hcju/isabelleps huggingface dataset:
ipresCons_fromMOD_fst: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "ipresCons h hA (fromMOD MOD)" shows "(wls s X \<longrightarrow> fst (h X) = X) \<and> (wlsAbs (us, s') A \<longrightarrow> fst (hA A) = A)"
ENR_imp_locally_path_connected: fixes S :: "'a set" assumes "ENR S" shows "locally path_connected S"
S'_def: shows "S' = - {z. Re z = 0 \<and> Im z \<le> 0}"
mat_multrow_def: fixes mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b"   and A :: "'b mat" shows "mat_multrow_gen mul k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
comp_L_1: shows "Maps.seq \<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>"
simulation_this: shows "simulation R (stutter_extend A) (stutter_extend B)"
sign_simps_4: fixes ae :: "'a"   and be :: "'a"   and ce :: "'a" shows "ae * be * ce = ae * (be * ce)"
inject_2: fixes x2 :: "('c, 'd) atm"   and y2 :: "('c, 'd) atm" shows "(Neg x2 = Neg y2) = (x2 = y2)"
ball_conj_distrib: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>A. P x \<and> Q x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
meet_join_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
sup: fixes j :: "'b"   and I :: "'b set" assumes "j \<notin> I"   and "P (comm_monoid_add.sum (\<squnion>) bot f I) (comm_monoid_add.sum (\<squnion>) bot g I)" shows "P (f j \<squnion> comm_monoid_add.sum (\<squnion>) bot f I) (g j \<squnion> comm_monoid_add.sum (\<squnion>) bot g I)"
listrel_subset: fixes r :: "('g \<times> 'g) set"   and A :: "'g set" assumes "r \<subseteq> A \<times> A" shows "listrel r \<subseteq> lists A \<times> lists A"
indspaceD_range': fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and x :: "('f, 'g) aezfun" assumes "f \<in> indspace G FH V" shows "f x \<in> V"
fpxs_nth_inject: fixes x :: "'b fpxs"   and y :: "'b fpxs" shows "(fpxs_nth x = fpxs_nth y) = (x = y)"
intros_2: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
list_succ_rotate: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs" shows "list_succ (rotate n xs) = list_succ xs"
conv_involutive: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv (conv x) = x"
partn_lst_def: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and \<alpha> :: "V list"   and n :: "nat" shows "Partitions.partn_lst r B \<alpha> n \<equiv> \<forall>f\<in>[B]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}. \<exists>i<length \<alpha>. \<exists>H\<subseteq>B. ordertype H r = \<alpha> ! i \<and> f ` [H]\<^bsup>n\<^esup> \<subseteq> {i}"
wcode_jump2: fixes lnb :: "nat"   and ires :: "cell list"   and rs :: "nat"   and rnb :: "nat" shows "\<exists>stp ln rn. steps0 (t_twice_len + 14 + t_fourtimes_len, Bk # Bk # Bk \<up> lnb @ Oc # ires, Oc \<up> Suc (4 * rs + 4) @ Bk \<up> rnb) t_wcode_main stp = (Suc 0, Bk # Bk \<up> ln @ Oc # ires, Bk # Oc \<up> Suc (4 * rs + 4) @ Bk \<up> rn)"
line_suitable_junctors_1: fixes \<A> :: "'b set"   and F :: "'b formula" shows "line_suitable \<A> (\<^bold>\<not> F) = line_suitable \<A> F"
dom_simp: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "C.VV.dom f = C.VxV.dom f"
simps_32: fixes x2 :: "'a" shows "set2_sum3 (In2 x2) = {x2}"
zone_set_def: fixes Z :: "('a \<Rightarrow> 'b) set"   and r :: "'a list" shows "zone_set Z r = {[r\<rightarrow>(0::'b)]u |u. u \<in> Z}"
rule_sub_lem_1_f: fixes v :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v]" shows "[\<^bold>\<box>\<psi> \<^bold>\<equiv> \<^bold>\<box>\<chi> in v]"
strict_monoI: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "class.order less_eq less"   and "\<And>x y. less x y \<Longrightarrow> f x < f y" shows "order.strict_mono less f"
bounded_linear_inner_left: fixes b :: "'b" shows "bounded_linear (\<lambda>a. a \<bullet> b)"
F_is_totient_k: shows "F k = of_nat (totient k)"
Top_Left_Zero: fixes P :: "'a hrel" assumes "P is \<H>" shows "\<^bold>\<top> ;; P = \<^bold>\<top>"
split_beta: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
