hcju/isabelleps huggingface dataset:
init_unique_mor: fixes a1 :: "'d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool"   and dummy :: "'d"   and a2 :: "'d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool"   and B1 :: "'e set"   and B2 :: "'f set"   and s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f"   and f1 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'e"   and f2 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'f"   and g1 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'e"   and g2 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'f" assumes "a1 \<in> car_init1 dummy"   and "a2 \<in> car_init2 dummy"   and "mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) B1 B2 s1 s2 f1 f2"   and "mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) B1 B2 s1 s2 g1 g2" shows "f1 a1 = g1 a1 \<and> f2 a2 = g2 a2"
slpa_get_return_edges: fixes cs :: "'edge list"   and as :: "'edge list" assumes "same_level_path_aux cs as"   and "cs \<noteq> []"   and "upd_cs cs as = []"   and "\<forall>xs ys. as = xs @ ys \<and> ys \<noteq> [] \<longrightarrow> upd_cs cs xs \<noteq> []" shows "last as \<in> get_return_edges (last cs)"
cis_arg_mult: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 * z2 \<noteq> 0" shows "cis (arg (z1 * z2)) = cis (arg z1 + arg z2)"
ide_base: shows "ide (g \<star> f\<^sup>*)"
upd_oo_def: fixes f :: "'b array Heap"   and i :: "nat"   and x :: "'b"   and a :: "'b array" shows "upd_oo f i x a \<equiv> Array.len a \<bind> (\<lambda>l. if i < l then Array.upd i x a else f)"
tc_not_0: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "tc p \<noteq> (0::'c)"
IH: shows "T_isort xs \<le> (length xs + 1)\<^sup>2"
same_fw_behaviour_one_equi_2: fixes x :: "'a word"   and y :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" shows "same_fw_behaviour_one x y c rs = same_fw_behaviour_one y x c rs"
Cond: fixes G :: "prog"   and s0 :: "st"   and e0 :: "expr"   and b :: "val"   and s1 :: "state"   and e1 :: "expr"   and e2 :: "expr"   and v :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<rightarrow> s2"
NSDERIV_cmult: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a"   and c :: "'a" assumes "NSDERIV f x :> D" shows "NSDERIV (\<lambda>x. c * f x) x :> c * D"
da_elim_cases_43: fixes Env :: "env"   and B :: "lname set"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A" obtains C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C__" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1__" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2__" and "nrm A = B \<union> assigns_if True (c ? e1 : e2) \<inter> assigns_if False (c ? e1 : e2)" and "brk A = (\<lambda>l. UNIV)"   | C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "\<not> Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2" and "nrm A = nrm E1 \<inter> nrm E2" and "brk A = (\<lambda>l. UNIV)"
emeasure_pair_measure: fixes X :: "('c \<times> 'a) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M M) X = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N"
gfp_Kleene_iter: fixes f :: "'b \<Rightarrow> 'b"   and k :: "nat" assumes "mono f"   and "(f ^^ Suc k) top = (f ^^ k) top" shows "gfp f = (f ^^ k) top"
igSubstIGVar1_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
is_tangent_def: fixes p :: "'a point"   and q :: "'a point" shows "is_tangent p q = (p \<noteq> Infinity \<and> p = q \<and> p \<noteq> opp q)"
has_ell2_norm_def: fixes x :: "'c \<Rightarrow> complex" shows "has_ell2_norm x = bdd_above (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite)"
sees_method_is_class: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "is_class P C"
Collect_case_prod_in_rel_leE: fixes X :: "('d \<times> 'e) set"   and Y :: "('d \<times> 'e) set" assumes "X \<subseteq> {(x, y). in_rel Y x y}" obtains "X \<subseteq> Y"
E3: shows "(\<Sum>x\<in>set init. \<Sum>i<length qs. ALG' A qs init i x) = (\<Sum>x\<in>set init. \<Sum>y\<in>set init. \<Sum>i | i < length qs \<and> qs ! i = y. ALG' A qs init i x)"
L: shows "L \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
strong_confluentpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "strong_confluentp r"   and "r x y"   and "r x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>=\<^sup>= z u"
coprime1: shows "coprime u (u\<^sup>2 - v\<^sup>2)"
