hcju/isabelleps huggingface dataset:
rel_eq: shows "rel = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left I \<le> \<tau> \<sigma> ne - t then [rel] else []) auxlist')) {}"
update_convs_5: fixes num' :: "nat \<Rightarrow> nat"   and id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "num_update num' \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num' num, sid = sid, \<dots> = more\<rparr>"
mult_minus1_right: fixes one :: "'c"   and times :: "'c \<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 z :: "'c" assumes "class.ring_1 one times plus zero minus uminus" shows "times z (uminus one) = uminus z"
sum_power2_le_zero_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (plus (power.power one times x 2) (power.power one times y 2)) zero = (x = zero \<and> y = zero)"
EphK_Nonce: fixes X :: "ephkey"   and X' :: "nonce_t" shows "EphK X \<noteq> Nonce X'"
set_disconnected_nodes_get_parent_is_l_set_disconnected_nodes_get_parent: shows "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs Shadow_DOM.get_parent_locs"
subgroupE_1: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "subgroup H G" shows "H \<subseteq> carrier G"
elims: fixes x :: "'b tree"   and y :: "real" assumes "local.\<Phi> x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = local.\<Phi> l + local.\<Phi> r + local.\<phi> l r"
moebius_inverse: fixes a :: "'a"   and d :: "'a"   and b :: "'a"   and c :: "'a"   and z :: "'a" assumes "a * d \<noteq> b * c"   and "c * z + d \<noteq> (0::'a)" shows "moebius d (- b) (- c) a (moebius a b c d z) = z"
\<psi>_2: fixes p :: "'a" assumes "p \<in> carrier" shows "\<exists>c\<in>I. domain (\<psi> p) \<subseteq> \<X> c"
set_iterator_image_foldli_conv: fixes g :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "set_iterator_image g (foldli xs) = foldli (map g xs)"
eq_resource_on_resource_of_oracleI: fixes S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and r1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and r2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and s1 :: "'a"   and s2 :: "'b" assumes "rel_fun S (rel_fun (eq_on A) (rel_spmf (rel_prod (=) S))) r1 r2"   and "S s1 s2" shows "A \<turnstile>\<^sub>R RES r1 s1 \<sim> RES r2 s2"
simps_2: fixes P :: "'c"   and x :: "'d"   and \<Pi> :: "('c, 'd) chain"   and n :: "nat" shows "nthtriple (\<lbrace> P \<rbrace> \<cdot> x \<cdot> \<Pi>) (Suc n) = nthtriple \<Pi> n"
mat_of_row_dim_1: fixes y :: "'b vec" shows "dim_row (mat_of_row y) = 1"
DPR: shows "Domainp R = arith_ops_record.DP ops"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Sum_Type.Inl x \<Rightarrow> f1 x | Sum_Type.Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Sum_Type.Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Sum_Type.Inr x2 \<and> \<not> P (f2 x2))))"
7: shows "(top * e * (w\<^sup>T \<sqinter> - v * - v\<^sup>T \<sqinter> w\<^sup>\<star> * e\<^sup>T * top)\<^sup>\<star>)\<^sup>T \<le> w\<^sup>\<star> * e\<^sup>T * top"
Termination: fixes cc :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "terminationPseudo 1 cc Q"
image_is_empty: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set" shows "(f ` A = {}) = (A = {})"
f2: shows "\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> rng_to_rng_of_frac (x \<otimes> y) = rng_to_rng_of_frac x \<otimes>\<^bsub>rec_rng_of_frac\<^esub> rng_to_rng_of_frac y"
induct: fixes H :: "(Dom \<times> ('c, 'd) Expr) set"   and x1 :: "('c, 'd) Expr"   and x2 :: "Dom"   and P :: "('c, 'd) Expr \<Rightarrow> Dom \<Rightarrow> bool" assumes "H \<turnstile>\<^bsub>\<E>\<^esub> x1 : x2"   and "\<And>v d. P (Const v) d"   and "\<And>x d. DA x = d \<Longrightarrow> P (Var x) d"   and "\<And>d e. (d, e) \<in> H \<Longrightarrow> P e d"   and "\<And>arglist dl d f. \<forall>i<length arglist. (H \<turnstile>\<^bsub>\<E>\<^esub> arglist ! i : dl ! i \<and> P (arglist ! i) (dl ! i)) \<and> dl ! i \<le> d \<Longrightarrow> P (Op f arglist) d" shows "P x1 x2"
underS_I: fixes i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set" assumes "i \<noteq> j"   and "(i, j) \<in> R" shows "i \<in> underS R j"
