hcju/isabelleps huggingface dataset:
trg_src: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.trg (Span.src \<mu>) = Span.src \<mu>"
sum_component_iarray: fixes S :: "'a set"   and i :: "nat"   and f :: "'a \<Rightarrow> 'b iarray" assumes "\<forall>x\<in>S. i < IArray.length (f x)"   and "finite S"   and "S \<noteq> {}" shows "sum f S !! i = (\<Sum>x\<in>S. f x !! i)"
rel_converterD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and conv1 :: "('a, 'c, 'e, 'g) converter"   and conv2 :: "('b, 'd, 'f, 'h) converter" assumes "rel_converter A B C R conv1 conv2" shows "rel_fun A (rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) (run_converter conv1) (run_converter conv2)"
conga__lea456123: 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"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong D E F A B C"
sum_dcharacter_atMost_le: fixes m :: "nat" assumes "\<chi>\<^sub>0 \<noteq> \<chi>\<^sub>0" shows "cmod (sum \<chi>\<^sub>0 {..m}) \<le> real (totient n)"
kgdd1_points_index_group_block: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "(\<B> + mset_set \<G>) index ps = 1"
simps_2: fixes \<sigma> :: "'tp" shows "GE_wtPsym (Guard \<sigma>) = (\<not> unprot \<sigma>)"
Span_min: fixes Us :: "'a list"   and E :: "'a set" assumes "set Us \<subseteq> carrier R"   and "subgroup E (add_monoid R)"   and "K <#> set Us \<subseteq> E" shows "Span K Us \<subseteq> E"
li_le_dim_2: fixes A :: "rat vec set" assumes "gs.fin_dim"   and "A \<subseteq> Rn"   and "gs.lin_indpt A" shows "card A \<le> gs.dim"
Mul_Count: fixes n :: "nat" shows "\<turnstile> Mul_Count n  \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> Blacks \<acute>Ma \<subseteq> \<acute>bc \<and> (\<acute>Safe \<or> \<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>l < \<acute>q \<and> (\<acute>q \<le> \<acute>Queue \<or> \<acute>obc \<subset> Blacks \<acute>M)) \<and> \<acute>q < n + 1\<rbrace>"
fsubsetD: fixes A :: "'m fset"   and B :: "'m fset"   and c :: "'m" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
spr_repSim_def: shows "spr_repSim \<equiv> snd"
axes: fixes u :: "complex"   and v :: "complex"   and w :: "complex" assumes "is_real u"   and "0 < Re u"   and "Re u < 1"   and "is_real v"   and "- 1 < Re v"   and "Re v < 1"   and "Re u \<noteq> Re v"   and "is_imag w"   and "0 \<le> Im w"   and "Im w < 1" shows "P 0\<^sub>h (of_complex u) (of_complex v) (of_complex w)"
no_collision_delta: shows "u \<delta> < other.s \<delta>"
either_contains_arg2: fixes t' :: "'b ttree"   and t :: "'b ttree" shows "paths t' \<subseteq> paths (t \<oplus>\<oplus> t')"
phiDefNodes_def: fixes g :: "'g"   and v :: "'var" shows "phiDefNodes g v \<equiv> fold (\<union>) (concat (map (\<lambda>n. if v \<in> uses g n then [phiDefNodes_aux g v (\<alpha>n g) n] else []) (\<alpha>n g))) {}"
sem_eq: fixes I :: "('a, 'b, 'c) interp"   and J :: "('a, 'b, 'c) interp" assumes "Iagree I J (SIGF \<phi>)" shows "fml_sem I \<phi> = fml_sem J \<phi>"
strict_trans: fixes b :: "nat list"   and a :: "nat list"   and c :: "nat list" assumes "b <\<^sub>v a"   and "c <\<^sub>v b" shows "c <\<^sub>v a"
eq_defined_reflD_2: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and a :: "'a" assumes "class.Eq_def eq" shows "(TT = eq\<cdot>a\<cdot>a) = (a \<noteq> \<bottom>)"
is_cartesian_closed_category: shows "cartesian_closed_category local.comp"
fps_inverse_gp: shows "inverse (Abs_fps (\<lambda>n. 1::'a)) = Abs_fps (\<lambda>n. if n = 0 then 1::'a else if n = 1 then - (1::'a) else (0::'a))"
iTILL_card: fixes n :: "nat" shows "card [\<dots>n] = Suc n"
productive_imp_produces_Max_atom: fixes C :: "'a literal multiset" assumes "production C \<noteq> {}" shows "production C = {Max (atms_of C)}"
list_cases_Cons_snoc: fixes xs :: "'b list" obtains "xs = []"   | x :: "'b" where "xs = [x__]"   | x :: "'b" and ys :: "'b list" and y :: "'b" where "xs = x # ys @ [y]"
