hcju/isabelleps huggingface dataset:
antisym_conv: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "y \<le>\<^sub>o\<^sub>w x" shows "(x \<le>\<^sub>o\<^sub>w y) = (x = y)"
NSDERIV_quotient: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and d :: "'a"   and g :: "'a \<Rightarrow> 'a"   and e :: "'a" assumes "NSDERIV f x :> d"   and "NSDERIV g x :> e"   and "g x \<noteq> (0::'a)" shows "NSDERIV (\<lambda>y. f y / g y) x :> (d * g x - e * f x) / g x ^ Suc (Suc 0)"
preserves_comp_2: fixes A1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<times> 'c \<Rightarrow> 'd"   and f2' :: "'c"   and f2 :: "'c" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
length_partition_by_median_8: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "1 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length l < length ps"
lm074: fixes XX :: "'a set set" shows "\<Union> (finestpart ` XX) \<subseteq> finestpart (\<Union> XX)"
assms: shows "matrix_idempotent_semiring.coreflexive f"
Bin1OpRed2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and bop :: "bop" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val v \<guillemotleft>bop\<guillemotright> e,s\<rangle> -ta\<rightarrow> \<langle>Val v \<guillemotleft>bop\<guillemotright> e',s'\<rangle>"
MWLsSteps_det_cases_4: fixes \<iota> :: "nat"   and b :: "'exp"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "\<alpha> = []" and "p = Some c1" and "m' = m" and "BMap (E b m)"   | "\<alpha> = []" and "p = Some c2" and "m' = m" and "\<not> BMap (E b m)"
shows_string_append: fixes x :: "char list"   and r :: "char list"   and s :: "char list" shows "shows_string x (r @ s) = shows_string x r @ s"
trace_in_merge_at: fixes t' :: "real"   and \<omega>' :: "(real \<times> 'a) stream"   and ss :: "'a set"   and t :: "real"   and x :: "'a"   and \<omega> :: "(real \<times> 'a) stream" assumes "wf_times t' \<omega>'" shows "trace_in ss t x (merge_at \<omega> t' \<omega>') = (if t < t' then trace_in ss t x \<omega> else \<exists>y. trace_in {y} t' x \<omega> \<and> trace_in ss t y \<omega>')"
injective_eq_1d_open_map_UNIV: fixes S :: "real set"   and f :: "real \<Rightarrow> real" assumes "continuous_on S f"   and "is_interval S" shows "inj_on f S = (\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open (f ` T))"
lexordp_eq_trans: fixes xs :: "'d word list"   and ys :: "'d word list"   and zs :: "'d word list" assumes "signed.lexordp_eq xs ys"   and "signed.lexordp_eq ys zs" shows "signed.lexordp_eq xs zs"
sel_2: fixes x1 :: "'d"   and x2 :: "'b" shows "snd (x1, x2) = x2"
surjec_generator: 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 H :: "'e set" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "surjec\<^bsub>M,N\<^esub> f"   and "generator R M H" shows "generator R N (f ` H)"
loc_srj_lm_1: fixes x :: "nat" shows "nat_to_sch (Suc (Suc x)) = (let u = mod7 (c_fst (Suc (Suc x))); v = c_snd (Suc (Suc x)); v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
naturality: fixes f :: "'d" assumes "D.arr f" shows "D.\<ll>' (D.cod f) \<cdot>\<^sub>D D.map f = (\<I>\<^sub>D \<otimes>\<^sub>D f) \<cdot>\<^sub>D D.\<ll>' (D.dom f)"
assms_1: fixes e :: "'a uprod" assumes "e \<in> F" shows "proper_uprod e"
Derives1_sentence1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b" shows "CFG.is_sentence \<NN> \<TT> a"
normf: shows "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
