hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "('a, 'b, 'c) Formula.formula" shows "Formula.formula.Rep_formula (Formula.Not x) = Formula.Not\<^sub>\<alpha> (Formula.formula.Rep_formula x)"
simps: fixes s :: "'state"   and a :: "'act"   and ou :: "'out"   and s' :: "'state" shows "validTrans (Trans s a ou s') = (step s a = (ou, s'))"
subgroup_generated2: fixes S :: "'a set" shows "subgroup_generated (subgroup_generated G S) S = subgroup_generated G S"
times_aezdeltafun_aezdeltafun: fixes b1 :: "'a"   and a1 :: "'b"   and b2 :: "'a"   and a2 :: "'b" shows "b1 \<delta>\<delta> a1 * (b2 \<delta>\<delta> a2) = (b1 * b2) \<delta>\<delta> (a1 + a2)"
even_plus_one_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a one) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
Red_I_\<G>_def: fixes N :: "'f set" shows "no_labels.Red_I_\<G> N = (\<Inter>q\<in>Q. no_labels.Red_I_\<G>_q q N)"
span_eq_iff: fixes s :: "('c \<Rightarrow> real) set" assumes "s \<subseteq> dest.diff_fun_space" shows "(dest.diff_fun_space.m1.span s = s) = dest.diff_fun_space.m1.subspace s"
l1_obs_iagreement_Resp: shows "oreach l1 \<subseteq> l2_iagreement_Resp"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "src (g \<cdot> f) = src g \<cdot> src f"
theorem1_correctness: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and i :: "nat"   and e' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater>" obtains eP' :: "Syntax.term" and eV' :: "Syntax.term" and \<pi> :: "Syntax.term list" where "\<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater>" and "\<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>[], eV'\<ggreater>" and "{$$} \<turnstile> e', eP', eV' : \<tau>"
siso_transT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and t :: "'state" assumes "siso c"   and "(c, s) \<rightarrow>t s'"   and "s \<approx> t" shows "\<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'"
nat_zminus_int: fixes n :: "nat" shows "nat (- int n) = 0"
poly_eq_iff: fixes p :: "'a Polynomial.poly"   and q :: "'a Polynomial.poly" shows "(p = q) = (\<forall>n. poly.coeff p n = poly.coeff q n)"
finvimage_iff1: fixes a :: "V"   and r :: "V"   and A :: "V" shows "(a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A) = (a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A))"
snd_convol': fixes f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and x :: "'f" shows "snd (<f , g> x) = g x"
partition_on_quotient: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "equiv A r" shows "partition_on A (A // r)"
inject_1: fixes x21 :: "nat"   and x22 :: "'b list"   and x23 :: "'b list"   and x24 :: "'b list"   and x25 :: "'b list"   and y21 :: "nat"   and y22 :: "'b list"   and y23 :: "'b list"   and y24 :: "'b list"   and y25 :: "'b list" shows "(Rev x21 x22 x23 x24 x25 = Rev y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
ide_in_hom_1: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>f : src f \<rightarrow> trg f\<guillemotright>"
interchange: fixes h :: "'a"   and g :: "'a"   and h' :: "'a"   and g' :: "'a" assumes "C.seq h g"   and "C.seq h' g'" shows "C.tensor h h' \<cdot> C.tensor g g' = C.tensor (h \<cdot> g) (h' \<cdot> g')"
allUses_def: fixes g :: "'g"   and n :: "'node" shows "step.allUses g n \<equiv> u_g g n \<union> step.phiUses g n"
l7_13: fixes A :: "'p"   and P' :: "'p"   and P :: "'p"   and Q' :: "'p"   and Q :: "'p" assumes "A Midpoint P' P"   and "A Midpoint Q' Q" shows "Cong P Q P' Q'"
analytically_valid_y: fixes s :: "(real \<times> real) set"   and F :: "real \<times> real \<Rightarrow> 'a" assumes "analytically_valid s F i" shows "(\<lambda>x. integral UNIV (\<lambda>y. partial_vector_derivative F i (y, x) * indicator s (y, x))) \<in> borel_measurable lborel"
