hcju/isabelleps huggingface dataset:
elims: fixes x :: "'b fundef"   and xa :: "nat"   and xb :: "'b"   and y :: "'b fundef" assumes "rewrite_fundef_body x xa xb = y" obtains xs :: "'b list" and ar :: "nat" and n :: "nat" and x :: "'b" where "xa__ = Fundef xs ar" and "xa = n" and "xb = x" and "y = Fundef (rewrite xs n x) ar"
assms_2: shows "cosh (f x) \<noteq> (0::'a)"
knows_Says: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows A (Says A B X # evs) = insert X (knows A evs)"
floor_less_cancel: fixes x :: "'d"   and y :: "'d" assumes "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor>" shows "x < y"
is_associative_weak_composition: shows "associative_weak_composition (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D"
pre_zeta_aux_eq: fixes m :: "nat"   and n :: "nat"   and a :: "real"   and s :: "complex" assumes "m \<le> n"   and "0 < a"   and "- 2 * real m < Re s" shows "pre_zeta_aux m a s = pre_zeta_aux n a s"
component_secure_prot_def: fixes n :: "'lbl"   and P :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set"   and Sec :: "('fun, 'var) Term.term set"   and attack :: "'fun" shows "component_secure_prot n P Sec attack \<equiv> \<forall>\<A>\<in>P. suffix [(ln n, send\<langle>Fun attack []\<rangle>\<^sub>s\<^sub>t)] \<A> \<longrightarrow> (\<forall>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<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>) \<longrightarrow> \<not> \<I>\<^sub>\<tau> \<Turnstile> \<langle>proj_unl n \<A>\<rangle> \<and> (\<forall>\<A>'. prefix \<A>' \<A> \<longrightarrow> (\<forall>t\<in>Sec - declassified\<^sub>l\<^sub>s\<^sub>t \<A>' \<I>\<^sub>\<tau>. \<not> \<I>\<^sub>\<tau> \<Turnstile> \<langle>proj_unl n \<A>' @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rangle>)))"
rmge2_def: fixes e :: "nat" shows "rmge2 e \<equiv> if \<forall>i<e_length e. e_nth e i < 2 then Some 0 else Some (e_nth e (GREATEST i. i < e_length e \<and> 2 \<le> e_nth e i))"
le_RES_nofailI: fixes a :: "'a nres"   and x :: "'a set" assumes "a \<le> RES x" shows "nofail a"
mul_instr_sub1_def: fixes instr_name :: "sparc_operation"   and result :: "word32" shows "mul_instr_sub1 instr_name result \<equiv> if instr_name \<in> {arith_type SMULcc, arith_type UMULcc} then do psr_val \<leftarrow> gets (cpu_reg_val PSR); new_n_val \<leftarrow> gets (\<lambda>s. ucast (result >> 31)); new_z_val \<leftarrow> gets (\<lambda>s. if result = 0 then 1 else 0); new_v_val \<leftarrow> gets (\<lambda>s. 0); new_c_val \<leftarrow> gets (\<lambda>s. 0); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_icc new_n_val new_z_val new_v_val new_c_val psr_val); write_cpu new_psr_val PSR; return () od else return ()"
index_together: fixes G :: "'a set"   and x :: "'a"   and y :: "'a" assumes "G \<in> \<G>"   and "x \<in> G"   and "y \<in> G"   and "x \<noteq> y" shows "\<B> index {x, y} = 0"
shiftr_eq_drop_bit: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftr drop_bit a n = drop_bit n a"
code_unique_ref: fixes us :: "'a list list" assumes "us \<in> lists \<langle>\<C>\<rangle>" shows "Ref \<C> us = Dec \<C> concat us"
Fvars_ground_fm_aux: fixes A :: "fm"   and B :: "name set" assumes "Fvars A \<subseteq> B" shows "ground_fm_aux A (atom ` B)"
eq_order_le: fixes x :: "'b"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = y"   and "order r" shows "x \<sqsubseteq>\<^bsub>r\<^esub> y"
diagonal_to_Smith_i_dvd: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and c :: "nat" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "\<forall>a b. to_nat a \<in> insert i (set xs) \<and> to_nat a = to_nat b \<longrightarrow> A $ from_nat c $ from_nat c dvd A $ a $ b"   and "c \<notin> set xs"   and "c < min (nrows A) (ncols A)"   and "distinct xs" shows "A $ from_nat c $ from_nat c dvd diagonal_to_Smith_i xs A i bezout $ from_nat i $ from_nat i"
simps: fixes g :: "'g"   and a :: "'node list" shows "local.path g a = ((\<exists>n. a = [n] \<and> n \<in> set (\<alpha>n g) \<and> invar g) \<or> (\<exists>ns n'. a = n' # ns \<and> local.path g ns \<and> n' \<in> set (predecessors g (hd ns))))"