IH: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat"   and t2 :: "pdf_type"   and e :: "cexpr" assumes "\<Gamma>(x := t1) \<turnstile>\<^sub>c e' : t2"   and "\<Gamma> \<turnstile>\<^sub>c e : t1" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_comp_aux x e e' : t2"
shift_var_set_Un: shows "shift_var_set V \<union> Suc ` V' = shift_var_set (V \<union> V')"
kronecker_samples_def: fixes n :: "nat" shows "kronecker_samples n \<equiv> let min = - int (n div 2) in [min..min + int n]"
Kcknows_2: shows "s \<in> toSet (snd ec)"
subst_mk_scheme: fixes S :: "nat \<Rightarrow> typ"   and t :: "typ" shows "$ S (mk_scheme t) = mk_scheme ($ S t)"
find_adds_SomeD2: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "find_adds fs u = Some f" shows "f \<noteq> 0"
n_mult_idem: fixes x :: "'a" shows "!x \<cdot> !x = !x"
ide_prod: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "ide (a \<otimes> b)"
vector_sneg_minus1: fixes x :: "('a, 'b) vec" shows "- x = - (1::'a) *s x"
Rep_assn_inject: fixes x :: "assn"   and y :: "assn" shows "(Rep_assn x = Rep_assn y) = (x = y)"
edge_preserving_atomic: fixes h1 :: "('e \<times> 'f) set"   and e1 :: "('g \<times> 'e \<times> 'e) set"   and e2 :: "('g \<times> 'f \<times> 'f) set"   and v1 :: "'e"   and v1' :: "'f"   and v2 :: "'e"   and v2' :: "'f"   and k :: "'g" assumes "edge_preserving h1 e1 e2"   and "(v1, v1') \<in> h1"   and "(v2, v2') \<in> h1"   and "(k, v1, v2) \<in> e1" shows "(k, v1', v2') \<in> e2"
assms_2: fixes k' :: "'a" shows "snd (f (k', 0::'b)) = (0::'c)"
whileLoop_results_simps: fixes x :: "'c \<times> 'd"   and y :: "('c \<times> 'd) option"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" shows "((Some x, y) \<in> whileLoop_results C B) = ((\<exists>r s. x = (r, s) \<and> y = Some (r, s) \<and> \<not> C r s) \<or> (\<exists>r s. x = (r, s) \<and> y = None \<and> C r s \<and> snd (B r s)) \<or> (\<exists>r s. x = (r, s) \<and> C r s \<and> (\<exists>r' s'. (r', s') \<in> fst (B r s) \<and> (Some (r', s'), y) \<in> whileLoop_results C B)))"
simps_9: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and i :: "nat"   and \<phi> :: "atom fm"   and var :: "nat" shows "liftmap f (AllN i \<phi>) var = AllN i (liftmap f \<phi> (var + i))"
init_Accessible: shows "dfa.init Canon.DFA \<in> MN.accessible"
subset: shows "\<forall>clause. clause el formula0 \<longrightarrow> clause el formula"
brn_gt_0: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and c :: "('c, 'd, 'a) cmd" assumes "PL cval"   and "proper c" shows "0 < PL.brn c"
f_expand_take_mod: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" assumes "n mod k = 0" shows "xs \<odot> k \<down> n = xs \<down> (n div k) \<odot> k"
is_scc_unique: fixes E :: "('b \<times> 'b) set"   and scc :: "'b set"   and scc' :: "'b set"   and v :: "'b" assumes "is_scc E scc"   and "is_scc E scc'"   and "v \<in> scc"   and "v \<in> scc'" shows "scc = scc'"
e: assumes "\<forall>f\<in>set fs. irreducible G f"   and "foldr (\<otimes>) fs \<one> = a" shows "P"
r_zero: fixes x :: "'a" shows "x + (0::'a) = x"
verit_sko_forall: fixes P :: "'a \<Rightarrow> bool" shows "(\<forall>x. P x) = P (SOME x. \<not> P x)"
unit_closed: shows "stk1.zero_stalk U \<in> stk1.carrier_stalk"
circ_decompose_9: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>+"
flopped_half_apartments_gf: shows "folding_g.Y = f \<turnstile> X"
LI_soundness_single: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto> (S\<^sub>2, \<theta>\<^sub>2)"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>2,\<theta>\<^sub>2\<rangle>" shows "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>"
uncurry_apply: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "uncurry f (a, b) = f a b"
division_ring_divide_zero: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a zero = zero"
subprob_space_def: fixes M :: "'c measure" shows "subprob_space M \<equiv> finite_measure M \<and> subprob_space_axioms M"
