hcju/isabelleps huggingface dataset:
H_assign: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" shows "rel_kat.H \<lceil>\<lambda>s. P (s(v := e s))\<rceil> (v ::= e) \<lceil>P\<rceil>"
finite_keys: fixes f :: "'c \<Rightarrow>\<^sub>0 'd" shows "finite (Poly_Mapping.keys f)"
redT_updLs_iff: fixes ls :: "('d, 'c) locks"   and t :: "'c"   and las :: "'d lock_actions"   and l :: "'d" shows "redT_updLs ls t las $ l = upd_locks (ls $ l) t (las $ l)"
\<A>_2: shows "typing_cond\<^sub>s\<^sub>s\<^sub>t (unlabel \<A>)"
rcoset_rel_def: fixes A :: "'a set" shows "rcoset_rel A \<equiv> {(x, y). x - y \<in> A}"
hmsetmset_diff: fixes M :: "hmultiset"   and N :: "hmultiset" shows "hmsetmset (M - N) = hmsetmset M - hmsetmset N"
a1: shows "s' = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
t_2: shows "\<forall>x\<in>fv t. \<exists>a. \<Gamma>\<^sub>v x = Var (prot_atom.Atom a)"
extend_1_evals: fixes 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"   and es :: "char list exp list"   and t :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es'' :: "char list exp list"   and t'' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and t' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,t\<rangle> [\<rightarrow>] \<langle>es'',t''\<rangle>"   and "P \<turnstile> \<langle>es'',t''\<rangle> [\<Rightarrow>] \<langle>es',t'\<rangle>" shows "P \<turnstile> \<langle>es,t\<rangle> [\<Rightarrow>] \<langle>es',t'\<rangle>"
assms_1: shows "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"
II: shows "II \<in> UNpart01 cl dl P - {BrnFT cl dl}"
rep_eq: fixes x :: "'b" shows "(*\<^sub>V) (vector_to_cblinfun x) = (\<lambda>\<phi>. one_dim_iso \<phi> *\<^sub>C x)"
check_prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "check_prefix xs ys \<equiv> ebind (CHECK (length xs \<le> length ys) ()) (\<lambda>_. ebind (EWHILEIT (\<lambda>i. i \<le> length xs \<and> take i xs = take i ys) (\<lambda>i. i < length xs) (\<lambda>i. ebind (EASSERT (i < length xs \<and> i < length ys)) (\<lambda>_. ebind (CHECK (xs ! i = ys ! i) ()) (\<lambda>_. ERETURN (i + 1)))) 0) (\<lambda>_. ERETURN ()))"
normal_form_disjunct_def: fixes \<phi> :: "String.literal ltln"   and M :: "String.literal ltln list"   and N :: "String.literal ltln list" shows "normal_form_disjunct \<phi> M N \<equiv> flatten_sigma_2_list \<phi> M and\<^sub>n (and_list (map (\<lambda>\<psi>. G\<^sub>n (F\<^sub>n flatten_sigma_1_list \<psi> N)) M) and\<^sub>n and_list (map (\<lambda>\<psi>. F\<^sub>n (G\<^sub>n flatten_pi_1_list \<psi> M)) N))"
trans_join: fixes r :: "('a \<times> 'a) set" shows "trans r = (\<forall>(x, y1)\<in>r. \<forall>(y2, z)\<in>r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
weakPsiCongResPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<doteq> Q"   and "x \<sharp> \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<doteq> \<lparr>\<nu>x\<rparr>Q"
RedSFAcc: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and v :: "val"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "sh D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh),True\<rangle> \<rightarrow> \<langle>Val v,(h, l, sh),False\<rangle>"
demonic_demonic: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'a \<Rightarrow> bool" shows "[: r :] \<circ> [: r' :] = [: r OO r' :]"
no_\<tau>moves1_def: shows "no_\<tau>moves1 \<equiv> \<lambda>s1 t. wset s1 t = None \<and> (\<exists>x. thr s1 t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> r1.silent_move t (x, shr s1) (x', m')))"
characteristic_weak_formula_supp_aux: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "supp (Abs_bset {indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q |Q. \<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q}) \<subseteq> supp P"
