hcju/isabelleps huggingface dataset:
pairs_uminus_closed: fixes x :: "'b regular"   and y :: "'b dense Filters.filter" shows "stone_phi.pairs_uminus (x, y) \<in> stone_phi.pairs"
DERIV_inverse_function: fixes f :: "real \<Rightarrow> real"   and D :: "real"   and g :: "real \<Rightarrow> real"   and x :: "real"   and a :: "real"   and b :: "real" assumes "(f has_real_derivative D) (at (g x))"   and "D \<noteq> 0"   and "a < x"   and "x < b"   and "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y"   and "isCont g x" shows "(g has_real_derivative inverse D) (at x)"
spannable_def: fixes x :: "'a" shows "spannable x \<equiv> \<exists>y. spanning x y"
obj_sumE: fixes s :: "'c + 'd"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
comp_params_index_pair: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and x :: "'a"   and y :: "'a" assumes "bibd point_set block_collection u_block_size index"   and "{x, y} \<subseteq> point_set"   and "x \<noteq> y" shows "int (incidence_system.complement_blocks point_set block_collection index {x, y}) = int (size block_collection) + index - 2 * (index * (int (card point_set) - 1) div (u_block_size - 1))"
f0: fixes x :: "real" assumes "0 < x"   and "x < a" shows "(f0 has_real_derivative f' x) (at x)"
prefixToElementContainsTrailElement: fixes e :: "'b"   and M :: "('b \<times> bool) list" assumes "e \<in> set (elements M)" shows "e \<in> set (elements (prefixToElement e M))"
strict_trans: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a <s b"   and "b <s c" shows "a <s c"
LABEL_simps_2: fixes inp :: "nat"   and ct :: "(char list \<times> nat \<times> unit list) list"   and outp :: "nat"   and P :: "'c" shows "Case_Labeling.CTXT inp ct outp P \<equiv> P"
simps_1: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and more :: "'d" shows "(\<plusminus>\<^bsub>\<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, \<dots> = more\<rparr>\<^esub>) = pop"
inverse: shows "(R ===> R) (arith_ops_record.inverse ops) inverse_class.inverse"
signed_take_bit_Suc_minus_bit1: fixes n :: "nat"   and k :: "num" shows "signed_take_bit (Suc n) (- numeral (num.Bit1 k)) = signed_take_bit n (- numeral k - 1) * 2 + 1"
Lp_1: fixes p :: "real"   and M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "1 \<le> p" shows "eNorm (\<LL> (ennreal p) M) f = (if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)) else \<infinity>)"
struct_specD2: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "punit.struct_spec sel ap ab compl" shows "punit.ap_spec ap"
simps_5: fixes uu :: "dir list"   and uv :: "dir"   and h :: "('c, 'd) hd" shows "emb_step_at uu uv (Hd h) = Hd h"
enat_min_eq_0_iff: fixes a :: "enat"   and b :: "enat" shows "(min a b = 0) = (a = 0 \<or> b = 0)"
Const_conv_monom: fixes c :: "'b" shows "Const c = MPoly_Type.monom 0 c"
heun_remainder1_def: fixes x :: "real \<Rightarrow> 'c"   and f :: "real \<times> 'c \<Rightarrow> 'b"   and f' :: "real \<times> 'c \<Rightarrow> (real \<times> 'b) \<Rightarrow>\<^sub>L 'b"   and f'' :: "real \<times> 'c \<Rightarrow> (real \<times> 'b) \<Rightarrow>\<^sub>L (real \<times> 'b) \<Rightarrow>\<^sub>L 'b"   and t :: "real"   and h :: "real"   and s :: "real" shows "heun_remainder1 x f f' f'' t h s = (h ^ 3 / 6) *\<^sub>R (f'' (h * s + t, x (h * s + t)) $ (1, f (h * s + t, x (h * s + t))) $ (1, f (h * s + t, x (h * s + t))) + f' (h * s + t, x (h * s + t)) $ (0, f' (h * s + t, x (h * s + t)) $ (1, f (h * s + t, x (h * s + t)))))"
jordan_matrix_char_poly: fixes n_as :: "(nat \<times> 'a) list" shows "char_poly (jordan_matrix n_as) = (\<Prod>(n, a)\<leftarrow>n_as. [:- a, 1::'a:] ^ n)"
