hcju/isabelleps huggingface dataset:
LeastI: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and P :: "'h \<Rightarrow> bool"   and k :: "'h" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
that: fixes x :: "'a"   and ys :: "'a list"   and y :: "'a" assumes "xs = x # ys @ [y]" shows "thesis"
simps_3: fixes f1 :: "'b"   and f2 :: "'b" shows "(case Accept of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2) = f1"
less_uexpr_def: fixes P :: "('c, 'd) uexpr"   and Q :: "('c, 'd) uexpr" shows "(P < Q) = (P \<le> Q \<and> \<not> Q \<le> P)"
cycle_free_diag_equiv: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len m i i xs) = cycle_free m n"
simps_13: fixes m :: "nat"   and p :: "fm" shows "bound m (A p) = bound (Suc m) p"
tllist_of_llist_inverse: fixes b :: "'b"   and xs :: "'a llist" shows "llist_of_tllist (tllist_of_llist b xs) = xs"
dom_m_fmdrop: fixes C :: "'c"   and N :: "('c, 'd) fmap" shows "dom_m (fmdrop C N) = remove1_mset C (dom_m N)"
assms_1: shows "x \<in> zero_space\<^sub>N N"
simps_30: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> Stmt.If x51 x52 x53"
rows_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "rows A = {row i A |i. i \<in> UNIV}"
vec_times_polys_to_mat: fixes ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and ts :: "'d list"   and v :: "'e vec" assumes "Keys (set ps) \<subseteq> set ts"   and "v \<in> carrier_vec (length ps)" shows "row_to_poly ts (v \<^sub>v* polys_to_mat ts ps) = sum_list (map2 (\<cdot>) (list_of_vec v) ps)"
cases_simp: fixes P :: "bool"   and Q :: "bool" shows "((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q)) = Q"
Mul_Mutator: fixes j :: "nat"   and n :: "nat" assumes "0 \<le> j \<and> j < n" shows "\<turnstile> Mul_Mutator j n  \<lbrace>False\<rbrace>"
elims: fixes x :: "real_alg_2"   and y :: "rat option" assumes "to_rat_2 x = y" obtains r :: "rat" where "x = Rational r" and "y = Some r"   | n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai" and "y = None"
prod_encode_snd_inj: fixes i :: "nat" shows "inj (\<lambda>j. prod_encode (i, j))"
wf_flatten_PLUS: fixes X :: "'c rexp set"   and wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "finite X" shows "alphabet.wf wf_atom n (PLUS (sorted_list_of_set X)) = (\<forall>r\<in>X. alphabet.wf wf_atom n r)"
llist_of_eq_LNil_conv: fixes xs :: "'a list" shows "(llist_of xs = LNil) = (xs = [])"
partitionListRight: fixes xs :: "'d list"   and ys :: "'d list"   and xs' :: "'d list"   and y :: "'d"   and ys' :: "'d list" assumes "xs @ ys = xs' @ y # ys'"   and "y mem ys"   and "distinct (xs @ ys)" obtains zs :: "'d list" where "xs' = xs @ zs" and "ys = zs @ y # ys'"
preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "S.dom (Y a f) = Y a (Cop.dom f)"
assms_1: shows "snd ` set txs \<subseteq> var"
enumerator_correctness: fixes m :: "nat"   and nmax :: "nat"   and ls :: "nat list" assumes "2 < m"   and "1 < nmax"   and "ls \<in> set (enumerator m nmax)" shows "incrIndexList ls m nmax"
out_sub12_1: shows "outside (path_image c1 \<union> path_image c2) \<subseteq> outside (path_image c1 \<union> path_image c)"
wpt_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp\<^sub>t c Q = (\<lambda>l s. \<exists>t. (c, s) \<Rightarrow> t \<and> Q l t)"
diff_zero: fixes a :: "'a" shows "a - (0::'a) = a"
case_list_compute_def: fixes l :: "'b list"   and a :: "'a"   and f :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a" shows "case_list_compute l a f = (case l of [] \<Rightarrow> a | x # xa \<Rightarrow> f x xa)"
normalize_eq_0_iff: 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" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = zero) = (a = zero)"
ds_ord_sl_ord: fixes x :: "'a"   and y :: "'a" shows "domain_semigroup.ds_ord d (\<cdot>) (d x) (d y) = (d x \<cdot> d y = d x)"
snd_sylow_axioms: shows "snd_sylow G p a m"
invertible_commutes_with_inverse: fixes a :: "'a" assumes "f.source.additive.invertible a"   and "a \<in> A" shows "f (f.source.additive.inverse a) = JB.additive.inverse (f a)"
\<theta>_simps_2: shows "src \<theta>' = src f'"
