hcju/isabelleps huggingface dataset:
Abs_bit0_induct: fixes P :: "'a bit0 \<Rightarrow> bool"   and x :: "'a bit0" assumes "\<And>y. y \<in> {0..<2 * int CARD('a)} \<Longrightarrow> P (Abs_bit0 y)" shows "P x"
part_complement_reversal: fixes y :: "i"   and x :: "i" assumes "CEMC"   and "y \<noteq> u"   and "P x y" shows "P (\<midarrow> y) (\<midarrow> x)"
action_loc_aux_simps_3: fixes ad' :: "'a"   and al' :: "addr_loc"   and P :: "'b prog"   and ad :: "'a"   and al :: "addr_loc"   and v :: "'a val" shows "((ad', al') \<in> action_loc_aux P (NormalAction (ReadMem ad al v))) = (ad = ad' \<and> al = al')"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* xvec"   and "distinct xvec" shows "Prop (P \<parallel> Q')"
isubexpr_typing: fixes e1 :: "exp"   and e0 :: "exp"   and CT :: "nat \<Rightarrow> classDef option"   and C :: "nat" assumes "e1 \<in> isubexprs(e0)"   and "CT;Map.empty \<turnstile> e0 : C" shows "\<exists>D. CT;Map.empty \<turnstile> e1 : D"
idempotent_one_closed: shows "top \<sqinter> top = top"
add_mset_add_mset_same_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a, A = a, B) = (A = B)"
accs_is_ranked: fixes t :: "'L tree"   and q :: "'Q" assumes "accs \<delta> t q" shows "t \<in> ranked_trees A"
part_equivpI: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<exists>x. R x x"   and "symp R"   and "transp R" shows "part_equivp R"
dg_1_def: fixes \<aa> :: "V"   and \<ff> :: "V" shows "dg_1 \<aa> \<ff> = [ZFC_in_HOL.set {\<aa>}, ZFC_in_HOL.set {\<ff>}, ZFC_in_HOL.set {\<langle>\<ff>, \<aa>\<rangle>}, ZFC_in_HOL.set {\<langle>\<ff>, \<aa>\<rangle>}]\<^sub>\<circ>"
cdiv: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "o[F](\<lambda>x. f x / c) = o[F](f)"
xtab: shows "compP\<^sub>2 P\<^sub>1,C,M \<rhd> compxE\<^sub>2 body 0 (length []) / {..<length (compE\<^sub>2 body)},length []"
globally_sound_modes_compatible: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" assumes "globally_sound_mode_use (cms, mem)" shows "compatible_modes (map snd cms)"
inner_diff_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (minus y z) = inner x y - inner x z"
mndet_refine_FD: fixes a :: "'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" assumes "a \<in> A" shows "mndet A P \<le> (a \<rightarrow> P a)"
init_segment_antisym: fixes S :: "nat set"   and T :: "nat set" assumes "init_segment S T"   and "init_segment T S" shows "S = T"
integral_restrict_Int: fixes T :: "'c set"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "integral T (\<lambda>x. if x \<in> S then f x else (0::'b)) = integral (S \<inter> T) f"
assms: shows "run_one_step d i (s, vs, ves, $Binop_i x261 x262) = (s', vs', res)"
add_divide_eq_if_simps_2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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 :: "'b"   and z :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus (divide a z) b = (if z = zero then b else divide (plus a (times b z)) z)"
finter_assoc: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |\<inter>| B |\<inter>| C = A |\<inter>| (B |\<inter>| C)"
valid_def: fixes t :: "'a tree" shows "local.valid t = (well_shaped t \<and> local.well_valued0 t \<and> (t \<noteq> Leaf \<longrightarrow> val t \<noteq> None))"
poly_resultant_zero: fixes p :: "'a poly poly"   and q :: "'a poly poly"   and x :: "'a"   and y :: "'a" assumes "0 < degree p \<or> 0 < degree q"   and "poly2 p x y = (0::'a)"   and "poly2 q x y = (0::'a)" shows "poly (resultant p q) x = (0::'a)"
uinvarm_def: fixes G :: "('a, 'b) graph"   and v0 :: "'a"   and v :: "'a"   and wl :: "'a set"   and res :: "'a \<Rightarrow> ('a \<times> 'b \<times> 'a) list option"   and it :: "('b \<times> 'a) set"   and \<sigma> :: "('a \<Rightarrow> 'b infty option) \<times> ('a \<Rightarrow> (('a \<times> 'b \<times> 'a) list \<times> 'b) option)" assumes "Dijkstra G v0" shows "Dijkstra.uinvarm G v0 v wl res it \<sigma> \<equiv> Dijkstra.uinvar G v0 v wl res it (Dijkstra.\<alpha>s \<sigma>) \<and> Dijkstra.dinvarm \<sigma>"