finite_insert: fixes a :: "'a::type"   and A :: "'a::type set" shows "finite (insert (a::'a::type) (A::'a::type set)) = finite A"
wf_config_def: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and s :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and T :: "ty" shows "P,E,s \<turnstile> e : T \<surd> \<equiv> P,E \<turnstile> s \<surd> \<and> iconf (shp s) e \<and> P,E,hp s,shp s \<turnstile> e : T"
bit_push_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (push_bit m a) n = (m \<le> n \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (n - m))"
assms_3: shows "Q_I_neg = filterQueryNeg I Qn"
C_def: shows "C = (\<lambda>H'. {\<chi>' \<in> characters (G\<lparr>carrier := H'\<rparr>). \<forall>x\<in>H. \<chi>' x = \<chi> x})"
labeled_bot_entailed: shows "entails_\<G> (lhd Ns) {(B, active)}"
type_update_select_length2: fixes cts :: "ct list"   and tm :: "checker_type" assumes "type_update_select (TopType cts) = tm"   and "length cts = 2"   and "tm \<noteq> Bot" shows "\<exists>t1 t2. cts = [t1, t2] \<and> ct_eq t2 (TSome T_i32) \<and> tm = TopType [t1]"
intro: fixes M :: "'a \<Rightarrow> 'b measure" assumes "\<And>i. sigma_finite_measure (M i)" shows "product_sigma_finite M"
compl_le_compl_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (uminus x) (uminus y) = less_eq y x"
Tpows_v'_def: shows "Tpows_v' = map (\<lambda>S. S v) Tpows'"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>j. P (C j)"   and "\<And>n. P (Bound n)"   and "\<And>t. P t \<Longrightarrow> P (Neg t)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (Add t s)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (Sub t s)"   and "\<And>i t. (i \<noteq> 0 \<Longrightarrow> P t) \<Longrightarrow> P (Mul i t)"   and "\<And>t. P t \<Longrightarrow> P (Floor t)"   and "\<And>n c t. \<lbrakk>c = 0 \<Longrightarrow> P t; c \<noteq> 0 \<Longrightarrow> P t\<rbrakk> \<Longrightarrow> P (CN n c t)"   and "\<And>c t s. P (Add (Mul c (Floor t)) s) \<Longrightarrow> P (CF c t s)" shows "P a0"
step_12: shows "\<alpha> = Cantor_sum \<beta>s ns"
prems_2: shows "s \<le>ns concat (ws \<cdot> [a])"
Nd: fixes n :: "nat" assumes "Nd \<le> n" shows "M + K + 3 * deltaG TYPE('a) \<le> dist basepoint ((f ^^ n) basepoint)"
aposs_na_poss: fixes n :: "ant" assumes "n \<noteq> \<infinity>"   and "0 < n" shows "0 < na n"
pmaxsimpchain: fixes xs :: "'a set list" assumes "pgallery xs" shows "folding_g.gallery xs"
iunwinding_implies_view_partitioned1: shows "iview_partitioned"
axioms_1: fixes x :: "'d \<Longrightarrow> 'e" assumes "mwb_lens x" shows "weak_lens x"
group_isomorphisms_homology_group_prod_deformation: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and p :: "int" assumes "homotopic_with (\<lambda>x. True) X X id f"   and "f ` topspace X \<subseteq> S" obtains H :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set" and K :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set" where "subgroup H (homology_group p (subtopology X S))" and "subgroup K (homology_group p (subtopology X S))" and "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p (subtopology X S)\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) H \<times>\<times> subgroup_generated (homology_group p (subtopology X S)) K) (homology_group p (subtopology X S))" and "hom_boundary (p + 1) X S \<in> Group.iso (relative_homology_group (p + 1) X S) (subgroup_generated (homology_group p (subtopology X S)) H)" and "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) K) (homology_group p X)"