ag_inv_inj: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e"   and y :: "'e" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<noteq> y" shows "-\<^sub>a\<^bsub>A\<^esub> x \<noteq> -\<^sub>a\<^bsub>A\<^esub> y"
simps_3: fixes carrier' :: "'c set \<Rightarrow> 'c set"   and carrier :: "'c set"   and more :: "'d" shows "carrier_update carrier' \<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier' carrier, \<dots> = more\<rparr>"
dist_induces_open: fixes U :: "fbint set" shows "open U = (\<forall>x\<in>U. \<exists>e>0. \<forall>y. fb.dist x y < e \<longrightarrow> y \<in> U)"
step: shows "\<Gamma> \<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"
equiintegrable_on_null: fixes a :: "'a"   and b :: "'a"   and F :: "('a \<Rightarrow> 'b) set" assumes "content (cbox a b) = 0" shows "F equiintegrable_on cbox a b"
that: fixes pth1 :: "('b \<times> 'a) list"   and e :: "'b \<times> 'a"   and pth2 :: "('b \<times> 'a) list" assumes "pth = pth1 @ [e] @ pth2"   and "snd e = ps"   and "ps \<notin> snd ` set pth1" shows "thesis"
simps_21: fixes x31 :: "acom"   and x32 :: "acom"   and x51 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x52 :: "char list"   and x53 :: "bexp"   and x54 :: "acom" shows "x31;; x32 \<noteq> {x51/x52} WHILE x53 DO x54"
symp_tllist_all2: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "symp R"   and "symp S" shows "symp (tllist_all2 R S)"
stream_all_Stream: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and X :: "'b stream" shows "stream_all P (x ## X) = (P x \<and> stream_all P X)"
map_def: fixes f :: "'c" shows "\<epsilon>'.map f = (if C.arr f then \<epsilon>' (C.cod f) \<cdot>\<^sub>C (F \<circ> G) f else C.null)"
wpo_with_assms_axioms: shows "wpo_with_assms weight_S weight_NS (\<lambda>f g. (pr_strict f g, pr_weak f g)) (\<lambda>(f, n). n = 0 \<and> least f) full_status False (\<lambda>f. False)"
bij_perm_rem: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c" shows "bij (perm_rem x f) = bij f"
card_orbit_eq: fixes f :: "'b perm"   and a :: "'b" shows "card (orbit f a) = order f a"
RETURN_refine_iff: fixes x :: "'c"   and R :: "('c \<times> 'd) set"   and y :: "'d" shows "(RETURN x \<le> \<Down> R (RETURN y)) = ((x, y) \<in> R)"
nonempty_imp_ex_extreme: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "well_related_set A less_eq"   and "X \<subseteq> A"   and "X \<noteq> {}" shows "\<exists>e. extreme X (\<lambda>x y. less_eq y x) e"
assoc'_is_natural_2: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] = \<a>\<^sup>-\<^sup>1[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> (\<mu> \<star> \<nu> \<star> \<tau>)"
deny: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'a match_expr"   and p :: "'b"   and a :: "action"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option" assumes "Semantics.matches \<gamma> m p"   and "a = Drop \<or> a = Reject" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
mapping_of: fixes x :: "('a, 'b) pp" shows "pp.mapping_of x \<in> UNIV"
Tietze_periodic_interval: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f"   and "f a = f b" obtains g :: "real \<Rightarrow> real" where "continuous_on UNIV g" and "\<And>x. x \<in> {a..b} \<Longrightarrow> g x = f x" and "\<And>x. g (x + (b - a)) = g x"
mono_mndet_T_process: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq>\<^sub>T Q x" shows "mndet A P \<sqsubseteq>\<^sub>T mndet A Q"
red_rtrancl_\<tau>_heapD_inv: fixes t :: "'t"   and s :: "'x2 \<times> 'm2"   and s' :: "'x2 \<times> 'm2"   and wfs :: "'t \<Rightarrow> 'x2 \<times> 'm2 \<Rightarrow> bool" assumes "r2.silent_moves t s s'"   and "wfs t s" shows "snd s' = snd s"
Isubst_fresh: fixes x :: "name"   and y :: "name"   and t1 :: "trmI"   and t2 :: "trmI" assumes "x \<sharp> [y].t1"   and "x \<sharp> t2" shows "x \<sharp> t1[y::=t2]"
dom_comp: fixes g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.seq g f" shows "Cop_S.dom (Cop_S.comp g f) = Cop_S.dom f"
