hcju/isabelleps huggingface dataset:
floor_log_divide_eq: fixes i :: "real"   and j :: "real"   and p :: "real" assumes "0 < i"   and "0 < j"   and "1 < p" shows "\<lfloor>log p (i / j)\<rfloor> = \<lfloor>log p i\<rfloor> - \<lfloor>log p j\<rfloor> - (if j * p powr real_of_int (\<lfloor>log p i\<rfloor> - \<lfloor>log p j\<rfloor>) \<le> i then 0 else 1)"
insert_J_eq: shows "insert b J = {i. i < dim_col A \<and> i \<in> insert b J}"
euler_product_zeta: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / of_nat p powr s) else 1) \<longlonglongrightarrow> zeta s"
HC_intros_6: fixes G :: "'b formula"   and F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (G::'b formula) \<^bold>\<rightarrow> (F::'b formula) \<^bold>\<or> G"
handle_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and vs :: "val list"   and ls :: "val list"   and pc :: "nat"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" shows "handle P C M a h vs ls pc frs = find_handler P a h ((vs, ls, C, M, pc) # frs)"
Aexp_def: fixes e :: "exp" shows "Aexp e = (\<Lambda> a. fst (cccExp e\<cdot>a))"
hequiv_refl: fixes i :: "'b"   and H :: "(('c, 'b) fm list \<times> 'b) set" assumes "i \<in> names H" shows "hequiv H i i"
max_cw: fixes l :: "'b" assumes "l \<in> L" shows "cw_len l \<le> max_len"
radical_Int_subset: fixes A :: "'b set"   and B :: "'b set" shows "\<surd>(A \<inter> B) \<subseteq> \<surd>A \<inter> \<surd>B"
l_def: shows "l = (\<lambda>y. lim (\<lambda>n. u y n / real n))"
int_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" shows "U.int \<xi> T = M.int \<xi> T"
brouwer_weak: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "compact S"   and "convex S"   and "interior S \<noteq> {}"   and "continuous_on S f"   and "f ` S \<subseteq> S" obtains x :: "'a" where "x \<in> S" and "f x = x"
Ssegment_sub: fixes D :: "'c Order"   and a :: "'c" shows "Ssegment D a \<subseteq> carrier D"
Res: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> Rel'" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"
det_R: shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
cong: fixes n :: "nat" shows "fs_int.D n \<equiv> fs_int.D n"
not_one_less_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
pathstart_circlepath: fixes z :: "complex"   and r :: "real" shows "pathstart (circlepath z r) = z + complex_of_real r"
arctan_0_1_bounds': fixes y :: "float"   and n :: "nat"   and prec :: "nat" assumes "0 \<le> real_of_float y"   and "real_of_float y \<le> 1"   and "even n" shows "arctan (sqrt (real_of_float y)) \<in> {sqrt (real_of_float y) * real_of_float (lb_arctan_horner prec n 1 y)..sqrt (real_of_float y) * real_of_float (ub_arctan_horner prec (Suc n) 1 y)}"
inv_naturality: fixes f :: "'c" assumes "C.arr f" shows "(F \<circ> G) f \<cdot>\<^sub>C C.inv (\<epsilon>'.map (C.dom f)) = C.inv (\<epsilon>'.map (C.cod f)) \<cdot>\<^sub>C C.map f"
j_def: shows "j \<equiv> \<lambda>z. if 0 \<le> Im (snd z) then h (fst z, Arg2pi (snd z) / (2 * pi)) else h (fst z, 1 - Arg2pi (cnj (snd z)) / (2 * pi))"
that: fixes R :: "('a \<times> 'a) set" assumes "wf R"   and "\<And>s. fst g s \<Longrightarrow> (snd (snd g s), s) \<in> R" shows "thesis"
A: shows "D x \<in> sinks I D u (xs @ [x])"
cUNION_assoc: fixes g :: "'e \<Rightarrow> 'd cset"   and f :: "'f \<Rightarrow> 'e cset"   and A :: "'f cset" shows "cUnion (cimage g (cUnion (cimage f A))) = cUnion (cimage (\<lambda>x. cUnion (cimage g (f x))) A)"
ac_notin_Diff: fixes x :: "'a"   and A :: "'a antichain"   and B :: "'a antichain" assumes "x \<notin>\<^sub>A A - B" shows "x \<notin>\<^sub>A A \<or> x \<in>\<^sub>A B"
lim: fixes x :: "(real, 'n) vec" assumes "x \<in> g ` S" shows "(\<lambda>n. h n x) \<longlonglongrightarrow> f x"
ordinal_of_nat_of_ordinal: fixes x :: "ordinal" assumes "x < \<omega>" shows "ordinal_of_nat (inv ordinal_of_nat x) = x"
1: shows "esize (\<Squnion> C) = \<Squnion> (esize ` C)"
rep_eq: fixes x :: "'f"   and xa :: "('f, 'e) generator" shows "generator (removeAll_trans x xa) = removeAll_raw x (generator xa)"
