hcju/isabelleps huggingface dataset:
ext_inject: fixes tag :: "obj_tag"   and "values" :: "vn \<Rightarrow> val option"   and more :: "'a"   and tag' :: "obj_tag"   and values' :: "vn \<Rightarrow> val option"   and more' :: "'a" shows "(\<lparr>tag = tag, values = values, \<dots> = more\<rparr> = \<lparr>tag = tag', values = values', \<dots> = more'\<rparr>) = (tag = tag' \<and> values = values' \<and> more = more')"
cong_mod_left: 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 division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo b a) c a = unique_euclidean_semiring.cong modulo b c a"
upd_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 c :: "('a, 'c, 'd) Stmt"   and upd :: "'a ModeUpd"   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 "(((c@[upd], mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((c, sifum_lang.update_modes upd mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
triangle_in_hom_1: shows "\<guillemotleft>(S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S S.\<a>' (S.UP f) (S.UP g) (S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) : S.UP f \<star>\<^sub>S S.src (S.UP f) \<Rightarrow>\<^sub>S S.trg (S.UP f) \<star>\<^sub>S S.UP f\<guillemotright>"
infdist_almost_attained: fixes x :: "'b"   and X :: "'b set"   and a :: "real" assumes "infdist x X < a"   and "X \<noteq> {}" shows "\<exists>y\<in>X. dist x y < a"
assms_3: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> pmdl G"   and "f \<in> dgrad_p_set d m"   and "f \<noteq> 0" shows "\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f"
set_rev: fixes xs :: "'e list" shows "set (rev xs) = set xs"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
simps_4: fixes as :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "isVal (Let as e) = False"
Celarent: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A E B"   and "B Q C" shows "A E C"
fds_scale_1: fixes f :: "'b fds" shows "fds_scale 1 f = f"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "splice x xa = y" obtains ys :: "'b list" where "xa__ = []" and "xa = ys__" and "y = ys__"   | x :: "'b" and xs :: "'b list" and ys :: "'b list" where "xa__ = x # xs" and "xa = ys" and "y = x # splice ys xs"
local_state_of_finite_restricted_global_state_is_finite_2: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r' :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and r :: "'b" assumes "s r' = Some ls"   and "finite (LID\<^sub>G (s(r := None)))"   and "r \<noteq> r'" shows "finite (LID\<^sub>L ls)"
\<mu>: shows "\<guillemotleft>\<mu> : g \<Rightarrow> g'\<guillemotright> \<and> S.UP \<mu> = \<nu>"
alt_nondet_def: fixes return :: "'f \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and union :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and m1 :: "('d, 'e) nondetT"   and m2 :: "('d, 'e) nondetT" shows "nondetM_base.alt_nondet return bind union m1 m2 = NondetT (bind (run_nondet m1) (\<lambda>A. bind (run_nondet m2) (\<lambda>B. return (union A B))))"
exec_gpv1_def: shows "exec_gpv1 = exec_gpv"
s_in_type: shows "(left r, x) \<in> {r. fst r \<le> snd r}"
val_Bool_neg: fixes b :: "tr" assumes "defined b" shows "\<lbrakk>neg\<cdot>b\<rbrakk> = (\<not> \<lbrakk>b\<rbrakk>)"
assms_2: shows "(st, st') \<in> p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)"
Abs_Int1_axioms: shows "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"
diff_minus_eq_add: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
f0: shows "i \<in> {0..2 ^ (n + 1)} \<and> j = 0"
Cons_path: fixes g :: "'g"   and ns :: "'node list"   and n' :: "'node" assumes "local.path g ns"   and "n' \<in> set (predecessors g (hd ns))" shows "local.path g (n' # ns)"
vector_component_7: fixes r :: "real"   and x :: "('a, 'b) vec"   and i :: "'b" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
coeff_of_sum_diff_degree1: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q < deg R p" shows "(p \<oplus>\<^bsub>UP R\<^esub> q) (deg R p) = p (deg R p)"
VV_impl2: fixes v :: "'a"   and p :: "Player" assumes "v \<in> G'.VV p**" shows "v \<notin> G'.VV p"
add_col_sub_row_diff_ev_part_old: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat n n"   and "i \<le> j"   and "i \<noteq> 0"   and "i < n"   and "j < n"   and "i' < n"   and "j' < n"   and "j' < j \<or> j' = j \<and> i \<le> i'"   and "local.inv_part local.diff_ev A i j"   and "local.inv_all local.uppert A" shows "local.diff_ev (add_col_sub_row a (i - 1) j A) i' j'"