inv_\<nu>'\<nu>_r: shows "\<guillemotleft>local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) : g \<star> w' \<star> w \<Rightarrow> g \<star> src f\<guillemotright>"
IH_1: fixes zs :: "'a list"   and n :: "nat"   and ys :: "'a list" assumes "(zs, n) \<in> set_pmf (rquicksort R ys)"   and "length ys < length xs"   and "set ys \<subseteq> set xs" shows "Linorder_Relations.sorted_wrt R zs"
bi_unique_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
Cons: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "P x y"   and "list_all2 P xs ys"   and "Q xs ys" shows "Q (x # xs) (y # ys)"
r_def: shows "r = (if conv_radius f = \<infinity> then 1 else real_of_ereal (conv_radius f) / 2)"
n_Elem_NatInterval: fixes q :: "nat"   and m :: "nat"   and n :: "nat" assumes "q \<le> m" shows "Elem (nat2Nat (n + q)) (NatInterval n m)"
statEqEnt: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Phi> :: "'e" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'"   and "SImp \<Psi> \<Phi>" shows "SImp \<Psi>' \<Phi>"
null_sets_completion_subset: fixes B :: "'a set"   and A :: "'a set"   and M :: "'a measure" assumes "B \<subseteq> A"   and "A \<in> null_sets (completion M)" shows "B \<in> null_sets (completion M)"
assms: shows "\<forall>x<y. a * x\<^sup>2 + b * x + c = 0"
filter_empty_ivls_impl_mem_clw_rel_ivl_rel_iff: fixes xs :: "('b \<times> 'b) list"   and X :: "'b set" shows "((filter_empty_ivls_impl (\<le>) xs, X) \<in> clw_rel (\<langle>rnv_rel\<rangle>ivl_rel)) = ((xs, X) \<in> clw_rel (\<langle>rnv_rel\<rangle>ivl_rel))"
ex_simps_1: fixes P :: "'b \<Rightarrow> bool"   and Q :: "bool" shows "(\<exists>x. P x \<and> Q) = ((\<exists>x. P x) \<and> Q)"
prems_7: shows "Suc n < mod_type_class.to_nat a"
lexext_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "lexext gt (xs @ [x]) xs"
abelian_subgroup_a_kernel: shows "abelian_subgroup (a_kernel G H h) G"
quasi_inverse_antipar_1: fixes f :: "'a" assumes "equivalence_map f" shows "src f\<^sup>~ = trg f"
sum_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "sum\<^sub>M f = (\<lambda>(i, j). if i = hd enum_class.enum \<and> j = i then ab_semigroup_add_0.sum_0 (+) (bot + bot) (\<lambda>k. ab_semigroup_add_0.sum_0 (+) (bot + bot) (\<lambda>l. f (k, l)) {l. True}) {k. True} else bot + bot)"
prj_simps_10: shows "C.arr fgh.Prj\<^sub>1"
rep_eq: fixes x :: "nat"   and xa :: "float interval" shows "bounds_of_interval (sqrt_float_interval x xa) = (case bounds_of_interval xa of (lx, ux) \<Rightarrow> (lb_sqrt x lx, ub_sqrt x ux))"
sel_1: fixes xs :: "'a llist"   and b :: "'b" assumes "lnull xs" shows "terminal (tllist_of_llist b xs) = (case xs of LNil \<Rightarrow> b)"
reduced_ifex: fixes b :: "'b" shows "reduced (ifex_of b) {}"
non_neg_mat_st_mat: fixes A :: "'a st_mat" shows "non_neg_mat (st_mat A)"
strict_lunit: fixes f :: "'b" assumes "D.ide f" shows "\<l>\<^sub>D[f] = f"
dual_Inf: fixes A :: "'b set"   and sup :: "'b" shows "is_Inf (dual ` A) (dual sup) = is_Sup A sup"
list_all_zip_map2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and as :: "'a list"   and f :: "'c \<Rightarrow> 'b"   and bs :: "'c list" shows "list_all_zip P as (map f bs) = list_all_zip (\<lambda>a b. P a (f b)) as bs"
diff_single_trivial: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<notin># M" shows "remove1_mset x M = M"
igSwapIGAbs_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
matrix_row_length: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and nr :: "nat"   and nc :: "nat"   and M :: "'b list list" assumes "Matrix_Tensor.mult id f"   and "mat nr nc M" shows "mat (mult.row_length M) (length M) M"
Nat_le_diff_conv2_same: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i \<le> i - j" shows "j = 0"
closed_approachable: fixes S :: "'b set"   and x :: "'b" assumes "closed S" shows "(\<forall>e>0. \<exists>y\<in>S. dist y x < e) = (x \<in> S)"