less_imp_le: fixes x :: "'a Word.word"   and y :: "'a Word.word" assumes "x <s y" shows "x \<le>s y"
good_set_def: shows "good_set = {x \<in> space M. \<forall>n. f ((T ^^ Suc n) x) = f ((T ^^ n) x)}"
valid_states_nempty: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "finite PROB" shows "\<exists>s. s \<in> valid_states PROB"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_cons_subset: fixes F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list"   and f :: "('d, 'c) Term.term \<times> ('e, 'c) Term.term" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (f # F)"
C_def: shows "C = Beta (- a - 1 / 2) (1 / 2) / 2"
igbg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "\<langle>Rm, Rv\<rangle>igbg_impl_rel_eext \<equiv> \<langle>Rm, Rv \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel\<rangle>gen_igbg_impl_rel_eext"
full_dim_span: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n"   and "finite S"   and "vectorspace.dim class_ring (vs (local.span S)) = card S" shows "lin_indpt S"
diagonal_to_Smith_PQ_def: fixes A :: "(('a, 'c) vec, 'b) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_PQ A bezout = diagonal_to_Smith_aux_PQ [0..<min (nrows A) (ncols A) - 1] bezout (mat (1::'a), A, mat (1::'a))"
while_def: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "while b c s = the (while_option b c s)"
assms_3: shows "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"
sum_over_permutations_insert: fixes S :: "'a set"   and a :: "'a"   and f :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "finite S"   and "a \<notin> S" shows "sum f {p. p permutes insert a S} = (\<Sum>b\<in>insert a S. \<Sum>q\<in>{p. p permutes S}. f (Fun.swap a b id \<circ> q))"
intruder_deduct_restricted_induct: fixes M :: "('fun, 'var) Term.term set"   and Q :: "('fun, 'var) Term.term \<Rightarrow> bool"   and t :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> (('fun, 'var) Term.term \<Rightarrow> bool) \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t"   and "\<And>t. t \<in> M \<Longrightarrow> P M Q t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r t; \<And>t. t \<in> set T \<Longrightarrow> P M Q t; Q (Fun f T)\<rbrakk> \<Longrightarrow> P M Q (Fun f T)"   and "\<And>t K T t\<^sub>i. \<lbrakk>\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t; P M Q t; Ana t = (K, T); \<And>k. k \<in> set K \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r k; \<And>k. k \<in> set K \<Longrightarrow> P M Q k; t\<^sub>i \<in> set T\<rbrakk> \<Longrightarrow> P M Q t\<^sub>i" shows "P M Q t"
rev_is_rev_conv: fixes xs :: "'g list"   and ys :: "'g list" shows "(rev xs = rev ys) = (xs = ys)"
returnOk_def: shows "returnOk \<equiv> return \<circ> Inr"
suffix_suffix: fixes m :: "nat"   and k :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "suffix m (suffix k x) = suffix (k + m) x"
intro: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and G :: "('b, 'c, 'e, 'f) functor"   and u :: "'b \<Rightarrow> 'f" assumes "Functor F : AA \<longrightarrow> BB"   and "Functor G : AA \<longrightarrow> BB"   and "u \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub>"   and "u \<in> extensional (Ob\<^bsub>AA\<^esub>)"   and "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. u A \<in> Hom\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A G \<^bsub>\<o>\<^esub> A"   and "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. \<forall>B\<in>Ob\<^bsub>AA\<^esub>. \<forall>f\<in>Hom\<^bsub>AA\<^esub> A B. G \<^bsub>\<a>\<^esub> f \<bullet>\<^bsub>BB\<^esub> u A = u B \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f" shows "natural_transformation_axioms AA BB F G u"