distinct_remdups_sorted: fixes xs :: "'a list" assumes "sorted xs" shows "distinct (remdups_sorted xs)"
rational_boxes: fixes e :: "real"   and x :: "'a" assumes "0 < e" shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
iter_winden_step_ivl_termination: fixes c0 :: "com" shows "\<exists>c. iter_widen (step_ivl \<top>) (\<bottom>\<^sub>c c0) = Some c"
Max_insert2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> less_eq b a" shows "linorder.Max less_eq (insert a A) = a"
UP_mapsto: shows "replete_setcat.UP \<in> UNIV \<rightarrow> Collect (category.terminal replete_setcat.comp)"
witness_4: fixes b :: "'c" assumes "b \<in> T" shows "i b \<in> S"
higher_differentiable_on_sin: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. sin (f x)) n"
fst3_simp: fixes a :: "'b"   and b :: "'c"   and c :: "'d" shows "fst3 (a, b, c) = a"
openin_map_topology: fixes p :: "'c \<Rightarrow> 'b"   and X :: "'c topology" shows "openin (map_topology p X) = (\<lambda>U. U \<subseteq> p ` topspace X \<and> openin X (p -` U \<inter> topspace X))"
simps_6: fixes n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "lang n (Times r s) = lang n r @@ lang n s"
range_preflect_0: shows "surj (preflect (0::'b))"
eval_wp_AC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (a \<Squnion> b) P = (\<lambda>s. max (wp a P s) (wp b P s))"
less_eqE: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f"   and b :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'f" where "b = plus a c"
obtains_step: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r' :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" shows "obtains s r e s' r' ((l, i) # t) = (\<exists>(s'', T)|\<in>|possible_steps e s' r' l i. obtains s r e s'' (evaluate_updates T i r') t)"
binary_functor_axioms: shows "binary_functor (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) E.map"
negInfCase: assumes "w = NEG_INF" shows "\<exists>r. Rep_bword bw \<equiv>\<^sub>E r"
two_hrealpow_ge_one: fixes n :: "nat" shows "1 \<le> 2 ^ n"
is_expansion_aux_exp_neg: fixes F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux F f basis"   and "basis_wf basis"   and "ms_exp_gt 0 (ms_aux_hd_exp F)" shows "is_expansion_aux (powser_ms_aux' exp_series_stream F) (\<lambda>x. exp (f x)) basis"
im_sheaf_def: fixes V :: "'a set set" shows "local.im_sheaf V \<equiv> \<O> identity Spec \<^sup>\<inverse> Spec V"
lQuot_lists: fixes c :: "'b"   and S :: "'b set" assumes "c \<in> S" shows "lQuot c (lists S) = lists S"
Domainp_rel: fixes R1 :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'k \<Rightarrow> bool" shows "Domainp (rell_G R1 R2) = pred_G (Domainp R1) (Domainp R2)"
zero_literal_def: shows "STR '''' \<equiv> Abs_literal []"
foldr_disj_map_rlfm2: fixes f :: "'a \<Rightarrow> fm"   and xs :: "'a list" assumes "\<forall>n. isrlfm (f n)" shows "isrlfm (foldr MIR.disj (map f xs) F)"
divide_less_cancel: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a c) (divide b c) = ((less zero c \<longrightarrow> less a b) \<and> (less c zero \<longrightarrow> less b a) \<and> c \<noteq> zero)"
dmb_le_dbm_entry_bound_inf: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<preceq> b"   and "a = \<infinity>" shows "b = \<infinity>"
vector_smult_hom: fixes x :: "real"   and v :: "(real, 'b) Finite_Cartesian_Product.vec" shows "of_real x *s map_vector of_real v = map_vector of_real (x *s v)"
CondTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
block_size_gt_0: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set" assumes "design point_set block_collection"   and "bl \<in># block_collection" shows "0 < card bl"