monad_prob_prob: shows "monad_prob return_pmf (\<bind>) (\<bind>)"
bdd_below_def: fixes A :: "'a set" shows "bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<le> x)"
reduced_word_forI_length: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and as :: "'b list"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero"   and "as \<in> lists A"   and "monoid_add.sum_list plus zero as = a"   and "length as = monoid_add.word_length plus zero A a" shows "monoid_add.reduced_word_for plus zero A a as"
statEqEnt: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Phi> :: "'c" assumes "\<Psi> \<simeq> \<Psi>'"   and "\<Psi> \<turnstile> \<Phi>" shows "\<Psi>' \<turnstile> \<Phi>"
lemDirectionJoining: fixes p :: "'a Point"   and v :: "'a Vector" shows "from p to p \<leadsto> v = v"
prems_1: shows "(\<sigma>', coll') \<in> cbig P \<sigma>"
order_iff: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "less_eq_prod a b = (a = inf_prod a b)"
intros_7: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and c :: "String.literal"   and T :: "typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "const_type (sig \<Theta>) (const_of_class c) = Some (itselfT Core.aT \<rightarrow> propT)"   and "wf_type (sig \<Theta>) T"   and "has_sort (osig (sig \<Theta>)) T (insert c full_sort)" shows "\<Theta>,\<Gamma> \<turnstile> mk_of_class T c"
gd: fixes x :: "real" assumes "x \<in> {c..b} - T" shows "g differentiable at x within {c..b}"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(DBVar x1 = DBVar y1) = (x1 = y1)"
i: shows "ideal_generated {a, b} = {y. \<exists>f. (\<Sum>i\<in>{a, b}. f i * i) = y}"
mHom_eq_ker: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and a :: "'e"   and b :: "'e" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "a \<in> carrier M"   and "b \<in> carrier M"   and "a \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> b \<in> ker\<^bsub>M,N\<^esub> f" shows "f a = f b"
fun_upd_twist: fixes a :: "'a"   and c :: "'a"   and m :: "'a \<Rightarrow> 'b"   and b :: "'b"   and d :: "'b" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
iT_Mult_iprev: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" shows "iprev (n * k) (I \<otimes> k) = iprev n I * k"
given_quot_alt: fixes f :: "nat"   and l :: "nat" assumes "0 < f" shows "(l * f + f - Suc 0) div f = l"
add_pow_rdistr_int: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "a \<otimes> [k] \<cdot> b = [k] \<cdot> (a \<otimes> b)"
redT_upds_preserves_lock_thread_ok: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option"   and t :: "'b"   and xw :: "'c \<times> 'a \<Rightarrow>f nat"   and tas :: "('b, 'c, 'd) new_thread_action list"   and las :: "'a lock_actions"   and xw' :: "'c \<times> 'a \<Rightarrow>f nat" assumes "lock_thread_ok ls ts"   and "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "lock_thread_ok (redT_updLs ls t las) (redT_updTs ts tas(t \<mapsto> xw'))"
simps_15: fixes v :: "fm"   and va :: "fm" shows "disjuncts (Iff v va) = [Iff v va]"
integral_divide_zero: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and c :: "'c" shows "LINT x|M. f x / c = integral\<^sup>L M f / c"
sp_spec_subj_subj_def: fixes a :: "partition_id_t"   and b :: "partition_id_t" shows "Policy.sp_spec_subj_subj a b \<equiv> \<exists>f m1 m2. Policy.sp_spec_subj_obj a (FILEP f) m1 \<and> Policy.sp_spec_subj_obj b (FILEP f) m2"
t_ins_setp_t_ins_set_eq: fixes X :: "'b t_type" shows "t_ins_setp X = (\<lambda>x. x \<in> t_ins_set X)"
Un_UNIV_left: fixes B :: "'c set" shows "UNIV \<union> B = UNIV"
assms: shows "\<FF> : \<AA> \<mapsto>\<^bsub>dg_GRPH \<alpha>\<^esub> \<BB>"
can_def: fixes u :: "'a E.term"   and t :: "'a E.term" shows "can u t \<equiv> \<lbrace>E.Inv (u\<^bold>\<down>) \<^bold>\<cdot> t\<^bold>\<down>\<rbrace>"
hom_finsum: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'c"   and A :: "'g set" assumes "ring_hom_ring R S h"   and "f \<in> A \<rightarrow> carrier R" shows "h (finsum R f A) = finsum S (h \<circ> f) A"
