hcju/isabelleps huggingface dataset:
LLs: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t2 :: "'b" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "Syntax_Arith.LLs var Var FvarsT Fvars subst eql cnj imp exi fls pls t1 t2 \<in> fmla"
f: shows "\<bar>f x - (a * x + b)\<bar> \<le> d"
mk_ps_chain_preserves_length: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" shows "chainlen (mk_ps_chain \<Pi> \<pi>) = chainlen \<Pi> + length \<pi>"
rep_eq: fixes x :: "nat"   and xa :: "uint8" shows "Rep_uint8 (drop_bit x xa) = drop_bit x (Rep_uint8 xa)"
exp_strong_induct: fixes P :: "'b \<Rightarrow> exp \<Rightarrow> bool"   and c :: "'b"   and exp :: "exp" assumes "\<And>var c. P c (Var var)"   and "\<And>exp var c. (\<And>c. P c exp) \<Longrightarrow> P c (App exp var)"   and "\<And>\<Gamma> exp c. \<lbrakk>atom ` domA \<Gamma> \<sharp>* c; \<And>c x. x \<in> domA \<Gamma> \<Longrightarrow> P c (the (map_of \<Gamma> x)); \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Let \<Gamma> exp)"   and "\<And>var exp c. \<lbrakk>{atom var} \<sharp>* c; \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Lam [var]. exp)"   and "\<And>b c. P c (Bool b)"   and "\<And>scrut e1 e2 c. \<lbrakk>\<And>c. P c scrut; \<And>c. P c e1; \<And>c. P c e2\<rbrakk> \<Longrightarrow> P c (scrut ? e1 : e2)" shows "P c exp"
acomp_acomp: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "acomp a\<^sub>1 @ acomp a\<^sub>2 @ P \<Turnstile> cfs\<box>"   and "\<And>cfs. acomp a\<^sub>1 \<Turnstile> cfs\<box> \<Longrightarrow> apred a\<^sub>1 (cfs ! 0) (cfs ! (length cfs - 1))"   and "\<And>cfs. acomp a\<^sub>2 \<Turnstile> cfs\<box> \<Longrightarrow> apred a\<^sub>2 (cfs ! 0) (cfs ! (length cfs - 1))" shows "case cfs ! 0 of (pc, s, stk) \<Rightarrow> pc = 0 \<and> (\<exists>k<length cfs. cfs ! k = (size (acomp a\<^sub>1 @ acomp a\<^sub>2), s, aval a\<^sub>2 s # aval a\<^sub>1 s # stk))"
wt_attack_if_tfr_attack: fixes \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "\<I> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle>"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "tfr\<^sub>s\<^sub>t S"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)" obtains \<I>\<^sub>\<tau> :: "'var \<Rightarrow> ('fun, 'var) Term.term" where "subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)" and "\<I>\<^sub>\<tau> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle>" and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>" and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)"
lemma_2_10_30: fixes b :: "'a"   and a :: "'a" shows "((b r\<rightarrow> a) l\<rightarrow> a) r\<rightarrow> a = b r\<rightarrow> a"
assms_1: shows "g\<cdot>\<bottom> = \<bottom>"
map_le_empty: fixes g :: "'a \<Rightarrow> 'b option" shows "Map.empty \<subseteq>\<^sub>m g"
simps: fixes \<Delta> :: "('a, 'b) network_scheme"   and a1 :: "'a antiparallel_edges.vertex"   and a2 :: "'a antiparallel_edges.vertex" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.edg \<Delta> a1 a2 = ((\<exists>x y. a1 = antiparallel_edges.Vertex x \<and> a2 = antiparallel_edges.Edge x y \<and> edge \<Delta> x y) \<or> (\<exists>x y. a1 = antiparallel_edges.Edge x y \<and> a2 = antiparallel_edges.Vertex y \<and> edge \<Delta> x y))"
unsandwich_right_inverse: fixes A :: "'o"   and u :: "'o \<Rightarrow> 'a set_arrow" assumes "A \<in> Ob"   and "u : Hom(A,_) \<Rightarrow> F in Func (AA , Set )" shows "\<sigma>(A,\<sigma>\<^sup>\<leftarrow>(A,u)) = u"
strict_sorted_iff: fixes l :: "'a word list" shows "signed.strict_sorted l = (signed.sorted l \<and> distinct l)"
simps_2: fixes \<A> :: "'a ltln set" shows "\<A> \<Turnstile>\<^sub>P false\<^sub>n = False"