eq_resource_on_sym: fixes A :: "'y set"   and res' :: "('y, 'z) resource"   and res :: "('y, 'z) resource" assumes "A \<turnstile>\<^sub>R res' \<sim> res" shows "A \<turnstile>\<^sub>R res \<sim> res'"
term_ok_app_eqD: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "term_ok \<Theta> (s $ t)" shows "term_ok \<Theta> s \<and> term_ok \<Theta> t"
no_plus_overflow_unat_size: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> x + y) = (unat x + unat y < 2 ^ size x)"
KC2_secure_lemma: fixes U :: "msg"   and KC3 :: "nat"   and C :: "agent"   and N :: "nat"   and KC2 :: "nat"   and X :: "msg"   and evs :: "event list" assumes "U = Crypt KC3 \<lbrace>Agent C, Nonce N, Key KC2, X\<rbrace>"   and "U \<in> parts (knows Spy evs)"   and "evs \<in> set_cr"   and "Nonce N \<notin> analz (knows Spy evs)" shows "\<exists>k i W. Says (Cardholder k) (CA i) \<lbrace>U, W\<rbrace> \<in> set evs \<and> Cardholder k \<notin> bad \<and> CA i \<notin> bad"
Gf': shows "\<guillemotleft>G f' : Go (C.cod f) \<rightarrow>\<^sub>D Go (C.cod f')\<guillemotright> \<and> f' \<cdot>\<^sub>C \<epsilon>o (C.cod f) = \<epsilon>o (C.cod f') \<cdot>\<^sub>C F (G f')"
rawpsubst_compose_freshVar: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list"   and us :: "'var list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)"   and "\<And>i j. \<lbrakk>i < j; j < length txs\<rbrakk> \<Longrightarrow> snd (txs ! j) \<notin> FvarsT (fst (txs ! i))"   and "set us \<subseteq> var"   and "set us \<inter> Fvars \<phi> = {}"   and "set us \<inter> \<Union> (FvarsT ` fst ` set txs) = {}"   and "set us \<inter> snd ` set txs = {}"   and "length us = length txs"   and "distinct us" shows "rawpsubst (rawpsubst \<phi> (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubst \<phi> txs"
u: shows "ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
exp_ind: fixes x :: "nat"   and a :: "'a" shows "a \<up> Suc x = a \<up> x @ [a]"
computation_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration" shows "computation channel trans send recv init final \<equiv> distributed_system channel \<and> computation_axioms channel trans send recv init final"
M_def: shows "M = sylvester_mat (poly_x_minus_y p) (poly_lift q)"
Subgroup_imp_Subring: fixes H :: "'g set" assumes "Supgroup.Subgroup H" shows "Ring1 (aezfun_setspan H) \<and> aezfun_setspan H \<subseteq> Supgroup.group_ring"
assms_4: shows "prv (imp \<phi> (neg \<phi>1))"
Synchronized1Throw2Null: fixes xs :: "'addr val list"   and V :: "nat"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and ad :: "'addr"   and h :: "'heap" assumes "xs ! V = Null"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Throw ad,(h, xs)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),(h, xs)\<rangle>"
get_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "DocumentClass.type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_disconnected_nodes document_ptr"
invertible_def: fixes u :: "'b" assumes "u \<in> M'" shows "invertible' u = (\<exists>v\<in>M'. u \<cdot>' v = \<one>' \<and> v \<cdot>' u = \<one>')"
frac_frac: fixes x :: "'b" shows "frac (frac x) = frac x"
leq_antisym: fixes x :: "'c spmf"   and y :: "'c spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y x" shows "x = y"
simps: fixes p :: "prog"   and a :: "trace" shows "tracesp p a = (\<exists>\<sigma> \<sigma>'. a = \<lparr>initial = \<sigma>, final = \<sigma>'\<rparr> \<and> \<sigma> \<in> states p \<and> body p \<rhd> \<sigma> \<leadsto> Some \<sigma>')"
assert_def: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "assert P x = (if P x then return x else raise STR ''assert'')"
add_im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.add_im_sheaf S add_str f \<equiv> \<lambda>V. add_str (f \<^sup>\<inverse> S V)"