cong_unique_inverse_prime: fixes p :: "nat"   and x :: "nat" assumes "prime p"   and "0 < x"   and "x < p" shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = 1] (mod p)"
neg_divideR_less_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "c < 0" shows "(b /\<^sub>R c < a) = (c *\<^sub>R a < b)"
IH_4: assumes "avl r" shows "height (AVL.insert x r) = height r \<or> height (AVL.insert x r) = height r + 1"
last_t_fusion: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" assumes "Dioid_Models.last x = first y" shows "Dioid_Models.last (t_fusion x y) = Dioid_Models.last y"
simps_55: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x61 :: "poscheckvariant"   and x62 :: "('d, 'e) Term.term"   and x63 :: "('d, 'e) Term.term" shows "(case \<langle>x61: x62 \<in> x63\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f6 x61 x62 x63"
word_Suc_le: fixes x :: "'b word"   and k :: "'b word" assumes "x \<noteq> - 1" shows "(x + 1 \<le> k) = (x < k)"
simps: fixes a :: "'c"   and s :: "'d + 'c" shows "(a \<in> Basic_BNFs.setr s) = (\<exists>x. a = x \<and> s = Inr x)"
inj_on_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
ArityAnalysisLetSafeNoCard_axioms: shows "ArityAnalysisLetSafeNoCard Aexp Aheap"
Ueq': shows "inter_scheme l U' = concat [c', a', d', b'] @ interact as' bs'"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "cless_eq b a" shows "cless b a"
rHom_func: fixes f :: "'e \<Rightarrow> 'f"   and A :: "('e, 'g) Ring_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "f \<in> rHom A R" shows "f \<in> carrier A \<rightarrow> carrier R"
less: fixes a :: "nat"   and b :: "nat" assumes "a + b < aa__ + ba__" shows "P a b"
len_fills_subview: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "0 < \<parallel>len v ts c\<parallel> \<longrightarrow> (\<exists>v1 v2 v3 v'. v=v1\<parallel>v2 \<and> v2=v'\<parallel>v3 \<and> len v' ts c = ext v' \<and> \<parallel>len v' ts c\<parallel> = \<parallel>len v ts c\<parallel>)"
deg_pm_zero: shows "deg_pm 0 = (0::'d)"
select_convs_1: fixes counter :: "nat"   and discovered :: "'a \<Rightarrow> nat option"   and finished :: "'a \<Rightarrow> nat option"   and pending :: "('a \<times> 'a) set"   and stack :: "'a list"   and tree_edges :: "('a \<times> 'a) set"   and back_edges :: "('a \<times> 'a) set"   and cross_edges :: "('a \<times> 'a) set"   and more :: "'b" shows "counter \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = counter"
sfis_nn: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> sfis f M" shows "nonnegative f"
from_bool_eq_if': fixes P :: "bool"   and Q :: "bool" shows "((if P then 1 else 0) = from_bool Q) = (P = Q)"
ang_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'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.QCongA Bet Cong a"   and "a A B C" shows "a C B A"
simps_2: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and h :: "nat"   and r :: "('b \<times> nat) tree" shows "invar \<langle>l, (a, h), r\<rangle> = (invar l \<and> invar r \<and> h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = \<langle>lr, (b, h), rr\<rangle> \<and> h = lvl rr + 1)))"
infinite: fixes A :: "'b set"   and g :: "'b \<Rightarrow> bool" assumes "infinite A" shows "bool.F g A = True"
G_eq_suminf: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" shows "MC_syntax.G K x y = (\<Sum>i. ennreal (MC_syntax.p K x y i))"
\<alpha>_le_card: fixes G :: "nat set \<times> nat set set" assumes "finite (uverts G)" shows "\<alpha> G \<le> enat (card (uverts G))"
psubst_neg: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "psubst (neg \<phi>) txs = neg (psubst \<phi> txs)"
abs_eq: fixes x :: "'c \<Rightarrow> ('d, 'c) step" assumes "eq_onp productive x x" shows "lunstream' (Abs_lgenerator' x) = lunstream x"
P_H2_H2: shows "(P_H2, H2) = xj"
safe_distance_3r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.safe_distance_3r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2"