apply_cltn2_left_abs: fixes v :: "(real, 3) vec"   and C :: "cltn2" assumes "v \<noteq> 0" shows "apply_cltn2 (proj2_abs v) C = proj2_abs (v v* cltn2_rep C)"
special_soundness_alt: shows "special_soundness = (\<forall>h a e z e' z'. e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> h \<in> valid_pub \<longrightarrow> e \<noteq> e' \<longrightarrow> check h a e z \<and> check h a e' z' \<longrightarrow> \<A>ss h (a, e, z) (a, e', z') \<bind> (\<lambda>w'. return_spmf ((h, w') \<in> Rel)) = return_spmf True)"
P_D2_H2_H2_def: fixes A :: "(D2 \<times> H2) set" shows "P_D2_H2_H2 A = (if {(D1, H2), (D2, H2)} \<subseteq> A then {(D1, H2), (D2, H2)} else if (D1, H2) \<in> A then {(D1, H2)} else if (D2, H2) \<in> A then {(D2, H2)} else {})"
step_def: fixes s :: "'state_t"   and oa :: "'action_t option" shows "step s oa \<equiv> case oa of None \<Rightarrow> s | Some a \<Rightarrow> kstep s a"
icdi_path_swap: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>'"   and "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> =\<^bsub>j\<^esub> \<pi>'" shows "j icd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"
c_gt_half: shows "\<forall>\<^sub>F N in sequentially. real N / 2 < real (card {n \<in> {N..<2 * N}. c (n + 1) < c n})"
inject: fixes x2 :: "'e"   and y2 :: "'e" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
compl_coset: fixes xs :: "'a list" shows "- List.coset xs = set xs"
simps: fixes s :: "'s"   and m :: "'m" shows "local.put_option s (OptionT m) = OptionT (put s m)"
discont_null: shows "emeasure M {x. \<not> isCont f x} = 0"
matrix_to_iarray_fst_echelon_form_of_upt_k_det: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A" shows "fst (echelon_form_of_upt_k_det bezout (1::'c, A) k) = fst (echelon_form_of_upt_k_det_iarrays (1::'c, matrix_to_iarray A) k bezout)"
assignment_rhs\<^sub>e\<^sub>s\<^sub>t_finite: fixes A :: "('fun, 'var) extstrand_step list" shows "finite (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A)"
obtain_t_subset_points: obtains T :: "'a set" where "T \<subseteq> \<V>" and "int (card T) = 2" and "finite T"
setmonotone_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "setmonotone f = (\<forall>X. X \<subseteq> f X)"
DSourcesA91_L1: shows "DSources level1 sA91 = {sA81}"
zero_im_sheaf_def: shows "idim.zero_im_sheaf \<equiv> \<lambda>V. \<zero>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
uprod_exhaust: fixes x :: "'a uprod" obtains a :: "'a" and b :: "'a" where "x = Upair a b"
downsets_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.downsets less_eq = Fix (ord.downset_set less_eq)"
abelian_FactGroup: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" assumes "comm_group G"   and "subgroup N G" shows "comm_group (G Mod N)"
that: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> A; x \<noteq> []\<^sub>\<circ>\<rbrakk> \<Longrightarrow> f\<lparr>x\<rparr> \<in>\<^sub>\<circ> x" shows "thesis"
cases: fixes a :: "'a" assumes "is_num a" obtains "a = (1::'a)"   | x :: "'a" where "a = - x__" and "is_num x__"   | x :: "'a" and y :: "'a" where "a = x + y" and "is_num x" and "is_num y"
higher_deriv_const: fixes n :: "nat"   and c :: "'a" shows "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n = 0 then c else (0::'a))"
open_closed: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "open S = topological_space.closed open (- S)"
llexord_lappend_leftI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c llist"   and zs :: "'c llist"   and xs :: "'c llist" assumes "llexord r ys zs" shows "llexord r (lappend xs ys) (lappend xs zs)"
integrable_on_scaleR_iff: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "c \<noteq> 0" shows "((\<lambda>x. c *\<^sub>R f x) integrable_on S) = (f integrable_on S)"
map_add_eval_left: fixes x :: "'c"   and h :: "'c \<Rightarrow> 'd option"   and h' :: "'c \<Rightarrow> 'd option" assumes "x \<in> dom h"   and "h \<bottom> h'" shows "(h ++ h') x = h x"
eAbs_simp2: fixes xs :: "'varSort"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and X :: "'c"   and x :: "'var" assumes "\<forall>s. \<not> (isInBar (xs, s) \<and> igWls MOD s X)" shows "eAbs MOD xs x (OK X) = ERR"