that: assumes "r a b"   and "a \<noteq> b" shows "thesis"
reachable_init: fixes s :: "'e"   and A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> reachable A I"
Upper_is_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "is_closed (Upper L A)"
roofed_circ_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S = RF\<^bsub>\<Gamma>\<^esub> S - \<E>\<^bsub>\<Gamma>\<^esub> S"
commutative_squareE: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" obtains "arr f" and "arr g" and "arr h" and "arr k" and "cod f = cod g" and "local.dom h = local.dom k" and "local.dom f = cod h" and "local.dom g = cod k" and "f \<cdot> h = g \<cdot> k"
similar_bot: shows "\<bottom> \<triangleleft>\<triangleright> \<bottom>"
distinct_5: fixes a1 :: "'d"   and list :: "name list"   and a2 :: "'d" shows "a1\<lparr>\<nu>*list\<rparr>\<langle>a2\<rangle> \<noteq> \<tau>"
Rats_eq_range_of_rat_o_nat_to_rat_surj: shows "field_char_0.Rats (/) (1::'a) (+) (0::'a) (-) = range (field_char_0.of_rat (/) (1::'a) (+) (0::'a) (-) \<circ> nat_to_rat_surj)"
surjective: fixes r :: "('g, 'h, 'i, 'j, 'k, 'l) system_state_scheme" shows "r = \<lparr>GST = GST r, HST = HST r, \<dots> = system_state.more r\<rparr>"
elims_1: fixes x :: "'a \<times> nat"   and y :: "bool" assumes "is_valid_state_id x = y" obtains i :: "'a" and j :: "nat" where "x = (i, j)" and "y = (j \<le> length (events i))"
ord_0_nat: fixes n :: "nat" shows "ord 0 n = (if n = 1 then 1 else 0)"
lex_rep_nat_pp: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" shows "lex_pp (rep_nat_pp x) (rep_nat_pp y) = lex_pp x y"
xr_less_xa: shows "x ^ r - 1 < x ^ a - 1"
bij_betw_vertices: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexIsomorphism X Y f" shows "bij_betw f (\<Union> X) (\<Union> Y)"
parts_insert_Nonce: fixes N :: "nat"   and H :: "msg set" shows "parts (insert (Nonce N) H) = insert (Nonce N) (parts H)"
mult_map_scale_conv_mult: fixes s :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.map ((*) s) p = monomial s (0::'b) * p"
setVar'_pState_inv: fixes prog :: "program"   and p :: "pState"   and gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer"   and g :: "'a gState_scheme" assumes "pState_inv prog p" shows "pState_inv prog (snd (setVar' gl v idx val g p))"
b: fixes b :: "real" assumes "b \<in> B V f" shows "b \<le> y"
minus_less_iff: fixes a :: "'a"   and b :: "'a" shows "(- a < b) = (- b < a)"
prv_subst_imp_softSubst: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "t \<in> trm"   and "\<phi> \<in> fmla"   and "x \<notin> FvarsT t" shows "prv (imp (subst \<phi> t x) (softSubst \<phi> t x))"
ipresIGFreshAbs_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'j, 'k, 'a, 'l, 'g, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGFreshAbs hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresFreshAbs Delta hA MOD"
subtopology_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
prv_eql_subst_trm: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "prv (imp (eql t1 t2) (imp (subst \<phi> t1 x) (subst \<phi> t2 x)))"
assms_1: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
integrable_const_bound: fixes f :: "'c \<Rightarrow> 'd"   and B :: "real"   and M :: "'c pmf" assumes "AE x in measure_pmf M. norm (f x) \<le> B"   and "measure_pmf.random_variable M borel f" shows "integrable (measure_pmf M) f"
expolys: shows "rpoly (qs ! p1) x = 0 \<and> rpoly (qs ! p2) x = 0"
binary_code_axioms: shows "binary_code u\<^sub>1 u\<^sub>0"
singleActionChain: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