select_convs_1: fixes tag :: "obj_tag"   and "values" :: "vn \<Rightarrow> val option"   and more :: "'a" shows "tag \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = tag"
single_valued_Id_on: fixes A :: "'d set" shows "single_valued (Id_on A)"
results_gpv_Done: fixes \<Gamma> :: "('e, 'f) \<I>"   and x :: "'d" shows "results_gpv \<Gamma> (Generative_Probabilistic_Value.Done x) = {x}"
card_isolated_verts_da0: shows "card H.isolated_verts = card G.isolated_verts + card ({tail G a, head G a} \<inter> H.isolated_verts)"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(rev_I x = rev_I y) = (x = y)"
hom_dvd: fixes p :: "rat poly"   and q :: "rat poly" assumes "p dvd q" shows "map_poly of_rat p dvd map_poly of_rat q"
fpr_first_in_path: fixes P :: "cld list"   and ctx :: "ctx"   and fqn :: "fqn"   and path :: "(ctx \<times> cld) list"   and ctxcld :: "ctx \<times> cld" assumes "find_path_f P ctx (cl_fqn fqn) = Some path"   and "find_cld_f P ctx fqn = Some ctxcld" shows "ctxcld \<in> set path"
ex_distrib_or: fixes P :: "'a \<Rightarrow> assn"   and Q :: "assn" shows "(\<exists>\<^sub>Ax. P x \<or>\<^sub>A Q) = (\<exists>\<^sub>Ax. P x) \<or>\<^sub>A Q"
non_exec_assign_global: fixes \<sigma> :: "'a control_state_scheme"   and upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign_global upd rhs) M) = (upd (\<lambda>_. rhs \<sigma>) \<sigma> \<Turnstile> M)"
usubst_ode_ext: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and x :: "char"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<theta> :: "trm"   and I :: "interp" assumes "Uvariation (F 0) \<omega> (U \<union> {RVar x, DVar x})"   and "usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta> \<noteq> undeft" shows "solves_ODE I F x (the (usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta>)) = solves_ODE (USubst.adjoint \<sigma> I \<omega>) F x \<theta>"
even_push_bit_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (push_bit n a) = (n \<noteq> 0 \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
wf_mbufn_def: 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" shows "wf_mbufn i js Ps buf = list_all3 (\<lambda>P j xs. i \<le> j \<and> list_all2 P [i..<j] xs) Ps js buf"
singleton_conv2: fixes a :: "'b" shows "{x. a = x} = \<eta> a"
monoidal_functor_axioms: shows "monoidal_functor (\<cdot>\<^sub>C) C'.T C'.\<alpha> \<iota>\<^sub>C (\<cdot>\<^sub>D) D'.T D'.\<alpha> \<iota>\<^sub>D F (\<lambda>f. \<phi> (snd f, fst f))"
hin: fixes x :: "'a" assumes "x \<in> S" shows "h x \<in> sphere 0 1"
trivial_limit_at_bot_linorder: shows "at_bot \<noteq> bot"
bigMax: shows "wmin (wmin (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2)) (wmin (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2)) \<equiv>\<^sub>E min (min (rl1 * rl2) (ru1 * rl2)) (min (rl1 * ru2) (ru1 * ru2))"
a1: shows "snd a \<otimes> snd h [^] e = g' [^] z"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'d \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_reader return bind ask"   and "monad_state return bind get put"   and "monad_reader_state_axioms ask get put" shows "monad_reader_state return bind ask get put"
src_def: fixes \<mu> :: "'a" shows "src \<mu> = (if arr \<mu> then src\<^sub>B \<mu> else null)"
par_col_par: fixes C :: "'p"   and D' :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p" assumes "C \<noteq> D'"   and "A B Par C D"   and "Col C D D'" shows "A B Par C D'"
assms: fixes a :: "'a" assumes "a \<in> Field r" shows "f a = g a"