simps: fixes a1 :: "ccs"   and a2 :: "ccs" shows "a1 \<approx> a2 = (\<exists>P Q. a1 = P \<and> a2 = Q \<and> P \<leadsto>\<^sup>^<weakBisimulation> Q \<and> Q \<approx> P)"
im_jointfunTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'e"   and m :: "nat"   and g :: "nat \<Rightarrow> 'e" shows "jointfun n f m g ` {i. i \<le> n} = f ` {i. i \<le> n}"
hyps: fixes x :: "'a list" assumes "\<^bold>|x\<^bold>| < \<^bold>|xa__\<^bold>|"   and "x \<noteq> \<epsilon>" shows "\<exists>r k. x \<in>\<^sub>p r * \<and> k \<noteq> 0 \<and> x = r \<^sup>@ k"
alpha_qOp_iff: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and delta' :: "'h"   and inp' :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp' :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "(X #= qOp delta' inp' binp') = (\<exists>inp binp. X = qOp delta' inp binp \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
Brouwer_degree2_homeomorphic_map: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "homeomorphic_map (nsphere p) (nsphere p) f" shows "\<bar>Brouwer_degree2 p f\<bar> = 1"
4: shows "x\<^sup>T ; r \<cdot> (x ; 1 \<cdot> - r)\<^sup>T \<le> x\<^sup>\<star>"
winding_number_cong: fixes p :: "real \<Rightarrow> complex"   and q :: "real \<Rightarrow> complex"   and z :: "complex" assumes "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t" shows "winding_number p z = winding_number q z"
inj_idom_hom_axioms: shows "inj_idom_hom of_real"
A0id: shows "A0 = 0\<^sub>m (n - 1) 1"
trm_simp_5: fixes A :: "'b trm"   and B :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" assumes "SimplyTyped.Pair A B = SimplyTyped.Pair C D" shows "A = C"
i_join_i_expand_iT_Mult: fixes k :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "I \<noteq> {}" shows "f \<odot> k \<Join>\<^sub> (I \<otimes> k) = f \<Join>\<^sub> I"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'a" and x2 :: "'a list" and y1 :: "'b" and y2 :: "'b list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
sums_iff: fixes f :: "nat \<Rightarrow> 'c"   and x :: "'c" shows "f sums x = (summable f \<and> suminf f = x)"
in_set_takeD: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" assumes "x \<in> set (take n xs)" shows "x \<in> set xs"
fold_mset_single: fixes s :: "'a"   and x :: "'a" shows "fold_mset (\<^bold>*) s {#x#} = x \<^bold>* s"
scale_ms_aux_eq_MSLNil_iff: fixes x :: "real"   and xs :: "('a \<times> real) msllist" shows "(scale_ms_aux x xs = MSLNil) = (xs = MSLNil)"
abc: shows "a \<squnion>\<^bsub>f\<^esub> b \<squnion>\<^bsub>f\<^esub> c \<in> A"
arg_mult_eq: fixes z :: "complex"   and z1 :: "complex"   and z2 :: "complex" assumes "z * z1 \<noteq> 0"   and "z * z2 \<noteq> 0"   and "arg (z * z1) = arg (z * z2)" shows "arg z1 = arg z2"
separating_insert1: fixes a :: "'a"   and V :: "'a set"   and F :: "'a \<Rightarrow> 'b set" assumes "separating (insert a V) F" shows "separating V F"
minus_inv_sym_aux: fixes a :: "'b"   and b :: "'b" shows "(- {(a, b), (b, a)})\<inverse> = - {(a, b), (b, a)}"
assms_3: shows "sorted_list_of_set A = sorted_list_of_set B"
Deduct2_with_PseudoOrder_axioms: shows "Deduct2_with_PseudoOrder var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv Lq"
intro: assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus" shows "OFCLASS('b, bezout_ring_class)"
mod_eq_dvd_iff_poly: fixes x :: "'a mod_ring poly"   and n :: "'a mod_ring poly"   and y :: "'a mod_ring poly" shows "(x mod n = y mod n) = (n dvd x - y)"
INF_SUP_set: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "Inf ((\<lambda>B. Sup (g ` B)) ` A) = Sup ((\<lambda>B. Inf (g ` B)) ` {uu_. \<exists>f. uu_ = f ` A \<and> (\<forall>C\<in>A. f C \<in> C)})"
padic_int_simps_1: fixes p :: "int" shows "\<one>\<^bsub>padic_int p\<^esub> = padic_one p"
hom_add: fixes x :: "int poly"   and y :: "int poly" shows "of_int_poly (x + y) = of_int_poly x + of_int_poly y"
assms_5: fixes w :: "'action list" assumes "path w s"   and "A \<inter> {a. en a s} \<inter> set w = {}" shows "A \<inter> set w = {}"
update_convs_9: fixes traps' :: "Trap set \<Rightarrow> Trap set"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "traps_update traps' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps' traps, undef = undef, \<dots> = more\<rparr>"