1: shows "YFtorNT C f maps\<^bsub>CatExp (Op C) SET\<^esub> Hom\<^bsub>C\<^esub>[\<midarrow>,dom\<^bsub>C\<^esub> f] to Hom\<^bsub>C\<^esub>[\<midarrow>,cod\<^bsub>C\<^esub> f]"
cut_le_absorb: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t \<down>\<le> t = I \<down>\<le> t"
sorted_a: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and x :: "'a extended"   and l :: "nat"   and y :: "'a extended"   and u :: "nat" assumes "integrate_insert_commute M a s"   and "idx s x = Inr l"   and "idx s y = Inr u" shows "(l \<le> u) = (a x \<le> a y)"
l3_trans_def: shows "l3_trans \<equiv> \<Union>M X Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B X \<union> l3_step3 Ra A B X \<union> l3_step4 Rb A B X \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
sup2: fixes B :: "'l"   and k :: "'l"   and b :: "'l"   and a :: "'l" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
mono_implies_arr: fixes f :: "'a" assumes "B.mono f" shows "B.arr f"
hyps: shows "f (sum_list xs) = sum_list (map f xs)"
abs_sgn: fixes k :: "'b" shows "\<bar>k\<bar> = k * sgn k"
term_induct_fresh: fixes P :: "('b, 'c, 'varSort, 'var, 'd) param"   and phi :: "('b, 'c, 'varSort, 'var, 'd) qTerm set \<Rightarrow> bool"   and phiAbs :: "('b, 'c, 'varSort, 'var, 'd) qAbs set \<Rightarrow> bool"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and A :: "('b, 'c, 'varSort, 'var, 'd) qAbs set" assumes "goodPar P"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; x \<notin> varsOf P; \<And>Y. Y \<in> termsOf P \<Longrightarrow> fresh xs x Y; \<And>A. A \<in> absOf P \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envsOf P \<Longrightarrow> freshEnv xs x rho; phi X\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
hom_char: fixes f :: "'c"   and b :: "'c"   and a :: "'c" shows "Fop.Bop.in_hom f b a = Cop.in_hom f a b"
sorted_snd_def: fixes ps :: "(int \<times> int) list" shows "sorted_snd ps = sorted_wrt (\<lambda>p\<^sub>0 p\<^sub>1. snd p\<^sub>0 \<le> snd p\<^sub>1) ps"
asm: shows "x \<in> (mdss ! i) AsmNoWrite"
intro: fixes p :: "nat \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>j. p j" shows "infinitely_many p"
P_new: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new" shows "v0 \<leadsto>P_new\<leadsto>\<^bsub>G\<^esub> last P_new"
h: shows "mat (length subsets) (length signs) (\<lambda>(i, j). z (subsets ! i) (signs ! j)) = mat (length subsets) (length signs) (\<lambda>(i, y). map (\<lambda>index_list. map (z index_list) signs) subsets ! i ! y)"
tendsto_minus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and F :: "'b filter" assumes "class.topological_group_add minus plus zero uminus open"   and "filterlim f (topological_space.nhds open a) F" shows "LIM x F. uminus (f x) :> topological_space.nhds open (uminus a)"
le_add_same_cancel1: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
path_concatenation_end_points: fixes x :: "'a"   and y :: "'a" assumes "end_points x = start_points y"   and "x ; 1 \<cdot> y\<^sup>T ; 1 = 0" shows "end_points (x + y) = end_points y"
elims: fixes x :: "((('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set) set"   and y :: "(('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set set" assumes "combine_pairs' x = y" obtains P :: "((('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set) set" where "x = P" and "y = (\<Union> (fst ` P), snd ` P)"
event_can_go_back_if_no_sender: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "can_occur ev' c'"   and "\<not> isRecvMarker ev'"   and "\<not> isSend ev" shows "can_occur ev' c"
normal_invE_2: fixes N :: "'a set set"   and x :: "'a set"   and h :: "'a set" assumes "N \<lhd> G Mod N"   and "x \<in> carrier (G Mod N)"   and "h \<in> N" shows "x \<otimes>\<^bsub>G Mod N\<^esub> h \<otimes>\<^bsub>G Mod N\<^esub> inv\<^bsub>G Mod N\<^esub> x \<in> N"
simps: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and a1 :: "('a, 'i, 'j) state"   and a2 :: "('f, 'i, 'j) state" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>" shows "std_inca_simulation.match Fstd_get Finca_get \<DD>\<ee>\<II>\<nn>\<ll> a1 a2 = (\<exists>F1 F2 H st. a1 = State F1 H st \<and> a2 = State F2 H st \<and> std_inca_simulation.rel_fundefs \<DD>\<ee>\<II>\<nn>\<ll> (Fstd_get F1) (Finca_get F2))"
