hcju/isabelleps huggingface dataset:
wt'_wt: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e :' T" shows "P,E,h \<turnstile> e : T"
Nom: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and p :: "('c, 'b) fm"   and i :: "'b"   and j :: "'b" assumes "Hintikka A H"   and "\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A"   and "p at i in' H"   and "Nom j at i in' H" shows "p at j in' H"
dsg4: fixes y :: "'a"   and x :: "'a" shows "ar ar y \<cdot> ar ar x = ar ar x \<cdot> ar ar y"
simps_3: fixes v :: "jvm_prog"   and vb :: "nat option"   and vd :: "heap"   and vg :: "sheap" shows "exec (v, vb, vd, [], vg) = None"
conversepD: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'b" assumes "r\<inverse>\<inverse> b a" shows "r a b"
ldropn_inf_llist: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "ldropn n (inf_llist f) = inf_llist (\<lambda>m. f (m + n))"
hyps_3: assumes "\<And>x. x \<in> F \<Longrightarrow> 0 < f x" shows "log a (prod f F) = sum (log a \<circ> f) F"
lower_rank_of_def: fixes carrier' :: "'a set" shows "\<E>.lower_rank_of carrier' \<equiv> Min {card B |B. \<E>.basis_in carrier' B}"
sig_red_singleI: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "rep_list f \<noteq> 0"   and "t + punit.lt (rep_list f) \<in> keys (rep_list p)"   and "q = p - monom_mult (lookup (rep_list p) (t + punit.lt (rep_list f)) / punit.lc (rep_list f)) t f"   and "ord_term_lin.is_le_rel sing_reg"   and "ordered_powerprod_lin.is_le_rel top_tail"   and "sing_reg (t \<oplus> lt f) (lt p)"   and "top_tail (t + punit.lt (rep_list f)) (punit.lt (rep_list p))" shows "sig_red_single sing_reg top_tail p q f t"
InvariantImpliedLiteralsAfterUnitPropagate: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and uClause :: "Literal list"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "formulaEntailsClause F uClause"   and "isUnitClause uClause uLiteral (elements M)"   and "M' = M @ [(uLiteral, False)]" shows "InvariantImpliedLiterals F M'"
convergence_ext: fixes xs :: "('a \<times> ('a, 'b) operation) list"   and ys :: "('a \<times> ('a, 'b) operation) list"   and s :: "'b \<Rightarrow> 'a set \<times> 'a set" assumes "set xs = set ys"   and "hb.concurrent_ops_commute xs"   and "hb.concurrent_ops_commute ys"   and "distinct xs"   and "distinct ys"   and "hb.hb_consistent xs"   and "hb.hb_consistent ys" shows "hb.apply_operations xs s = hb.apply_operations ys s"
wf_mbufn_in_set_Mini: fixes i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and buf :: "event_data option list set list list" assumes "wf_mbufn i js Ps buf"   and "[] \<in> set buf" shows "Mini i js = i"
rename_ensures: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "(rename h F \<in> h ` A ensures h ` B) = (F \<in> A ensures B)"
hyp: fixes d :: "nat"   and dir :: "dir" assumes "d < length p" shows "set (gridgen (child p dir d) {d' \<in> ds. d' \<le> d} l) = lgrid (child p dir d) {d' \<in> ds. d' \<le> d} (level p + Suc l)"
last_changes_implies_send_when_msgs_nonempty: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "msgs (S t i) cid \<noteq> []"   and "msgs (S t (i + 1)) cid \<noteq> []"   and "last (msgs (S t i) cid) = Marker"   and "last (msgs (S t (i + 1)) cid) \<noteq> Marker"   and "channel cid = Some (p, q)" shows "\<exists>u u' m. t ! i = Send cid p q u u' m"
weighted_FO_resolution_prover_with_size_timestamp_factors_axioms: shows "weighted_FO_resolution_prover_with_size_timestamp_factors S (\<cdot>a) id_subst (\<odot>) renamings_apart atm_of_atms mgu less_atm timestamp_factor"
filterlim_at_bot_le: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" shows "filterlim f at_bot F = (\<forall>Z\<le>c. \<forall>\<^sub>F x in F. f x \<le> Z)"