forwards: assumes "csa \<in> consistent_sign_vectors_r fss UNIV"   and "\<not> has_no_zeros csa" shows "csa \<in> set (find_sgas_aux fss)"
measurable_from_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "subalgebra M F"   and "f \<in> F \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
counit_simps_1: shows "seq \<tau> ((h \<star> \<l>[k]) \<cdot> (h \<star> \<xi> \<star> k) \<cdot> (h \<star> \<a>\<^sup>-\<^sup>1[f, g, k]) \<cdot> \<a>[h, f, g \<star> k])"
normalize_bseq: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" shows "Language.normalize (bseq c1 c2) = sequence bseq (flatten (Language.normalize c1) @ flatten (Language.normalize c2))"
fp_cop_F_unfold: fixes ds :: "'d set" shows "fp_cop_F ds = cop_F ds (fp_cop_F ds)"
write0_def: fixes a :: "'a"   and P :: "'a process" shows "a \<rightarrow> P \<equiv> Mprefix {a} (\<lambda>x. P)"
sub2_wit: fixes us :: "'a vec list"   and ws :: "'a vec list"   and m :: "nat"   and i :: "nat"   and wits :: "'a list list"   and vvs :: "'a vec list" assumes "set us \<subseteq> carrier_vec n"   and "set ws \<subseteq> carrier_vec n"   and "length us + length ws = m"   and "ws = map ((!) fs) [i..<m]"   and "us = map gso (rev [0..<i])"   and "\<And>j. j < m \<Longrightarrow> fs ! j \<in> carrier_vec n"   and "m \<le> n"   and "sub2_wit us ws = (wits, vvs)" shows "gram_schmidt_sub2 n us ws = vvs \<and> vvs = map gso [i..<m] \<and> wits = map (\<lambda>i. map (\<mu> i) [0..<i]) [i..<m]"
simps_2: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, receive\<langle>t\<rangle>) = (l, send\<langle>t\<rangle>)"
bounded_check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and \<psi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" assumes "#\<^sub>V \<AA> = idx"   and "WS1S_Alt.bounded.check_eqv idx \<phi> \<psi>" shows "WS1S_Alt.sat\<^sub>b \<AA> \<phi> = WS1S_Alt.sat\<^sub>b \<AA> \<psi>"
SEM_def: fixes c :: "'a com"   and S :: "'a set" shows "SEM c S \<equiv> \<Union> (sem c ` S)"
sup_dual_def: fixes x :: "'a dual"   and y :: "'a dual" shows "x \<squnion> y = \<partial> (\<partial>\<^sup>- x \<sqinter> \<partial>\<^sup>- y)"
bounded_hashcode_nat_bounds: fixes n :: "nat"   and a :: "'c" assumes "1 < n" shows "bounded_hashcode_nat n a < n"
nat_pow_pow: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier P" shows "(x [^]\<^bsub>P\<^esub> n) [^]\<^bsub>P\<^esub> m = x [^]\<^bsub>P\<^esub> (n * m)"
f_def: fixes u :: "nat"   and v :: "nat" shows "f u v = c_diff (c_union u v) (c_union (c_diff u v) (c_diff v u))"
hom_div: fixes x :: "'a"   and y :: "'a" shows "hom (x div y) = hom x div hom y"
wf_cdecl_def: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (D, C) \<notin> (subcls1 P)\<^sup>* \<and> (\<forall>(M, b, Ts, T, m)\<in>set ms. \<forall>D' b' Ts' T' m'. P \<turnstile> D sees M, b' :  Ts'\<rightarrow>T' = m' in D' \<longrightarrow> b = b' \<and> P \<turnstile> Ts' [\<le>] Ts \<and> subtype P T T')) \<and> wf_clinit ms"
mes_rel: fixes n :: "nat" shows "(\<lambda>x. F n x * indicator (\<Union>n. (T ^^ n) --` A) x) \<in> borel_measurable M"
ordinal_mono_div: fixes y :: "ordinal" shows "mono (\<lambda>x. x div y)"
is_expansion_aux_MSLCons: fixes c :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux (MSLCons (c, e) xs) f basis" obtains b :: "real \<Rightarrow> real" and basisa :: "(real \<Rightarrow> real) list" where "basis = b # basisa" and "is_expansion_aux xs (\<lambda>x. f x - eval c x * b x powr e) (b # basisa)" and "is_expansion c basisa" and "\<And>e'. e < e' \<Longrightarrow> f \<in> o(\<lambda>x. b x powr e')" and "ms_exp_gt e (ms_aux_hd_exp xs)"
restrict_apply': fixes x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "x \<in> A" shows "restrict f A x = f x"
ipurge_tr_length: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list" shows "length (ipurge_tr I D u xs) \<le> length xs"