LeastI2_order: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and x :: "'d"   and Q :: "'d \<Rightarrow> bool" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y"   and "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> less_eq x y\<rbrakk> \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
x0: fixes x :: "real" assumes "x0 \<le> x" shows "\<exists>p. prime p \<and> real p \<in> {x<..(1 + \<epsilon>' / \<alpha>) * x}"
currentD_OUT: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and b :: "'k" assumes "current \<Gamma> f"   and "b \<in> B \<Gamma>" shows "d_OUT f b = 0"
FinPow_def: fixes xs :: "'b fset" shows "FinPow xs = Abs_fset (Abs_fset ` Pow \<langle>xs\<rangle>\<^sub>f)"
star_def_2: fixes x :: "'a" shows "x\<^sup>\<star> = top"
pdg_dghm_up_is_dghm: fixes \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" assumes "digraph \<alpha> \<CC>"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> \<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i" shows "dghm_up I \<AA> \<CC> \<phi> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_prod I \<AA>"
assms_4: shows "\<forall>a\<in>allAllocations ({seller} \<union> N) (set \<Omega>). \<forall>X\<in>Range a. b (seller, X) = (0::'b)"
associative_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "associative f \<equiv> \<forall>x y z. f (f x y) z = f x (f y z)"
rref_suc: shows "reduced_row_echelon_form_upt_k A (Suc k)"
right_minus_eq: fixes minus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and plus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero :: "'h"   and uminus :: "'h \<Rightarrow> 'h"   and a :: "'h"   and b :: "'h" assumes "class.group_add minus plus zero uminus" shows "(minus a b = zero) = (a = b)"
conv: fixes e :: "real" assumes "0 < e" shows "\<exists>N. \<forall>n x. N \<le> n \<longrightarrow> x \<in> S \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
Gcd_insert: fixes a :: "'a"   and A :: "'a set" shows "Gcd (insert a A) = gcd a (Gcd A)"
Rep_p_point_inverse: fixes x :: "p_point" shows "Abs_p_point (Rep_p_point x) = x"
show_l_zero: fixes a :: "rat"   and x :: "rat" shows "(a + x = x) = (a = 0)"
coprime_m_def: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.coprime_m m f g = (\<forall>h. poly_mod.dvdm m h f \<longrightarrow> poly_mod.dvdm m h g \<longrightarrow> poly_mod.dvdm m h 1)"
Cons: fixes t :: "('a, 'b) tm" shows "position_of t (p @ [d1]) = position_of (emb_step_at (p @ [opp d1] @ q) d2 t) (p @ [d1])"
prems_1: shows "\<lbrakk>M; \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)"
assms_2: shows "set (Abs_State.dom S) \<subseteq> X"
f_inv_into_f: fixes y :: "'f"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
lift_assn_def: fixes R :: "('c \<times> 'd) set"   and \<Phi> :: "'d \<Rightarrow> bool"   and s :: "'c" shows "lift_assn R \<Phi> s \<equiv> \<exists>s'. (s, s') \<in> R \<and> \<Phi> s'"
mkKripke_simps_1: fixes ws :: "'b set"   and rels :: "'c \<Rightarrow> ('b \<times> 'b) set"   and val :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "worlds (mkKripke ws rels val) = ws"
mahler_approximation_def: fixes d :: "nat"   and f :: "int poly" shows "local.mahler_approximation d f = local.mahler_approximation_main (d * d) ((- 1) ^ degree f) f (- 1) 0 2"
B: shows "0\<^sub>m (dim_row A) (dim_col (diag_block_mat As)) \<in> carrier_mat (dim_row A) (dim_col (diag_block_mat As))"
thetaWhile_Sretr: shows "thetaWhile \<subseteq> Sretr (thetaWhile \<union> Sbis)"
assms_1: fixes B :: "'a tensor" assumes "B \<in> set BS" shows "cprank_max1 B"
eval_wp_DC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (a \<Sqinter> b) P = (\<lambda>s. min (wp a P s) (wp b P s))"
assms_5: fixes i :: "nat" assumes "i < n" shows "(0::'a) \<le> u $ i"
left_adjoint_preserved_by_iso': fixes f :: "'a"   and f' :: "'a" assumes "is_left_adjoint f"   and "f \<cong> f'" shows "is_left_adjoint f'"
af_trans_imp_wf: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "almost_full_on P A"   and "transp_on P A" shows "wfp_on (strict P) A"
map_fun_def: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i"   and h :: "'j \<Rightarrow> 'k" shows "map_fun f g h = g \<circ> h \<circ> f"
