hcju/isabelleps huggingface dataset:
trcl_concat: fixes c :: "'a"   and w1 :: "'b list"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set"   and w2 :: "'b list"   and c'' :: "'a" assumes "(c, w1, c') \<in> trcl t"   and "(c', w2, c'') \<in> trcl t" shows "(c, w1 @ w2, c'') \<in> trcl t"
fds_ln_mult: fixes f :: "'a fds"   and g :: "'a fds"   and l' :: "'a"   and l'' :: "'a"   and l :: "'a" assumes "fds_nth f 1 \<noteq> (0::'a)"   and "fds_nth g 1 \<noteq> (0::'a)"   and "l' + l'' = l" shows "fds_ln l (f * g) = fds_ln l' f + fds_ln l'' g"
substitution_in_allVars: fixes g :: "'g" assumes "redundant g" shows "substitution g \<in> allVars g"
ln_upper_7_def: fixes x :: "real" shows "ln_upper_7 x \<equiv> (3 * x ^ 3 + 131 * x\<^sup>2 + 239 * x + 47) * (x - 1) / (12 * (4 * x ^ 3 + 18 * x\<^sup>2 + 12 * x + 1))"
assms_1: shows "p \<noteq> 0 \<or> 0 < n"
f_nonneg: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "0 \<le> f x"
RETURN_ref_SPECD: fixes c :: "'b"   and R :: "('b \<times> 'c) set"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "RETURN c \<le> \<Down> R (SPEC \<Phi>)" obtains a :: "'c" where "(c, a) \<in> R" and "\<Phi> a"
gen_llength_cons'_def: fixes g :: "('d, 'c) lgenerator'" shows "gen_llength_cons' g = gen_llength_cons (lgenerator g)"
ListRed2: fixes es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>Val v # es,s\<rangle> [\<rightarrow>] \<langle>Val v # es',s'\<rangle>"
expectation_X_Y: shows "\<mu> = (\<Sum>i\<in>I. expectation (X i))"
assms_2: shows "x \<le> 3 / 2 * ln 2 * real n"
strip_while_eq_Nil: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(strip_while P xs = []) = (\<forall>x\<in>set xs. P x)"
dual_map_def: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d \<Rightarrow> real" assumes "real_vector_space_pair_on S T" shows "dual_map S f y = restrict0 S (\<lambda>x. y (f x))"
that: fixes as :: "'edge list" assumes "n -as\<rightarrow>* n'" shows "thesis"
cross_ratio_commute_13: fixes z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" shows "cross_ratio z u v w = reciprocal (cross_ratio v u z w)"
VE_empty_cntxt: fixes v :: "('a, 'b, 'c) val"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(VE v = \<E> [e]) = (\<E> = \<box> \<and> VE v = e)"
bi_uniqueDl: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h"   and z :: "'g" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
valid: fixes x :: "nat"   and i :: "'a"   and b :: "'b" shows "(Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<longrightarrow> (i, Geq x b) \<in> as) \<and> (Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> (i, Leq x b) \<in> as)"
6: fixes l :: "nat"   and e :: "expr" shows "l \<le> snd (compile l e)"
simps_2: fixes x :: "'a"   and y :: "'a"   and s :: "'c"   and b :: "('a, 'c) rbt" shows "rbt_del_from_left x Empty y s b = Branch R (rbt_del x Empty) y s b"
Inf_less_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and S :: "'b set"   and a :: "'b" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf S) a = (\<exists>x\<in>S. less x a)"
mono_strict_invE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less x y"
closure_of_memI: fixes S :: "('d, 'e) eq_object_scheme"   and a :: "'d"   and A :: "'d set" assumes "a .\<in>\<^bsub>S\<^esub> A"   and "a \<in> carrier S" shows "a \<in> closure_of\<^bsub>S\<^esub> A"
multi_count_eq: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(\<forall>a. count M a = count N a) = (M = N)"
abs_ast_initial_state_def: shows "abs_ast_initial_state \<equiv> map_of (zip [0..<length astI] astI)"
star_omega_3: fixes x :: "'a" shows "(x\<^sup>\<star>)\<^sup>\<omega> = \<top>"
valid_path_edges': fixes v :: "'a"   and w :: "'a"   and Ps :: "'a llist" assumes "valid_path (LCons v (LCons w Ps))" shows "v \<rightarrow> w"
in_hom: shows "\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>"
le_plus_\<I>_iff: fixes \<I>1 :: "('a, 'c) \<I>"   and \<I>2 :: "('b, 'd) \<I>"   and \<I>1' :: "('a, 'c) \<I>"   and \<I>2' :: "('b, 'd) \<I>" shows "(\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<le> \<I>1' \<oplus>\<^sub>\<I> \<I>2') = (\<I>1 \<le> \<I>1' \<and> \<I>2 \<le> \<I>2')"
mod_less_eq_dividend: fixes m :: "nat"   and n :: "nat" shows "m mod n \<le> m"
assms_5: shows "\<b> ps (Suc i) \<le> d"
assms_1: shows "x \<notin>\<^sub>\<circ> A"