MtransC_Ex_parTrace: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and cf :: "('b, 'c) com \<times> 'a"   and cf' :: "('b, 'c) com \<times> 'a" assumes "PL_Indis indis"   and "PL.MtransC tval aval cf cf'" shows "\<exists>cfl. PL_Indis.parTrace tval aval cfl \<and> hd cfl = cf \<and> last cfl = cf'"
freeIn_list_conj: fixes F :: "atom fm list"   and var :: "nat" assumes "\<forall>f\<in>set F. freeIn var f" shows "freeIn var (list_conj F)"
420: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - - (- x \<sqinter> y) = - - (- x \<sqinter> y)"
cases: fixes x :: "('d, 'c) trie" obtains vo :: "'c option" and kts :: "('d \<times> ('d, 'c) trie) list" where "x = Trie vo kts"
hyperpow_eq_one: fixes n :: "nat star" shows "1 pow n = 1"
simps_6: fixes x31 :: "comb"   and x32 :: "comb" shows "S \<noteq> x31 \<bullet> x32"
irc_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and a :: "'a" assumes "irc_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "a \<notin> f (insert a B)" shows "f (insert a B) = f B"
dual_attract: shows "\<forall>p\<in>A. \<forall>q\<in>A. \<forall>x\<in>A. p \<sim> q \<longrightarrow> q \<sqsubseteq> x \<longrightarrow> p \<sqsubseteq> x"
cases: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a1 :: "('g, 'i, 'k) gpv"   and a2 :: "('h, 'j, 'l) gpv" assumes "rel_gpv'' A C R a1 a2" obtains gpv :: "('g, 'i, 'k) gpv" and gpv' :: "('h, 'j, 'l) gpv" where "a1 = gpv" and "a2 = gpv'" and "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<pi> \<star> \<rho>)"
presSubstAbs_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSubstAbs h hA (fromMOD MOD)" shows "presSubstAbs (fromIMor h) (fromIMorAbs hA) MOD"
gfp_exp_nneg: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "nneg (gfp_exp t)"
assms_3: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "I \<subseteq> A"   and "Q I a b" shows "a \<in> A"
unlessD: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A unless B" shows "F \<in> A - B co A \<union> B"
dual_additive_def: fixes f :: "'a \<Rightarrow> 'a" shows "dual_additive f \<equiv> \<forall>x y. f (x \<squnion> y) = f x \<sqinter> f y"
ext_wf_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_wf_axioms ext \<equiv> \<forall>gt. wfP (\<lambda>x y. gt y x) \<longrightarrow> wfP (\<lambda>xs ys. ext gt ys xs)"
2: shows "set_of_map (map_of (AList.delete (fst p) ps)) = set (AList.clearjunk (AList.delete (fst p) ps))"
pref_eq: fixes X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'b" assumes "|~ X = Y" shows "X = Y"
assms_1: shows "is_valid_problem_sas_plus \<Psi>"
simps: fixes callee :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e, 'f, 'g) gpv"   and gpv :: "('d, 'i, 'h) gpv"   and s :: "'e" shows "inline1 callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
select_convs_2: fixes lenf :: "nat"   and front :: "'b list"   and status :: "'b status"   and rear :: "'b list"   and lenr :: "nat"   and more :: "'c" shows "front \<lparr>lenf = lenf, front = front, status = status, rear = rear, lenr = lenr, \<dots> = more\<rparr> = front"
timpl_closure'_def: fixes TI :: "('j set \<times> 'j set) set" shows "timpl_closure' TI \<equiv> (timpl_closure'_step TI)\<^sup>*"
assms_3: shows "S t i \<turnstile> ev \<mapsto> S t (Suc i)"
reflection: shows "rel_reflects_binary_pred (indRelTEQ TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
simps_4: fixes x21 :: "'a"   and x22 :: "'b"   and x23 :: "'c"   and x1 :: "'a" shows "State x21 x22 x23 \<noteq> Idle x1"
raw_match'_restriction: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and restriction :: "(variable \<times> String.literal set) set" assumes "raw_match' T U subs = Some subs'"   and "tvsT T \<subseteq> restriction" shows "raw_match' T U (subs |` restriction) = Some (subs' |` restriction)"