hcju/isabelleps huggingface dataset:
mtf2_forward_effect3: fixes q :: "'b"   and xs :: "'b list"   and n :: "nat"   and i :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q - n \<le> i \<and> i < index xs q" shows "index (mtf2 n q xs) (xs ! i) = Suc (index xs (xs ! i)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q"
tl_drop: fixes n :: "nat"   and xs :: "'a list" shows "tl (drop n xs) = drop n (tl xs)"
elims_3: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)"   and xb :: "nat set" assumes "\<not> x \<Turnstile>\<^sub>b xa \<parallel> xb" obtains v :: "nat \<Rightarrow> 'a" and lb :: "nat \<Rightarrow> 'a option" and ub :: "nat \<Rightarrow> 'a option" and S :: "nat set" where "x = v" and "xa = (lb, ub)" and "xb = S" and "\<not> (\<forall>x\<in>S. in_bounds x v (lb, ub))"
D: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
countable_enum_cases: fixes S :: "'f set" assumes "countable S" obtains f :: "'f \<Rightarrow> nat" where "finite S" and "bij_betw f__ S {..<card S}"   | f :: "'f \<Rightarrow> nat" where "infinite S" and "bij_betw f S UNIV"
nfa_Reverse_nfa: shows "nfa (Reverse_nfa Canon.DFA)"
nth_list_of: fixes xs :: "'a llist" assumes "lfinite xs" shows "(!) (list_of xs) = lnth xs"
sbh_step_preserves_load_tmps_bound: fixes "is" :: "instr list"   and \<O> :: "nat \<Rightarrow> nat option"   and \<D> :: "'a memref list"   and \<theta> :: "nat \<Rightarrow> nat"   and sb :: "bool"   and \<S> :: "nat set"   and m :: "(nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and is' :: "instr list"   and \<O>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "'a memref list"   and \<theta>' :: "nat \<Rightarrow> nat"   and sb' :: "bool"   and \<S>' :: "nat set"   and m' :: "(nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "(is, \<O>, \<D>, \<theta>, sb, \<S>, m) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<O>', \<D>', \<theta>', sb', \<S>', m')"   and "\<forall>i\<in>load_tmps is. i < n" shows "\<forall>i\<in>load_tmps is'. i < n"
prod_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C f0"   and "partial_magma.arr C f1" shows "partial_magma.cod C (elementary_category_with_binary_products.prod C pr0 pr1 f0 f1) = elementary_category_with_binary_products.prod C pr0 pr1 (partial_magma.cod C f0) (partial_magma.cod C f1)"
FSwSbImorph_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'h, 'j, 'm) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'i, 'k, 'n) model_scheme" shows "FSwSbImorph h hA MOD MOD' \<equiv> FSwImorph h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
precise_hassoc_map_assn: shows "precise hassoc_map_assn"
neq_dynM_statM: shows "dynM \<noteq> statM"
alw_S: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg" shows "almost_everywhere (R_G.T cfg) (pred_stream (\<lambda>s. s \<in> \<S>))"
distinct_1: fixes x21 :: "ctxt"   and x22 :: "trm" shows "\<diamond> \<noteq> x21 \<^sup>\<bullet> x22"
meval_modes_respect_priv: fixes gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and n :: "nat"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "conc.meval_abv gc\<^sub>C n gc\<^sub>C'"   and "length (fst gc\<^sub>C) = length cms"   and "modes_respect_priv (map snd (fst gc\<^sub>C))" shows "modes_respect_priv (map snd (fst gc\<^sub>C'))"
fib_work_final_def: shows "fib_work_final \<equiv> fst (fix_syn (\<lambda>(fib_work_final, fib_f_final). (smap\<cdot>fib_f_final\<cdot>nats, Nat_case\<cdot>1\<cdot>(Nat_case\<cdot>1\<cdot>(\<Lambda> n'. fib_work_final !! n' + fib_work_final !! (n' + 1))))))"
set_list_bind: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'c list" shows "set (List.bind xs f) = (\<Union>x\<in>set xs. set (f x))"
Heine_Borel_imp_Bolzano_Weierstrass: fixes s :: "'b set"   and t :: "'b set" assumes "compact s"   and "infinite t"   and "t \<subseteq> s" shows "\<exists>x\<in>s. x islimpt t"