consistent_sign_vectors_r_def: fixes qs :: "real poly list"   and S :: "real set" shows "consistent_sign_vectors_r qs S = signs_at qs ` S"
the_apartment_iso_int_im: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set"   and x :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'"   and "x \<in> A \<inter> A'" shows "the_apartment_iso A A' ` x = x"
assms_2: fixes v :: "'b" assumes "x = Some v" shows "fs v \<le> fs' v"
chain_def: fixes C :: "'a set" shows "chain_on (\<sqsubset>) C A = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x))"
ofilter_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. under r a \<subseteq> A)"
assms_1: fixes i :: "'a" shows "openin (T i) (X i)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "g \<cdot>\<^sub>S f = (if right a f \<and> right a g \<and> cod f = local.dom g then g \<cdot> f else null)"
all_nth_imp_all_set: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>i<length xs. P (xs ! i)"   and "x \<in> set xs" shows "P x"
stream_dec_enc_Inr: fixes n :: "nat"   and FO :: "nat set"   and w :: "'a list"   and I :: "(nat + nat set) list"   and i :: "nat"   and P' :: "nat set"   and P :: "nat set" assumes "stream_dec n FO (stream_enc (w, I)) ! i = Inr P'"   and "I ! i = Inr P"   and "i \<notin> FO"   and "i < n"   and "length I = n" shows "P = P'"
set_of_minus: fixes X :: "'b interval"   and Y :: "'b interval" shows "set_of (X - Y) = {x - y |x y. x \<in>\<^sub>i X \<and> y \<in>\<^sub>i Y}"
bool_to_ternary_simps_2: fixes Y :: "bool" shows "(bool_to_ternary Y = TernaryFalse) = (\<not> Y)"
axioms_2: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "Group.group G"
step: fixes gpv :: "('a, 'call, 'ret) gpv"   and gpv' :: "('a, 'call, 'ret) gpv" assumes "R gpv gpv'" shows "rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv' \<or> (\<exists>gpv2 gpv2' f f'. gpv = gpv2 \<bind> f \<and> gpv' = gpv2' \<bind> f' \<and> rel_gpv (\<lambda>x y. R (f x) (f' y)) (=) gpv2 gpv2')))) (the_gpv gpv) (the_gpv gpv')"
check_cs_def: fixes check :: "'c \<Rightarrow> 'a list + 'c"   and s :: "'b \<times> 'c" shows "check_cs check s = sum_wrap (\<lambda>cs. check) s"
map_poly_hom_monom: fixes a :: "'a poly"   and i :: "nat" shows "map_poly (map_poly hom) (monom a i) = monom (map_poly hom a) i"
compl_connD: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and m :: "nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "compl_conn compl"   and "dickson_grading d"   and "fst ` set gs \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis (fst ` set gs)"   and "fst ` set bs \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "unique_idx (gs @ bs) data"   and "(p, q) \<in> set sps"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (compl gs bs (ps -- sps) sps data))) (fst p) (fst q)"
convex_polygon_Cons2: fixes x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and zs :: "((real \<times> real) \<times> real \<times> real) list" shows "convex_polygon (x # y # zs) = (fst x = snd (last (y # zs)) \<and> convex_polychain (x # y # zs))"
hta_prodWR_def: fixes H1 :: "('a, 'c, 'd) hashedTa_scheme"   and H2 :: "('b, 'c, 'e) hashedTa_scheme" shows "hta_prodWR H1 H2 \<equiv> init_hta (hhh_cart.cart (hta_Qi H1) (hta_Qi H2)) (\<delta>_prod_h (hta_\<delta> H1) (hta_\<delta> H2))"
arr_dom: fixes f :: "'a1" assumes "C1.arr f" shows "C1.arr (C1.dom f)"
special_subgroup_eq_sum_list: fixes S :: "'w set" assumes "order.greater_eq S S" shows "\<langle>S\<rangle> = (\<Union>ss\<in>lists S. {sum_list ss})"
measure_Un3: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B - Sigma_Algebra.measure M (A \<inter> B)"
restrict_def: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
remove_def: fixes x :: "'b"   and A :: "'b set" shows "Set.remove x A = A - {x}"
r_inv_ex: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<exists>y\<in>carrier V. x \<oplus>\<^bsub>V\<^esub> y = \<zero>\<^bsub>V\<^esub>"
homotopic_with_equal: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and X :: "'a topology"   and Y :: "'b topology" assumes "P f"   and "P g"   and "continuous_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "homotopic_with P X Y f g"
LeftDerivationLadder_deriv_n_bound: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and b :: "symbol list"   and index :: "nat" assumes "LeftDerivationLadder a D L b"   and "index < length L" shows "deriv_n (L ! index) \<le> length D"