choiceSet_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "choiceSet A r \<equiv> {x \<in> A. \<forall>y\<in>A. x \<^bsub>r\<^esub>\<preceq> y}"
iterates_above_Field: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "a \<in> Field leq" shows "x \<in> Field leq"
inf_bot_right: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and x :: "'d" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
rel_cong: fixes x :: "'b rexp"   and ya :: "'b rexp"   and y :: "'c rexp"   and xa :: "'c rexp"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> atoms ya; yb \<in> atoms xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_rexp R x y = rel_rexp Ra ya xa"
vrestr_intersec: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" shows "f --` (A \<inter> B) = f --` A \<inter> f --` B"
T2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>\<box>[P]_v = \<box>[P]_v"
r_cancel_one': fixes x :: "nat \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "a \<in> carrier P" shows "(x = a \<oplus>\<^bsub>P\<^esub> x) = (a = \<zero>\<^bsub>P\<^esub>)"
surjective: fixes r :: "'b mhead_scheme" shows "r = \<lparr>access = access r, static = static r, pars = pars r, resT = resT r, \<dots> = mhead.more r\<rparr>"
assms_3: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<psi>) rel2"
pair: fixes x :: "i"   and y :: "i" assumes "GEM1" shows "\<exists>v. (\<forall>w. w = x \<or> w = y \<longrightarrow> P w v) \<and> (\<forall>w. P w v \<longrightarrow> (\<exists>z. (z = x \<or> z = y) \<and> O w z))"
ordermap_eq_iff: fixes x :: "'c"   and A :: "'c set"   and y :: "'c"   and r :: "('c \<times> 'c) set" assumes "x \<in> A"   and "y \<in> A"   and "wf r"   and "total_on A r"   and "small A" shows "(ordermap A r x = ordermap A r y) = (x = y)"
nat_pow_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "([^]\<^bsub>Ring_Divisibility.mult_of R\<^esub>) = ([^]\<^bsub>R\<^esub>)"
cexpr_sem_cexpr_subst_aux: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat"   and e :: "cexpr"   and e' :: "cexpr" shows "cexpr_sem \<sigma> (cexpr_subst x e e') = cexpr_sem (insert_var x \<sigma> (cexpr_sem \<sigma> e)) e'"
minGraphProps_4: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "f \<in> \<F> g"
ccSup_mono: fixes B :: "'a set"   and A :: "'a set" assumes "countable B"   and "countable A"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b" shows "Sup A \<le> Sup B"
tranclpD: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
AxiomC: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> M" shows "M \<turnstile>\<^sub>c t"
less_imp_le: fixes x :: "'a option \<Rightarrow>\<^sub>0 nat"   and y :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord_strict x y" shows "extended_ord x y"
compact_uniformly_equicontinuous: fixes S :: "'a set"   and \<F> :: "('a \<Rightarrow> 'b) set"   and e :: "real" assumes "compact S"   and "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>f\<in>\<F>. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
simple_function_sum: fixes P :: "'b set"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)" shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
valid_ReqsIFS: shows "valid_reqs (get_IFS M)"
adj_sub_ode: fixes \<sigma> :: "('b, 'c, 'a) subst"   and x1 :: "('b, 'a) ODE"   and x2 :: "('b, 'c, 'a) formula" shows "(\<Union>i\<in>{i |i. Inl i \<in> SIGO x1}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP (EvolveODE x1 x2)))"
fun_rel_sv: fixes Ra :: "('b \<times> 'a) set"   and Rv :: "('c \<times> 'd) set" assumes "Range Ra = UNIV"   and "single_valued Rv" shows "single_valued (Ra \<rightarrow> Rv)"
l0: fixes s :: "complex" assumes "c < Re s" shows "((\<lambda>x. f x - g x) has_laplace 0) s"
conversepD: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'c" assumes "r\<inverse>\<inverse> b a" shows "r a b"
favorites_subset_alts: fixes i :: "'agent" shows "favorites R i \<subseteq> alts"
permute_nat_def: fixes p :: "perm"   and n :: "nat" shows "p \<bullet> n = n"
list_all_simps_2: fixes P :: "'c \<Rightarrow> bool" shows "list_all P [] = True"