R_implies_SR_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "R \<V>' Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "SR \<V> Tr\<^bsub>ES\<^esub>"
ccomp_bigstep: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and stk :: "int list" assumes "(c, s) \<Rightarrow> t" shows "ccomp c \<turnstile> (0, s, stk) \<rightarrow>* (size (ccomp c), t, stk)"
dfa_in_ST: fixes u :: "'a list" shows "dfa.nextl MS (dfa.init MS) u \<in> ST.nextl {q0} u"
assms_1: shows "M' \<equiv> reset M n (v c) d"
inv: fixes J :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'a"   and i :: "'a" assumes "function_to_finmap J f f'"   and "i \<in> J" shows "f' (f i) = i"
continuous_within_sequentially: fixes a :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous (at a within s) f = (\<forall>x. (\<forall>n. x n \<in> s) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
fls_base_factor_to_fps_right_inverse_nonzero: fixes y :: "'b"   and f :: "'b fls" assumes "y \<noteq> (0::'b)" shows "fls_base_factor_to_fps (fls_right_inverse f y) = fps_right_inverse (fls_base_factor_to_fps f) y"
mker_submodule: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "submodule R M (ker\<^bsub>M,N\<^esub> f)"
lim: fixes x :: "'n" assumes "x \<in> S" shows "(\<lambda>k. f k x) \<longlonglongrightarrow> g x"
mapping_of_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'c" shows "mapping_of (monom m a) = Poly_Mapping.single m a"
distinct_20: fixes x51 :: "trm"   and x52 :: "POPLmarkRecord.type"   and x21 :: "POPLmarkRecord.type"   and x22 :: "trm" shows "x51 \<bullet>\<^sub>\<tau> x52 \<noteq> (\<lambda>:x21. x22)"
standard_open_is_zariski_open: fixes x :: "'a" assumes "x \<in> R" shows "is_zariski_open \<D>(x)"
comp_char: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "g \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f = (if J_C.seq g f then J_C.MkArr (J_C.Dom f) (J_C.Cod g) (vertical_composite.map (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map f) (J_C.Map g)) else J_C.null)"
card_eq_1_singleton: fixes A :: "'b set" assumes "card A = 1" obtains x :: "'b" where "A = {x}"
unsafe_asciis_of_literal_def: fixes xs :: "String.literal" shows "unsafe_asciis_of_literal xs = String.asciis_of_literal xs"
dfs_reachable_invar_def: fixes D :: "'a \<Rightarrow> bool"   and V :: "'a set"   and W :: "'a set"   and brk :: "bool" shows "local.dfs_reachable_invar D V W brk = (set w\<^sub>0 \<subseteq> W \<union> V \<and> W \<union> V \<subseteq> (E_of_succ succ)\<^sup>* `` set w\<^sub>0 \<and> E_of_succ succ `` V \<subseteq> W \<union> V \<and> Collect D \<inter> V = {} \<and> (brk \<longrightarrow> Collect D \<inter> (E_of_succ succ)\<^sup>* `` set w\<^sub>0 \<noteq> {}))"
link_struct: fixes hsx :: "'b tree"   and x :: "'b"   and hsy :: "'b tree"   and y :: "'b"   and hs :: "'b tree" shows "\<exists>l a. link \<langle>hsx, x, \<langle>hsy, y, hs\<rangle>\<rangle> = \<langle>l, a, hs\<rangle>"
nAct_mono_back: fixes c :: "'id"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat" assumes "\<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle> \<le> \<langle>c #\<^bsub>enat n'\<^esub>inf_llist t\<rangle>" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
prj1_simps_1: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.arr \<p>\<^sub>1[f, g]"
tcomp_pre_def: fixes Q :: "'e \<Rightarrow> bool"   and T :: "('f \<times> 'e) set"   and P :: "'f \<Rightarrow> bool" shows "tcomp_pre Q T P \<equiv> \<lambda>a. Q a \<and> (\<forall>a'. (a', a) \<in> T \<longrightarrow> P a')"
part_transitivity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "PM"   and "P x y"   and "P y z" shows "P x z"
pfa_capacity_def: shows "pfa_capacity \<equiv> \<lambda>(a, n). Array.len a"
fls_unit_base_subdegree_power_2: fixes f :: "'b fls"   and y :: "'b"   and n :: "nat" assumes "f $$ fls_subdegree f * y = (1::'b)" shows "fls_subdegree (f ^ n) = int n * fls_subdegree f"