diff_numeral_special_10: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus one) one = uminus (numeral.numeral one plus (num.Bit0 num.One))"
differentiable_const: fixes a :: "'d"   and F :: "'c filter" shows "(\<lambda>z. a) differentiable F"
union_of_mono: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and Q' :: "'b set \<Rightarrow> bool" assumes "(P union_of Q) S"   and "\<And>x. Q x \<Longrightarrow> Q' x" shows "(P union_of Q') S"
integral_mult_left_zero: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and c :: "'c" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
t: shows "t \<in> Basis \<rightarrow> {0<..<1}"
notin_List_update: fixes a :: "'b"   and as :: "'b list"   and h :: "'b \<Rightarrow> 'b ref"   and y :: "'b ref"   and x :: "'b ref" assumes "a \<notin> set as" shows "List (h(a := y)) x as = List h x as"
move_right1_def: shows "move_right1 \<equiv> \<lambda>(l, r, buf). if r < length buf then ASSERT (l < length buf) \<bind> (\<lambda>_. RETURN (l + 1, r + 1, buf[l := buf ! r])) else RETURN (l, r, buf)"
quasinorm_intersection_commute: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 \<inter>\<^sub>N N2 = N2 \<inter>\<^sub>N N1"
widen_null: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "P \<turnstile> NT \<le> Class C"
ImpR: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta> \<down> Suc n"
binary_endofunctor_axioms: shows "binary_endofunctor (\<cdot>) T"
adjointFO_def: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "adjointFO I \<sigma> \<nu> = \<lparr>Functions = \<lambda>f. case f of Inl f' \<Rightarrow> Functions I f' | Inr f' \<Rightarrow> \<lambda>_. dterm_sem I (\<sigma> f') \<nu>, Predicates = Predicates I, Contexts = Contexts I, Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
stake_szip: fixes n :: "nat"   and s1 :: "'c stream"   and s2 :: "'d stream" shows "stake n (s1 ||| s2) = stake n s1 || stake n s2"
card_eq_dim: fixes B :: "'a set"   and V :: "'a set" assumes "B \<subseteq> V"   and "card B = dim V"   and "finite B" shows "independent B = (V \<subseteq> span B)"
conv_def_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linear_aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "conv x = x"
additive_wp_SetPC: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> supp (p s) \<Longrightarrow> Transformers.additive (wp (a x))"   and "\<And>s. finite (supp (p s))" shows "Transformers.additive (wp (SetPC a p))"
nat_mult_power_less_eq: fixes b :: "nat"   and a :: "nat"   and n :: "nat"   and m :: "nat" assumes "0 < b" shows "(a * b ^ n < b ^ m) = (a < b ^ (m - n))"
monotone_ldrop: shows "monotone (rel_prod (=) (\<sqsubseteq>)) (\<sqsubseteq>) (\<lambda>(x, y). ldrop x y)"
P: shows "P = (\<lambda>(s, l, r). s = 10)"
onorm_bound: fixes b :: "real"   and f :: "'a \<Rightarrow> 'b" assumes "0 \<le> b"   and "\<And>x. norm (f x) \<le> b * norm x" shows "onorm f \<le> b"
scene_override_unit: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on \<bottom>\<^sub>S = S\<^sub>1"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>L 'e" shows "norm x = onorm (blinfun_apply x)"
xX_simp5: shows "E \<union> backflows stateful \<union> backflows X - (backflows (X - {x}) - E) = E \<union> backflows stateful \<union> backflows {x}"
assign_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and x :: "'a"   and e :: "'c"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((x \<leftarrow> e, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stop \<and> mds = mds' \<and> mem' = mem(x := eval\<^sub>A mem e)"
multiplicity_decompose: shows "\<not> p dvd x div p ^ multiplicity p x"
hyps: assumes "Q \<sqsubseteq> R \<sqinter> (Q ;; P)" shows "Q \<sqsubseteq> R ;; power.power II (;;) P n"
supports_finite: fixes S :: "atom set"   and x :: "'e" assumes "S supports x"   and "finite S" shows "finite (supp x)"
red_not_leaf: fixes t :: "('a, 'b) rbt" assumes "cl t = R" shows "t \<noteq> Leaf"
rtranclp_converseI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "r\<^sup>*\<^sup>* y x" shows "r\<inverse>\<inverse>\<^sup>*\<^sup>* x y"
axioms_2: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd" assumes "bidirected_digraph G arev" shows "bidirected_digraph_axioms G arev"