comp_def: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f = (if (A1.arr (fst f) \<and> A2.arr (snd f)) \<and> (A1.arr (fst g) \<and> A2.arr (snd g)) \<and> (if A1.arr (fst f) \<and> A2.arr (snd f) then (A1.cod (fst f), A2.cod (snd f)) else (A1.null, A2.null)) = (if A1.arr (fst g) \<and> A2.arr (snd g) then (A1.dom (fst g), A2.dom (snd g)) else (A1.null, A2.null)) then (fst g \<cdot>\<^sub>A\<^sub>1 fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f) else (A1.null, A2.null))"
fbox_temp_dyn: fixes a :: "real"   and U :: "(real, 4) vec \<Rightarrow> real set"   and L :: "real"   and G :: "(real, 4) vec \<Rightarrow> bool"   and Q :: "(real, 4) vec \<Rightarrow> bool" assumes "0 < a"   and "\<And>s. s \<in> UNIV \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> UNIV" shows "|x\<acute>=\<lambda>t. temp_vec_field a L & G on U UNIV @ 0] Q = (\<lambda>s. s \<in> UNIV \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (temp_flow a L \<tau> s)) \<longrightarrow> Q (temp_flow a L t s)))"
assms: shows "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. p = q \<longrightarrow> i = j"
Exe0I: fixes p :: "\<o>"   and v :: "i" assumes "\<exists>r. Some r = d\<^sub>0 p \<and> ex0 r v" shows "[\<lparr>p\<rparr> in v]"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> B.VV.arr f" shows "fst f \<star>\<^sub>B snd f = B.null"
sum_set_simps_2: fixes x :: "'b" shows "Basic_BNFs.setl (Inr x) = {}"
hyps: fixes ll :: "('p, 'ba) label list el_step list"   and u :: "'n"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "length ll \<le> length wwl"   and "(([u], {#}), ll, s', c') \<in> trcl (ntrp fg)"   and "atU U (add_mset s' c')" shows "\<exists>Ml Me h. (u, Ml, Me, h) \<in> RU_cs fg U \<and> Ml \<subseteq> mon_loc fg ll \<and> Me \<subseteq> mon_env fg ll \<and> h \<le> \<alpha>ah (map (\<alpha>nl fg) ll)"
invertible_inverse_inverse: fixes u :: "'a set" assumes "quotient.invertible u"   and "u \<in> Partition" shows "quotient.inverse (quotient.inverse u) = u"
xy_prop: shows "x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)"
one_integer_mod_group: fixes n :: "nat" shows "\<one>\<^bsub>integer_mod_group n\<^esub> = 0"
W_def: shows "W = ker (S \<down> V) \<inter> V"
that: fixes C :: "('a, 'b) ctxt" assumes "C \<noteq> \<box>"   and "s = C\<langle>t\<rangle>" shows "thesis"
SMP_Union: fixes f :: "'a \<Rightarrow> ('fun, 'var) Term.term set"   and M :: "'a set" shows "SMP (\<Union> (f ` M)) = (\<Union>m\<in>M. SMP (f m))"
total: shows "total_on {x \<in> X. \<exists>c. I x = Intv c} r"
assms_1: shows "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
hom_pow: fixes x :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier R" shows "\<phi> (x [^]\<^bsub>R\<^esub> n) = \<phi> x [^]\<^bsub>S\<^esub> n"
thes_WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "\<down>\<^sub>s (WHILE b DO c, s) = s"
merge_Some_iff: fixes m :: "('d \<times> 'c) list"   and n :: "('d \<times> 'c) list"   and k :: "'d"   and x :: "'c" shows "(map_of (merge m n) k = Some x) = (map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)"
Max_mono: fixes M :: "'t set"   and N :: "'t set" assumes "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "ord_term_lin.Max M \<preceq>\<^sub>t ord_term_lin.Max N"
simps_1: fixes x :: "'a"   and y :: "'a" shows "(Some x \<le> Some y) = (x \<le> y)"
sup_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = Collect ((\<lambda>x. x \<in> A) \<squnion> (\<lambda>x. x \<in> B))"
succ_tran_correct: fixes G :: "'b list_graph"   and v :: "'b" shows "FiniteGraph.succ_tran (list_graph_to_graph G) v = set (FiniteListGraph.succ_tran G v)"
nodes_it_correct: fixes g :: "'G" assumes "invar g" shows "set_iterator (nodes_it g) (nodes (\<alpha> g))"
is_pointermap_impl_def: fixes b :: "'b pointermap"   and bi :: "'b pointermap_impl" shows "is_pointermap_impl b bi \<equiv> is_array_list (entries b) (entriesi bi) * is_hashmap (getentry b) (getentryi bi)"
case_cong_weak: fixes M :: "'a + 'b"   and M' :: "'a + 'b"   and f1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c" assumes "M = M'" shows "(case M of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case M' of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x)"