pure_T_conv: fixes x :: "'b tree"   and f :: "('b \<Rightarrow> 'a) tree" shows "pure (\<lambda>x f. f x) \<diamondop> x \<diamondop> f = f \<diamondop> x"
diff: fixes x :: "'a"   and y :: "'a" shows "f' (x - y) = f' x - f' y"
power_in: fixes x :: "'b list"   and G :: "'b list set"   and k :: "nat" assumes "x \<in> \<langle>G\<rangle>" shows "x \<^sup>@ k \<in> \<langle>G\<rangle>"
CastRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow> \<langle>Cast C e',s',b'\<rangle>"
first_countable_basis: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.first_countable_topology open" shows "\<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
\<pi>1_def: shows "\<pi>1 \<equiv> stake (Suc i) \<pi> @- sconst (toSink (fst (\<pi> !! i)), L (toSink (fst (\<pi> !! i))))"
1: shows "action_needed_vars a s \<subseteq> system_needed_vars PROB s"
Bang_1: fixes P :: "pi"   and Rs :: "residual" assumes "P \<parallel> !P \<Longrightarrow>\<^sub>l Rs" shows "!P \<Longrightarrow>\<^sub>l Rs"
fdia_star_unfold: fixes y :: "'a"   and x :: "'a" shows " |(1::'a)\<rangle> y +  |x\<rangle>  |x\<^sup>\<star>\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
equiintegrable_on_integrable: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "F equiintegrable_on I"   and "f \<in> F" shows "f integrable_on I"
downshift_upshift: fixes P :: "'v" shows "downshift (upshift P) = P"
intro: 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>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>"
l_2: shows "(w \<circ> a) \<longlonglongrightarrow> liminf (w \<circ> r)"
elliptic_imp_not_parabolic_loxodromic_1: fixes f :: "'b \<Rightarrow> 'b" assumes "elliptic_isometry f" shows "\<not> parabolic_isometry f"
mult_vm_vm: fixes v :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "v *\<^sub>v\<^sub>m A *\<^sub>v\<^sub>m B = v *\<^sub>v\<^sub>m (A *\<^sub>m\<^sub>m B)"
eq_refl: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
hyps: assumes "n \<le> m" shows "\<exists>y. cls_val_process Mkt pf m y < 0"
true_conj_zero_2: fixes P :: "'a upred" shows "(false \<and> P) = false"
block_unique: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "a \<in> A"   and "A \<in> P"   and "a \<in> B"   and "B \<in> P" shows "A = B"
emeasure_subadditive_countably: fixes f :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range f \<subseteq> sets M" shows "emeasure M (\<Union> (range f)) \<le> (\<Sum>i. emeasure M (f i))"
prems_3: shows "\<Gamma>\<turnstile> \<langle>Catch c1' c2',Normal s\<rangle> \<Rightarrow>\<notin>range Fault"
exteq_wrap_Id: fixes f :: "recf"   and a :: "nat"   and n :: "nat" assumes "recfn a f" shows "f \<simeq> wrap_Id f n"
Suc: fixes n :: "nat" shows "omega_sum_aux (Suc n) [] = 0"
simps_3: fixes x31 :: "'b expr'"   and x32 :: "'b expr'"   and y31 :: "'b expr'"   and y32 :: "'b expr'" shows "(Add' x31 x32 = Add' y31 y32) = (x31 = y31 \<and> x32 = y32)"
longest_Lyndon_suf_suf: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "LynSuf w \<le>s w"
le_infI2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and x :: "'b"   and a :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq b x" shows "less_eq (inf a b) x"
const_map_ran: fixes x :: "'c"   and v :: "'c"   and S :: "'d set" shows "(x \<in> ran (const_map v S)) = (S \<noteq> {} \<and> x = v)"
DIAndCutP1_def: shows "DIAndCutP1 = ([[$\<alpha> vid1]]Pc pid1)"
simps_4: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and i :: "nat"   and p :: "'d"   and q :: "'d"   and s :: "'e"   and s' :: "'e"   and msg :: "'f"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Send i p q s s' msg) c' = (distributed_system.can_occur channel trans send recv (Send i p q s s' msg) c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c' i = msgs c i @ [Msg msg] \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> no_snapshot_change c c')"