D_hyp: shows "D \<in> \<G>_Fset N - Red_F_G (\<G>_Fset N)"
inprod_cvec_bilinear2: fixes z' :: "complex \<times> complex"   and k :: "complex"   and z :: "complex \<times> complex"   and w :: "complex \<times> complex" assumes "z' = k *\<^sub>s\<^sub>v z" shows "\<langle>w,z'\<rangle> = k * \<langle>w,z\<rangle>"
SN_on_subset1: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('a \<times> 'a) set" assumes "SN_on r A"   and "s \<subseteq> r" shows "SN_on s A"
eSuc: fixes n' :: "enat" assumes "n = eSuc n'" shows "thesis"
pochhammer_code: fixes a :: "'a"   and n :: "nat" shows "pochhammer a n = (if n = 0 then 1::'a else fold_atLeastAtMost_nat (\<lambda>n. (*) (a + of_nat n)) 0 (n - 1) (1::'a))"
assms_2: shows "prefs_from_table_wf agents alts xs"
CAs0'_def: shows "CAs0' = CAs0 \<cdot>\<cdot>cl \<rho>s"
power2_norm_eq_inner: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(norm x)\<^sup>2 = inner x x"
zero_eq_of_rat_iff: fixes a :: "rat" shows "((0::'a) = of_rat a) = (0 = a)"
DirProd_list_carrier_memI: fixes gs :: "'c list"   and Gs :: "('c, 'd) monoid_scheme list" assumes "length gs = length Gs"   and "\<And>i. i < length Gs \<Longrightarrow> gs ! i \<in> carrier (Gs ! i)" shows "gs \<in> carrier (DirProd_list Gs)"
F_D_part: fixes P :: "'b process" shows "F P = {(s, x). s \<in> D P} \<union> {(s, x). s \<notin> D P \<and> (s, x) \<in> F P}"
Cond: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P\<^sub>1 :: "('e, 'd, 'f) ann"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P\<^sub>2 :: "('e, 'd, 'f) ann"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and r :: "'e set"   and b :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q, A"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q, A"   and "r \<inter> b \<subseteq> pre P\<^sub>1"   and "r \<inter> - b \<subseteq> pre P\<^sub>2" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnBin r P\<^sub>1 P\<^sub>2) (Cond b c\<^sub>1 c\<^sub>2) Q, A"
part_product_identity: fixes x :: "i"   and y :: "i" assumes "P x y" shows "x \<otimes> y = x"
iter_dom_properties: fixes ss' :: "nat list list"   and w' :: "nat list" shows "iter f step start (unstables r step start) = (ss', w') \<longrightarrow> wf_dom ss' w' \<and> stables r step ss' \<and> start [\<sqsubseteq>\<^bsub>r\<^esub>] ss' \<and> (\<forall>ts\<in>list n A. start [\<sqsubseteq>\<^bsub>r\<^esub>] ts \<and> stables r step ts \<longrightarrow> ss' [\<sqsubseteq>\<^bsub>r\<^esub>] ts)"
pre_pure_rule: fixes P :: "assn"   and b :: "bool"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" shows "(\<not> <P * \<up> b> f <Q>) = (\<not> <P> f <Q> \<and> b)"
prim_invar2_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" shows "prim_invar2 Q \<pi> \<equiv> prim_invar2_init Q \<pi> \<or> prim_invar2_ctd Q \<pi>"
Lyndon_mono_last_longest: fixes ws :: "'a list list" assumes "ws \<noteq> \<epsilon>"   and "Lyndon_mono ws" shows "LynSuf (concat ws) = last ws"
take_ext: fixes k :: "nat"   and xs' :: "'c list"   and xs :: "'c list" assumes "\<forall>i<k. xs' ! i = xs ! i"   and "k < length xs"   and "k < length xs'" shows "take k xs' = take k xs"
Ak: shows "A ^\<^sub>m k \<in> carrier_mat n n"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'d set" assumes "n \<le> card S" obtains T :: "'d set" where "T \<subseteq> S" and "card T = n" and "finite T"
nested_propos_4: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "nested_propos (af_letter_opt \<phi> \<nu>) \<subseteq> nested_propos \<phi>"
project_extend_Join: fixes F :: "'a program"   and G :: "'c program" shows "project h UNIV (extend h F \<squnion> G) = F \<squnion> project h UNIV G"
find_index_le_size: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "find_index P xs \<le> length xs"
elims_2: fixes x :: "'a \<times> 'a"   and xa :: "'a \<times> 'a" assumes "my_int_rel x xa" obtains xb :: "'a" and y :: "'a" and u :: "'a" and v :: "'a" where "x = (xb, y)" and "xa = (u, v)" and "xb + v = u + y"