IntI: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
h1b_helper_leq: fixes leq :: "(real \<times> real \<times> real) list" assumes "\<forall>(a, b, c)\<in>set leq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0" shows "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set leq. a * x\<^sup>2 + b * x + c \<le> 0"
remove1_def: shows "remove1 \<equiv> \<lambda>uu uua. rec_list (\<lambda>x. []) (\<lambda>y xs xsa x. if x = y then xs else y # xsa x) uua uu"
a2: shows "bdd_above (L2_set (cmod \<circ> x) ` Collect finite)"
assms_3: shows "(I, J) = getIJ Qp Qn V"
rel_interior_ereal_semiline: fixes a :: "ereal" shows "rel_interior {y. a \<le> ereal y} = {y. a < ereal y}"
integrable_spike_finite: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "f integrable_on T" shows "g integrable_on T"
constrains_weaken: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B co B'"
null_sets_completion_iff2: fixes A :: "'a set"   and M :: "'a measure" shows "(A \<in> null_sets (completion M)) = (\<exists>N'\<in>null_sets M. A \<subseteq> N')"
InrI: fixes b :: "'a"   and B :: "'a set"   and A :: "'b set" assumes "b \<in> B" shows "Inr b \<in> A <+> B"
powr_half_sqrt: fixes x :: "real" assumes "0 \<le> x" shows "x powr (1 / 2) = sqrt x"
vequalityE: fixes A :: "V"   and B :: "V" assumes "A = B" obtains "A \<subseteq>\<^sub>\<circ> B" and "B \<subseteq>\<^sub>\<circ> A"
lcoset_refl: fixes G :: "'b set"   and a :: "'b" assumes "Group G" shows "a \<in> a +o G"
is_map: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "map_in_bicategory V H \<a> \<i> src trg f" shows "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g"
prv_LLq_suc_Num_pls_Num: fixes t :: "'trm"   and n :: "nat" assumes "t \<in> atrm" shows "prv (LLq (suc (Num n)) (pls (suc t) (Num n)))"
monad_prob_envT': assumes "monad_prob Monad_Overloading.return Monad_Overloading.bind sample" shows "monad_prob Monad_Overloading.return Monad_Overloading.bind sample"
HOrcvdMsgs_def: fixes A :: "('e, 'g, 'f, 'h) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'e"   and HO :: "'e set"   and cfg :: "'e \<Rightarrow> 'g" shows "HOrcvdMsgs A r p HO cfg \<equiv> \<lambda>q. if q \<in> HO then Some (sendMsg A r q p (cfg q)) else None"
show_law_real: fixes r :: "real" shows "show_law showsp_real r"
bl_to_bin_lt2p_aux: fixes bs :: "bool list" shows "bl_to_bin_aux bs 1 < (1 + 1) * 2 ^ length bs"
bind_scale_spmf: fixes r :: "real"   and p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf" assumes "r \<le> 1" shows "scale_spmf r p \<bind> f = p \<bind> (\<lambda>x. scale_spmf r (f x))"
hyps: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (s, l, r) B stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps0 (s + off, l, r) (A @ shift B off) stp = (s' + off, l', r')"
mod_exE: fixes P :: "'a \<Rightarrow> assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> (\<exists>\<^sub>Ax. P x)" obtains x :: "'a" where "h \<Turnstile> P x"
simps_2: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and k :: "nat" shows "alternate f g (Suc k) = alternate g f k \<circ> f"
CleqC': shows "P \<turnstile> C \<preceq>\<^sup>* C'"
prems: shows "([:- a, 1::'a:] * p) \<circ>\<^sub>p q \<noteq> 0"
nonneg_Ax: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "le_vec 0 (A *v x)"
vec_times_mat_eqD_3: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "[y\<^sub>v*A]=c" shows "dim_row A\<^sup>T = dim_vec c"
Lcm_dvd_nat: fixes M :: "nat set"   and n :: "nat" assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