quotient_hom: fixes G :: "'c set"   and T :: "'c \<Rightarrow> 'd"   and H :: "'c set" assumes "GroupHom G T"   and "Group H \<and> order.greater_eq G H"   and "Group.normal G H"   and "order.greater_eq (ker T \<inter> G) H" shows "GroupHom (Group.quotient_group G H) (restrict0 (quotientfun T \<circ> the_inv_into (G // Restr (lcoset_rel H) G) (BinOpSetGroup.Abs_G_perm (G // Restr (lcoset_rel H) G) (+))) (BinOpSetGroup.Abs_G_perm (G // Restr (lcoset_rel H) G) (+) ` G // Restr (lcoset_rel H) G))"
supportE_def: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat" shows "supportE P = {x. \<exists>l1 l2 s. (\<forall>y. y \<noteq> x \<longrightarrow> l1 y = l2 y) \<and> P l1 s \<noteq> P l2 s}"
dg_1_is_arrE: fixes \<aa> :: "V"   and \<ff> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b" obtains "a = \<aa>" and "b = \<aa>" and "f = \<ff>"
singleton_distinct_path: fixes v :: "'v"   and w :: "'w"   and v' :: "'v" assumes "(v, w, v') \<in> E" shows "is_trail v [(v, w, v')] v'"
atMost_eq_iff: fixes x :: "'b"   and y :: "'b" shows "({..x} = {..y}) = (x = y)"
assms_16: assumes "cands \<noteq> []" shows "d < r"
s0_init_def: shows "s0_init \<equiv> l1_secrecy"
mod_in_carrier: fixes m :: "int"   and a :: "int" assumes "residues m" shows "a mod m \<in> carrier (residue_ring m)"
subprob_space_null_measure_iff: fixes M :: "'c measure" shows "subprob_space (null_measure M) = (space M \<noteq> {})"
inner_rot_pos_move_base: fixes x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "'a" assumes "0 < (x - y) \<bullet> rot (a - b)"   and "y \<in> {a--b}" shows "0 < (x - a) \<bullet> rot (a - b)"
subcls_is_class: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "(subcls1 P)\<^sup>+\<^sup>+ C D" shows "is_class P C"
f_0: fixes x :: "'s"   and y :: "'s" shows "f x y 0 = (if x = y then 1 else 0)"
counit_simps_4: shows "src\<^sub>D \<psi> = trg\<^sub>D (F f)"
SpecStuck: fixes s :: "'a"   and r :: "('a \<times> 'a) set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Stuck"
reachable_snapshot_inv_black_heap_no_grey_refs_dequeue_Mutate: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and m' :: "'h"   and r :: "'g"   and f :: "'e"   and opt_r' :: "'g option"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws"   and "black_heap s"   and "no_grey_refs s"   and "valid_refs_inv s" shows "mut_m.reachable_snapshot_inv m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := ws)\<rparr>))"
assms: shows "((f, l), p, f', l') \<in> trcl pr_algo_lts'"
cont_fun_lub_apply: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "cont luba orda (fun_lub lubb) (fun_ord ordb) f = (\<forall>x. cont luba orda lubb ordb (\<lambda>y. f y x))"
kcomp_klift: fixes f :: "'c \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set" shows "f \<circ>\<^sub>K g = g\<^sup>\<dagger> \<circ> f"
exhaust: fixes y :: "enat" obtains "y = 0"   | x2 :: "enat" where "y = eSuc x2"
word_uint_eq_iff: fixes a :: "'b word"   and b :: "'b word" shows "(a = b) = (uint a = uint b)"
fix_eq: fixes F :: "'b \<rightarrow> 'b" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
Sigma_empty_iff: fixes I :: "'e set"   and X :: "'e \<Rightarrow> 'f set" shows "(Sigma I X = {}) = (\<forall>i\<in>I. X i = {})"
a_uniq: shows "\<forall>b. b \<in> S \<and> (\<forall>c\<in>S - {b}. norm (v - b) < norm (v - c)) \<longrightarrow> b = a"
Inf_FL_to_Inf_F: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Inf_FL :: "('a \<times> 'c) inference set"   and \<iota>\<^sub>F\<^sub>L :: "('a \<times> 'c) inference" assumes "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F Inf_FL"   and "\<iota>\<^sub>F\<^sub>L \<in> Inf_FL" shows "Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L)) \<in> Inf_F"