rel_chain_drop: fixes l :: "'b list"   and R :: "('b \<times> 'b) set"   and n :: "nat" assumes "rel_chain l R" shows "rel_chain (drop n l) R"
rel_refl_strong: fixes x :: "'c + 'd"   and R1a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> setl x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> setr x \<Longrightarrow> R2a z2 z2" shows "rel_sum R1a R2a x x"
dvd_smult_iff: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" assumes "a \<noteq> (0::'b)" shows "(p dvd smult a q) = (p dvd q)"
nth_default_take: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list"   and m :: "nat" shows "nth_default x (take n xs) m = (if m < n then nth_default x xs m else x)"
pelims_3: fixes x :: "'a val" assumes "\<not> is_Bool x"   and "is_Bool_dom x" obtains "x = Unit" and "is_Bool_dom Unit"   | "x = Null" and "is_Bool_dom Null"   | v :: "32 word" where "x = Intg v__" and "is_Bool_dom (Intg v__)"   | v :: "'a" where "x = Addr v" and "is_Bool_dom (Addr v)"
Qw_cmpser: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "w_cmpser G (Suc n) f" shows "Gchain n (Qw_cmpser G f)"
component_of_term_of_pair: fixes t :: "'b"   and k :: "'a" shows "pprod.component_of_term (t, k) = k"
dom_trm_vars: fixes t :: "'a trm"   and C :: "'a literal set" assumes "dom_trm t C" shows "vars_of t \<subseteq> vars_of_cl C"
setmult_rcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"   and "K \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"   and "x \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "H <#>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> (K #>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x) = H <#>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> K #>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x"
assms_3: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "X xs f basis" shows "xs = MSLNil \<and> (\<forall>\<^sub>F x in at_top. f x = 0) \<and> length basis = Suc (expansion_level TYPE('a)) \<or> (\<exists>xs' b e basis' C. xs = MSLCons (C, e) xs' \<and> basis = b # basis' \<and> ms_closure X xs' (\<lambda>x. f x - eval C x * b x powr e) (b # basis') \<and> is_expansion C basis' \<and> (\<forall>e'>e. f \<in> o(\<lambda>x. b x powr e')) \<and> ms_exp_gt e (ms_aux_hd_exp xs'))"
InvariantUniqAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state" shows "let state' = applyUnitPropagate state in InvariantUniq (getM state')"
cartesian_closed_category_axioms: shows "cartesian_closed_category local.comp"
eval_ternary_DeMorgan_1: fixes a :: "ternaryvalue"   and b :: "ternaryvalue" shows "eval_ternary_Not (eval_ternary_And a b) = eval_ternary_Or (eval_ternary_Not a) (eval_ternary_Not b)"
restrict_extensional: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set" shows "restrict f A \<in> extensional A"
finite_field_mult_group_has_gen: assumes "finite (carrier R)" shows "\<exists>a\<in>carrier (mult_of R). carrier (mult_of R) = {a [^] i |i. i \<in> UNIV}"
parallel_reduces_diamond': fixes A :: "'a trm"   and C :: "'a trm"   and B :: "'a trm" assumes "A >>\<^sup>* C"   and "A >> B" shows "\<exists>D. B >>\<^sup>* D \<and> C >> D"
linear_leq: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and p :: "real mpoly"   and v :: "real" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "aEval (Leq p) (L[var := B / C]) = aEval (linear_substitution var b c (Leq p)) (L[var := v])"
single_is_union: fixes a :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" shows "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
map_ident: fixes t :: "'c dual" shows "map_dual (\<lambda>x. x) t = t"