p_q_1: shows "local.my_U p \<in> {0..1}"
filterlim_at_top_at_left: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "\<And>x y. \<lbrakk>Q x; Q y; x \<le> y\<rbrakk> \<Longrightarrow> f x \<le> f y"   and "\<And>x. P x \<Longrightarrow> f (g x) = x"   and "\<And>x. P x \<Longrightarrow> Q (g x)"   and "eventually Q (at_left a)"   and "\<And>b. Q b \<Longrightarrow> b < a"   and "eventually P at_top" shows "filterlim f at_top (at_left a)"
ev: shows "\<forall>\<^sub>\<infinity>n. Abs_fps (\<lambda>m. coeff (p $ m) n) = 0"
fold_insert2: fixes A :: "name set"   and x :: "name"   and z :: "tm" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>i. subst i (F i)) z (insert x A) = Finite_Set.fold (\<lambda>i. subst i (F i)) (subst x (F x) z) A"
simps_11: fixes t :: "Core.term"   and u :: "Core.term" shows "eta_reducible (t $ u) = (eta_reducible t \<or> eta_reducible u)"
dvd_multiplicity_eq: 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 x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "dvd.dvd times x y = (\<forall>p. factorial_semiring.multiplicity times one p x \<le> factorial_semiring.multiplicity times one p y)"
in_bigtheta_zero_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f \<in> \<Theta>[F](\<lambda>x. 0::'b)) = (\<forall>\<^sub>F x in F. f x = (0::'b))"
fine_division_exists: fixes g :: "'c \<Rightarrow> 'c set"   and a :: "'c"   and b :: "'c" assumes "gauge g" obtains p :: "('c \<times> 'c set) set" where "p tagged_division_of cbox a b" and "g fine p"
SigmaD2: fixes a :: "'a"   and b :: "'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
fps_times_def: shows "(*) = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i = 0..n. f $ i * g $ (n - i)))"
integral_has_vector_derivative_continuous_at: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and x :: "real"   and S :: "real set" assumes "f integrable_on {a..b}"   and "x \<in> {a..b} - S"   and "finite S"   and "continuous (at x within {a..b} - S) f" shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b} - S)"
comparator_lexicographic: fixes c1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and c2 :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b" assumes "comparator c1"   and "comparator c2"   and "\<And>x y. \<lbrakk>f x = f y; g x = g y\<rbrakk> \<Longrightarrow> x = y" shows "comparator (\<lambda>x y. case c1 (f x) (f y) of Eq \<Rightarrow> c2 (g x) (g y) | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
alw_destR: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "alw \<phi> \<pi>l" shows "alw \<phi> (map stl \<pi>l)"
assms_2: fixes xs :: "'a list" assumes "xs \<in> XS" shows "distinct xs"
P5: shows "(Bet A P P' \<or> Bet A P' P) \<and> Cong A P' X Y"
prems_2: shows "nonempty_lasttreebal (Node tts tt)"
unit_vwb_lens: shows "vwb_lens 0\<^sub>L"
sigma_sets_generated_ring_eq: shows "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
a_subid_aux: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> y \<le> y"
le_add_diff_inverse: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "b + (a - b) = a"
uses_finite: fixes g :: "'g"   and n :: "'node" shows "finite ((usesOf \<circ> uses) g n)"
simps_1: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and s' :: "'b" shows "LTS.path step s [] s' = (s' = s)"
subst_prim: fixes x :: "nat"   and v :: "exp"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp" shows "subst x v (EPrim f e1 e2) = EPrim f (subst x v e1) (subst x v e2)"
InsInitE: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c :: "stmt"   and e :: "expr"   and Q :: "'b assn" shows "G,A\<turnstile>{Normal P} InsInitE c e-\<succ> {Q}"
fMax_in: fixes A :: "'a fset" assumes "A \<noteq> {||}" shows "fMax A |\<in>| A"
WellOrderedStrategies_axioms: shows "WellOrderedStrategies G (winning_region p) p good r"
YFtorNTCompDef: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and g :: "'e" assumes "LSCategory C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "YFtorNT C f \<approx>>\<^bsub>CatExp (Op C) SET\<^esub> YFtorNT C g"
SpanD_lincomb_arb_len_coeffs: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set ms \<subseteq> V" shows "R_scalar_mult.RSpan UNIV smult ms = {scalar_mult.lincomb smult rs ms |rs. set rs \<subseteq> UNIV}"