lens_unit_prod_sublens_2: fixes X :: "'d \<Longrightarrow> 'e" shows "0\<^sub>L +\<^sub>L X \<subseteq>\<^sub>L X"
simps_2: fixes left :: "'c \<Rightarrow> 'f \<Rightarrow> ('a \<times> 'c, 'd, 'e) gpv"   and right :: "'c \<Rightarrow> 'g \<Rightarrow> ('b \<times> 'c, 'd, 'e) gpv"   and s :: "'c"   and x :: "'g" shows "plus_intercept left right s (Inr x) = map_gpv (apfst Inr) id (right s x)"
TS_immediate_prefix_containment: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "0 < i" shows "vs @ butlast xs \<in> TS M2 M1 \<Omega> V m i"
last_prog_hd_prog: fixes p :: "'b"   and xs :: "'b memref list" shows "last_prog (hd_prog p xs) xs = last_prog p xs"
tso_store_invD_3: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and m :: "'a"   and fl :: "bool"   and ws :: "('b, 'c, 'd) mem_store_action list" assumes "tso_store_inv s" shows "sys_mem_store_buffers (mutator m) s \<noteq> mw_fA fl # ws"
intro: fixes f :: "'d \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'd"   and X :: "'d set" assumes "c1_on_open f f' X" shows "c1_on_open_euclidean f f' X"
rtc_implies_srtc: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "r\<^sup>*\<^sup>* a b" shows "relation.srtc r a b"
poly_root_factor_1: fixes r :: "'c"   and q :: "'c poly"   and k :: "'c" shows "(poly ([:r, 1::'c:] * q) k = (0::'c)) = (k = - r \<or> poly q k = (0::'c))"
ss_wt_Par_UNlift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and II :: "nat set"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "II \<in> UNpart cl P"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)"   and "\<And>n I. \<lbrakk>n < length cl; I \<in> P n\<rbrakk> \<Longrightarrow> sum (wt (cl ! n) s) I = sum (wt (dl ! n) t) (F n I)"   and "s \<approx> t" shows "sum (wt (Par cl) s) II = sum (wt (Par dl) t) (UNlift cl dl P F II)"
components_are_iso: fixes a :: "'a" assumes "Cop.ide a" shows "Cop.iso (Cop.map a)"
hcomp_ide_isomorphic: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.isomorphic g h"   and "src\<^sub>D f = trg\<^sub>D g" shows "D.isomorphic (f \<star>\<^sub>D g) (f \<star>\<^sub>D h)"
bnd: fixes x :: "real" assumes "a < x"   and "x < b" shows "norm (f' x) \<le> \<phi>' x"
conseq_extract_state_indep_prop: fixes P :: "'d set"   and R :: "bool"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. R"   and "R \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
q_def: shows "q = (\<lambda>f. \<Sum>h\<in>hom_components p. snd (q0 h) f)"
top_dres_def: shows "top \<equiv> dFAILi"
dependent_live_vars_Use_cases: fixes V :: "'var"   and as' :: "'edge list"   and as :: "'edge list"   and n' :: "'node"   and n :: "'node" assumes "(V, as', as) \<in> dependent_live_vars n'"   and "n -as\<rightarrow>* n'" shows "\<exists>nx as''. as = as' @ as'' \<and> n -as'\<rightarrow>* nx \<and> nx -as''\<rightarrow>\<^sub>d* n' \<and> V \<in> Use nx \<and> (\<forall>n''\<in>set (sourcenodes as'). V \<notin> Def n'')"
HT'_partial_eq_HT: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT'_partial \<pi> P (\<lambda>_. c) Q = HT_partial \<pi> P c Q"
simps_20: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "rec_finite_4 f1 f2 f3 f4 a\<^sub>4 = f4"
assms: assumes "PROP P"   and "CONSTRAINT_SLOT (True)" shows "PROP Q"
subst_apply_term_empty: fixes t :: "('c, 'd) Term.term" shows "t \<cdot> Var = t"
simps_1: shows "normalize_match MatchAny = [MatchAny]"
distinct_set_conv_list: fixes xs :: "'a list"   and g :: "'a \<Rightarrow> 'b" assumes "distinct xs" shows "prod g (set xs) = prod_list (map g xs)"