cases: fixes \<Gamma> :: "('d, 'e) \<I>"   and a :: "('f, 'd, 'e) gpv" assumes "\<Gamma> \<turnstile>g a \<surd>" obtains gpv :: "('f, 'd, ('f, 'd, 'e) rpv) generat spmf" where "a = GPV gpv" and "\<And>out c. IO out c \<in> set_spmf gpv \<Longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>)"
label_wrtD': fixes X :: "'b set set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplex X"   and "ChamberComplex.label_wrt X B f"   and "SimplicialComplex.maxsimp X C" shows "bij_betw f C B"
llength_lnths_ile: fixes xs :: "'b llist"   and A :: "nat set" shows "llength (lselect A xs) \<le> llength xs"
is_unit_power_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one" shows "dvd.dvd times (power.power one times a n) one = (dvd.dvd times a one \<or> n = 0)"
enat_ord_code_6: fixes q :: "enat" shows "(\<infinity> < q) = False"
simps_1: shows "is_bal ET\<^sub>0 = True"
subterm_if_in_strand_ik: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "\<exists>t'. receive\<langle>t'\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> t \<sqsubseteq> t'"
trancl_into_trancl2: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>+" shows "(a, c) \<in> r\<^sup>+"
SeqAbstFormP_Ex: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm"   and x' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, k1, x, x', v, i)"   and "atom k \<sharp> (s1, k1, x, x', v, i)" shows "{SeqAbstFormP v (SUCC i) x x' s1 k1, OrdP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i (Q_Ex x) (Q_Ex x') (Var s) (Var k)))"
simps_3: fixes e :: "(name, hf) finfun"   and A :: "fm"   and B :: "fm" shows "eval_fm e (A OR B) = (eval_fm e A \<or> eval_fm e B)"
simps: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a"   and e :: "'b"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "'a"   and a2 :: "'b" assumes "peano zero succ" shows "peano.Rec zero succ e r a1 a2 = (a1 = zero \<and> a2 = e \<or> (\<exists>m n. a1 = succ m \<and> a2 = r m n \<and> peano.Rec zero succ e r m n))"
unsat_farkas_coefficients: fixes cs :: "constraint set" assumes "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s cs"   and "finite cs" shows "\<exists>C. farkas_coefficients cs C"
var_app_typesE: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and ts :: "dB list"   and T :: "LambdaType.type" assumes "e \<turnstile> dB.Var i \<degree>\<degree> ts : T" obtains Ts :: "LambdaType.type list" where "e \<turnstile> dB.Var i : Ts \<Rrightarrow> T" and "e \<tturnstile> ts : Ts"
has_derivative_sin: fixes g :: "'c \<Rightarrow> real"   and g' :: "'c \<Rightarrow> real"   and x :: "'c"   and s :: "'c set" assumes "(g has_derivative g') (at x within s)" shows "((\<lambda>x. sin (g x)) has_derivative (\<lambda>xa. g' xa * cos (g x))) (at x within s)"
interval_integral_FTC2: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "a \<le> c"   and "c \<le> b"   and "continuous_on {a..b} f"   and "a \<le> x"   and "x \<le> b" shows "((\<lambda>x. interval_lebesgue_integral lborel (ereal c) (ereal x) f) has_vector_derivative f x) (at x within {a..b})"
closure_closure: fixes S :: "'b set" shows "closure (closure S) = closure S"
swapidseq_ext_of_permutation: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S"   and "finite S" shows "\<exists>n. swapidseq_ext S n p"
Gt_lt_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y = Gt) = cmp.lt y x"
cases: fixes a1 :: "'Var \<Rightarrow> 'BExp set option"   and a2 :: "'BExp"   and a3 :: "'BExp set" assumes "a1 \<turnstile>\<^sub>b a2 \<in> a3 " obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and e :: "'BExp" where "a1 = \<Gamma>" and "a2 = e" and "a3 = \<Union> (to_total \<Gamma> ` bexp_vars e)"
exhaust: fixes y :: "g_action" obtains "y = Any"   | "y = Count"
filter_min_cases: fixes x :: "'a"   and xs :: "'a list" assumes "transp rel"   and "x \<in> set xs" obtains "x \<in> set (local.filter_min xs)"   | y :: "'a" where "y \<in> set (local.filter_min xs)" and "x \<notin> set (local.filter_min xs)" and "rel y x"