val_def: shows "f.val \<equiv> sum (augmentingFlow p) (cf.outgoing s) - sum (augmentingFlow p) (cf.incoming s)"
Sup_le_iff: fixes A :: "'d set"   and b :: "'d" shows "(\<Sqinter>A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
nfoldli_simps_1: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli [] c f s = RETURN s"
wt_ifnz: fixes \<Gamma> :: "(nat \<times> ty) list"   and e1 :: "exp"   and n :: "nat"   and e2 :: "exp"   and B :: "ty"   and e3 :: "exp" assumes "\<Gamma> \<turnstile> e1 : TNat n"   and "n \<noteq> 0"   and "\<Gamma> \<turnstile> e2 : B" shows "\<Gamma> \<turnstile> EIf e1 e2 e3 : B"
Cons_4: shows "unitary_schur_decomposition A (e # es) = (C, P, Q)"
assms: shows "a \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
b_spg: shows "base ds' p \<in> sparsegrid' dm"
mult_isol: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (times_prod' z x) (times_prod' z y)"
head_in_verts: fixes G :: "('e, 'f) pre_digraph"   and e :: "'f" assumes "wf_digraph G"   and "e \<in> arcs G" shows "head G e \<in> verts G"
the_inv_f_f: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "inj f" shows "the_inv f (f x) = x"
evaln_var_eq: fixes G :: "prog"   and s :: "state"   and t :: "Term.var"   and n :: "nat"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In2 t\<succ>\<midarrow>n\<rightarrow> (w, s') = (\<exists>vf. w = In2 vf \<and> G\<turnstile>s \<midarrow>t=\<succ>vf\<midarrow>n\<rightarrow> s')"
ux: shows "x - real_of_int k * (2 * pi) \<le> real_of_float (float_plus_up prec ux (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_up prec (ub_pi prec) else float_round_down prec (lb_pi prec))))"
plossless_gpv_try_gpvI: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and gpv' :: "('j, 'h, 'i) gpv" assumes "pfinite_gpv \<I> gpv"   and "\<not> colossless_gpv \<I> gpv \<Longrightarrow> plossless_gpv \<I> gpv'" shows "plossless_gpv \<I> (TRY gpv ELSE gpv')"
smult_neg: fixes r :: "'r"   and m :: "'m" assumes "r \<in> R"   and "m \<in> M" shows "r \<cdot> - m = - (r \<cdot> m)"
simps_6: fixes p :: "fm"   and q :: "fm" shows "qfree (Imp p q) = (qfree p \<and> qfree q)"
dvd_smult: fixes p :: "'b poly"   and q :: "'b poly"   and a :: "'b" assumes "p dvd q" shows "p dvd Polynomial.smult a q"
intAf: shows "integrable M (\<lambda>x. indicat_real A (T x) * f x)"
ivl_def: shows "ivl = ivl_of_aform_err p X"
update_convs_1: fixes newOID' :: "(CTypeId \<Rightarrow> nat) \<Rightarrow> CTypeId \<Rightarrow> nat"   and newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "newOID_update newOID' \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = \<lparr>newOID = newOID' newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr>"
assms_3: shows "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV"
fg_start_round_later: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and n :: "nat"   and p :: "'f"   and r :: "nat"   and r' :: "nat" assumes "fg_run A rho HOs SHOs coords"   and "round (rho n) p = r"   and "r < r'" shows "n < fg_start_round rho p r'"
subcls_irrefl: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "C \<noteq> D"
cases: fixes a1 :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and a2 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and a3 :: "nat"   and a4 :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and a5 :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.meval_abv (a1, a2) a3 (a4, a5)" obtains cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list" and n :: "nat" and mem :: "'Var\<^sub>C \<Rightarrow> 'Val" and cm' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)" and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val" where "a1 = cms" and "a2 = mem" and "a3 = n" and "a4 = cms[n := cm']" and "a5 = mem'" and "(cms ! n, mem) \<leadsto>\<^sub>C (cm', mem')" and "n < length cms"
mop_def: fixes P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f nres" shows "mop P f \<equiv> \<lambda>x. ASSERT (P x) \<bind> (\<lambda>_. f x)"