varsPB_def: fixes \<Phi> :: "('e, 'f) lit list set" shows "varsPB \<Phi> = \<Union> {varsC c |c. c \<in> \<Phi>}"
assms: shows "Q \<Longrightarrow>\<alpha> \<prec> Q'"
axiom_impall_def: fixes x :: "String.literal"   and p :: "fol fm" shows "axiom_impall x p \<equiv> if \<not> free_in x p then Thm (Imp p (Forall x p)) else Thm Truth"
WOC_iff_indRelRTPO_is_reduction_correspondence_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(weakly_operational_corresponding (TRel\<^sup>*) \<and> weak_reduction_correspondence_simulation (TRel\<^sup>+) Target) = weak_reduction_correspondence_simulation (indRelRTPO TRel) (STCal Source Target)"
eSuc_Sup: fixes A :: "enat set" assumes "A \<noteq> {}" shows "eSuc (Sup A) = Sup (eSuc ` A)"
strict_monoI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "order_class.strict_mono f"
vertex_colorings_def: fixes G :: "nat set \<times> nat set set" shows "vertex_colorings G \<equiv> {C. \<Union> C = uverts G \<and> (\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<and> (\<forall>c\<in>C. c \<noteq> {} \<and> (\<forall>u\<in>c. \<forall>v\<in>c. {u, v} \<notin> uedges G))}"
finite_glitOfC: fixes c :: "('fsym, 'psym) lit list" shows "finite (glitOfC c)"
atLeastSucLessThan_greaterThanLessThan: fixes l :: "nat"   and u :: "nat" shows "{Suc l..<u} = {l<..<u}"
sup_continuous_mono: fixes F :: "'b \<Rightarrow> 'c" assumes "sup_continuous F" shows "mono F"
gbinomial_Beta: fixes z :: "'a"   and n :: "nat" assumes "z + (1::'a) \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "z gchoose n = inverse ((z + (1::'a)) * Beta (z - of_nat n + (1::'a)) (of_nat n + (1::'a)))"
until_introL: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<phi> \<pi>l"   and "Shallow.until \<phi> \<psi> (map stl \<pi>l)" shows "Shallow.until \<phi> \<psi> \<pi>l"
inverse_arrows_def: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "VV.inverse_arrows f g \<equiv> VV.ide (VV.comp g f) \<and> VV.ide (VV.comp f g)"
arcs_ends: shows "arcs_ends H = pairself fv ` arcs_ends G"
S_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "(proj2_abs v \<in> S) = (v \<bullet> (M *v v) = 0)"
assms_2: shows "(b, cs) \<in> group_by f as"
b': shows "\<forall>r'\<in>R'. rightPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
seqp_receiveTE: fixes \<xi> :: "'e"   and l :: "'h"   and u\<^sub>m\<^sub>s\<^sub>g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains msg :: "'f" where "a = receive msg" and "\<xi>' = u\<^sub>m\<^sub>s\<^sub>g msg \<xi>" and "q = p"
tSplice_jkbpC: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "{t, t'} \<subseteq> SPR.jkbpC"   and "envObs a (tFirst t) = envObs a (tFirst t')"   and "tObsC t = tObsC t'" shows "t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t' \<in> SPR.jkbpC"
Complex_scale1: fixes a :: "real"   and b :: "real"   and c :: "real" shows "Complex (a * b) (a * c) = cor a * Complex b c"
simps_2: fixes x :: "'b list"   and xs :: "'b list list" shows "concat (x # xs) = x @ concat xs"
us_facts_4: shows "set us \<inter> snd ` set txs = {}"
contracting_fun_def: fixes g :: "'b \<Rightarrow> 'b" shows "contracting_fun g \<equiv> mono g \<and> g (0::'b) = (0::'b) \<and> (\<forall>n. g n \<le> n)"
ex_in_set_tree: fixes t :: "'b tree"   and x :: "'b" assumes "t \<noteq> \<langle>\<rangle>"   and "bst t" shows "\<exists>x'\<in>set_tree t. splay x' t = splay x t \<and> T_splay x' t = T_splay x t"
pair_measure_eq_bind: shows "M1 \<Otimes>\<^sub>M M2 = M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. return (M1 \<Otimes>\<^sub>M M2) (x, y)))"