T_inorder2_Node: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and xs :: "'b list" shows "T_inorder2 \<langle>l, x, r\<rangle> xs = T_inorder2 r xs + T_inorder2 l (x # Root_Balanced_Tree.inorder2 r xs) + 1"
permute_swap_cancel: fixes a :: "atom"   and b :: "atom"   and x :: "'a" shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
gamma_s_Top: shows "\<gamma>\<^sub>s \<top> = UNIV"
compl_inf_bot: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) x = bot"
All_mapping_update_iff: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "All_mapping (update k v m) P = (P k v \<and> All_mapping m (\<lambda>k' v'. k = k' \<or> P k' v'))"
wp: shows "wp\<^sub>1 c1 (wp\<^sub>1 c2 Q) l s"
tuple: shows "\<guillemotleft>tuple f g : local.dom f \<rightarrow> local.prod (cod f) (cod g)\<guillemotright>"
chamberD_simplex: fixes x :: "'a set" assumes "domain.chamber x" shows "x \<in> X"
indexu_def: fixes s :: "('a, 'b) state" shows "\<I>\<^sub>u s = fst \<circ> the \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>u s)"
cannot_y: fixes r :: "(att \<times> char list, att, nat) env"   and r' :: "(att \<times> char list, att, nat) env" assumes "Q r"   and "r \<midarrow>y\<rightarrow> r'" shows "False"
wordinterval_CIDR_split_existential: fixes x :: "'b word"   and w :: "'b wordinterval" assumes "x \<in> wordinterval_to_set w" shows "\<exists>s. s \<in> set (wordinterval_CIDR_split_prefixmatch w) \<and> x \<in> prefix_to_wordset s"
io_11: shows "frontier inside = path_image (complex_of \<circ> c)"
assms_2: shows "p (deg R p) [^] n \<noteq> \<zero>"
comp_converter_assoc: fixes conv1 :: "('m, 'n, 's, 't) converter"   and conv2 :: "('s, 't, 'q, 'r) converter"   and conv3 :: "('q, 'r, 'o, 'p) converter" shows "(conv1 \<odot> conv2) \<odot> conv3 = conv1 \<odot> conv2 \<odot> conv3"
inducts: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" assumes "not_final_thread s t" obtains x :: "'x" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(x__, ln__)\<rfloor>" and "\<not> final x__"   | x :: "'x" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, lna__)\<rfloor>" and "lna__ \<noteq> no_wait_locks"   | x :: "'x" and w :: "'h wait_set_status" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "wset s t = \<lfloor>w\<rfloor>"
tvs_bind_fv2_subset: fixes a :: "variable"   and \<tau> :: "typ"   and i :: "nat"   and B :: "Core.term" shows "tvs (bind_fv2 (a, \<tau>) i B) \<subseteq> tvs B"
inject_1: fixes x5 :: "String.literal"   and y5 :: "String.literal" shows "(Class x5 = Class y5) = (x5 = y5)"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid_axioms G"
disjDomGD_def: fixes p :: "('c \<Rightarrow> 'd option) list" shows "disjDomGD p = disjDom (butlast p)"
B: shows "sinks I D u (x # xs) = sinks I D u xs"
n_mult_lb2': fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<le> !y"
UST: shows "(\<lambda>x. if x \<in> S' then g x else h x) ` U \<subseteq> S \<union> T"
A_dim_intro_1: shows "\<not> (dim_row A = 0 \<or> dim_col A = 0)"
iso_sym: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
x_to_y: shows "from x to y = \<lparr>tdir = tval y - tval x, xdir = 0::'a, ydir = 0::'a, zdir = 0::'a\<rparr>"
fmap_def: shows "fmap = COERCE((udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'g \<rightarrow> udom\<cdot>'g, ('e \<rightarrow> 'f) \<rightarrow> 'e\<cdot>'g \<rightarrow> 'f\<cdot>'g)\<cdot>fmapU"
atom_components_eq_iff: fixes a :: "atom"   and b :: "atom" shows "(a = b) = (sort_of a = sort_of b \<and> nat_of a = nat_of b)"
funpow_increasing: fixes m :: "nat"   and n :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "m \<le> n"   and "mono f" shows "(f ^^ n) \<top> \<le> (f ^^ m) \<top>"