simps: fixes a :: "hf" shows "coding_hf a = ((\<exists>x. a = x \<and> (\<exists>i. x = ord_of i)) \<or> (\<exists>x y. a = \<langle>x, y\<rangle> \<and> coding_hf x \<and> coding_hf y))"
dvd_minus_iff: fixes x :: "'a"   and y :: "'a" shows "(x dvd - y) = (x dvd y)"
gd_nat_term_to_pair_unit: fixes cmp_term :: "'b nat_term_order" shows "gd_nat_term to_pair_unit fst cmp_term"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "del_min2 \<langle>l, x, r\<rangle> = merge l r"
set_lebesgue_integral_eq_integral_2: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S" shows "set_lebesgue_integral lebesgue S f = integral S f"
Poly_snoc_zero: fixes as :: "'b list" shows "Poly (as @ [0::'b]) = Poly as"
disj_cong: fixes P :: "bool"   and P' :: "bool"   and Q :: "bool"   and Q' :: "bool" assumes "P = P'"   and "\<not> P' \<Longrightarrow> Q = Q'" shows "(P \<or> Q) = (P' \<or> Q')"
subst_frees: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and env :: "(name, 'a) fmap"   and t :: "'a" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "fmpred (\<lambda>_ t. pre_term.closed_except frees t {||}) env" shows "frees (subst t env) = frees t |-| fmdom env"
infdist_proper_attained: fixes C :: "'b set"   and x :: "'b" assumes "proper C"   and "C \<noteq> {}" shows "\<exists>c\<in>C. infdist x C = dist x c"
alist_split_set_2: fixes xs :: "('c, 'd) bintree option list" shows "set (prios xs) = snd ` set (elements xs)"
L_While_unfold: fixes b :: "bexp"   and c :: "com"   and X :: "char list set" shows "L (WHILE b DO c) X = vars b \<union> X \<union> L c (L (WHILE b DO c) X)"
T_def: shows "T \<equiv> {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
assms_2: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
erH_mem: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and p :: "'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "p \<in> carrier R" shows "erH R S X A B Y h p \<in> carrier A"
intro: fixes b :: "real"   and e :: "real"   and hb :: "real" assumes "0 < b"   and "b < 1"   and "0 < e"   and "0 < hb" shows "akra_bazzi_asymptotics_bep b e hb"
hyps_2: fixes V :: "char list" assumes "(l(V__ \<mapsto> w')) V = None"   and "(E(V__ \<mapsto> T')) V = \<lfloor>T\<rfloor>"   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> T casts v' to v' "
assms: shows "affine hull B = affine hull V \<and> \<not> affine_dependent B"
inf_fres: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> x = \<Sqinter> {z. y \<le> z \<leftarrow> x}"
count_union: fixes M :: "'c multiset"   and N :: "'c multiset"   and a :: "'c" shows "count (M + N) a = count M a + count N a"
smult_distrib_left_diff: fixes r :: "'f"   and m :: "'v"   and n :: "'v" assumes "r \<in> UNIV"   and "m \<in> V"   and "n \<in> V" shows "r \<cdot> (m - n) = r \<cdot> m - r \<cdot> n"
nn_integral_indicator: fixes A :: "'d set"   and M :: "'d measure" assumes "A \<in> sets M" shows "integral\<^sup>N M (indicator A) = emeasure M A"
distinct_27: fixes x3 :: "fresh_t"   and x61 :: "msg"   and x62 :: "msg" shows "Nonce x3 \<noteq> \<lbrace>x61, x62\<rbrace>"
partition_on_all_in_part_eq_part: fixes A :: "'c set"   and P :: "'c set set"   and X' :: "'c set" assumes "partition_on A P"   and "X' \<in> P" shows "{x \<in> A. (THE X. x \<in> X \<and> X \<in> P) = X'} = X'"
is_rerr_asm: fixes \<xi>' :: "'a state_scheme"   and \<xi> :: "'a state_scheme" assumes "\<xi>' \<in> is_rerr \<xi>" shows "\<exists>dests' sip'. msg \<xi> = Rerr dests' sip' \<and> \<xi>' = \<xi>\<lparr>dests := dests', sip := sip'\<rparr>"
right_unit: fixes a :: "('a \<times> 'a) set" assumes "a \<in> carrier_local_ring_at" shows "add_local_ring_at a zero_local_ring_at = a"
frameResPres: fixes F :: "'b frame"   and G :: "'b frame"   and x :: "name" assumes "F \<simeq>\<^sub>F G" shows "\<lparr>\<nu>x\<rparr>F \<simeq>\<^sub>F \<lparr>\<nu>x\<rparr>G"
fp: shows "\<forall>x\<in>F. \<bar>f x\<bar> \<le> p x"
k_def: shows "k = div_op (H $$ (0, 2)) (H $$ (0, 0))"
