hcju/isabelleps huggingface dataset:
integrable_on_superset: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and t :: "'c set" assumes "f integrable_on S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'d)"   and "S \<subseteq> t" shows "f integrable_on t"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y + z) = x \<cdot> y + x \<cdot> z"
ineM_def: fixes sP :: "specID"   and M :: "chanID set"   and E :: "Expression" shows "ineM sP M E \<equiv> \<exists>ch. ch \<in> ins sP \<and> ch \<in> M \<and> exprChannel ch E"
eint_pos_int_times_ge: fixes c :: "eint"   and n :: "int" assumes "0 \<le> c"   and "0 < n" shows "c \<le> eint n * c"
elims_2: fixes x :: "checker_type"   and xa :: "t list" assumes "c_types_agree x xa" obtains ts :: "t list" and ts' :: "t list" where "x = Type ts__" and "xa = ts'__" and "ts__ = ts'__"   | ts :: "ct list" and ts' :: "t list" where "x = TopType ts" and "xa = ts'" and "ct_suffix ts (to_ct_list ts')"
of_nat_le_numeral_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "nat"   and i :: "num"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (semiring_1.of_nat one plus zero x) (power.power one times (numeral.numeral one plus i) n) = (x \<le> numeral i ^ n)"
bdd_below_is_intervalI: fixes I :: "real set"   and a :: "real"   and b :: "real" assumes "is_interval I"   and "a \<le> b"   and "a \<notin> I"   and "b \<in> I" shows "bdd_below I"
span_subspace: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "module scale"   and "A \<subseteq> B"   and "B \<subseteq> module.span scale A"   and "module.subspace scale B" shows "module.span scale A = B"
Max_le_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Max less_eq A) x = (\<forall>a\<in>A. less_eq a x)"
ring_simprules_2: fixes R :: "('e, 'f) ring_scheme" assumes "ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
well_valued0_append_butlast_lchild: fixes t :: "'b tree"   and as :: "'b list"   and x :: "'b" assumes "well_shaped t"   and "well_valued0 as t" shows "well_valued0 (butlast as @ [last as + x]) (lchild t)"
Y_delta_def: shows "Y_delta \<equiv> AbsN 0 (AbsN 1 (App (Var 0) (App (Var 1) (Var 1))))"
S5n_rels_closed: fixes M :: "('b, 'c, 'd) KripkeStructure"   and a :: "'b"   and X :: "'d set" assumes "S5n M" shows "relations M a `` relations M a `` X \<subseteq> relations M a `` X"
round_robin_start_def: fixes n0 :: "nat"   and t :: "'t" shows "round_robin_start n0 t = (queue_enqueue t (queue_empty ()), n0)"
ifex_ite_rel_bf: fixes fi :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and i :: "'b ifex"   and ft :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and t :: "'b ifex"   and fe :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and e :: "'b ifex" assumes "(fi, i) \<in> bf_ifex_rel"   and "(ft, t) \<in> bf_ifex_rel"   and "(fe, e) \<in> bf_ifex_rel" shows "(bf_ite fi ft fe, ifex_ite i t e) \<in> bf_ifex_rel"
3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>s\<^sub>t ` {dual\<^sub>s\<^sub>t \<A>}))"
mono: fixes x :: "('b, 'call, 'ret) gpv \<times> 's" shows "spmf.mono_body (\<lambda>exec_gpv. case x of (c, s) \<Rightarrow> the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y)))"
slice1_ddep_slice1: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice1 n"   and "n s-V\<rightarrow>\<^sub>d\<^sub>d n'" shows "nx \<in> sum_SDG_slice1 n'"
angles'_1: shows "angle b1 a1 c1 = angle b2 a2 c2"