RedDynCast: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and 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"   and E :: "char list \<Rightarrow> ty option"   and Cs :: "char list list" assumes "hp s a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> Path D to C via Cs' "   and "P \<turnstile> Path D to C unique" shows "P,E \<turnstile> \<langle>Cast C (ref (a, Cs)),s\<rangle> \<rightarrow> \<langle>ref (a, Cs'),s\<rangle>"
scale_scale_on: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> S" shows "a *\<^sub>R b *\<^sub>R x = (a * b) *\<^sub>R x"
emeasure_pair_measure_Times: fixes A :: "'a set"   and N :: "'a measure"   and B :: "val set" assumes "A \<in> sets N"   and "B \<in> sets (stock_measure (val_type v2))" shows "emeasure (N \<Otimes>\<^sub>M stock_measure (val_type v2)) (A \<times> B) = emeasure N A * emeasure (stock_measure (val_type v2)) B"
beta2_def: shows "beta2 \<equiv> $pc2 = #b \<and> pc2$ = #g \<and> y$ = Suc<$y> \<and> Unchanged (x, sem, pc1)"
simps_4: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Or \<phi> \<psi>) = (MFOTL.sat \<sigma> v i \<phi> \<or> MFOTL.sat \<sigma> v i \<psi>)"
star_class_defs_7: shows "(div) \<equiv> *f2* (div)"
wf_in_rel: fixes R :: "('c \<times> 'c) set" assumes "wf R" shows "wfP (in_rel R)"
prod_sum_PiE: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (B x)" shows "(\<Prod>x\<in>A. sum (f x) (B x)) = (\<Sum>g\<in>Pi\<^sub>E A B. \<Prod>x\<in>A. f x (g x))"
path_pred_of_se_Assign: fixes c :: "('c, 'd) conf"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and c' :: "('c, 'd) conf" assumes "se c (Assign v e) c'" shows "\<exists>sv. conjunct (pred c') = (\<lambda>\<sigma>. conjunct (pred c) \<sigma> \<and> \<sigma> sv = adapt_aexp e (store c) \<sigma>)"
eq_UNIV_iff: fixes S :: "'a set" shows "(S = UNIV) = (\<forall>x. x \<in> S)"
step_fold_def: fixes g :: "'f \<Rightarrow> (String.literal, nat) assoc_list \<times> 'g \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> 'h \<Rightarrow> 'e list \<times> edgeIndex \<times> (String.literal, nat) assoc_list"   and steps :: "'f list"   and lbls :: "(String.literal, nat) assoc_list"   and pri :: "'g"   and pos :: "nat"   and nxt :: "edgeIndex"   and onxt :: "edgeIndex option"   and iB :: "'h" shows "step_fold g steps lbls pri pos nxt onxt iB = (let (uu_, nxt, lbls, s) = step_fold' g steps lbls pri pos nxt onxt iB in (s, nxt, lbls))"
scaleR_right: fixes a :: "'b \<Rightarrow>\<^sub>L 'a"   and r :: "real"   and b :: "'b" shows "blinfun_apply a (r *\<^sub>R b) = r *\<^sub>R blinfun_apply a b"
f1: shows "4 \<le> rk {A', B', P, R, a, A, B}"
llength_llist_of: fixes xs :: "'a list" shows "llength (llist_of xs) = enat (length xs)"
unit_eqI: fixes a :: "'c"   and \<mu> :: "'d" assumes "C'.obj a"   and "\<guillemotleft>\<mu> : F'.map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>"   and "D.iso \<mu>"   and "\<mu> \<cdot>\<^sub>D D'.\<i> (F'.map\<^sub>0 a) = (F (C'.\<i> a) \<cdot>\<^sub>D \<Phi> (snd (a, a), fst (a, a))) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D\<^sup>o\<^sup>p \<mu>)" shows "\<mu> = F'.unit a"
etran_or_ctran: fixes x :: "('b com option \<times> 'b) list"   and m :: "nat"   and i :: "nat" assumes "x \<in> cptn"   and "m \<le> length x"   and "\<And>i. Suc i < m \<Longrightarrow> (x ! i, x ! Suc i) \<notin> ctran"   and "Suc i < m" shows "x ! i -e\<rightarrow> x ! Suc i"
bind_spmf_def: fixes x :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
\<delta>_prod_2sng_3: fixes r1 :: "('b, 'a) ta_rule"   and r2 :: "('c, 'a) ta_rule" assumes "rhsl r1 = rhsl r2"   and "length (rhsq r1) = length (rhsq r2)" shows "\<delta>_prod {r1} {r2} = {r_prod r1 r2}"
zero_complex_interval_def: shows "0 \<equiv> Complex_Interval 0 0"
order_refl: fixes x :: "'d word" shows "x \<le>s x"
uniformly_Cauchy_onI: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" shows "uniformly_Cauchy_on X f"
sum_1: fixes f :: "1 \<Rightarrow> 'c" shows "sum f UNIV = f 1"
3: shows "r\<^sup>T ; x \<cdot> x\<^sup>T\<^sup>+ ; r = x\<^sup>T\<^sup>+ ; r \<cdot> 1 ; r\<^sup>T ; x"
algebraic_uminus: fixes x :: "'a" assumes "algebraic x" shows "algebraic (- x)"
density_RN_deriv_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "density M (RN_deriv M (density M f)) = density M f"
abc_ex_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