measurable_count_space_eq2: fixes A :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "finite A" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
rep_eq: shows "poly.coeff Q' = (\<lambda>k. if n < k then 0 else (- 1) ^ (n - k) * e' (n - k))"
max_id: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.non_strict_order less_eq"   and "less_eq y x" shows "ord.max less_eq x y = x"
prefeq_reflection: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "|~ x = y" shows "x \<equiv> y"
unfold_token_run_eq: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "\<MM>.token_run x (Suc n) = \<up>step (\<UU>.token_run x n) (w n)"
r3: shows "right (len v1 ts c) \<le> right (ext v1)"
butlast_last: fixes lm1 :: "'a list"   and aa :: "nat"   and n :: "nat"   and lm2 :: "'a list"   and lm3 :: "'a list"   and lm4 :: "'a list" assumes "length lm1 = aa" shows "(lm1 @ (0::'a) \<up> n @ last lm2 # lm3 @ butlast lm2 @ (0::'a) # lm4) ! (aa + n) = last lm2"
LPDi_Test_1: fixes i :: "nat" shows "LPDi i (MSkip 0) \<subseteq> {MSkip 0}"
properfactor_of_zero_2: fixes b :: "'a list" assumes "b \<in> carrier (K [X])" shows "properfactor (K [X]) b \<zero>\<^bsub>K [X]\<^esub> = (b \<noteq> \<zero>\<^bsub>K [X]\<^esub>)"
rec_conj_def: shows "rec_conj = Cn 2 rec_sg [Cn 2 rec_mult [recf.id 2 0, recf.id 2 1]]"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and sum3 :: "('b, 'c, 'd) sum3" shows "P (case sum3 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = ((\<forall>x1. sum3 = In1 x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum3 = In2 x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. sum3 = In3 x3 \<longrightarrow> P (f3 x3)))"
hom_mult_eq_zero: fixes x :: "'a poly"   and y :: "'a poly"   and a :: "'a" assumes "x * y = 0" shows "poly x a * poly y a = (0::'a)"
total_ES2_C2_inter_Upsilon2_inter_N1_inter_Delta1: shows "EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub>)"
1: shows "punit.lc (rep_list p) \<noteq> (0::'b)"
in_snd_sym_preproc_auxE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set (snd (sym_preproc_aux gs ks (vs, fs)))" obtains "p \<in> set fs"   | g :: "'t \<Rightarrow>\<^sub>0 'c" and t :: "'a" where "g \<in> set gs" and "p = monom_mult (1::'c) t g"
rm_vars_comp: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and vs :: "'c set"   and t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "range_vars \<delta> \<inter> vs = {}" shows "t \<cdot> rm_vars vs (\<delta> \<circ>\<^sub>s \<theta>) = t \<cdot> rm_vars vs \<delta> \<circ>\<^sub>s rm_vars vs \<theta>"
all_coarser_partitions_with_list_equivalence: fixes Ps :: "'b set list list"   and elem :: "'b" assumes "\<forall>P\<in>set Ps. distinct P" shows "set (map set (all_coarser_partitions_with_list elem Ps)) = all_coarser_partitions_with elem (set (map set Ps))"
simps: fixes n :: "nat" shows "Discrete.log n = (if n < 2 then 0 else Suc (Discrete.log (n div 2)))"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" shows "hindrance_by \<Gamma> f \<epsilon> = (\<exists>a. a \<in> A \<Gamma> \<and> a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<and> d_OUT f a < weight \<Gamma> a \<and> \<epsilon> < weight \<Gamma> a - d_OUT f a)"
u_def: shows "u = (\<lambda>x. min (h x) 0)"
xb: shows "x \<le> Min {y \<in> S. x \<le> y}"
mult_add_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v\<^sub>1 \<in> carrier_vec nc"   and "v\<^sub>2 \<in> carrier_vec nc" shows "A *\<^sub>v (v\<^sub>1 + v\<^sub>2) = A *\<^sub>v v\<^sub>1 + A *\<^sub>v v\<^sub>2"
distinct_list_single_elem_decomp: shows "{xs. set xs \<subseteq> {0::'b} \<and> distinct xs} = {[], [0::'b]}"
npOne: fixes n :: "nat" shows "1\<^sub>r^\<^bsup>R n\<^esup> = 1\<^sub>r"
wellformed_finite: fixes G :: "nat set \<times> nat set set" assumes "finite (uverts G)"   and "uwellformed G" shows "finite_graph G"
that: fixes pre :: "'a list"   and suf :: "'a list" assumes "qs = pre @ [a] @ suf"   and "\<And>x. x \<in> S \<Longrightarrow> x \<notin> set suf"   and "Lxy pre S = as" shows "thesis"