is_correct_localState_Pointwise_Output_Shrink_def: fixes pointwise_shrink :: "'d list \<Rightarrow> 'd"   and localState :: "'d \<Rightarrow> 'e" shows "is_correct_localState_Pointwise_Output_Shrink pointwise_shrink localState \<equiv> \<forall>cs. cs \<noteq> [] \<longrightarrow> localState (pointwise_shrink cs) = localState (last cs)"
simps_4: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Inverse a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)"
count_space_def: fixes \<Omega> :: "'b set" shows "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then of_nat (card A) else \<infinity>)"
IH2: assumes "\<exists>pn. p2 \<in> subterms (\<Gamma> pn)"   and "((\<sigma>, p2), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
distinct: fixes xs :: "'a llist" assumes "xs \<in> Y" shows "ldistinct xs"
normal_poly_def: fixes p :: "'a poly" shows "normal_poly p \<equiv> p \<noteq> 0 \<and> (\<forall>i. (0::'a) \<le> coeff p i) \<and> (\<forall>i. coeff p i * coeff p (i + 2) \<le> (coeff p (i + 1))\<^sup>2) \<and> (\<forall>i j k. i \<le> j \<longrightarrow> j \<le> k \<longrightarrow> (0::'a) < coeff p i \<longrightarrow> (0::'a) < coeff p k \<longrightarrow> (0::'a) < coeff p j)"
monoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
hmset_pair_decompose_less_eq: fixes m1 :: "hmultiset"   and m2 :: "hmultiset" assumes "m1 \<le> m2" shows "\<exists>k n1 n2. m1 = k + n1 \<and> m2 = k + n2 \<and> (head_\<omega> n1 < head_\<omega> n2 \<or> n1 = 0 \<and> n2 = 0)"
unsat_subs_unsat: fixes c1 :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf" assumes "\<not> Conf.sat c1"   and "c2 \<sqsubseteq> c1" shows "\<not> Conf.sat c2"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'g" shows "replicate (n + m) x = replicate n x @ replicate m x"
swap0_swap0: fixes n :: "nat"   and i :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 'a" shows "swap0 n i (swap0 n i x) = x"
intros: fixes rest :: "('r, 'u, 's, 'v, 't, 'w, 'r) rest_scheme"   and \<I>_adv :: "('s, 't) \<I>"   and I :: "'r \<Rightarrow> bool"   and \<I>_usr :: "('v, 'w) \<I>" assumes "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_adv rest s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'"   and "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_usr rest s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'"   and "I (rinit rest)" shows "WT_rest \<I>_adv \<I>_usr I rest"
induct: fixes \<theta> :: "nat \<Rightarrow> nat option"   and x1 :: "stmt \<times> nat"   and x2 :: "(stmt \<times> nat) \<times> instr list"   and P :: "stmt \<times> nat \<Rightarrow> (stmt \<times> nat) \<times> instr list \<Rightarrow> bool" assumes "\<theta>\<turnstile> x1 \<rightarrow>\<^sub>s x2"   and "\<And>a volatile e A L R W t. \<forall>sop. a \<noteq> Tmp sop \<Longrightarrow> P (Assign volatile a e A L R W, t) ((Assign volatile (Tmp (eval_expr t a)) e A L R W, t + used_tmps a), issue_expr t a)"   and "\<And>D volatile a e A L R W t. D \<subseteq> dom \<theta> \<Longrightarrow> P (Assign volatile (Tmp (D, a)) e A L R W, t) ((Skip, t + used_tmps e), issue_expr t e @ [Write volatile (a \<theta>) (eval_expr t e) (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"   and "\<And>a c\<^sub>e s\<^sub>e A L R W t. \<forall>sop. a \<noteq> Tmp sop \<Longrightarrow> P (CAS a c\<^sub>e s\<^sub>e A L R W, t) ((CAS (Tmp (eval_expr t a)) c\<^sub>e s\<^sub>e A L R W, t + used_tmps a), issue_expr t a)"   and "\<And>c\<^sub>e D\<^sub>a a s\<^sub>e A L R W t. \<forall>sop. c\<^sub>e \<noteq> Tmp sop \<Longrightarrow> P (CAS (Tmp (D\<^sub>a, a)) c\<^sub>e s\<^sub>e A L R W, t) ((CAS (Tmp (D\<^sub>a, a)) (Tmp (eval_expr t c\<^sub>e)) s\<^sub>e A