bounded_derivative_imp_lipschitz: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and C :: "real" assumes "\<And>x. x \<in> X \<Longrightarrow> (f has_derivative f' x) (at x within X)"   and "convex X"   and "\<And>x. x \<in> X \<Longrightarrow> onorm (f' x) \<le> C"   and "0 \<le> C" shows "C-lipschitz_on X f"
a: shows "x ; 1 \<cdot> (x\<^sup>T ; 1 + y ; 1) \<cdot> y\<^sup>T ; 1 = 0 \<and> start_points x \<cdot> end_points y = 0"
f: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
b: shows "(d::'a::refinement_lattice) \<sqsubseteq> (j::'a::refinement_lattice) \<parallel> d // j"
mem_UnitsD: fixes u :: "'a" assumes "u \<in> Units" shows "invertible u \<and> u \<in> M"
simps_19: fixes P :: "nat" shows "bound0 (NClosed P) = True"
bicategorical_language_axioms: shows "bicategorical_language (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B"
same_symbol: shows "\<gamma> ! j = \<gamma>' ! (j - length \<delta>)"
rrk_is_preserved: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec" assumes "invertible P" shows "row_rank A = row_rank (P ** A)"
simps_2: fixes s :: "'a list"   and r :: "'a"   and rs :: "'a list"   and a :: "nat \<times> nat list"   and as :: "(nat \<times> nat list) list" shows "T\<^sub>p s (r # rs) (a # as) = t\<^sub>p s r a + T\<^sub>p (step s r a) rs as"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "b \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "c \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "(a = b \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> c) = (b = a \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> c)"
tag_in_interval: fixes p :: "('b \<times> 'b set) set"   and i :: "'b set"   and x :: "'b"   and k :: "'b set" assumes "p tagged_division_of i"   and "(x, k) \<in> p" shows "x \<in> i"
skip: fixes m :: "'id \<Rightarrow> 'val" shows "SRabbr skip m [] m"
psi_double_lemma: fixes n :: "nat" assumes "1200 \<le> n" shows "real n / 6 \<le> psi n - psi (n div 2)"
distinct_1: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form" shows "Bound x11 x12 x13 x14 \<noteq> Assign x21 x22 x23"
R1: shows "row_echelon_form (gauss_jordan_single A)"
bsmap_filter_xlate: fixes P :: "(('c, 'd) BsSkewElem, 'd) SkewBinomialHeapStruc.SkewBinomialTree \<Rightarrow> bool"   and l :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (filter (\<lambda>x. P (bsmapt x)) l) = filter P (map bsmapt l)"
i2: shows "ideal_generated {a, b} = ideal_generated {d'}"
Init_subset_reachable: fixes F :: "'b program" shows "Init F \<subseteq> reachable F"
P_def: fixes s :: "'s"   and n :: "nat" shows "P s n = emeasure (T (simple ct s)) {x \<in> space (T (simple ct s)). ((\<lambda>P xs. HLD S2 xs \<or> HLD S1 xs \<and> nxt P xs) ^^ n) (\<lambda>x. False) (s ## x)}"
set_indets_pp: fixes p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'b" shows "set (indets_pp p) = indets (Poly_Mapping.map_key PP p)"
FunctorMapsTo: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme"   and f :: "'i"   and X :: "'g"   and Y :: "'g" assumes "Ftor F : A \<longrightarrow> B"   and "f maps\<^bsub>A\<^esub> X to Y" shows "F ## f maps\<^bsub>B\<^esub> F @@ X to F @@ Y"
call_zero_iff: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(\<forall>u. cinner x u = 0) = (x = zero)"
index_mat_multrow_mat_2: fixes n :: "nat"   and k :: "nat"   and a :: "'b" shows "dim_row (multrow_mat n k a) = n"
integral_has_vector_derivative': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "continuous_on {a..b} f"   and "x \<in> {a..b}" shows "((\<lambda>u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})"
inter_mset_empty_distrib_right: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" shows "(A \<inter># (B + C) = {#}) = (A \<inter># B = {#} \<and> A \<inter># C = {#})"
uniform: fixes bl :: "'a set" assumes "bl \<in># \<B>'" shows "int (card bl) = \<k>"