S3RPC_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S3 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> $Enabled (<RPCNext crCh rmCh rst p>_(r p))"
PCF_consts_rel_POR: shows "PCF_consts_rel (unlr POR.delta)"
keys_sum_list_subset: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list" shows "keys (sum_list ps) \<subseteq> Keys (set ps)"
writerT_eq_iff: fixes a :: "'c\<cdot>('d, 'e) writerT"   and b :: "'c\<cdot>('d, 'e) writerT" shows "(a = b) = (runWriterT\<cdot>a = runWriterT\<cdot>b)"
le_cases: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" obtains "ko.le ko x y"   | "ko.le ko y x"
F3: shows "val_Zp (f' \<bullet> \<alpha>) = val_Zp (f' \<bullet> a)"
expectation_gpv_fixp_induct: fixes P :: "(('h, 'i, 'j) gpv \<Rightarrow> ennreal) \<Rightarrow> bool"   and fail :: "ennreal"   and \<I> :: "('i, 'j) \<I>"   and f :: "'h \<Rightarrow> ennreal" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>expectation_gpv'. \<lbrakk>\<And>gpv. expectation_gpv' gpv \<le> expectation_gpv fail \<I> f gpv; P expectation_gpv'\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv' (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None))" shows "P (expectation_gpv fail \<I> f)"
restrict_merge: fixes zI :: "'a option list"   and I :: "nat set"   and z :: "'a option list"   and zJ :: "'a option list"   and J :: "nat set"   and A :: "nat set"   and X :: "'a option list set"   and n :: "nat" assumes "zI = restrict I z"   and "zJ = restrict J z"   and "restrict (A \<inter> I) zI \<in> restrict I ` X"   and "restrict (A \<inter> J) zJ \<in> restrict J ` Set.filter (isSameIntersection zI (A \<inter> I)) X"   and "z = merge zJ zI"   and "table n A X"   and "A \<subseteq> I \<union> J"   and "1 \<le> card (A \<inter> I)"   and "wf_set n (I \<union> J)"   and "wf_tuple n (I \<union> J) z" shows "restrict A z \<in> X"
cinner_ge_zero: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "0 \<le> cinner x x"
add_assocD_var2: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "ps_prod_D x y \<and> ps_prod_D (ps_prod_times x y) z" shows "ps_prod_D y z \<and> ps_prod_D x (ps_prod_times y z)"
distinct_19: fixes x3 :: "'a formula"   and x41 :: "'a formula"   and x42 :: "'a formula" shows "\<^bold>\<not> x3 \<noteq> x41 \<^bold>\<and> x42"
pos_minus_divide_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq (uminus (divide b c)) a = less_eq (uminus b) (times a c)"
compat_last_chart: fixes b :: "'b" assumes "b \<in> Basis" shows "\<infinity>-smooth_compat last_chart (basis_chart b)"
incrs_def: shows "incrs \<equiv> \<lambda>h t x. incr h t x + s h t x"
lprefix_llist_of: fixes xs :: "'c list"   and ys :: "'c list" shows "lprefix (llist_of xs) (llist_of ys) = prefix xs ys"
instance_of\<^sub>l_def: fixes l\<^sub>1 :: "fterm literal"   and l\<^sub>2 :: "fterm literal" shows "instance_of\<^sub>l l\<^sub>1 l\<^sub>2 = (\<exists>\<sigma>. l\<^sub>1 = l\<^sub>2 \<cdot>\<^sub>l \<sigma>)"
cong: shows "\<I>_real,\<I>_ideal \<turnstile>\<^sub>C sim \<sim> sim'"
graph_union_vertices: fixes G\<^sub>1 :: "('f, 'e) labeled_graph"   and G\<^sub>2 :: "('f, 'e) labeled_graph" shows "vertices (graph_union G\<^sub>1 G\<^sub>2) = vertices G\<^sub>1 \<union> vertices G\<^sub>2"
reduce_column_aux_preserves: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list"   and i :: "nat" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs"   and "i \<notin> set xs"   and "i < m" shows "Matrix.row K i = Matrix.row K_init i"