L R W, t + used_tmps c\<^sub>e), issue_expr t c\<^sub>e)"   and "\<And>D\<^sub>a D\<^sub>c t s\<^sub>e D f a c A L R W. \<lbrakk>D\<^sub>a \<subseteq> dom \<theta>; D\<^sub>c \<subseteq> dom \<theta>; eval_expr t s\<^sub>e = (D, f)\<rbrakk> \<Longrightarrow> P (CAS (Tmp (D\<^sub>a, a)) (Tmp (D\<^sub>c, c)) s\<^sub>e A L R W, t) ((Skip, Suc (t + used_tmps s\<^sub>e)), issue_expr t s\<^sub>e @ [RMW (a \<theta>) (t + used_tmps s\<^sub>e) (D, f) (\<lambda>\<theta>. the (\<theta> (t + used_tmps s\<^sub>e)) = c \<theta>) (\<lambda>v\<^sub>1 v\<^sub>2. v\<^sub>1) (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"   and "\<And>s\<^sub>1 t s\<^sub>1' t' is s\<^sub>2. \<lbrakk>\<theta>\<turnstile> (s\<^sub>1, t) \<rightarrow>\<^sub>s ((s\<^sub>1', t'), is); P (s\<^sub>1, t) ((s\<^sub>1', t'), is)\<rbrakk> \<Longrightarrow> P (Seq s\<^sub>1 s\<^sub>2, t) ((Seq s\<^sub>1' s\<^sub>2, t'), is)"   and "\<And>s\<^sub>2 t. P (Seq Skip s\<^sub>2, t) ((s\<^sub>2, t), [])"   and "\<And>e s\<^sub>1 s\<^sub>2 t. \<forall>sop. e \<noteq> Tmp sop \<Longrightarrow> P (Cond e s\<^sub>1 s\<^sub>2, t) ((Cond (Tmp (eval_expr t e)) s\<^sub>1 s\<^sub>2, t + used_tmps e), issue_expr t e)"   and "\<And>D e s\<^sub>1 s\<^sub>2 t. \<lbrakk>D \<subseteq> dom \<theta>; isTrue (e \<theta>)\<rbrakk> \<Longrightarrow> P (Cond (Tmp (D, e)) s\<^sub>1 s\<^sub>2, t) ((s\<^sub>1, t), [])"   and "\<And>D e s\<^sub>1 s\<^sub>2 t. \<lbrakk>D \<subseteq> dom \<theta>; \<not> isTrue (e \<theta>)\<rbrakk> \<Longrightarrow> P (Cond (Tmp (D, e)) s\<^sub>1 s\<^sub>2, t) ((s\<^sub>2, t), [])"   and "\<And>e s t. P (While e s, t) ((Cond e (Seq s (While e s)) Skip, t), [])"   and "\<And>A L R W t. P (SGhost A L R W, t) ((Skip, t), [Ghost (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"   and "\<And>t. P (SFence, t) ((Skip, t), [Fence])" shows "P x1 x2"
disjoint_def: fixes a :: "'a"   and b :: "'a" shows "a ## b \<equiv> a \<oplus> b \<noteq> None"
hyps: fixes s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" assumes "length s + length t < length sa__ + length ta__"   and "(s, t) \<in> greek_less r" shows "(inv_greek s, inv_greek t) \<in> greek_less r"
bounded_subset_cball: fixes S :: "'b set" shows "bounded S = (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
T6: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[\<circle>F]_v"
Lp_le_1_1: fixes p :: "real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "0 < p"   and "p \<le> 1" shows "eNorm (\<LL> (ennreal p) M) f = (if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)) else \<infinity>)"
not_Cons_self2: fixes x :: "'k"   and xs :: "'k list" shows "x # xs \<noteq> xs"
cardeq: fixes x :: "nat" assumes "x \<in> {0..<Suc k}" shows "card {y \<in> {0..<Suc (Suc k)}. y < x} = card {0..<x}"
pp: shows "b = lincomb (ls p) (a ` Is p) \<and> r \<in> Is p \<and> ls p (a r) < (0::'a) \<and> (\<forall>h'. h' \<in> Is p \<longrightarrow> h' < r \<longrightarrow> (0::'a) \<le> ls p (a h')) \<and> cs p \<in> carrier_vec n \<and> local.span (a ` (Is p - {r})) = {x \<in> carrier_vec n. cs p \<bullet> x = (0::'a)} \<and> cs p \<bullet> b < (0::'a) \<and> ks p < m \<and> cs p \<bullet> a (ks p) < (0::'a) \<and> (\<forall>k'<ks p. (0::'a) \<le> cs p \<bullet> a k') \<and> Is (Suc p) = insert (ks p) (Is p - {r})"
get_ancestors_locs_def: shows "get_ancestors_locs = get_parent_locs"
simps_15: fixes x21 :: "'e"   and x22 :: "'e list" shows "set (x21 # x22) = insert x21 (set x22)"