j2: shows "j < dim_col A + dim_col (0\<^sub>m 0 (dim_col B))"
Max_def: shows "Max \<equiv> semilattice_set.F max"
simps: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" shows "a1 \<rightarrow>*c a2 = ((\<exists>c s. a1 = (c, s) \<and> a2 = (c, s)) \<or> (\<exists>c s c' s' c'' s''. a1 = (c, s) \<and> a2 = (c'', s'') \<and> (c, s) \<rightarrow>*c (c', s') \<and> (c', s') \<rightarrow>c (c'', s'')))"
hide_un_D: fixes A :: "'a set"   and P :: "'a process"   and B :: "'a set" assumes "finite A" shows "D ((P \ A) \ B) = D (P \ (A \<union> B))"
Push_def: shows "Old_Datatype.Push \<equiv> case_nat"
semiring_simprules_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y \<in> carrier R"
inject: fixes x1 :: "'b"   and x2 :: "'d"   and y1 :: "'b"   and y2 :: "'d" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
SFAssInit: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields"   and sh'' :: "sheap" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>Val v',(h', l', sh')\<rangle>"   and "sh' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs' = sfs(F \<mapsto> v)"   and "sh'' = sh'(D \<mapsto> (sfs', i))" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h', l', sh'')\<rangle>"
star_n_le: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "(star_n X \<le> star_n Y) = (\<forall>\<^sub>F n in \<U>. X n \<le> Y n)"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'a set" assumes "n \<le> card S" obtains T :: "'a set" where "T \<subseteq> S" and "card T = n" and "finite T"
alldeg0: shows "\<forall>p\<in>set qs. degree p = 0"
sel_4: fixes x :: "quat"   and y :: "quat" shows "Im3 (x + y) = Im3 x + Im3 y"
c_manifold_order_le: fixes l :: "enat" assumes "l \<le> l" shows "c_manifold charts1 l"
filterlim_at_top_gt: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and c :: "'d" shows "filterlim f at_top F = (\<forall>Z>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
assms: shows "fold_const_fa (Mult a b) = Num x"
h_def: shows "h \<equiv> \<lambda>x. f x powr g x * (g' * ln (f x) + f' * g x / f x)"
Says_Server_message_form: fixes A :: "agent"   and nb' :: "msg"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>nb', Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace>, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "K \<notin> range shrK"
rev_HM: shows "bidel_arc.HM G M rev.a' = rev.HM"
card_vimage_inj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set" assumes "inj f"   and "A \<subseteq> range f" shows "card (f -` A) = card A"
compare_EQ_iff_eq_TT: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b" assumes "class.Ord_linear eq compare" shows "(compare\<cdot>x\<cdot>y = EQ) = (eq\<cdot>x\<cdot>y = TT)"
alt_fail1: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'e" assumes "monad_fail_alt return bind fail alt" shows "alt fail m = m"
distinct_252: fixes x12 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Prj2 x12 \<noteq> Prj1 x11"
map_2: fixes f :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "map_option f \<lfloor>x2\<rfloor> = \<lfloor>f x2\<rfloor>"
\<psi>: shows "\<psi> \<in> Left_a.hom f g \<and> a \<star> \<psi> = \<phi>"
s_def: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and i :: "nat" shows "s c t i = (THE c'. trace c (take i t) c')"
polynomial_is_Zp_continuous: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)" shows "is_Zp_continuous ((\<bullet>) f)"
measurable_compose: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'d \<Rightarrow> 'e"   and L :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M L"
division_of_trivial: fixes s :: "'b set set" shows "(s division_of {}) = (s = {})"
psubset_subset_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subset> B"   and "B \<subseteq> C" shows "A \<subset> C"
ge_plus_game_left: fixes y :: "game"   and z :: "game"   and x :: "game" shows "ge_game (y, z) = ge_game (plus_game x y, plus_game x z)"
cInf_greatest: fixes X :: "'b multiset set"   and z :: "'b multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "z \<subseteq># Inf X"
invalid_def: shows "invalid \<equiv> \<lambda>\<tau>. \<bottom>"
sync_def: fixes P :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "P \<lbrakk>A\<rbrakk> Q \<equiv> Abs_process ({(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}, {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)})"