less_floor_iff: fixes z :: "int"   and x :: "'d" shows "(z < \<lfloor>x\<rfloor>) = (of_int z + (1::'d) \<le> x)"
span_subspace: fixes A :: "('a, 'b) vec set"   and B :: "('a, 'b) vec set" assumes "A \<subseteq> B"   and "B \<subseteq> vec.span A"   and "vec.subspace B" shows "vec.span A = B"
maxdeg_max: fixes A :: "(('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'e) set"   and p :: "('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'e" assumes "finite A"   and "p \<in> A" shows "poly_deg p \<le> maxdeg A"
flat_ord_def: fixes b :: "'g"   and x :: "'g"   and y :: "'g" shows "flat_ord b x y = (x = b \<or> x = y)"
asym: fixes b :: "'d"   and a :: "'d" assumes "b < a"   and "a < b" shows "False"
FrameRuleSound: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3 {P} C { Q}" shows "\<Turnstile>\<^sub>3 {P \<and>* F} C { Q \<and>* F}"
1: shows "(norm a)\<^sup>2 + b\<^sup>2 = 1"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "B.arr f"
exposed_face_of_refl_eq: fixes S :: "'b set" shows "(S exposed_face_of S) = convex S"
nn_integral_point_measure: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "finite {a \<in> A. 0 < f a \<and> 0 < g a}" shows "integral\<^sup>N (point_measure A f) g = (\<Sum>a | a \<in> A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
lemma_4_5_a: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c"   and "b \<le> c" shows "a \<squnion>1 b \<le> c"
simps_2: fixes l :: "'c tree"   and x :: "'c"   and d :: "bool"   and r :: "'c tree" shows "all_distinct (Node l x d r) = ((d \<or> x \<notin> set_of l \<and> x \<notin> set_of r) \<and> set_of l \<inter> set_of r = {} \<and> all_distinct l \<and> all_distinct r)"
in_span_listE: fixes v :: "'a vec"   and vs :: "'a vec list" assumes "v \<in> span_list vs" obtains c :: "nat \<Rightarrow> 'a" where "True" and "v = lincomb_list c vs"
rel_pmf_OO_trans: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r :: "'e pmf" assumes "rel_pmf R p q"   and "rel_pmf S q r" shows "rel_pmf (R OO S) p r"
assms: fixes c :: "('test, 'atom, 'choice) cmd" assumes "c \<in> set cl" shows "siso c"
singleton_fset_is: fixes M :: "'a fset"   and y :: "'a" assumes "fBall M (\<lambda>x. x = y)"   and "M \<noteq> {||}" shows "M = {|y|}"
thm_noncont_propos_3: fixes v :: "i" shows "[NonContingent p\<^sub>0 in v]"
0: shows "g \<in> injections ({x} \<union> X) Y"
wfP_minimalE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "wfP r"   and "P x" obtains z :: "'a" where "P z" and "r\<^sup>*\<^sup>* z x" and "\<And>y. r y z \<Longrightarrow> \<not> P y"
distinguishing_weak_formula_distinguishes: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P \<equiv>\<cdot> Q" shows "distinguishing_weak_formula P Q distinguishes P from Q"
finite_measure_Union: fixes A :: "real set"   and B :: "real set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
1: fixes u :: "'b stream"   and v :: "'a stream" assumes "stream_all2 R u v" shows "pred_stream (Rangep R) v"
shortest_paths_locale_step1_axioms: shows "shortest_paths_locale_step1 G s num parent_edge dist"
right_uniqueI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x y; A x z\<rbrakk> \<Longrightarrow> y = z" shows "right_unique A"
fdisjnt_union_left: fixes A :: "'a fset"   and M :: "'a fset"   and B :: "'a fset" assumes "fdisjnt A M"   and "fdisjnt B M" shows "fdisjnt (A |\<union>| B) M"
prob_compl: fixes A :: "'a set" assumes "A \<in> PS.events" shows "PS.prob (space P - A) = 1 - PS.prob A"
tendsto_Zfun_iff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter" shows "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
distinct_mset_dom: fixes N :: "('a, 'b) fmap" shows "distinct_mset (dom_m N)"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "MIR.num" shows "nozerocoeff (CN n c t) = (c \<noteq> 0 \<and> nozerocoeff t)"
zero_le_imp_eq_int_raw: fixes k :: "nat \<times> nat" assumes "less_int_raw (0, 0) k" shows "\<exists>n>0. k \<approx> int_of_nat_raw n"