Skip: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn" shows "G,A\<turnstile>{Normal (P\<leftarrow>\<diamondsuit>)} .Skip. {P}"
mod_ex_dist: fixes P :: "'b \<Rightarrow> assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> (\<exists>\<^sub>Ax. P x)) = (\<exists>x. h \<Turnstile> P x)"
dvd_times_left_cancel_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times a b) (times a c) = dvd.dvd times b c"
times_Box_def: fixes x :: "'a Box"   and y :: "'a Box" shows "x * y \<equiv> bliftM2 (\<Lambda> a b. a * b)\<cdot>x\<cdot>y"
assms_1: fixes ys :: "'a list"   and zs :: "'b list" assumes "as = []"   and "bs = ys"   and "cs = zs" shows "P"
rep_eq: fixes x :: "((real, 'e) vec, 'f) vec" shows "blinfun_apply (blinfun_of_vmatrix x) = (*v) x"
lookup_eq: fixes t :: "('c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (inorder t)" shows "lookup t x = AList_Upd_Del.map_of (inorder t) x"
assms_1: shows "dim_col A + dim_col B = dim_vec v"
pickQFresh_card_of: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood" shows "pickQFresh xs V XS \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs (pickQFresh xs V XS) X \<and> qFresh xs (pickQFresh xs V XS) X)"
ide_compE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C (C g f)" obtains "partial_magma.arr C (C g f)" and "partial_magma.arr C (C f g)" and "C g f = partial_magma.dom C f" and "C g f = partial_magma.cod C g"
nat_to_string_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>d. is_digit d \<Longrightarrow> P d"   and "\<And>n. \<lbrakk>\<not> is_digit n; P (n div 10); P (n mod 10)\<rbrakk> \<Longrightarrow> P n" shows "P n"
sf_subset_insert: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "sf \<phi> \<subseteq> insert \<phi> X"   and "\<psi> \<in> subfrmlsn \<phi>"   and "\<phi> \<noteq> \<psi>" shows "sf \<psi> \<subseteq> X"
monomial_0D: fixes t :: "'b"   and c :: "'c" assumes "monomial c t = 0" shows "c = (0::'c)"
euler_increment: fixes f :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and h :: "real"   and t :: "real"   and x :: "'e" shows "euler_increment f h t x = f t x"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>) = (\<exists>!a. a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr> \<and> \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b)"
subst_floatarith_Num: fixes s :: "nat \<Rightarrow> floatarith"   and fa :: "floatarith"   and y :: "float" assumes "subst_floatarith s fa = Num y" obtains "fa = Num y"   | n :: "nat" where "fa = Var n" and "Num y = s n"
rep_eq: fixes x :: "'b set_dlist" shows "DList_Set.null x = List.null (list_of_dlist x)"
log_exp2_le: fixes n :: "nat" assumes "0 < n" shows "2 ^ local.log n \<le> n"
in_set_with_P: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "P x" shows "x \<in> {y. P y}"
fst_eq_Domain: fixes R :: "('a \<times> 'b) set" shows "fst ` R = Domain R"
pq_ex: shows "(\<exists>x. Ifm bbs (x # bs) p) = (\<exists>x. Ifm bbs (x # bs) (fst (unit p)))"
euler_err_fas_def: fixes X0 :: "'a list"   and h :: "floatarith"   and CX :: "floatarith list" shows "euler_err_fas X0 h CX = map (euler_err_fas_nth X0 h CX) [0..<length X0]"
Sup_dual_def_var: fixes X :: "'a set" shows "- \<Squnion> X = \<Sqinter> (uminus ` X)"
empty_set_diff_single: fixes A :: "'a set"   and x :: "'a" assumes "A - {x} = {}" shows "A = {} \<or> A = {x}"
image_normal_form_iff: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]" shows "(p \<in> normal_form F ` P[X]) = (p \<in> P[X] \<and> \<not> punit.is_red (punit.reduced_GB F) p)"
prime_nat_naiveI: fixes p :: "nat" assumes "2 \<le> p"   and "\<And>n. n dvd p \<Longrightarrow> n = 1 \<or> n = p" shows "prime p"
sum_inverses_conv_fraction: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> (0::'b)"   and "finite A" shows "(\<Sum>x\<in>A. (1::'b) / f x) = (\<Sum>x\<in>A. prod f (A - {x})) / prod f A"
numneg_nb: fixes t :: "MIR.num" assumes "numbound0 t" shows "numbound0 (numneg t)"
1: shows "finsum R f (insert a A) = f a \<oplus> finsum R f A"