spn_simps_4: 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 \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "trg (bicategory_of_spans.spn V H \<a> \<i> src trg \<mu>) = src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))"
collapse: fixes hmultiset :: "hmultiset" shows "HMSet (hmsetmset hmultiset) = hmultiset"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "('b, 'c) mapping" shows "rep (Mapping.filter x xa) = (\<lambda>k. case rep xa k of None \<Rightarrow> None | Some v \<Rightarrow> if x k v then Some v else None)"
cond_def: fixes g :: "'g" shows "cond g \<equiv> \<lambda>(u, p). uninst.redundant (uses(g := u)) (phis(g := p)) g"
reduced_word_for_sum_list: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as" shows "monoid_add.sum_list plus zero as = a"
kruskal'_def: fixes L :: "(nat \<times> int \<times> nat) list" shows "kruskal' L = kruskal (return L) (\<lambda>(u, w, v). return (u, v)) ()"
read_only_read_witness: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "a \<in> read_only_reads \<O> sb" shows "\<exists>xs ys t v. sb = xs @ Read\<^sub>s\<^sub>b False a t v # ys \<and> a \<in> read_only (share xs \<S>) \<and> a \<notin> read_only_reads \<O> xs"
il: shows "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"
sup_None2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and x :: "'b option" shows "None \<squnion>\<^bsub>Opt.sup f\<^esub> x = OK x"
fps_of_ratfps_inverse: fixes x :: "'b ratfps" shows "fps_of_ratfps (inverse x) = inverse (fps_of_ratfps x)"
simps_1: fixes r :: "real" shows "real_of_ereal (ereal r) = r"
word_power_increasing: fixes x :: "nat"   and y :: "nat" assumes "2 ^ x < 2 ^ y"   and "x < LENGTH('b)"   and "y < LENGTH('b)" shows "x < y"
valid_eval_match: fixes env :: "v sem_env"   and s :: "'a state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "evaluate_match True env s v pes err_v (eval_match' env v pes err_v s)"
consr: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "list_all2 Q xs ys"   and "Q x y"   and "P xs ys"   and "suffix (x # xs) as"   and "suffix (y # ys) bs" shows "P (x # xs) (y # ys)"
avg_correct: fixes s :: "'s" assumes "invar s" shows "avg s = average (\<alpha> s)"
le_fun_any_inv: fixes t :: "(val \<times> val) fset"   and v :: "val" assumes "VFun t \<sqsubseteq> v" obtains t2 :: "(val \<times> val) fset" where "v = VFun t2" and "fset t \<subseteq> fset t2"
Cantors_paradox: fixes A :: "'a set" shows "\<nexists>f. f ` A = Pow A"
monoI: fixes f :: "'a \<Rightarrow> 'd" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "mono f"
val_of_divide: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "x \<in> carrier (padic_int p)"   and "y \<in> nonzero (padic_int p)"   and "padic_integers.val_Zp p y \<le> padic_integers.val_Zp p x" shows "padic_integers.val_Zp p (padic_integers.divide p x y) = padic_integers.val_Zp p x - padic_integers.val_Zp p y"
uboundmax: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real" assumes "C \<subseteq> PR"   and "np \<le> card C + khl" shows "\<exists>i\<in>C. Max (reduce f PR) \<le> f i"
BisT_matchT_TMT_rev: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>T d" shows "matchT_TMT d c"
path_connectedin_imp_connectedin: fixes X :: "'a topology"   and S :: "'a set" assumes "path_connectedin X S" shows "connectedin X S"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "'b mpoly" shows "mapping_of (lowerPoly x xa xb) = lowerPoly\<^sub>0 x xa (mapping_of xb)"
hoare_dgr_correctness: fixes D :: "I \<times> I \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "I \<Rightarrow> 'a"   and Q :: "I \<Rightarrow> 'a" assumes "dmono D"   and "Hoare_dgr P D Q" shows "\<Turnstile> P{| pt D |}Q"
zeta_def: shows "zeta = truncate_up p ((d_min' + d_max') / 2)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(TNil x1 = TNil y1) = (x1 = y1)"
lq_pref: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v" shows "u \<cdot> u\<inverse>\<^sup>>v = v"
NewC: fixes E :: "env"   and C :: "qtname"   and dt :: "loc \<Rightarrow> ty option" assumes "is_acc_class (prg E) (pkg E) C" shows "E,dt\<Turnstile>NewC C\<Colon>-Class C"