gen_elim: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and cs :: "'b list"   and w :: "'c" assumes "gen P cs w" obtains "\<And>c. c \<in> set cs \<Longrightarrow> P c w"
H: fixes D :: "'a set"   and C :: "'a set" assumes "D \<subseteq> carrier"   and "C \<subseteq> carrier"   and "D dense"   and "C closed"   and "S = D \<inter> C" shows "R"
simps_16: fixes x21 :: "'h"   and x22 :: "'i" shows "generat_pures (IO x21 x22) = {}"
simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ac :: "'b list"   and x :: "'b" shows "ord.quicksort_acc less ac [x] = x # ac"
when_2: fixes P :: "bool"   and zero :: "'b"   and a :: "'b" assumes "\<not> P" shows "zero.when zero a P = zero"
infinite: fixes A :: "'e set"   and g :: "'e \<Rightarrow> 'f" assumes "infinite A" shows "prod g A = (1::'f)"
residual_network_sel_1: fixes f :: "'v node \<times> 'v node \<Rightarrow> ennreal"   and x :: "'v node"   and y :: "'v node" shows "edge (\<Delta>'.residual_network f) x y = (edge \<Delta>' x y \<or> edge \<Delta>' y x \<and> y \<noteq> source \<Delta>')"
sigma_sets_mono': fixes A :: "'b set set"   and B :: "'b set set"   and X :: "'b set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
psubstt_closed_1: fixes i :: "nat"   and f :: "'b \<Rightarrow> 'a"   and t :: "'b FOL_Fitting.term" shows "closedt i (psubstt f t) = closedt i t"
orderE: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
long_chain_betw: 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"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.chain_with betw a b c X" shows "betw a b c"
ntr_valid_preserve: fixes c :: "'n list multiset"   and ww :: "('p, 'ba) label list list"   and c' :: "'n list multiset" assumes "(c, ww, c') \<in> trcl (ntr fg)"   and "valid fg c" shows "valid fg c'"
ct_list_eq_ts_conv_eq: fixes ts :: "t list"   and ts' :: "t list" shows "ct_list_eq (to_ct_list ts) (to_ct_list ts') = (ts = ts')"
fact: shows "U \<subseteq> fst (usubstappp \<sigma> U \<alpha>)"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a) hd" assumes "e \<in> vars_hd a" obtains "a = Var e"
simps_7: fixes x :: "octo"   and y :: "octo" shows "Im6 (x + y) = Im6 x + Im6 y"
int_numeral_bitAND_num: fixes n :: "num"   and m :: "num" shows "numeral n AND numeral m = (case bitAND_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> numeral n')"
boundOutputPar1Dest: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' (Q \<parallel> R)"   and "xvec \<sharp>* R"   and "yvec \<sharp>* R" obtains T :: "('d, 'e, 'f) psi" where "P = T \<parallel> R" and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"
conjunct_def: fixes E :: "(('a \<Rightarrow> 'b) \<Rightarrow> bool) set" shows "conjunct E \<equiv> \<lambda>\<sigma>. \<forall>e\<in>E. e \<sigma>"
floor_less_cancel: fixes x :: "'a"   and y :: "'a" assumes "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor>" shows "x < y"
convert_atom_list_change: fixes xs' :: "real list"   and var :: "nat"   and L :: "atom list"   and x :: "real"   and \<Gamma> :: "real list"   and x' :: "real" assumes "length xs' = var" shows "convert_atom_list var L (xs' @ x # \<Gamma>) = convert_atom_list var L (xs' @ x' # \<Gamma>)"
ldrop_conv_fixp: shows "\<lambda>xs n. ldropn n xs \<equiv> llist_lift.fixp_fun (\<lambda>ldrop xs. case xs of LNil \<Rightarrow> \<lambda>_. LNil | LCons x xs \<Rightarrow> \<lambda>n. if n = 0 then LCons x xs else ldrop xs (n - 1))"
Server_trusts_WL4: fixes B' :: "agent"   and A :: "agent"   and B :: "agent"   and NB :: "nat"   and evs :: "event list" assumes "Says B' Server \<lbrace>Agent A, Agent B, Crypt (shrK A) (Nonce NB)\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> woolam" shows "\<exists>B. Says A B (Crypt (shrK A) (Nonce NB)) \<in> set evs"