reach_in_beh: fixes s :: "'a"   and T :: "('a, 'b) TS_scheme" assumes "s \<in> reach T" shows "\<exists>b\<in>beh T. s \<in> set b"
non_speculative_lappend: fixes obs :: "('a, 'b) obs_event action llist"   and P :: "'c prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> 'a val set"   and obs' :: "('a, 'b) obs_event action llist" assumes "lfinite obs" shows "non_speculative P vs (lappend obs obs') = (non_speculative P vs obs \<and> non_speculative P (w_values P vs (list_of obs)) obs')"
plus_inj_eq: fixes x :: "nat"   and q :: "nat"   and x' :: "nat"   and y :: "nat" assumes "x < q"   and "x' < q"   and "(y + x) mod q = (y + x') mod q" shows "x = x'"
arith_simps_64: fixes n :: "num" shows "\<bar>numeral n\<bar> = numeral n"
fin_subst2: fixes M :: "trm"   and y :: "name"   and x :: "name"   and P :: "trm"   and c :: "coname" assumes "fin M y"   and "x \<noteq> y"   and "y \<sharp> P"   and "M \<noteq> Ax y c" shows "fin (M{c:=x.P}) y"
nsp: assumes "is_leveln_ftree 0 s"   and "is_measured_ftree s"   and "\<not> p i"   and "p (i + gmft s)"   and "nsplitTree p i s = (l1, nd, r1)" shows "is_leveln_ftree 0 l1 \<and> is_measured_ftree l1 \<and> is_leveln_ftree 0 r1 \<and> is_measured_ftree r1 \<and> is_leveln_node 0 nd \<and> is_measured_node nd"
isCont_bounded: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
lm010: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R = Outside' {x} R +< {x} \<times> R `` {x}"
distinct_list_of_dlist: fixes dxs :: "'d dlist" shows "distinct (list_of_dlist dxs)"
Gcd_def: fixes A :: "'a set" shows "normalization_euclidean_semiring_class.Gcd A = normalization_euclidean_semiring_class.Lcm {d. \<forall>a\<in>A. d dvd a}"
inf_sup_ord_4: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<le> y"
ord_conv_Gcd: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "group.ord G x = Gcd {n. x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>}"
ground_inference_saturated_def: fixes S :: "'a eclause set" shows "ground_inference_saturated S = (\<forall>C P \<sigma> C'. derivable C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> grounding_set P \<sigma> \<longrightarrow> redundant_inference C S P \<sigma>)"
f'_def: shows "f' = (\<lambda>_. gpv')"
rb_aux_inv1_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "rb_aux_inv1 bs = (set bs \<subseteq> dgrad_sig_set' (length fs) dgrad \<and> 0 \<notin> rep_list ` set bs \<and> sorted_wrt (\<lambda>x y. lt y \<prec>\<^sub>t lt x) bs \<and> (\<forall>i<length bs. \<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set (drop (Suc i) bs)) (bs ! i)) \<and> (\<forall>i<length bs. (\<exists>j<length fs. lt (bs ! i) = lt (monomial (1::'b) (term_of_pair (0::'a, j))) \<and> punit.lt (rep_list (bs ! i)) \<preceq> punit.lt (rep_list (monomial (1::'b) (term_of_pair (0::'a, j))))) \<or> (\<exists>p\<in>set bs. \<exists>q\<in>set bs. is_regular_spair p q \<and> rep_list (spair p q) \<noteq> 0 \<and> lt (bs ! i) = lt (spair p q) \<and> punit.lt (rep_list (bs ! i)) \<preceq> punit.lt (rep_list (spair p q)))) \<and> (\<forall>i<length bs. is_RB_upt dgrad rword (set (drop (Suc i) bs)) (lt (bs ! i))))"
integer_eq_iff: fixes k :: "integer"   and l :: "integer" shows "(k = l) = (int_of_integer k = int_of_integer l)"
UN_E: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'d" where "x \<in> A" and "b \<in> B x"
T_init_V_eq: fixes s :: "'b"   and o :: "'a"   and as :: "'b list"   and os :: "'a list"   and o' :: "'a" shows "emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'} = emeasure (T (s, o')) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}"
z: shows "\<forall>w. P w z = (P w x \<and> \<not> O w y)"