F_def: fixes i :: "nat" shows "F i = {x \<in> space M. enat i < f x}"
simps_10: fixes v :: "MIR.num" shows "qfree (Lt v) = True"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "NetsCollected (x # xs) = ((first_bothNet x \<noteq> firstList xs \<longrightarrow> (\<forall>a\<in>set xs. first_bothNet x \<noteq> first_bothNet a)) \<and> NetsCollected xs)"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> l'.cf.E" shows "l'.cf (u, v) = (0::'a)"
subgraph_def: fixes G\<^sub>1 :: "('b, 'c) labeled_graph"   and G\<^sub>2 :: "('b, 'c) labeled_graph" shows "subgraph G\<^sub>1 G\<^sub>2 = (graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2)"
eq_snd_iff: fixes b :: "'b"   and p :: "'c \<times> 'b" shows "(b = time p) = (\<exists>a. p = (a, b))"
TESL_interp_decreases_setinc: fixes \<Phi> :: "'b TESL_atomic list"   and \<Phi>' :: "'b TESL_atomic list" assumes "set \<Phi> \<subseteq> set \<Phi>'" shows "\<lbrakk>\<lbrakk> \<Phi>' \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<subseteq> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
relcomp_Image: fixes X :: "('e \<times> 'f) set"   and Y :: "('f \<times> 'd) set"   and Z :: "'e set" shows "(X O Y) `` Z = Y `` X `` Z"
ipurge_ref_aux_empty: fixes u :: "'c"   and I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'d list"   and X :: "'d set" assumes "u \<in> sinks_aux I D U xs"   and "\<And>x. x \<in> X \<Longrightarrow> (u, D x) \<in> I" shows "ipurge_ref_aux I D U xs X = {}"
lclss_Liminf_commute: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "Liminf_llist (lmap lclss_of_state Sts) = lclss_of_state (Liminf_state Sts)"
P2I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "recfn 2 r"   and "\<And>x y. eval r [x, y] = f x y" shows "f \<in> \<P>\<^sup>2"
exec_to_exec_mark_guards_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and f :: "'f"   and x :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f" shows "\<exists>f'. \<Gamma>\<turnstile> \<langle>mark_guards x c,s\<rangle> \<Rightarrow> Fault f'"
gcd_factorial_commute: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.gcd_factorial zero times one normalize a b = factorial_semiring.gcd_factorial zero times one normalize b a"
sylvester_index_mat: fixes i :: "nat"   and p :: "'a poly"   and q :: "'a poly"   and j :: "nat" assumes "i < degree p + degree q"   and "j < degree p + degree q" shows "sylvester_mat p q $$ (i, j) = (if i < degree q then if i \<le> j \<and> j - i \<le> degree p then coeff p (degree p + i - j) else (0::'a) else if i - degree q \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'a))"
next_thread_def: fixes seed :: "natural \<times> natural"   and active :: "'s_t" shows "next_thread seed active = (let ts = thr'_to_list active in if ts = [] then None else \<lfloor>Random.select (thr'_to_list active) seed\<rfloor>)"
lens_comp_assoc: fixes X :: "'d \<Longrightarrow> 'f"   and Y :: "'f \<Longrightarrow> 'g"   and Z :: "'g \<Longrightarrow> 'e" shows "X ;\<^sub>L (Y ;\<^sub>L Z) = X ;\<^sub>L Y ;\<^sub>L Z"
assms: shows "list_all (finite \<circ> NBA.nodes) AA"
join_comm: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "x \<squnion>\<^bsub>L\<^esub> y = y \<squnion>\<^bsub>L\<^esub> x"
psi_eq_nth_prefix: fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat"   and b :: "nat"   and x :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>= b"   and "x < e_length b" shows "eval (total_cons.r_psi s) [j, x] \<down>= e_nth b x"