replicate_mset_msubseteq_iff: fixes m :: "nat"   and a :: "'b"   and n :: "nat"   and b :: "'b" shows "(replicate_mset m a \<subseteq># replicate_mset n b) = (m = 0 \<or> a = b \<and> m \<le> n)"
elementary_interval: fixes a :: "'c"   and b :: "'c" obtains p :: "'c set set" where "p division_of cbox a b"
card_lists_length_eq: fixes A :: "'g set"   and n :: "nat" assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs = n} = card A ^ n"
val_of_nonzero_4: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p"   and "f \<noteq> padic_zero p"   and "prime p" shows "f (nat (padic_val p f)) = 0"
simps_1: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and x :: "'b" assumes "Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var" shows "ground_heads.ground_heads ground_heads_var (Var x) = ground_heads_var x"
assms: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> rel" shows "qSkel Y = qSkel X"
multiple_block_in: fixes n :: "nat"   and b :: "'a set"   and na :: "nat" assumes "0 < n"   and "b \<in># multiple_blocks na" shows "b \<in># mult_sys.multiple_blocks na n"
fps_deriv_mult_const_right: fixes f :: "'a fps"   and c :: "'a" shows "fps_deriv (f * fps_const c) = fps_deriv f * fps_const c"
echelon_foldl_condition3: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and m :: "'f"   and k :: "nat" assumes "is_bezout_ext bezout"   and "A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "\<forall>m. is_zero_row_upt_k m k A"   and "echelon_form_upt_k A k"   and "k < ncols A" shows "mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (bezout_iterate (interchange_rows A (0::'f) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d))) (nrows A - Suc 0) (0::'f) (mod_type_class.from_nat k) bezout)) = 0"
fresh_8: fixes a :: "W.tvar"   and x2 :: "W.var"   and x3 :: "trm"   and x1 :: "trm" shows "a \<sharp> Let x2 be x1 in x3 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
zmultiset_nonemptyE: fixes A :: "'b zmultiset" assumes "A \<noteq> {#}\<^sub>z" obtains x :: "'b" where "x \<in>#\<^sub>z A"
dirichlet_prod_assoc_aux2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "dirichlet_prod (dirichlet_prod f g) h n = (\<Sum>(a, b, c)\<in>{(a, b, c). a * b * c = n}. f a * g b * h c)"
D2_UNION: fixes P :: "D2 \<Rightarrow> 'd set" shows "\<Union> (range P) = \<Union> (P ` {D1, D2})"
c': fixes i :: "nat" shows "c' i = fps_to_fls (c'' i)"
abs_eq_iff: fixes x :: "('a, 'b, 'c, 'd) Tree"   and y :: "('a, 'b, 'c, 'd) Tree" shows "(abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
inj_iff: fixes f :: "'b \<Rightarrow> 'c" shows "inj f = (inv f \<circ> f = id)"
new_lift_1: fixes c :: "'a"   and t :: "'a FOL_Fitting.term" assumes "new_term c t" shows "new_term c (liftt t)"
P: shows "\<not> nat (padic_val p (a \<oplus>\<^bsub>padic_int p\<^esub> b)) < min (nat (padic_val p a)) (nat (padic_val p b))"
asms_2: fixes x :: "'a" shows "Af__ x x \<and> Q"
assms_6: fixes e :: "char list"   and ts :: "'a trm list" shows "P (Pred e ts)"
chkargs_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var" shows "rpo_optim.chkargs gt t s = (\<forall>s'\<in>set (args s). gt t s')"
filter_empty_mset: fixes P :: "'a \<Rightarrow> bool" shows "filter_mset P {#} = {#}"
monotoneD: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i"   and x :: "'h"   and y :: "'h" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
extreme_qfp: fixes c :: "'a" assumes "extreme_bound A (\<sqsubseteq>) C c"   and "\<forall>q\<in>A. \<forall>x\<in>A. f q \<sim> q \<longrightarrow> x \<sqsubseteq> f q \<longrightarrow> x \<sqsubseteq> q"   and "monotone_on A (\<sqsubseteq>) (\<sqsubseteq>) f" shows "extreme {q \<in> A. f q \<sim> q \<or> f q = q} (\<lambda>x y. y \<sqsubseteq> x) c"