le_join_disj: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.SL_top le join Top"   and "le x y \<or> le x z" shows "le x (join y z)"
sorted_list_of_fset_simps_1: fixes S :: "'b fset" shows "set (sorted_list_of_fset S) = fset S"
le_less: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
un_MSLCons2_def: fixes msllist :: "'a msllist" shows "un_MSLCons2 msllist = (case msllist of MSLCons x21 x22 \<Rightarrow> x22)"
mgenerator2Tr1: fixes n :: "nat"   and j :: "nat"   and k :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "0 < n"   and "j \<le> n"   and "k \<le> n"   and "distinct_pds K n P" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n) = (\<gamma>\<^bsub>k j\<^esub>) *\<^sub>a (\<delta>\<^bsub>j j\<^esub>)"
simps_3: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and a :: "fol" shows "semantics e f g (Atom a) = (case a of Rl i l \<Rightarrow> if i = STR ''='' \<and> length2 l then semantics_term e f (hd l) = semantics_term e f (hd (tl l)) else g i (semantics_list e f l))"
special_soundness: shows "Schnorr_\<Sigma>.special_soundness"
rep_eq: fixes x :: "int"   and xa :: "float" shows "real_of_float (float_down x xa) = round_down x (real_of_float xa)"
bind_return: fixes f :: "'a Heap" shows "f \<bind> return = f"
FDERIV_floatariths_nth: fixes i :: "nat"   and fas :: "floatarith list"   and xs :: "nat list"   and ds :: "floatarith list" assumes "i < length fas" shows "FDERIV_floatariths fas xs ds ! i = FDERIV_floatarith (fas ! i) xs ds"
sublist_iff_sublist_at: fixes xs :: "'b list"   and ys :: "'b list" shows "sublist xs ys = (\<exists>i. sublist_at xs ys i)"
rep_eq: fixes x :: "('a, 'b) lifted_boolean_algebra"   and xa :: "('a, 'b) lifted_boolean_algebra" shows "Rep_lifted_boolean_algebra (x \<squnion> xa) = (\<lambda>f. Rep_lifted_boolean_algebra x f \<squnion> Rep_lifted_boolean_algebra xa f)"
S\<^sub>r_S1: shows "S\<^sub>r \<subseteq> S1"
distinct_37: fixes x31 :: "'b"   and x32 :: "'a FOL_Fitting.term list"   and x7 :: "('a, 'b) form" shows "Pred x31 x32 \<noteq> Neg x7"
zero_less_iff_neq_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
bcontfun\<^sub>N_def: shows "bcontfun\<^sub>N = quasinorm_of (1, \<lambda>f. if f \<in> bcontfun then ennreal (norm (Bcontfun f)) else \<infinity>)"
weakening_10: shows "Option.is_none (memory \<C>) = Option.is_none (inst.mem (inst s ! i))"
MGU_is_mgu_singleton: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" shows "MGU \<theta> t u = is_mgu \<theta> {(t, u)}"
prod_mset_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one \<equiv> comm_monoid_mset.F times one"
I_qe_dlo\<^sub>1_pretty: fixes as :: "atom list" assumes "\<forall>a\<in>set as. is_Less a \<and> depends\<^sub>d\<^sub>l\<^sub>o a" shows "\<forall>xs. DLO.I (qe_dlo\<^sub>1 as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>d\<^sub>l\<^sub>o a (x # xs))"
domains_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "B.domains (g \<cdot>\<^sub>B f) = B.domains f"
spr_submodel_aux: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) spr_simWorld" assumes "t \<in> SPR.jkbpC"   and "s \<in> worlds (spr_simMCt t)" shows "gen_model spr_simMC s = gen_model (spr_simMCt t) s"
cri_less_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a \<times> 'c"   and y :: "'a \<times> 'c" assumes "dataflow_topology summary results_in" shows "dataflow_topology.cri_less summary results_in x y = (dataflow_topology.cri_less_eq summary results_in x y \<and> x \<noteq> y)"
pderiv_deg_0: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "deg R f = 0" shows "UP_cring.pderiv R f = \<zero>\<^bsub>UP R\<^esub>"