WEAK_ALL_def: fixes I :: "'c \<Rightarrow> bool"   and X :: "'c set"   and P :: "'c \<Rightarrow> bool nres" shows "WEAK_ALL\<^bsup>I\<^esup> X P = WHILE\<^bsup>\<lambda>(Y, b). b \<longrightarrow> (\<forall>x\<in>X - Y. I x)\<^esup> (\<lambda>(X, b). b \<and> X \<noteq> {}) (\<lambda>(X, b). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(x, X'). P x \<bind> (\<lambda>b'. RETURN (X', b' \<and> b))))) (X, True) \<bind> (\<lambda>(uu_, b). RETURN b)"
carrier_class: fixes x :: "'c" shows "x \<in> carrier cring_class_ops"
red_lcl_add: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and l\<^sub>0 :: "sfields" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "P \<turnstile> \<langle>e,(h, l\<^sub>0 ++ l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l\<^sub>0 ++ l', sh'),b'\<rangle>"
singletonI: fixes a :: "'b::type" shows "(a::'b::type) \<in> {a}"
intros_1: fixes p :: "'node"   and S :: "'node set" shows "reachable_through p S p"
RP_complete_if_fair_new_statement: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts"   and "FL.active_subset (lclss_of_state (lhd Sts)) = {}"   and "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}"   and "\<G>_Fset (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)) \<TTurnstile>e {{#}}" shows "{#} \<in> Q_of_state (Liminf_state Sts)"
assms_1: shows "unifier\<^sub>l\<^sub>s \<sigma> L"
assms_2: shows "acc {(x, y). x \<sqsubseteq> y}"
l_get_dom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
g: fixes y :: "nat"   and y' :: "nat"   and x :: "'a" assumes "y < y'" shows "g y x < g y' x"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier C"   and "y \<in> carrier C" shows "x \<otimes>\<^bsub>C\<^esub> y \<in> carrier C"
algorithmic_transitivity_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and u :: "trm" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T"   and "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> u : T"
rHom_add: fixes f :: "'d \<Rightarrow> 'e"   and A :: "('d, 'f) Ring_scheme"   and R :: "('e, 'g) Ring_scheme"   and x :: "'d"   and y :: "'d" assumes "f \<in> rHom A R"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "f (x \<plusminus>\<^bsub>A\<^esub> y) = f x \<plusminus>\<^bsub>R\<^esub> f y"
ddcc_ent_e: shows "set_mset DD \<union> CC \<Turnstile> {E}"
ANR_insert: fixes S :: "'a set"   and a :: "'a" assumes "ANR S"   and "closed S" shows "ANR (insert a S)"
zero_le_float: fixes a :: "int"   and b :: "int" shows "(0 \<le> float (a, b)) = (0 \<le> a)"
num_params_polynate: fixes p :: "'a poly" shows "num_params (polynate p) \<le> num_params p"
A: fixes M :: "real" shows "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at basepoint (u m) (u n)"
upd_globals_def: fixes upd :: "'a \<Rightarrow> 'a"   and s :: "('a, 'b) state_scheme" shows "upd_globals upd s = s\<lparr>globals := upd (globals s)\<rparr>"
finsum_cong': fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum G f A = finsum G g B"
simps_2: fixes x :: "'g"   and y :: "'g"   and xs :: "'g list" shows "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
full_imp_bal: fixes n :: "nat"   and t :: "'a tree23" assumes "full n t" shows "bal t"
wfp_on_finite: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "irreflp rel"   and "transp rel"   and "finite A" shows "wfp_on rel A"
sum_unfold_sum_mset: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and f :: "'l \<Rightarrow> 'k"   and A :: "'l set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f A = comm_monoid_add.sum_mset plus zero (image_mset f (mset_set A))"
bounded_open_segment: fixes a :: "'b"   and b :: "'b" shows "bounded (open_segment a b)"
interleave_cases: fixes a3 :: "'b list"   and a1 :: "'b list"   and a2 :: "'b list" assumes "a3 \<in> a1 \<otimes> a2" obtains "a1 = []" and "a2 = []" and "a3 = []"   | xs :: "'b list" and ys :: "'b list" and zs :: "'b list" and x :: "'b" where "a1 = x__ # xsb__" and "a2 = ysb__" and "a3 = x__ # zsb__" and "zsb__ \<in> xsb__ \<otimes> ysb__"   | xs :: "'b list" and ys :: "'b list" and zs :: "'b list" and x :: "'b" where "a1 = xs" and "a2 = x # ys" and "a3 = x # zs" and "zs \<in> xs \<otimes> ys"
tendsto_ennreal_iff: fixes f :: "'c \<Rightarrow> real"   and F :: "'c filter"   and x :: "real" assumes "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F = (f \<longlongrightarrow> x) F"
vd: shows "vertices (chain_sup S) = Domain (\<Union> (range g))"