Union_partJoin_L: fixes P :: "'a set set"   and Q :: "'a set set" shows "\<Union> P \<subseteq> \<Union> (partJoin P Q)"
reflexive: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] P"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "Ik.wtE \<xi> \<equiv> \<forall>x. intT (tpOfV x) (\<xi> x)"
characteristic_state_sim: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "finite A" shows "(\<sigma> 0 ~A~ \<tau> 0) = \<tau> \<Turnstile>\<^sub>p characteristic_formula A (\<sigma> 0)"
ln_less_self: fixes x :: "real" assumes "0 < x" shows "ln x < x"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
nsets_zero: fixes A :: "'a set" shows "[A]\<^bsup>0\<^esup> = {{}}"
has_derivative_power_int': fixes x :: "'a"   and n :: "int"   and S :: "'a set" assumes "x \<noteq> (0::'a)" shows "((\<lambda>x. x powi n) has_derivative (\<lambda>y. y * (of_int n * x powi (n - 1)))) (at x within S)"
assms_1: shows "angle a1 b1 c1 = angle a2 b2 c2"
subgroup_nonempty: fixes G :: "('a, 'b) monoid_scheme" shows "\<not> subgroup {} G"
span_base: fixes a :: "'t \<Rightarrow>\<^sub>0 'd"   and S :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "a \<in> S" shows "a \<in> pmdl S"
norm_normalizes: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" shows "normalized (DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n)"
subgroup_card_eq_order: fixes H :: "'a set" assumes "subgroup H G" shows "card H = order (G\<lparr>carrier := H\<rparr>)"
norm_sum_Pythagorean: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite I"   and "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I" shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
col_inangle2__out: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Bet A B C"   and "P InAngle A B C"   and "Q InAngle A B C"   and "Col B P Q" shows "B Out P Q"
seqstar_unfoldr: fixes x :: "'a" shows "(1::'a) + x\<^sup>; ; x \<le> x\<^sup>;"
arrI: fixes f :: "(bool, bool list) concrete_category.arr" assumes "f \<noteq> J.Null"   and "J.Dom f \<in> {False, True}"   and "J.Cod f \<in> {False, True}"   and "J.Map f \<in> Collect (J.G.path (J.Dom f) (J.Cod f))" shows "J.arr f"
intros_2: fixes t :: "('b, 'c, 'd, 'e) Tree" shows "(t, tNot t) \<in> Tree_wf"
Lim_cong_within: fixes a :: "'b"   and b :: "'b"   and x :: "'c"   and y :: "'c"   and S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "a = b"   and "x = y"   and "S = T"   and "\<And>x. \<lbrakk>x \<noteq> b; x \<in> T\<rbrakk> \<Longrightarrow> f x = g x" shows "(f \<longlongrightarrow> x) (at a within S) = (g \<longlongrightarrow> y) (at b within T)"
multiplicative_sum_divisors: shows "multiplicative_function (\<lambda>n. \<Sum>d | d dvd n. ind squarefree d)"
wls_Abs_inj: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x X') = (X = X')"
intros: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
nprv_impE01: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla"   and F :: "'fmla set"   and \<chi> :: "'fmla" assumes "imp \<phi> \<psi> \<in> F"   and "\<phi> \<in> F"   and "nprv (insert \<psi> F) \<chi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<chi> \<in> fmla" shows "nprv F \<chi>"
Infinitesimal_add_minus_approx_self: fixes y :: "'a star"   and x :: "'a star" assumes "y \<in> Infinitesimal" shows "x \<approx> x + - y"
lebesgue_measure_ball_2: fixes R :: "real"   and x :: "'c" assumes "0 \<le> R" shows "emeasure lborel (cball x R) = ennreal R ^ DIM('c) * emeasure lborel (cball (0::'c) 1)"