PAC_checker_PAC_checker_specification2: fixes A :: "nat set \<times> (nat, int mpoly) fmap"   and B :: "nat set \<times> int mpoly multiset"   and a :: "status"   and spec :: "int mpoly"   and st :: "(int mpoly, nat, nat) pac_step list" assumes "(A, B) \<in> polys_rel_full"   and "\<not> is_failed a"   and "a = FOUND \<Longrightarrow> spec \<in> pac_ideal (set_mset (snd B))"   and "\<Union> (vars ` set_mset (ran_m (snd A))) \<subseteq> fst B"   and "vars spec \<subseteq> fst B" shows "PAC_checker spec A a st \<le> \<Down> (status_rel \<times>\<^sub>r polys_rel_full) (PAC_checker_specification2 spec B)"
distinct_updates: fixes al :: "('c \<times> 'd) list"   and ks :: "'c list"   and vs :: "'d list" assumes "distinct (map fst al)" shows "distinct (map fst (updates ks vs al))"
normalize_Lcm: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "normalize (Lcm A) = Lcm A"
finite_verts: shows "finite (verts (with_proj (list_digraph G_list)))"
emeasure_completion: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
invert_opposite_sides_of_square: fixes f :: "'c"   and g :: "'c"   and h :: "'c"   and k :: "'c" assumes "C.seq f g"   and "f \<cdot>\<^sub>C g = h \<cdot>\<^sub>C k"   and "C.iso f"   and "C.iso k" shows "C.seq g (C.inv k) \<and> C.seq (C.inv f) h \<and> g \<cdot>\<^sub>C C.inv k = C.inv f \<cdot>\<^sub>C h"
enum_ex_finite_4_def: fixes P :: "finite_4 \<Rightarrow> bool" shows "enum_ex P = (P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4)"
chine_simps_3: shows "C.cod \<nu>.chine = \<nu>.cod.apex"
assms_2: shows "j\<^sub>1 = encode f\<^sub>1"
list_of_def: fixes xs :: "'c llist" shows "list_of xs = (if lfinite xs then inv llist_of xs else undefined)"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_with_State_Idle_conv: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a"   and t1 :: "nat" assumes "0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "t1 \<in> [0\<dots>,k - Suc 0] \<oplus> t0"   and "State_Idle localState output_fun trans_fun (localState (s t1))"   and "output_fun (s t1) \<noteq> NoMsg" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<not> State_Idle localState output_fun trans_fun (localState (s t2)). t2 \<U> t1 [0\<dots>] \<oplus> t0. output_fun (s t1) = m \<and> State_Idle localState output_fun trans_fun (localState (s t1)))"
AL_delete_commutative: fixes M :: "('a \<times> 'b) list"   and c :: "'a"   and x :: "'a" shows "delete (delete M c) x = delete (delete M x) c"
Quotient3_refl2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and r :: "'a"   and s :: "'a" assumes "Quotient3 R Abs Rep"   and "R r s" shows "R s s"
b_def: shows "b = 8428294561696506782041394632 / 503593538783547230635598424135"
h: shows "Ipoly bs (head p) \<noteq> (0::'a)"
C_def: shows "C = (\<Sum>k<N. f k - g k)"
T_on'_append: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "Partial_Cost_Model.T_on' A s (xs @ ys) = Partial_Cost_Model.T_on' A s xs + Partial_Cost_Model.T_on' A (Partial_Cost_Model.config' A s xs) ys"
map_add_Some_iff: fixes m :: "'f \<Rightarrow> 'e option"   and n :: "'f \<Rightarrow> 'e option"   and k :: "'f"   and x :: "'e" shows "((m ++ n) k = \<lfloor>x\<rfloor>) = (n k = \<lfloor>x\<rfloor> \<or> n k = None \<and> m k = \<lfloor>x\<rfloor>)"
cross7_triple2: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "x \<times>\<^sub>7 y \<bullet> z = x \<bullet> y \<times>\<^sub>7 z"
PiE_iff: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
finite_enum_ext: fixes X :: "'a set"   and Y :: "'a set" assumes "\<And>i. i < card X \<Longrightarrow> enum X i = enum Y i"   and "finite X"   and "finite Y"   and "card X = card Y" shows "X = Y"