unsat_imp_se_unsat: fixes c :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and c' :: "('d, 'e) conf" assumes "se c l c'"   and "\<not> Conf.sat c" shows "\<not> Conf.sat c'"
ex_sup: fixes x :: "'a"   and y :: "'a" shows "\<exists>sup. is_sup x y sup"
gso_oc_projection_span_2: fixes i :: "nat" assumes "i < length fs" shows "is_oc_projection (gso i) (local.span (gso ` {0..<i})) (fs ! i)"
simps_10: fixes x21 :: "('a, 'b, 'c) expr"   and x22 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Apply x21 x22 \<noteq> VE x1"
decomp: fixes E :: "'b set"   and U :: "'b set"   and V :: "'b set"   and u1 :: "'b"   and u2 :: "'b"   and v1 :: "'b"   and v2 :: "'b" assumes "vectorspace E"   and "Subspace.subspace U E"   and "Subspace.subspace V E"   and "U \<inter> V = {0::'b}"   and "u1 \<in> U"   and "u2 \<in> U"   and "v1 \<in> V"   and "v2 \<in> V"   and "u1 + v1 = u2 + v2" shows "u1 = u2 \<and> v1 = v2"
wf_tuple_change_base: fixes n :: "nat"   and X :: "nat set"   and as :: "'a option list"   and Y :: "nat set" assumes "wf_tuple n X as"   and "join_mask n X = join_mask n Y" shows "wf_tuple n Y as"
times_iff_sum_squares: fixes f :: "real"   and g :: "real" shows "f * g = (f + g)\<^sup>2 / 4 - (f - g)\<^sup>2 / 4"
llength_ltake': fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "llength (ltake (enat n) xs) = enat n"
finsertI2: fixes a :: "'b"   and B :: "'b fset"   and b :: "'b" assumes "a |\<in>| B" shows "a |\<in>| finsert b B"
subst_single_eq: fixes t :: "nterm"   and s :: "name"   and t' :: "nterm" shows "subst_single t s t' = subst t (fmap_of_list [(s, t')])"
list_all2_elimR: fixes ys :: "'a list"   and y :: "'a"   and ys' :: "'a list"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'b list" assumes "ys = y # ys'"   and "list_all2 R xs ys" obtains x :: "'b" and xs' :: "'b list" where "xs = x # xs'" and "R x y" and "list_all2 R xs' ys'"
literal_subset_sandwich: fixes F :: "'b formula"   and C :: "'b literal set"   and R :: "'b literal set" assumes "is_lit_plus F"   and "cnf F = {C}"   and "R \<subseteq> C" shows "R = \<box> \<or> R = C"
mon_e_def: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and e :: "('h, 'k) label" shows "mon_e fg e = (case e of LCall p \<Rightarrow> mon fg p | _ \<Rightarrow> {})"
G'_reaches_V: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and V :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "(Subgraph_Node_Defs.E' E V)\<^sup>*\<^sup>* x y"   and "V x" shows "V y"
diff_cancel: fixes a :: "'e" shows "a - a = (0::'e)"
FiniteMessages: fixes cfg :: "('p, 'v, 's) configuration" assumes "finite Proc"   and "\<And>p s m. finite {v. v \<in># sends p s m}"   and "initReachable cfg" shows "finite {msg. msg \<in># msgs cfg}"
end_smult_assoc: fixes T :: "'v \<Rightarrow> 'v"   and a :: "'f"   and b :: "'f" assumes "range T \<subseteq> V" shows "(\<lambda>v. (a \<cdot>\<cdot> (\<cdot>) b) (T v)) = (a * b) \<cdot>\<cdot> T"
Theorem_15_match: fixes ds :: "'b set" shows "match (gfp_F ds) = cop ds"
fls_divide_X_power': fixes f :: "'a fls"   and n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X ^ n = fls_shift (int n) f"
Max_mono: fixes M :: "('x \<Rightarrow>\<^sub>0 nat) set"   and N :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "ordered_powerprod_lin.Max M \<preceq> ordered_powerprod_lin.Max N"
lbs_def: fixes R :: "'b set"   and S :: "'b set" shows "lbs R S = Collect (isLb R S)"
wiringD_eq: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and f :: "'i \<Rightarrow> 'k"   and g :: "'l \<Rightarrow> 'j" assumes "wiring \<I> \<I>' cnv (f, g)" shows "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"
holomorphic_fun_eq_0_on_ball: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and w :: "complex" assumes "f holomorphic_on ball z r"   and "w \<in> ball z r"   and "\<And>n. (deriv ^^ n) f z = 0" shows "f w = 0"
static: shows "statically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. no_labels.\<G>_Fset N1 \<Turnstile>G no_labels.\<G>_Fset N2) no_labels.Red_I_\<G> local.no_labels.Red_F_\<G>"