foldr_conv_fold: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list" shows "foldr f xs = fold f (rev xs)"
nat_filtration_singleton: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and w :: "bool stream" assumes "infinite_coin_toss_space p M"   and "pseudo_proj_True n w = w" shows "pseudo_proj_True n -` {w} \<in> sets (infinite_coin_toss_space.nat_filtration M n)"
xcpt_app_def: fixes i :: "instr"   and P :: "'b prog"   and pc :: "nat"   and mxs :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list" shows "xcpt_app i P pc mxs xt \<tau> = (\<forall>(f, t, C, h, d)\<in>set (relevant_entries P i pc xt). is_class P C \<and> d \<le> length (fst \<tau>) \<and> d < mxs)"
d: fixes x :: "'b"   and x' :: "'b" assumes "x \<in> cbox a b"   and "x' \<in> cbox a b"   and "dist x' x < d" shows "dist (f x') (f x) < e"
assms: fixes t :: "real" assumes "t \<in> S" shows "u t \<noteq> other.s t"
signed_take_bit_eq_if_negative: fixes a :: "'a"   and n :: "nat" assumes "bit a n" shows "signed_take_bit n a = take_bit n a OR NOT (mask n)"
not_lnull_sswSts: shows "\<not> lnull sswSts"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"   and "local.iso (f \<otimes> cod \<iota>)" shows "local.iso f"
dX_linear: fixes X :: "'x set"   and d :: "'d" assumes "inj_on Xd X" shows "Linear_order (dX X d \<times> dX X d)"
in_square_lemma': fixes i :: "nat"   and L :: "nat"   and n :: "nat"   and N :: "nat" assumes "i < L"   and "n < N" shows "i * N + n < N * L"
seq_type: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>'' :: "'Var set \<times> 'Var set"   and P'' :: "'BExp set" assumes "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>',\<S>',P'"   and "\<turnstile> \<Gamma>',\<S>',P' {c\<^sub>2} \<Gamma>'',\<S>'',P''" shows "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1 ;; c\<^sub>2} \<Gamma>'',\<S>'',P''"
adm_case_prod: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "adm (\<lambda>x. P x (fst (f x)) (snd (f x)))" shows "adm (\<lambda>x. case f x of (a, b) \<Rightarrow> P x a b)"
in_rel_Collect_case_prod_eq: fixes X :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "in_rel {(x, y). X x y} = X"
monoidal_category_axioms: shows "monoidal_category (\<cdot>\<^sub>C) T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C"
assms_4: shows "cnf \<Phi>' \<subseteq> cnf \<Phi>"
Qg_iop_closed: fixes N :: "'a set"   and A :: "'a set" assumes "G \<triangleright> N"   and "A \<in> set_rcs G N" shows "c_iop G N A \<in> set_rcs G N"
case_eq_if: fixes f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and option :: "'i option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
that: fixes X :: "(real \<times> real) \<times> real \<times> real"   and m' :: "real" assumes "X \<in> set Xs"   and "intersect_segment_xline prec X x = Some (m', M)"   and "\<And>X m' M'. \<lbrakk>X \<in> set Xs; intersect_segment_xline prec X x = Some (m', M')\<rbrakk> \<Longrightarrow> M' \<le> M" shows "thesis"
improve_lottery_def: fixes q :: "'alt pmf" shows "improve_lottery q = (let A = {r \<in> lotteries_on alts. q \<prec>[Pareto (SD \<circ> R)] r} in SOME r. r \<in> A \<and> \<not> (\<exists>r'\<in>A. set_pmf r' \<subset> set_pmf r))"
before_swaps: fixes "is" :: "nat list"   and xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct is"   and "\<forall>i\<in>set is. Suc i < length xs"   and "distinct xs"   and "i \<notin> set is"   and "i < j"   and "j < length xs" shows "swaps is xs ! i < swaps is xs ! j in xs"