set_shuffles: fixes zs :: "'g list"   and xs :: "'g list"   and ys :: "'g list" assumes "zs \<in> shuffles xs ys" shows "set zs = set xs \<union> set ys"
compatible_sym_iff: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.cpo below" shows "cpo.compatible below x y = cpo.compatible below y x"
elims: fixes x :: "nat"   and y :: "nat" assumes "oddpart x = y" obtains "x = 0" and "y = 0"   | v :: "nat" where "x = Suc v" and "y = (if odd (Suc v) then Suc v else oddpart (Suc v div 2))"
R26_d: shows "pmf (sds R26) d = 1 - pmf (sds R26) a"
NH\<^sub>N_def: fixes n :: "nat" shows "NH\<^sub>N n = \<integral>\<^sup>+ x. ennreal (real x) \<partial>measure_pmf (H\<^sub>N n)"
kuratowski_def: fixes X :: "'b set" shows "kuratowski X = (\<not> extremally_disconnected X \<and> \<not> open_unresolvable X)"
Rr: shows "rel_set \<phi> (Inr -` A1) (Inr -` A2)"
unitary_is_sq_length_preserving: fixes U :: "complex mat"   and v :: "complex vec" assumes "unitary U"   and "dim_vec v = dim_col U" shows "\<parallel>col_fst (U * |v\<rangle>)\<parallel>\<^sup>2 = \<parallel>v\<parallel>\<^sup>2"
document_ptr_kinds_M_defs: shows "document_ptr_kinds_M = Heap_Error_Monad.bind get_heap (\<lambda>h. return (sorted_list_of_set (fset (document_ptr_kinds h))))"
powr_neg_real_complex: fixes x :: "real"   and a :: "complex" shows "(- complex_of_real x) powr a = (- 1) powr (complex_of_real (sgn x) * a) * complex_of_real x powr a"
RetElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some vreturn" shows "False"
has_products_if_has_limits: fixes I :: "'a set" assumes "has_limits undefined"   and "I \<noteq> UNIV" shows "has_products I"
deg_pm_map_scale: fixes k :: "'b"   and t :: "'c \<Rightarrow>\<^sub>0 'b" shows "deg_pm (k \<cdot> t) = k * deg_pm t"
bounded_iff: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
irreducible_mult_unit_right: fixes a :: "'a"   and p :: "'a" assumes "a dvd (1::'a)" shows "irreducible (p * a) = irreducible p"
apartments_have_many_foldings1_3: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "C \<noteq> D"   and "C \<in> A"   and "D \<in> A" shows "restrict1 (canonical_retraction A C \<circ> canonical_retraction (supapartment D (some_third_chamber C D (C \<inter> D))) D) (\<Union> A) ` C = D"
simps_3: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (Eq (CN 0 c e)) = F"
div_mult_mult2: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
mcontI: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and luba :: "'a set \<Rightarrow> 'a"   and lubb :: "'b set \<Rightarrow> 'b" assumes "monotone orda ordb f"   and "cont luba orda lubb ordb f" shows "mcont luba orda lubb ordb f"
inf_sup_aci_7: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
bidel_arc_tail_conn_axioms: shows "bidel_arc_tail_conn G M a"
zero: shows "map_of (map (\<lambda>j. (poly P j, j)) js) 0 = None"
finite_has_minimal: fixes A :: "'a list set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. b \<le>lex m \<longrightarrow> m = b"
base_case: shows "Jacobi 2 p = (- 1) ^ (((nat p)\<^sup>2 - 1) div 8)"
g_2: fixes p :: "nat" assumes "p \<in> P" shows "ord (p ^ k p) (g p) = Carmichael (p ^ k p)"
lincomb_def: fixes a :: "rat vec \<Rightarrow> rat"   and A :: "rat vec set" shows "gs.lincomb a A = (\<Oplus>\<^bsub>gs.V\<^esub>v\<in>A. a v \<cdot>\<^sub>v v)"
restrict_spmf_def: fixes p :: "'a spmf"   and A :: "'a set" shows "p \<upharpoonleft> A = map_pmf (\<lambda>x. Option.bind x (\<lambda>y. if y \<in> A then Some y else None)) p"
public_const_deduct'_2: fixes c :: "'fun"   and M :: "('fun, 'var) Term.term set" assumes "arity c = 0"   and "public c" shows "M \<turnstile>\<^sub>c Fun c []"