axioms_1: fixes G :: "('a, 'b) graph" assumes "valid_unSimpGraph G" shows "valid_unMultigraph G"
induces_unit_counit_adjunction': shows "unit_counit_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta>.map \<epsilon>.map"
distproj_in_proj_set_1: fixes x :: "'c"   and A :: "'c set" assumes "proj_set x A \<noteq> {}" shows "distproj x A \<in> proj_set x A"
trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
verify_plan_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" shows "verify_plan problem \<pi>s = (if ast_problem.well_formed problem then if check_plan' problem \<pi>s then Inr () else Inl STR ''Invalid plan'' else Inl STR ''Problem not well formed'')"
pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option"   and Pa :: "'c \<Rightarrow> bool" assumes "pred_option P x"   and "\<And>z. \<lbrakk>z \<in> set_option x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_option Pa x"
small_big_mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> o[F](g1)"   and "f2 \<in> O[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
P3: shows "P3 \<in> carrier_mat (m - 1) (m - 1)"
am5_lem: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "rel_ad (x O y) O rel_ad (x O rel_ad y) = rel_ad x"
bi_unique_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
assms_1: shows "\<forall>k. j < k \<and> k < i \<longrightarrow> \<not> xs ! k < xs ! i"
assms: shows "\<tau>exec_1 P t \<sigma> \<sigma>'"
fin_def: fixes A :: "'e \<times> 'f \<times> 'd" shows "fin A = snd (snd A)"
deflation_axioms: fixes d :: "'a defl" shows "deflation (cast\<cdot>d)"
symcl_def: fixes Rel :: "('b \<times> 'b) set" shows "symcl Rel = Rel \<union> Rel\<inverse>"
f_def: fixes x :: "nat" shows "f x = c_pair n x"
ffun_upd_twice: fixes f :: "('a, 'b) ffun"   and x :: "'a"   and u :: "'b"   and v :: "'b" shows "f(x \<mapsto> u, x \<mapsto> v)\<^sub>f = f(x \<mapsto> v)\<^sub>f"
subclseq_trans: fixes A :: "qtname"   and B :: "qtname"   and G :: "prog"   and C :: "qtname" assumes "G\<turnstile>A\<preceq>\<^sub>C B"   and "G\<turnstile>B\<preceq>\<^sub>C C" shows "G\<turnstile>A\<preceq>\<^sub>C C"
beta_reduces_snd: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P \<rightarrow>\<beta>\<^sup>* P'" shows "Snd P \<rightarrow>\<beta>\<^sup>* Snd P'"
simps_1: fixes old :: "nat"   and v :: "nat"   and f :: "face" shows "natToVertexListRec old v f [] = []"
ParityGame_ext_def: fixes player0 :: "'c set"   and priority :: "'c \<Rightarrow> nat"   and more :: "'d" shows "\<lparr>player0 = player0, priority = priority, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons ParityGame_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple player0 priority) more"
add_nonneg_eq_0_iff: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "{#} \<subseteq># x"   and "{#} \<subseteq># y" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
D'_def: shows "D' \<equiv> {0<..<1} \<times> {0<..<1} \<times> {0<..<1}"
in_pdata_pairs_to_listE: fixes h :: "'t \<Rightarrow>\<^sub>0 'd"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" assumes "h \<in> set (pdata_pairs_to_list ps)" obtains f :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" and g :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" where "(f, g) \<in> set ps \<or> (g, f) \<in> set ps" and "h = monom_mult ((1::'d) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)"
mod_poly_eq: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "eucl_rel_poly x y (q, r)" shows "x mod y = r"
weak_matrix_match_def: fixes A1 :: "int poly fract list list"   and A2 :: "int poly fract list list"   and B1 :: "int poly fract list list" shows "weak_matrix_match A1 A2 B1 \<equiv> mat (rat_poly.row_length A1) (length A1) A1 \<and> mat (rat_poly.row_length A2) (length A2) A2 \<and> mat (rat_poly.row_length B1) 1 B1 \<and> A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> length A1 = rat_poly.row_length A2"
statI_prop: shows "\<forall>I. statT = IfaceT I \<longrightarrow> is_iface G I \<and> mode \<noteq> SuperM"