simps_1: fixes n :: "nat"   and l :: "nat" shows "T Ins (n, l) = (if real n + 1 \<le> f2 * real l then 1 else real n + 1)"
add_diff_cancel_left': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
pos_ord_nil: shows "\<forall>x y p. p \<noteq> [] \<longrightarrow> (p, []) \<in> pos_ord x y"
if_False: fixes x :: "'b"   and y :: "'b" shows "(if False then x else y) = y"
cf_h_cmp: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'g"   and j :: "nat" assumes "PolynRg A B Y"   and "pol_coeff S (n, f)"   and "h \<in> rHom S B"   and "j \<le> n" shows "snd (cf_h h (n, f)) j = cmp h f j"
vfinite_vinsert: fixes F :: "V"   and x :: "V" assumes "vfinite F" shows "vfinite (vinsert x F)"
has_bochner_integral_monotone_convergence: fixes M :: "'c measure"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and u :: "'c \<Rightarrow> real"   and x :: "real" assumes "\<And>i. integrable M (f i)"   and "AE x in M. incseq (\<lambda>n. f n x)"   and "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"   and "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"   and "u \<in> borel_measurable M" shows "has_bochner_integral M u x"
distinct_7: fixes x2 :: "('e, 'f, 'g) Tree"   and x3 :: "'f" shows "tNot x2 \<noteq> tPred x3"
prv_neg_exi_imp_all_neg: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "prv (imp (neg (exi x \<phi>)) (all x (neg \<phi>)))"
sifum_security_init_axioms: shows "sifum_security_init dma \<C>_vars eval"
induct: fixes x1 :: "CL"   and x2 :: "CL"   and P :: "CL \<Rightarrow> CL \<Rightarrow> bool" assumes "(x1, x2) \<in> red"   and "\<And>t t' u. \<lbrakk>(t, t') \<in> red; P t t'\<rbrakk> \<Longrightarrow> P (` t u) (` t' u)"   and "\<And>u u' t. \<lbrakk>(u, u') \<in> red; P u u'\<rbrakk> \<Longrightarrow> P (` t u) (` t u')"   and "\<And>x y z. P (` ` ` S x y z) (` ` x z ` y z)"   and "\<And>x y. P (` ` K x y) x"   and "\<And>x. P (` I x) x" shows "P x1 x2"
ass_1: assumes "boundary_chain (set l)"   and "valid_chain_list l"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "line_integral_exists F basis (rec_join l)"   and "finite basis" shows "\<forall>a\<in>set l. case a of (k, a) \<Rightarrow> line_integral_exists F basis a"
game2_b_def: shows "game2_b = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). let bad = r \<in> dom s1 in if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in spmf_of_set (nlists UNIV len) \<bind> (\<lambda>pad. let cipher = (r, pb [\<oplus>] pad) in exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) (s1(r \<mapsto> pad)) \<bind> (\<lambda>(b', s3). return_spmf (b' = b, bad)))) else coin_spmf \<bind> (\<lambda>b. return_spmf (b, bad))))"
F_def: shows "F = (\<lambda>x. (ln (d * x + b) - 1 / (c + 1)) * (d * x + b) powr (c + 1) / (d * (c + 1)))"
coprime_def': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b"   and q :: "'b" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.coprime times one p q \<equiv> \<forall>r. dvd.dvd times r p \<longrightarrow> dvd.dvd times r q \<longrightarrow> dvd.dvd times r one"
that: fixes \<F> :: "'a set set" assumes "finite \<F>"   and "int (card \<F>) + aff_dim S = int DIM('a)"   and "affine hull S = \<Inter> \<F>"   and "\<And>h. h \<in> \<F> \<Longrightarrow> \<exists>a b. a \<noteq> (0::'a) \<and> h = {x. a \<bullet> x = b}" shows "thesis"
borel_measurable_le: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
msetext_huet_def: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "msetext_huet gt ys xs = (let N = mset ys; M = mset xs in M \<noteq> N \<and> (\<forall>x. count N x < count M x \<longrightarrow> (\<exists>y. gt y x \<and> count M y < count N y)))"
proj2_points_define_line: fixes p :: "proj2"   and q :: "proj2" shows "\<exists>l. proj2_incident p l \<and> proj2_incident q l"
to_hma_from_hma\<^sub>m: fixes v :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.to_hma\<^sub>m (Mod_Type_Connect.from_hma\<^sub>m v) = v"
deduction_theorem: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"