induct: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a list" assumes "\<And>x. P x []"   and "\<And>x y xs. (x \<noteq> y \<Longrightarrow> P x xs) \<Longrightarrow> P x (y # xs)" shows "P a0 a1"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(BNat x1 = BNat y1) = (x1 = y1)"
waiting_switch_independent: shows "\<forall>n s. rwaiting (rcswitch n s) = rwaiting s"
fix_unique: shows "\<forall>v w. s v = v \<longrightarrow> s w = w \<longrightarrow> v \<noteq> w \<longrightarrow> s = id"
assms_5: shows "\<Prod>A dvd \<Prod>B"
CoCalls_eqI: fixes G :: "CoCalls"   and G' :: "CoCalls" assumes "\<And>x y. x--y\<in>G = x--y\<in>G'" shows "G = G'"
add_diff_eq: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
induct: fixes x :: "'e"   and g :: "'e \<Rightarrow> ('f, 'e) step"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> terminates_on g"   and "\<And>s. g s = Done \<Longrightarrow> P s"   and "\<And>s s'. \<lbrakk>g s = Skip s'; s' \<in> terminates_on g; P s'\<rbrakk> \<Longrightarrow> P s"   and "\<And>s a s'. \<lbrakk>g s = Yield a s'; s' \<in> terminates_on g; P s'\<rbrakk> \<Longrightarrow> P s" shows "P x"
star_subdist_var_2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<le> (x + y)\<^sup>\<star>"
refine: fixes n :: "nat" shows "(fw_impl n, fw_impl' n) \<in> (mtx_assn n)\<^sup>d \<rightarrow>\<^sub>a mtx_assn n"
all1_matrix_dim_1: fixes nr :: "nat"   and nc :: "nat" shows "dim_row (all1_matrix nr nc) = nr"
i_take_not_empty_conv: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "(f \<Down> n \<noteq> []) = (0 < n)"
PROTECT_def: fixes x :: "'a" shows "PROTECT x \<equiv> x"
apex_def: shows "Dom'.apex \<equiv> Maps.dom Dom'.leg0"
parts_pubkey_false: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt (SesK SK) (PubKey C) \<in> parts (used s)"   and "\<forall>n. (Owner n, SesKey SK) \<notin> s"   and "SesKey SK \<notin> spied s" shows "False"
Max_in: fixes A :: "'d DBMEntry set" assumes "finite A"   and "A \<noteq> {}" shows "linordered_monoid.Max A \<in> A"
alg_mult_eq_count_roots: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)" shows "alg_mult p = count (roots p)"
\<alpha>: shows "\<alpha> \<in> {0<..<pi}"
liftablep_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" shows "liftablep r f \<equiv> \<forall>x y. r x y \<longrightarrow> r (f x) (f y)"
maximal_independent_subset_extend: fixes S :: "'b set"   and V :: "'b set" assumes "S \<subseteq> U\<^sub>V\<^sub>S"   and "V \<subseteq> U\<^sub>V\<^sub>S"   and "S \<subseteq> V"   and "\<not> local.dependent S" obtains B :: "'b set" where "B \<subseteq> U\<^sub>V\<^sub>S" and "S \<subseteq> B" and "B \<subseteq> V" and "\<not> local.dependent B" and "V \<subseteq> local.span B"
imod_iless_divisor: fixes n :: "enat"   and m :: "enat" assumes "0 < n"   and "m \<noteq> \<infinity>" shows "m mod n < n"
assms_2: fixes x :: "'b" assumes "x \<in> s" shows "e \<inter> f x \<in> sets M"
assms_1: shows "valid ((Pr n f g, x # xs, []) # rest)"
MutuallyDistinct_def: fixes F :: "('a, 'b, 'c) seqauto set" shows "MutuallyDistinct F = (\<forall>a\<in>F. \<forall>b\<in>F. a \<noteq> b \<longrightarrow> States a \<inter> States b = {})"
prems_3: shows "mod_type_class.to_nat i \<le> Suc n"
almost_full_on_hom: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'c" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q (h x) (h y)"   and "almost_full_on P A" shows "almost_full_on Q (h ` A)"
simps_15: fixes x5 :: "char list" shows "Class x5 \<noteq> Boolean"
has_integral_integral_lborel: assumes "integrable lborel f" shows "(f has_integral integral\<^sup>L lborel f) UNIV"