vector_derivative_unique_within: fixes x :: "real"   and S :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and f'' :: "'a" assumes "at x within S \<noteq> bot"   and "(f has_vector_derivative f') (at x within S)"   and "(f has_vector_derivative f'') (at x within S)" shows "f' = f''"
step: fixes \<phi> :: "'k stream \<Rightarrow> bool"   and xs :: "'k stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
axioms_1: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h" shows "ring R"
un_TM2_def: fixes tm :: "'a tm" shows "un_TM2 tm = (case tm of TM x1 x2 \<Rightarrow> x2)"
div_exp: fixes x :: "'a"   and n :: "nat"   and y :: "'a" assumes "ff x ^ n / ff y ^ (n - 1) \<in> range ff" shows "ff (div_exp x y n) = ff x ^ n / ff y ^ (n - 1)"
f_range_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "f_range_on A f = (\<forall>B\<subseteq>A. f B \<subseteq> B)"
set_rows_eq: shows "set (rows B) = set (map of_int_hom.vec_hom fs_init)"
AGM_sel: fixes A :: "'a set"   and \<phi> :: "'a" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M A \<phi> \<equiv> if A .\<bottom>. \<phi> = {} then {A} else {B \<in> A .\<bottom>. \<phi>. A \<div> \<phi> \<subseteq> B}"
compl: fixes A :: "'a set" assumes "A \<in> M" shows "\<Omega> - A \<in> M"
s_in01: shows "s \<inter> {0..1} \<subseteq> {0..1}"
bheight_baliL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "bheight l = bheight r" shows "bheight (baliL l a r) = Suc (bheight l)"
lessE: fixes a :: "'b"   and b :: "'b" assumes "a < b" obtains c :: "'b" where "b = a + c" and "c \<noteq> (0::'b)"
to_woot_character_keeps_S: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message" shows "S (to_woot_character M m) = S m"
intros_1: fixes cfg1 :: "('p, 'v, 's) configuration" shows "reachable cfg1 cfg1"
S'_def: shows "S' = (MIN n\<in>{..<Suc (max N_L N_f')}. SOME s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>(f (x0 + x) n - f x0 n) / x - f' x0 n\<bar> < r / (3 * real (Suc (max N_L N_f')))))"
ring_simprules_8: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
order_less_le_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y \<le> z" shows "x < z"
respects_hcomp: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C g = trg\<^sub>C f" shows "(\<tau>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 (src\<^sub>C f)] = \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f))"
Col_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C \<or> Tarski_neutral_dimensionless.Col Bet A C B \<or> Tarski_neutral_dimensionless.Col Bet B A C \<or> Tarski_neutral_dimensionless.Col Bet B C A \<or> Tarski_neutral_dimensionless.Col Bet C A B \<or> Tarski_neutral_dimensionless.Col Bet C B A" shows "Tarski_neutral_dimensionless.Col Bet A B C"
Subst_all_def: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and xrhs :: "'b trm set" shows "Subst_all ES X xrhs \<equiv> {(Y, Subst yrhs X xrhs) |Y yrhs. (Y, yrhs) \<in> ES}"
neg_le_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq a (divide b c) = less_eq b (times a c)"
sba3_complement_inf_double_complement: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> - - y) = - (x \<sqinter> y)"
rep_eq: fixes x :: "'a interval list" shows "shows_list x = shows_list (map bounds_of_interval x)"
z_1: shows "z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<union> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f) \<union> path_image f \<union> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<union> closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2])"
L2: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (map f xs) = length xs"
cross_edge: fixes s :: "('v, 'es) state_scheme"   and u :: "'v"   and v :: "'v"   and s' :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "I s"   and "cond s"   and "\<not> is_break param s"   and "stack s \<noteq> []"   and "(u, v) \<in> pending s"   and "u = hd (stack s)"   and "v \<in> dom (discovered s)"   and "v \<in> dom (finished s)"   and "s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)" shows "on_cross_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