ab: shows "a * b = zerosign (of_bool (sign a \<noteq> sign b)) (IEEE.round To_nearest (valof a * valof b))"
InvariantImpliedLiteralsHoldsForPrefix: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and prefix :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "isPrefix prefix M" shows "InvariantImpliedLiterals F prefix"
new_tv_TVar_subst: fixes n :: "nat" shows "new_tv n TVar"
fmerge_commute: fixes A :: "'a fset"   and B :: "'a fset"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'm" shows "local.fmerge A (\<lambda>x. local.fmerge B (f x)) = local.fmerge B (\<lambda>y. local.fmerge A (\<lambda>x. f x y))"
func_space_is_vs: fixes S :: "'e set" shows "vectorspace K (func_space S)"
lnull_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "\<not> lnull (inf_llist f)"
helper: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x * (y * z) = y * x * z"
normalize_primitive_extract_def: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and f :: "'c negation_type list \<Rightarrow> 'c list"   and m :: "'b match_expr" shows "normalize_primitive_extract disc_sel C f m \<equiv> case primitive_extractor disc_sel m of (spts, rst) \<Rightarrow> map (\<lambda>spt. MatchAnd (Match (C spt)) rst) (f spts)"
"1_2": shows "fmdom' (fmrestrict_set {v \<in> fmdom' s2. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s2 v} s2) = {v \<in> fmdom' s2. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s2 v}"
psubst_ode: fixes I :: "('sf, 'sc, 'sz) interp"   and ODE :: "('sf, 'sz) ODE"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sc, 'sz) formula" assumes "is_interp I" shows "ODE_sem I ODE = ODE_sem (PFadjoint I \<sigma>) ODE"
card_option: shows "CARD('b option) = Suc CARD('b)"
tunit: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option" shows "\<Gamma> \<turnstile> Unit : TUnit"
that: fixes \<sigma> :: "nat \<Rightarrow> 'a option"   and \<zeta> :: "'b net_state" assumes "netgmap sr n = (\<sigma>, \<zeta>)"   and "dom \<sigma> = net_ips n" shows "thesis"
cnorm_le: fixes x :: "'c"   and y :: "'d" shows "(norm x \<le> norm y) = (\<langle>x, x\<rangle> \<le> \<langle>y, y\<rangle>)"
select_convs_1: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "name \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = name"
echelon_form_fold_Hermite_of_row_i: fixes A :: "(('d, 'e) vec, 'f) vec"   and ass :: "'d \<Rightarrow> 'd"   and res :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and k :: "nat" assumes "echelon_form A"   and "ass_function ass"   and "res_function res" shows "echelon_form (foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<k]))"
Mprefix_STOP: fixes P :: "'a \<Rightarrow> 'a process" shows "Mprefix {} P = STOP"
exhaust: fixes y :: "ground_action" obtains x1 :: "object atom formula" and x2 :: "object ast_effect" where "y = Ground_Action x1 x2"
card_Diff_subset: fixes B :: "'q set"   and A :: "'q set" assumes "finite B"   and "B \<subseteq> A" shows "card (A - B) = card A - card B"
rep_eq: fixes x :: "real"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (x *\<^sub>R xa) = (\<lambda>xb. x *\<^sub>R (xa *\<^sub>V xb))"
measurable_bind: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e measure"   and R :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "(\<lambda>x. g (fst x) (snd x)) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g x) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
assms: fixes x :: "'v" assumes "x \<notin> RF (TER g)" shows "d_OUT g x = 0"
fixp_Sup_le: shows "ccpo.fixp Sup (\<le>) = ccpo_class.fixp"
exhaust: fixes y :: "'g \<times> 'h" obtains a :: "'g" and b :: "'h" where "y = (a, b)"
map_indets_injI: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "inj (map_indets f)"