cos_one_sin_zero: fixes x :: "'b" assumes "cos x = (1::'b)" shows "sin x = (0::'b)"
pointwise_simp: fixes f :: "'b set \<Rightarrow> 'c set"   and X :: "'b set" assumes "pointwise f" shows "f X = \<Union> {f {x} |x. x \<in> X}"
nth_update_invalid: fixes i :: "nat"   and l :: "'a list"   and j :: "nat"   and x :: "'a" assumes "\<not> i < length l" shows "l[j := x] ! i = l ! i"
a1: fixes x :: "'a" assumes "x \<in># mset_set A" shows "count (mset_set A) x = 1"
assms_5: shows "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)"
equalmodstrick2: fixes a :: "nat"   and p :: "nat"   and b :: "nat" assumes "a mod p = b mod p" shows "pdifference a b mod p = 0"
e'_def: shows "\<forall>\<epsilon>>0. \<epsilon> < e' \<longrightarrow> (f has_contour_integral c * residue f z) (circlepath z \<epsilon>)"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and cs :: "'c set" shows "tr_filter (x # xs) cs = (if filter_chan_set x cs then x # tr_filter xs cs else tr_filter xs cs)"
span_alt: fixes B :: "'d set" shows "cspan B = {\<Sum>x | f x \<noteq> 0. f x *\<^sub>C x |f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
subst_lsst_unlabel_cons: fixes l :: "'f strand_label"   and b :: "('d, 'e) stateful_strand_step"   and A :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list"   and \<theta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "unlabel ((l, b) # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) # unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
Janiszewski_weak: fixes S :: "complex set"   and T :: "complex set"   and a :: "complex"   and b :: "complex" assumes "compact S"   and "compact T"   and "connected (S \<inter> T)"   and "connected_component (- S) a b"   and "connected_component (- T) a b" shows "connected_component (- (S \<union> T)) a b"
widen_Class_Iface: fixes G :: "prog"   and C :: "qtname"   and I :: "qtname" assumes "G\<turnstile>Class C\<preceq>Iface I" shows "G\<turnstile>C\<leadsto>I"
vspan: shows "lincomb l2 W \<in> local.span W"
simps_4: fixes m1 :: "'a match_expr"   and m2 :: "'a match_expr" shows "normalized_nnf_match (MatchAnd m1 m2) = (normalized_nnf_match m1 \<and> normalized_nnf_match m2)"
on_equator_meridian_is_K2_centre: fixes a :: "proj2" assumes "proj2_incident a equator"   and "proj2_incident a meridian" shows "a = K2_centre"
list_all2_lengthD: fixes P :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and xs :: "'k list"   and ys :: "'l list" assumes "list_all2 P xs ys" shows "length xs = length ys"
is_nth_power_0: fixes n :: "nat" assumes "0 < n" shows "is_nth_power n (0::'b)"
removable_singularities: fixes X :: "complex set"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "finite X"   and "X \<subseteq> interior S"   and "f holomorphic_on S - X"   and "\<And>z. z \<in> X \<Longrightarrow> f \<in> O[at z](\<lambda>_. 1)" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S - X. g z = f z)"
domain_block_nonnegative: fixes x :: "brick list" shows "0 \<le> domain_block x"
mem_val_alt_10_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 (pa AND 68719476732) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 (pa AND 68719476732) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 10 ((pa AND 68719476732) + 1) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 1) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 10 ((pa AND 68719476732) + 2) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 2) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 10 ((pa AND 68719476732) + 3) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 3) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
constrainsD: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and act :: "('b \<times> 'b) set"   and s :: "'b"   and s' :: "'b" assumes "F \<in> A co A'"   and "act \<in> Acts F"   and "(s, s') \<in> act"   and "s \<in> A" shows "s' \<in> A'"
gbinomial_0_1: fixes a :: "'c" shows "a gchoose 0 = (1::'c)"
inject_4: fixes x4 :: "'d rexp"   and y4 :: "'d rexp" shows "(Star x4 = Star y4) = (x4 = y4)"
gbinomial_pochhammer: fixes a :: "'a"   and k :: "nat" shows "a gchoose k = (- (1::'a)) ^ k * pochhammer (- a) k / fact k"