enum_DF_eq_iff: fixes j :: "nat"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "j \<le> k" shows "(enum (DF k i) j = enum (DF k i') j) = (i = i')"
splitAt_dist_ram10: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "a = fst (splitAt ram1 vs)"
\<ff>_is_max: fixes j :: "nat"   and s :: "'a list"   and b :: "'a list" assumes "j \<le> length s"   and "strict_border b (take j s)" shows "length b + 1 \<le> \<ff> s j"
downshift_add_zero: fixes P :: "'v" shows "downshift (add 0 P) = downshift P"
zeroseq_arctan_series: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda>n. 1 / real (n * 2 + 1) * x ^ (n * 2 + 1)) \<longlonglongrightarrow> 0"
Par: fixes P :: "pi"   and Q :: "pi"   and R :: "pi"   and T :: "pi" assumes "(P, Q) \<in> Rel"   and "(R, T) \<in> Rel'" shows "(P \<parallel> R, Q \<parallel> T) \<in> Rel''"
FG_constituents_irr: assumes "of_nat (card G) \<noteq> (0::'d)" shows "\<forall>U\<in>set ActingGroup.FG_constituents. IrrFinGroupRepresentation G (*) U"
card_seq: fixes x :: "nat" shows "\<forall>i. |i| = x \<longrightarrow> Rep_nat_int i = {} \<or> (\<exists>n. Rep_nat_int i = {n..n + (x - 1)})"
subst_cls_mset_def: fixes CC :: "'a literal multiset multiset"   and \<sigma> :: "'s" shows "CC \<cdot>cm \<sigma> = {#A \<cdot> \<sigma>. A \<in># CC#}"
content_decompose: fixes p :: "'a poly" obtains p' :: "'a poly" where "p = Polynomial.smult (content p) p'" and "content p' = (1::'a)"
C2l: fixes x :: "'a"   and y :: "'a" assumes "x = y \<cdot> x" shows "x = y\<^sup>\<star> \<cdot> x"
pfun_plus_commute: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "pdom f \<inter> pdom g = {}" shows "f + g = g + f"
simps_3: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and uv :: "'c"   and p :: "('b, 'a, 'c) Stmt"   and q :: "('b, 'a, 'c) Stmt" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.has_annotated_stop (Stmt.If uv p q) = (sifum_types.has_annotated_stop p \<or> sifum_types.has_annotated_stop q)"
concat_sing_pow: fixes a :: "'b list"   and k :: "nat" shows "concat ([a] \<^sup>@ k) = a \<^sup>@ k"
tabulation_data_axioms: shows "tabulation_data (\<cdot>) (\<star>) \<a> \<i> src trg r (\<rho> \<cdot> \<phi>) f g'"
S_def: shows "S \<equiv> \<lambda>n. \<Sum>i<n. complex_of_real (real i + a) powr - s"
A: shows "u \<in> range D \<inter> (- I) `` range D"
pw_rwof_step': fixes s' :: "'a" assumes "nofail (step s)"   and "rwof m0 cond step s"   and "cond s"   and "inres (step s) s'" shows "rwof m0 cond step s'"
that: fixes j :: "nat" assumes "j \<in> {k..i}"   and "\<pi> j = ipd (\<pi> k)" shows "thesis"
sinh_add: fixes x :: "'a"   and y :: "'a" shows "sinh (x + y) = sinh x * cosh y + cosh x * sinh y"
is_concrete_set_category: shows "concrete_set_category SC.comp (cardSuc (cmax |UNIV| natLeq)) UNIV SC.UP"
image_Int_subset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'b set" shows "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
simps_2: fixes s :: "'a \<Rightarrow> bool" shows "val_ifex Falseif s = False"
assms: fixes s :: "char list \<Rightarrow> int" shows "P' s \<le> P s"
topological_basis_imp_subbasis: fixes "open" :: "'a set \<Rightarrow> bool"   and B :: "'a set set" assumes "class.topological_space open"   and "topological_space.topological_basis open B" shows "open = generate_topology B"
frameImpE: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b"   and \<Phi> :: "'c" assumes "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> \<turnstile>\<^sub>F \<Phi>"   and "A\<^sub>F \<sharp>* \<Phi>" shows "\<Psi>\<^sub>F \<turnstile> \<Phi>"
add_order_relator: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.add_order plus zero (power.power zero plus a (monoid_add.add_order plus zero a)) = 1"
Partition_less: fixes m :: "nat"   and k :: "nat" assumes "m < k" shows "Partition m k = 0"