gfp_mono: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "\<And>Z. f Z \<le> g Z" shows "gfp f \<le> gfp g"
conductor_le_iff: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat" assumes "dcharacter n \<chi>" shows "(dcharacter.conductor n \<chi> \<le> a) = (\<exists>d\<le>a. dcharacter.induced_modulus n \<chi> d)"
card_points_less: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "|hom unity a| <o \<AA>"
simps_22: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Call i) ts = (if i < length (func_t \<C>) then case func_t \<C> ! i of tn _> tm \<Rightarrow> type_update ts (to_ct_list tn) (Type tm) else Bot)"
Gmult_assoc: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and g :: "'a"   and h :: "'a"   and v :: "'c" assumes "FGModule G smult V"   and "g \<in> G"   and "h \<in> G"   and "v \<in> V" shows "aezfun_scalar_mult.Gmult smult g (aezfun_scalar_mult.Gmult smult h v) = aezfun_scalar_mult.Gmult smult (g + h) v"
r_def: shows "r = (\<lambda>x. \<theta> x - \<psi> x)"
r_zero: fixes x :: "'a set" assumes "x \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> \<zero>\<^bsub>R Quot I\<^esub> = x"
of_natural_def: fixes R :: "('b, 'c) ring_scheme"   and n :: "nat" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> = ((\<oplus>\<^bsub>R\<^esub>) \<one>\<^bsub>R\<^esub> ^^ n) \<zero>\<^bsub>R\<^esub>"
card_extensional_funcset: fixes A :: "'b set"   and B :: "'c set" assumes "finite A" shows "card (A \<rightarrow>\<^sub>E B) = card B ^ card A"
iter_Suc2: fixes n :: "nat"   and s :: "'a" shows "iter (Suc n) s = iter n s \<bind> (\<lambda>s. if guard s then body s else return_spmf s)"
set_restriction_empty: fixes f :: "'a set \<Rightarrow> 'a set" assumes "set_restriction f" shows "f {} = {}"
Gets_imp_knows: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> bankerb_gets" shows "X \<in> knows B evs"
Bex_def: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
ring_times_x_0: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<cdot>\<^sub>r \<zero> = \<zero>"
monotonic: fixes P :: "pi"   and A :: "(pi \<times> pi) set"   and P' :: "pi"   and B :: "(pi \<times> pi) set" assumes "P \<leadsto>\<guillemotleft>A\<guillemotright> P'"   and "A \<subseteq> B" shows "P \<leadsto>\<guillemotleft>B\<guillemotright> P'"
false_alt_def: shows "false = \<guillemotleft>False\<guillemotright>"
endswith_split: fixes pat :: "[:'r:]" shows "endswith\<cdot>pat = snull oo csnd oo split\<cdot>pat"
certificate_valid_pubEK: fixes i :: "nat"   and EKi :: "nat"   and evs :: "event list" assumes "cert (CA i) EKi onlyEnc (priSK RCA) \<in> parts (knows Spy evs)"   and "evs \<in> set_mr" shows "EKi = pubEK (CA i)"
mult_zero_right: fixes a :: "real" shows "a \<cdot> (0::'a) = (0::'a)"
nxt_subset: fixes N :: "nat set"   and n :: "nat" shows "nxt N n \<subseteq> N"
rot_rot: fixes v :: "'a" shows "rot (rot v) = - v"
power2_eq_square: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "power.power one times a 2 = times a a"
assms: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
assms_7: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
map_fst_mk_fst: fixes k :: "'a"   and l :: "'b list" shows "map fst (map (Pair k) l) = replicate (length l) k"
ring_hom_in_hom_2: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<in> hom (add_monoid R) (add_monoid S)"
in_keys_mult_scalarE: fixes v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (p \<odot> q)" obtains t :: "'a" and u :: "'t" where "t \<in> keys p" and "u \<in> keys q" and "v = t \<oplus> u"
sum: fixes g :: "'c \<Rightarrow> complex"   and A :: "'c set"   and x :: "'b" shows "sum g A *\<^sub>C x = (\<Sum>xa\<in>A. g xa *\<^sub>C x)"
Posix_mkeps: fixes r :: "'b rexp" assumes "nullable r" shows "[] \<in> r \<rightarrow> mkeps r"
strip_eq_SKIP: fixes c :: "'c acom" shows "(strip c = com.SKIP) = (\<exists>P. c = SKIP {P})"
hom_zero: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring_hom_cring R S h" shows "h \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
stake_Suc: fixes n :: "nat"   and s :: "'b stream" shows "stake (Suc n) s = stake n s @ [s !! n]"
d_a_galois1: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" shows "(rdom x ; rel_ad y \<subseteq> rdom z) = (rdom x \<subseteq> rdom z \<union> rdom y)"
