>loc'. c_pts c1 loc' = (if loc' = loc then update_zmultiset (c_pts c0 loc') t n else c_pts c0 loc')" shows "\<exists>t'\<le>results_in u s. t' \<in>\<^sub>A frontier (c_imp c1 loc2)"
process_prop_5_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_5 P \<equiv> \<forall>xs x. xs \<in> snd P \<longrightarrow> xs @ [x] \<in> snd P"
cols_length: fixes A :: "'c mat" shows "length (cols A) = dim_col A"
map_iterator_rule_insert_P: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'d \<Rightarrow> 'e option"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "map_iterator it m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
Derives1_split: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> u i r v" shows "\<exists>x y. u = x @ [fst r] @ y \<and> v = x @ snd r @ y \<and> length x = i"
hadd_cancel_right: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "(x @+ y = x @+ z) = (y = z)"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'b fset"   and xb :: "'c fset" shows "rel_fset x xa xb = rel_set x (fset xa) (fset xb)"
hyps_7: shows "x\<^sub>3 = l [^] 2 \<ominus> x\<^sub>1 \<ominus> x\<^sub>2"
add_point_blocks_wf: fixes p :: "'a"   and bs :: "'a set set" shows "incidence_system (add_point p) (add_point_to_blocks p bs)"
c''_def: shows "\<C>'' = \<C>'\<lparr>local := local \<C>' @ t1s @ tfs, return := Some t2s\<rparr>"
vifunion_simps_1: fixes I :: "V"   and a :: "V"   and B :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> ((\<lambda>aa\<in>\<^sub>\<circ>I. vinsert a (B aa)) `\<^sub>\<circ> I) = (if I = 0 \<Rightarrow> 0 | otherwise \<Rightarrow> vinsert a (\<Union>\<^sub>\<circ> (VLambda I B `\<^sub>\<circ> I)))"
assms: fixes i :: "nat" assumes "i < length Gs" shows "Group.group (Gs ! i)"
weakMonotonic: fixes P :: "ccs"   and A :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and B :: "(ccs \<times> ccs) set" assumes "P \<leadsto>\<^sup>^<A> Q"   and "A \<subseteq> B" shows "P \<leadsto>\<^sup>^<B> Q"
disjnt_ge_max: fixes Y :: "'c set"   and X :: "'c set" assumes "finite Y"   and "\<And>x. x \<in> X \<Longrightarrow> Max Y < x" shows "disjnt X Y"
wf': shows "pref_profile_wf agents alts (R2 \<circ> \<pi>)"
IH: fixes y :: "nat"   and w :: "'person \<Rightarrow> 'color"   and p :: "'person" assumes "y < n"   and "w p = blue"   and "local.valid w"   and "w guru \<noteq> blue" shows "local.leaves y p w = (card (local.blues_seen w p) \<le> y)"
ni: shows "ni = nat \<parallel>fi\<parallel>\<^sup>2"
steps_imp_chainp': fixes n :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b" assumes "\<forall>i\<ge>n. \<exists>j\<ge>i. P (S i) (S j)" shows "\<exists>T. \<forall>i. P (T i) (T (Suc i))"
assms_5: fixes w :: "complex" assumes "w \<in> s" shows "f w = g w * (w - z)"
assms_2: shows "recognises_execution e s' (evaluate_updates t i r) trace"
simps: fixes K :: "'b set"   and q\<^sub>m :: "'b \<Rightarrow> 'c" shows "\<iota>\<^sub>\<times> K q\<^sub>m = (\<lambda>k. if k \<in> K then Some (q\<^sub>m k) else None)"
integrableD_3: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>"
nhdI: fixes U :: "'b set"   and T :: "'b set set"   and m :: "'b set"   and x :: "'b" assumes "U \<subseteq> carrier\<^bsub>T\<^esub>"   and "m open\<^bsub>T\<^esub>"   and "x \<in> m"   and "m \<subseteq> U" shows "U \<in> nhds\<^bsub>T\<^esub> x"
induct: fixes x1 :: "fterm literal set set"   and x2 :: "fterm literal set set"   and P :: "fterm literal set set \<Rightarrow> fterm literal set set \<Rightarrow> bool" assumes "mresolution_step x1 x2"   and "\<And>C\<^sub>1 Cs C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>. \<lbrakk>C\<^sub>1 \<in> Cs; C\<^sub>2 \<in> Cs; applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>\<rbrakk> \<Longrightarrow> P Cs (Cs \<union> {mresolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>})"   and "\<And>C Cs C'. \<lbrakk>C \<in> Cs; var_renaming_of C C'\<rbrakk> \<Longrightarrow> P Cs (Cs \<union> {C'})" shows "P x1 x2"
finite_subsets_at_top_finite: fixes A :: "'c set" assumes "finite A" shows "finite_subsets_at_top A = principal {A}"
det_singleton: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'c mat" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "A \<in> carrier_mat 1 1" shows "Determinant.det A = A $$ (0, 0)"
vrrestriction_vrange: shows "(xs #\<^sub>\<circ> x) \<restriction>\<^sup>r\<^sub>\<circ> \<R>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = xs #\<^sub>\<circ> x"
eq_on_refl: fixes s :: "'b \<Rightarrow> 'c"   and X :: "'b set" shows "s = s on X"
sum_atLeast_Suc_shift: fixes b :: "nat"   and a :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "0 < b"   and "a \<le> b" shows "sum f {Suc a..b} = (\<Sum>i = a..b - 1. f (Suc i))"
vars_Var: fixes i :: "nat" shows "vars (Var i) = (if (0::'a) = (1::'a) then {} else {i})"
simps_2: fixes f :: "nat \<Rightarrow> 'o"   and n :: "nat" shows "state f (Suc n) = nxt (f n) (state f n)"
approx_mult_HFinite: fixes a :: "'c star"   and b :: "'c star"   and c :: "'c star"   and d :: "'c star" assumes "a \<approx> b"   and "c \<approx> d"   and "b \<in> HFinite"   and "d \<in> HFinite" shows "a * c \<approx> b * d"
col_space_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "col_space A = vec.span (columns A)"
LIMSEQ_le: fixes X :: "nat \<Rightarrow> 'c"   and x :: "'c"   and Y :: "nat \<Rightarrow> 'c"   and y :: "'c" assumes "X \<longlonglongrightarrow> x"   and "Y \<longlonglongrightarrow> y"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> Y n" shows "x \<le> y"
i_le_a: shows "mod_type_class.from_nat i \<le> a"
psi_in_R2: shows "\<psi> \<in> \<R>\<^sup>2"
is_subset_MST_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and A :: "('a \<times> 'a) set" shows "is_subset_MST w g A \<equiv> \<exists>t. is_MST w g t \<and> A \<subseteq> edges t"
leadsTo_123: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> A \<longmapsto> A'" shows "\<exists>B. A \<subseteq> B \<and> F \<in> B \<longmapsto> A' \<and> F \<in> B - A' co B \<union> A'"
fresh_Vs: fixes a :: "atom" assumes "a \<sharp> p" shows "a \<sharp> Vs"
rel_frontier_eq_empty: fixes S :: "'b set" shows "(rel_frontier S = {}) = affine S"
extremum_unique: fixes a :: "'c" shows "(\<top> \<le> a) = (a = \<top>)"