Struct_axioms: shows "M.Struct wtFsym wtPsym arOf resOf intT intF intP"
lt_sum_le_Max: fixes f :: "'c \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and A :: "'c set" shows "lex_pm (i.lpp (sum f A)) (i.ordered_powerprod_lin.Max {i.lpp (f a) |a. a \<in> A})"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"   and "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf" shows "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"
rsp: shows "eq_onp (\<lambda>x. x \<in> multiset) (\<lambda>a. 0) (\<lambda>a. 0)"
simps_2: fixes l :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "cl_inv' (Node l c k v r) = (cl_inv l \<and> cl_inv r)"
intro: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le"   and "total_preorder_on_axioms carrier le" shows "total_preorder_on carrier le"
cr_ff: shows "set (coeffs cr) \<subseteq> range to_fract"
repL_def: fixes w :: "32 Word.word"   and r :: "real" shows "w \<equiv>\<^sub>L r \<equiv> \<exists>r'\<le>r. w \<equiv>\<^sub>E r'"
hyps_3: shows "\<forall>t\<in>{s<..<c}. \<not> P t"
wnwn: fixes w :: "complex" assumes "cmod (w - z) < d / 2"   and "cmod (w - z) < e / 2" shows "winding_number p w = winding_number \<gamma> w"
simps_5: fixes n :: "nat"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" shows "rexp_of n (FOr \<phi>\<^sub>1 \<phi>\<^sub>2) = rexp.Inter (Plus (rexp_of n \<phi>\<^sub>1) (rexp_of n \<phi>\<^sub>2)) (ENC n (FOV (FOr \<phi>\<^sub>1 \<phi>\<^sub>2)))"
cont_While_False: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom, 'choice) cmd"   and i :: "nat" assumes "tval tst s = False" shows "local.cont (While tst c) s i = Done"
Example2_lemma2: fixes j :: "nat"   and n :: "nat"   and b :: "nat \<Rightarrow> nat" assumes "j < n"   and "b j = 0" shows "Suc (sum b {0..<n}) = sum (b(j := Suc 0)) {0..<n}"
fut_spec_while_default: shows "fut_spec (while_default fut_default (\<lambda>(\<phi>, \<Phi>). \<phi> \<notin> set \<Phi>) fut_step (\<psi>, []))"
gen_dfs_fe_inv_imp_var: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a"   and it :: "'a set"   and V' :: "'a set"   and t :: "'a" assumes "gen_dfs_pre E U S V u0"   and "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 it V' False"   and "t \<in> it"   and "it \<subseteq> E `` {u0}"   and "t \<notin> V'" shows "(V', V) \<in> gen_dfs_var U"
eq_divide_eq_numeral_1: fixes w :: "num"   and b :: "'a"   and c :: "'a" shows "(numeral w = b / c) = (if c \<noteq> (0::'a) then numeral w * c = b else numeral w = (0::'a))"
simps: fixes S :: "'b set" shows "enum_set_to_list S = (if S = {} then [] else case enum_set_get_one enum_class.enum S of None \<Rightarrow> [] | Some a \<Rightarrow> a # enum_set_to_list (S - {a}))"
emeasure_eq_measure: fixes p :: "'c spmf"   and A :: "'c set" shows "emeasure (measure_spmf p) A = ennreal (Sigma_Algebra.measure (measure_spmf p) A)"
HInfinite_omega: shows "\<omega> \<in> HInfinite"
SubobjsR_subclass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "P \<turnstile> C \<preceq>\<^sup>* last Cs"
div_plus_div_distrib_dvd_left: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c a" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
assms_3: shows "ps\<^sub>+ \<noteq> []"
frefI: fixes P :: "'b \<Rightarrow> bool"   and R :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and S :: "('c \<times> 'd) set" assumes "\<And>x y. \<lbrakk>P y; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, g y) \<in> S" shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
fcard_mono: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| B" shows "fcard A \<le> fcard B"
insert_between_elements: fixes xs :: "'a list"   and pre :: "'a list"   and ref :: "'a"   and suf :: "'a list"   and e :: "'a" assumes "xs = pre @ ref # suf"   and "distinct xs"   and "\<And>i. i \<in> set xs \<Longrightarrow> i < e" shows "insert_rga xs (e, Some ref) = pre @ ref # e # suf"
abs_summable_partition: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'c \<Rightarrow> 'd set"   and I :: "'c set"   and T :: "'d set" assumes "\<And>i. f abs_summable_on S i"   and "(\<lambda>i. \<Sum>\<^sub>ax\<in>S i. norm (f x)) abs_summable_on I"   and "T \<subseteq> \<Union> (S ` I)" shows "f abs_summable_on T"
arity_sym_ne_infinity_if_\<delta>_gt_0: fixes f :: "'s" assumes "0 < \<delta>" shows "arity_sym f \<noteq> \<infinity>"