Pj_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "Pj G H = restrict ((\<bullet>\<^bsub>G\<^esub>) H) (carrier G)"
open_eball: fixes z :: "'a"   and r :: "ereal" shows "open (eball z r)"
has_vector_derivative_at_within: fixes f :: "real \<Rightarrow> 'e"   and f' :: "'e"   and x :: "real"   and S :: "real set" assumes "(f has_vector_derivative f') (at x)" shows "(f has_vector_derivative f') (at x within S)"
accessible_unsat_core: fixes s :: "('i, 'a) state" shows "\<U>\<^sub>c ` {s'. (s, s') \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*} = {\<U>\<^sub>c s}"
is_serial_solution_for_problem_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list" shows "is_serial_solution_for_problem \<Psi> \<psi> \<equiv> let I = \<Psi>\<^sub>I\<^sub>+; G = \<Psi>\<^sub>G\<^sub>+; ops = \<Psi>\<^sub>\<O>\<^sub>+ in G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I \<psi> \<and> list_all (\<lambda>op. ListMem op ops) \<psi>"
vcomp_in_hhom: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "seq \<nu> \<mu>"   and "src \<nu> = a"   and "trg \<nu> = b" shows "\<guillemotleft>\<nu> \<cdot> \<mu> : a \<rightarrow> b\<guillemotright>"
scaled: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real"   and c :: "real" assumes "vnm_utility carrier le u"   and "0 < c" shows "vnm_utility carrier le (\<lambda>x. c * u x)"
det_rw2: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> {f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}" shows "Determinant.det (mat\<^sub>r n n (\<lambda>i. col A (f i))) = (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i)))"
assms_9: shows "moebius_pt M2 z3 = w3"
eval\<kappa>_cases: fixes y :: "\<nu> option" assumes "y \<in> UNIV" obtains x :: "\<kappa>" where "y = eval\<kappa> x"
consistent_def: fixes \<Psi> :: "'a TESL_atomic list" shows "consistent \<Psi> \<equiv> \<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<noteq> {}"
ffold_cong: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b fset"   and s :: "'c"   and t :: "'c"   and B :: "'b fset" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"   and "s = t"   and "A = B" shows "ffold f s A = ffold g t B"
not_final_thread_final: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> final x" shows "not_final_thread s t"
fin: shows "finite {m. \<exists>lam. (m, lam) \<in> set n_as \<and> cmod lam = 1}"
simps_1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.LeftDerivation \<NN> \<TT> \<RR> a [] b = (a = b)"
finite_has_minimal2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq m a \<and> (\<forall>b\<in>A. less_eq b m \<longrightarrow> m = b)"
seq_hoare_r: fixes p :: "'a upred"   and Q\<^sub>1 :: "'a hrel"   and s :: "'a upred"   and Q\<^sub>2 :: "'a hrel"   and r :: "'a upred" assumes "\<lbrace>p\<rbrace> Q\<^sub>1 \<lbrace>s\<rbrace>\<^sub>u"   and "\<lbrace>s\<rbrace> Q\<^sub>2 \<lbrace>r\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> Q\<^sub>1 ;; Q\<^sub>2 \<lbrace>r\<rbrace>\<^sub>u"
of_rat_dense: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>q. x < real_of_rat q \<and> real_of_rat q < y"
simps_6: fixes uv :: "nat"   and C :: "acom" shows "strip ({uv} Ab C) = strip C"
simps: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'d + 'f"   and a2 :: "'e + 'g" shows "rel_sum R1 R2 a1 a2 = ((\<exists>a c. a1 = Inl a \<and> a2 = Inl c \<and> R1 a c) \<or> (\<exists>b d. a1 = Inr b \<and> a2 = Inr d \<and> R2 b d))"
top_edge_explicit: shows "(\<lambda>x. twoC (1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
lfp_unfold: fixes f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "lfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) = f (lfp f)"