fun_ord_def: fixes ord :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
ax: shows "Max {y \<in> S. y \<le> x} \<le> x"
coinduct: fixes X :: "('c, 'd) resource \<Rightarrow> ('e, 'f) resource \<Rightarrow> bool"   and x1 :: "('c, 'd) resource"   and x2 :: "('e, 'f) resource"   and A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>res1 res2. x1 = res1 \<and> x2 = res2 \<and> rel_fun A (rel_spmf (rel_prod B (\<lambda>uu_ uua_. X uu_ uua_ \<or> rel_resource A B uu_ uua_))) (run_resource res1) (run_resource res2)" shows "rel_resource A B x1 x2"
top_clocked_min_counter: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_top ck env s tp (s', r)"   and "ck" shows "evaluate_top ck env (update_clock (\<lambda>_. clock s - clock s') s) tp (update_clock (\<lambda>_. 0) s', r)"
separating_hyperplane_set_0: fixes S :: "'b set" assumes "convex S"   and "(0::'b) \<notin> S" shows "\<exists>a. a \<noteq> (0::'b) \<and> (\<forall>x\<in>S. 0 \<le> a \<bullet> x)"
produced_by_scan_step_trans: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat"   and Doc :: "character list"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "indexle k' u' k u"   and "LocalLexing.produced_by_scan_step \<TT> \<RR> \<SS> Lex Sel Doc x k' u'" shows "LocalLexing.produced_by_scan_step \<TT> \<RR> \<SS> Lex Sel Doc x k u"
TIP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
shift1_degree: fixes p :: "poly" shows "Reflected_Multivariate_Polynomial.degree (shift1 p) = 1 + Reflected_Multivariate_Polynomial.degree p"
stateinterpol_empty: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" shows "stateinterpol \<nu> \<omega> {} = \<omega>"
Phi_inj: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "is_inj (\<Phi> w x y z)"
consume_type_type: fixes xs :: "checker_type"   and cons :: "ct list"   and t_int :: "t list" assumes "consume xs cons = Type t_int" shows "\<exists>tn. xs = Type tn"
ket_vec_col: fixes v :: "complex Matrix.vec" shows "Matrix.col |v\<rangle> 0 = v"
continuous_map_componentwise: fixes X :: "'a topology"   and Y :: "'b \<Rightarrow> 'c topology"   and I :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "continuous_map X (product_topology Y I) f = (f ` topspace X \<subseteq> extensional I \<and> (\<forall>k\<in>I. continuous_map X (Y k) (\<lambda>x. f x k)))"
compatible_head: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "compatible G H" shows "head G = head H"
top_apply: fixes x :: "'h" shows "\<top> x = \<top>"
assms_1: shows "reaction \<Delta> s vl s1 vl1"
simps: fixes a :: "'b"   and s :: "'c + 'b" shows "(a \<in> Basic_BNFs.setr s) = (\<exists>x. a = x \<and> s = Inr x)"
embedded_serial_sas_plus_plan_operator_structure: fixes ops :: "('c, 'd) sas_plus_operator list"   and \<psi> :: "('c, 'd) sas_plus_operator list"   and \<Psi> :: "('c, 'd) sas_plus_problem" assumes "ops \<in> set (List_Supplement.embed \<psi>)" obtains op :: "('c, 'd) sas_plus_operator" where "op \<in> set \<psi>" and "map (sasp_op_to_strips \<Psi>) ops = [\<phi>\<^sub>O \<Psi> op]"
NE_R: fixes rules :: "'a stream" shows "i.R rules \<noteq> {}"
rep_eq: fixes x :: "'g \<Rightarrow> 'f"   and xa :: "'g cset" shows "rcset (cimage x xa) = x ` rcset xa"
less_eq_neg_nonpos: fixes 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 a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus a) = less_eq a zero"
filterlim_at_top_mirror: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" shows "filterlim f F at_top = (LIM x at_bot. f (- x) :> F)"