bigo_const_mult2: fixes c :: "'d"   and f :: "'c \<Rightarrow> 'd" shows "O(\<lambda>x. c * f x) \<subseteq> O(f)"
E1: shows "(\<Sum>i<length qs. E (map_pmf (\<lambda>(is, s). real (\<Sum>x\<in>set init. ALG x qs i (is, s))) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take i qs)))) = (\<Sum>i<length qs. sum (ALG' A qs init i) (set init))"
distinct_9: fixes x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom" shows "SKIP \<noteq> {x61} WHILE x62 DO x63"
matrix_mul_diag_matl: fixes f :: "'c \<Rightarrow> 'a"   and A :: "(('a, 'b) vec, 'c) vec" shows "diag_mat f ** A = (\<chi>i j. f i * A $ i $ j)"
cong_pseudo_reflexivity: shows "\<forall>a b. Cong a b b a"
g_def: shows "g \<equiv> dirichlet_inverse f (1::'a)"
l2product_ladd: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S"   and "h square_integrable S" shows "l2product S (\<lambda>x. f x + g x) h = l2product S f h + l2product S g h"
triangle_in_hom_4: shows "\<guillemotleft>\<r>[g] \<cdot> (g \<star> \<xi>) \<cdot> \<a>[g, f, g] \<cdot> (\<zeta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] : g \<Rightarrow> g\<guillemotright>"
elims_3: fixes x :: "real_max_bot"   and xa :: "real_max_bot" assumes "\<not> x \<le> xa" obtains uv_ :: "real" where "xa__ = R uv_" and "xa = MInfty"   | x :: "real" and y :: "real" where "xa__ = R x" and "xa = R y" and "\<not> x \<le> y"
anti_eq: fixes x :: "real" assumes "b \<le> a"   and "b < ereal x"   and "ereal x < a"   and "x \<notin> X" shows "f x = g x"
size1_ge0: fixes t :: "'b tree" shows "0 < size1 t"
DynProc: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and Q' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'d \<Rightarrow> 'f" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' s Z \<and> (\<forall>t. t \<in> Q' s Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' s Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"   and "\<forall>s\<in>P. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' s Z) Call (p s) (Q' s Z),(A' s Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"
IntI: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
optimize_matches_simple_ruleset: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr" assumes "simple_ruleset rs" shows "simple_ruleset (optimize_matches f rs)"
inject_4: fixes x1 :: "name"   and x2 :: "ccs"   and y1 :: "name"   and y2 :: "ccs" shows "(\<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>y1\<rparr>y2) = ([x1].x2 = [y1].y2)"
open_Int: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<inter> T)"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" shows "lrel R a b = ((a = LNil) = (b = LNil) \<and> (a \<noteq> LNil \<longrightarrow> b \<noteq> LNil \<longrightarrow> R (un_LCons1 a) (un_LCons1 b) \<and> lrel R (un_LCons2 a) (un_LCons2 b)))"
M_def: fixes k :: "nat" shows "gs2.M k = mat k k (\<lambda>(x, y). gs2.\<mu> x y)"
select_convs_2: fixes block :: "DiskBlock"   and proc :: "Proc"   and more :: "'a" shows "proc \<lparr>block = block, proc = proc, \<dots> = more\<rparr> = proc"
implements_represents: fixes jp :: "'a \<Rightarrow> ('cobs \<times> 'as option, 'aAct, 'b) Protocol" assumes "SPR.implements jp" shows "SPR.represents (SPR.jpTraces jp)"
sturm_rat_def: fixes p :: "rat poly" shows "sturm_rat p = sturm_aux_rat p (pderiv p)"
G_def: shows "G \<equiv> subsuming_states (map_of astG)"
count_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset"   and x :: "'a" shows "count (image_mset f A) x = sum (count A) (f -` {x} \<inter> set_mset A)"
frac_less2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and w :: "'c"   and z :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less_eq x y"   and "less zero w"   and "less w z" shows "less (divide x z) (divide y w)"
comp_eq_dest: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'d \<Rightarrow> 'b"   and d :: "'a \<Rightarrow> 'd"   and v :: "'a" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
E8: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>(F \<longrightarrow> G) \<longrightarrow> \<box>F \<longrightarrow> \<diamond>G"