cong_sum: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "\<And>x. x \<in> A \<Longrightarrow> unique_euclidean_semiring.cong modulo (f x) (g x) a" shows "unique_euclidean_semiring.cong modulo (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero g A) a"
norm_def: fixes t :: "'c local.term" shows "\<^bold>\<parallel>t\<^bold>\<parallel> = Inv (Cod t\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<cdot> Dom t\<^bold>\<down>"
match_list_complete: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list" assumes "match_list d P = None" shows "matchers (set P) = {}"
has_derivative_const: fixes c :: "'d"   and F :: "'c filter" shows "((\<lambda>x. c) has_derivative (\<lambda>x. 0::'d)) F"
subspace_cone_decomp: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "cone_decomp T ps" shows "phull.subspace T"
relbelow_on_def: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "relbelow_on A ord z rel a b \<equiv> a \<in> A \<and> b \<in> A \<and> rel a b \<and> ord a z \<and> ord b z"
return_bind: fixes x :: "'s"   and f :: "'s \<Rightarrow> 'm" shows "bind (return x) f = f x"
rbt_less_simps_1: fixes k :: "'a" shows "(Empty |\<guillemotleft> k) = True"
L2: shows "\<forall>x. (x \<in> B) = (g x \<in> C)"
coprime_power_right_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a (power.power one times b n) = (algebraic_semidom.coprime times one a b \<or> n = 0)"
divide_le_eq_1_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq (divide b a) one = less_eq b a"
smallstepsI_ps_eq: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and i :: "nat"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> I\<rightarrow>i \<lless>\<pi>', e'\<ggreater>" shows "\<pi> = \<pi>'"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Num of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f3"
Fvars_rawpsubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)"   and "\<forall>x\<in>snd ` set txs. \<forall>t\<in>fst ` set txs. x \<notin> FvarsT t" shows "Fvars (rawpsubst \<phi> txs) = Fvars \<phi> - snd ` set txs \<union> \<Union> {if x \<in> Fvars \<phi> then FvarsT t else {} |t x. (t, x) \<in> set txs}"
assms: shows "(\<lambda>x. ereal (Gromov_product_at a (u x) (v x))) \<longlonglongrightarrow> \<infinity>"
comm_ring_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.comm_ring plus zero minus uminus times \<equiv> class.ab_group_add plus zero minus uminus \<and> class.comm_semiring plus times"
simps_1: fixes r :: "'b routing_rule"   and rs :: "'b routing_rule list" shows "has_default_route (r # rs) = (routing_prefix r = 0 \<or> has_default_route rs)"
STCal_steps_2: fixes T :: "'procT"   and P' :: "('procS, 'procT) Proc" shows "TargetTerm T \<longmapsto>STCal Source Target* P' = (\<exists>T'. T' \<in>T P' \<and> T \<longmapsto>Target* T')"
simps_3: fixes K :: "nat"   and X :: "freemsg" shows "freeleft (CRYPT K X) = freeleft X"
l_get_tag_name_axioms: shows "l_get_tag_name type_wf get_tag_name get_tag_name_locs"
aezfun_setspan_proj_aezdeltafun: fixes A :: "'b set"   and b :: "'a"   and a :: "'b" shows "aezfun_setspan_proj A (b \<delta>\<delta> a) = ((\<delta>\<delta>) b \<down> A) a"
emptyset_is_bot: fixes S :: "'b set" shows "{} \<sqsubseteq> S"
ln_series: fixes x :: "real" assumes "0 < x"   and "x < 2" shows "ln x = (\<Sum>n. (- 1) ^ n * (1 / real (n + 1)) * (x - 1) ^ Suc n)"
"11_5": shows "b\<^sub>1 @- u'' =\<^sub>I u'"
lin_tss_Cons: shows "linearize tss = ts # linearize (tl_queue tss')"
simps_3: fixes x :: "'d"   and \<theta> :: "('b, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'c, 'd) formula" shows "PPsubst (DiffAssign x \<theta>) \<sigma> = DiffAssign x \<theta>"
upt_zero_length: fixes n :: "nat" shows "length [0..<n] = n"
compile_env_entry_def: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'n \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and x :: "'n \<times> ('b, 'o, 'fun, 'd, 'e) Inca.instr fundef" shows "compile_env_entry \<A> \<O> x = map_result id (Pair (fst x)) (compile_fundef \<A> (\<O> (fst x)) (snd x))"
bigtheta_trans2': fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "g \<in> \<Theta>[F](f)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> \<Theta>[F](h)"
suffs_cond2_conv: fixes ys :: "nat list"   and b :: "nat list"   and a :: "nat list" assumes "length ys = length b" shows "suffs (cond2 a b) b (ys, b \<bullet> ys) = ((\<forall>y\<in>set ys. y \<le> Max (set a)) \<and> subdprodr_impl a b ys)"