simps_2: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
integrable_diff: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x - g x)"
var_dist_subst_lt: fixes vs :: "nat list"   and i :: "nat"   and s :: "dB"   and t :: "dB" assumes "\<forall>v\<in>set vs. i < v" shows "var_dist vs s[t/i] = var_dist (map (\<lambda>v. v - 1) vs) (s[t/i])"
map_transform_Cons: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae ((x, e) # \<Gamma>) = (x, lift_transform t (ae x) e) # map_transform t ae \<Gamma>"
support_set_def: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "support_set p = {x. \<exists>n w. p x n w \<noteq> 0}"
intro: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list"   and A :: "'a set"   and T :: "'a"   and B :: "'a" assumes "SemilatAlg.mono r step (length \<tau>s) A"   and "pres_type step (length \<tau>s) A"   and "\<forall>pc<length \<tau>s. \<tau>s ! pc \<in> A \<and> \<tau>s ! pc \<noteq> T"   and "bounded step (length \<tau>s)"   and "B \<noteq> T" shows "lbvc_axioms A r T B step \<tau>s"
enter_subresultant_prs: fixes G2 :: "'a poly"   and G1 :: "'a poly" assumes "length (coeffs G2) \<le> length (coeffs G1)"   and "G2 \<noteq> 0" shows "\<exists>F n d f k b. subresultant_prs_locale2 F n d f k b G1 G2"
M_times_2: fixes m :: "int"   and x :: "int"   and y :: "int" shows "poly_mod.M m (x * poly_mod.M m y) = poly_mod.M m (x * y)"
p: fixes as :: "('b poly \<times> nat) set"   and p :: "'b poly" assumes "monic_factorization as p" shows "p = (\<Prod>(a, i)\<in>as. a ^ Suc i)"
prod_pow: fixes p :: "'a"   and n :: "nat" shows "(\<Prod>i = 0..<n. p) = p ^ n"
exhaust: fixes y :: "('a, 'b, 'c, 'd) edge_kind" obtains x1 :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" where "y = \<Up>x1"   | x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool" where "y = (x2)\<^sub>\<surd>"   | x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and x32 :: "'c" and x33 :: "'d" and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" where "y = x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34"   | x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and x42 :: "'d" and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" where "y = x41\<hookleftarrow>\<^bsub>x42\<^esub>x43"
one_not_zero: shows "(1::'a) \<noteq> (0::'a)"
three_neq_5: shows "\<^bold>3 \<noteq> \<^bold>1"
mono: fixes x :: "real"   and y :: "real" assumes "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f x \<le> f y"
finite_has_maximal2: fixes A :: "('a, 'b) Term.term set"   and a :: "('a, 'b) Term.term" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. a \<sqsubseteq> m \<and> (\<forall>b\<in>A. m \<sqsubseteq> b \<longrightarrow> m = b)"
change_env: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>"   and "FV e \<turnstile> \<rho> \<sqsubseteq> \<rho>'" shows "v \<in> E e \<rho>'"
N: fixes n :: "nat" assumes "N \<le> n" shows "P n"
cong_scalar_right: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "[b = c] (mod a)" shows "[b * d = c * d] (mod a)"
length_two_unique: assumes "length \<GG> = 2" shows "\<GG> = [{\<one>}, carrier G]"
lattice_of_def: fixes n :: "nat"   and fs :: "'a vec list" shows "vec_module.lattice_of n fs = range (\<lambda>c. abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs]))"
domain_eq: shows "Domainp (pcr_equal (=)) = (\<lambda>x. x \<in> {(=)})"
that: fixes x :: "'a" assumes "(THE x on U. P x) = Some x"   and "(THE x. x \<in> U \<and> P x) = x" shows "thesis"
emeasure_Un: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
simps_3: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and p :: "real mpoly" shows "linear_substitution var a b (Leq p) = (let d = MPoly_Type.degree p var; P = \<Sum>i = 0..<d + 1. isolate_variable_sparse p var i * a ^ i * b ^ (d - i) in Leq (P * b ^ (d mod 2)))"
bij_betw_def: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
Int_stable_pair_measure_generator: fixes A :: "'c measure"   and B :: "'d measure" shows "Int_stable {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B}"