hcju/isabelleps huggingface dataset:
monotone_if_bot: fixes bound :: "'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and bot :: "'c" assumes "\<And>x y. \<lbrakk>x \<sqsubseteq> y; \<not> x \<sqsubseteq> bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>x. \<not> x \<sqsubseteq> bound \<Longrightarrow> ord bot (f x)"   and "ord bot bot" shows "monotone (\<sqsubseteq>) ord (\<lambda>x. if x \<sqsubseteq> bound then bot else f x)"
presFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'c, 'e, 'f) Recursion.model_scheme" shows "presFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> fresh ys y X \<longrightarrow> gFresh MOD ys y X (h X)"
zmult_zle_mono1_neg: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "k \<le> 0" shows "j * k \<le> i * k"
intro: fixes R :: "'a set"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'c"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>r m. \<lbrakk>r \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> T (smult r m) = smult' r (T m)" shows "RModuleHom_axioms R smult M smult' T"
totient_conv_totient_aux2: fixes n :: "nat"   and xs :: "(nat \<times> nat) list" assumes "n \<noteq> 0"   and "set xs = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"   and "distinct xs" shows "totient n = totient_aux2 xs"
homh: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h" shows "h \<in> ring_hom R S"
4: fixes i :: "nat" assumes "i \<in> t" shows "enat i < esize s"
that: fixes u :: "'a list"   and v :: "'a list"   and k :: "nat" assumes "u \<cdot> v = x"   and "v \<cdot> u = y"   and "(u \<cdot> v) \<^sup>@ k \<cdot> u = z"   and "v \<noteq> \<epsilon>" shows "thesis"
rel_n_def: shows "rel_n \<equiv> \<lambda>X. Id \<inter> - X"
unlr_inf: fixes x :: "'a admS"   and y :: "'a admS" shows "unlr (inf x y) = unlr x \<inter> unlr y"
adm: shows "ccpo.admissible lub (\<lambda>x y. (x, y) \<in> leq) P"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi" shows "(\<Psi>', R, R) \<in> Rel"
ord_less_eq_trans: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "less a b"   and "b = c" shows "less a c"
cases: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b \<times> 'd"   and a2 :: "'c \<times> 'e" assumes "rel_prod R1 R2 a1 a2" obtains a :: "'b" and b :: "'c" and c :: "'d" and d :: "'e" where "a1 = (a, c)" and "a2 = (b, d)" and "R1 a b" and "R2 c d"
inf_left_commute: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type"   and y :: "'a::type"   and z :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) (inf (y::'a::type) (z::'a::type)) = inf y (inf x z)"
D_def: shows "\<F>C.D \<equiv> \<lambda>f. if \<F>C.arr f then \<F>C.mkarr \<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor> else null"
zero_def: shows "0 \<equiv> Abs_G (0::'a)"
split_beta: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (final_b prod)"
assms: shows "\<forall>A. isDiagonal A \<longrightarrow> (\<exists>P Q. invertible P \<and> invertible Q \<and> Smith_normal_form (P ** A ** Q))"
BSD_def: fixes \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "BSD \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ [c] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
not_le_imp_less: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
Nil_prefix: fixes xs :: "'a list" shows "[] \<le> xs"
rep_eq: fixes x :: "'b \<Rightarrow> 'a \<Rightarrow> complex" shows "cblinfun_apply (cblinfun_of_matrix x) = (\<lambda>xa. \<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. (j \<bullet>\<^sub>C xa * x i j) *\<^sub>C i)"
compact_imp_max_in_chain: fixes Y :: "nat \<Rightarrow> 'a" assumes "chain Y"   and "compact (\<Squnion>i. Y i)" shows "\<exists>i. max_in_chain i Y"
neq_top_trans: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> top"   and "x \<le> y" shows "x \<noteq> top"
leq_antisym: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "(x, y) \<in> leq"   and "(y, x) \<in> leq" shows "x = y"
B: shows "Esigma (a * u * q) = a * u * q + a * r * c"
lborel_distr_mult: fixes c :: "real" assumes "c \<noteq> 0" shows "distr lborel borel ((*) c) = density lborel (\<lambda>x. ennreal (inverse \<bar>c\<bar>))"
abs_of_nonneg: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq zero a" shows "abs a = a"
pp_dist_comp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> \<ominus> (x \<odot> y) = \<ominus> \<ominus> x \<odot> \<ominus> \<ominus> y"
gez: shows "A \<ge>\<^sub>m 0\<^sub>m n n"
siterate_smap: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "siterate f x = x ## smap f (siterate f x)"
preserves_arr: fixes f :: "'b" assumes "F.B.arr f" shows "G.B.arr (G f)"
finiteFinestpart: fixes X :: "'b set" shows "finite (finestpart X) = finite X"
Lcm_least_int: fixes A :: "int set"   and a :: "int" assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd a" shows "Lcm A dvd a"
homotopic_loops_def: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" shows "homotopic_loops s p q \<equiv> homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
beta_reds_into_equiv: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* t" shows "s \<leftrightarrow> t"
strict_trans2: fixes x :: "'o"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "lt x b a"   and "le x c b" shows "lt x c a"
finite_measure_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> events"   and "B \<in> events"   and "B \<subseteq> A" shows "prob (A - B) = prob A - prob B"
berlekamp_resulting_mat_basis: fixes u :: "'b mod_ring poly" shows "vectorspace.basis class_ring (Berlekamp_Type_Based.V u) (set (find_base_vectors (berlekamp_resulting_mat u)))"
semIntAll_prSwapAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSwapAll (semInt SEM) (semIntAbs SEM) SEM"
Int_greatest: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "C \<subseteq> A"   and "C \<subseteq> B" shows "C \<subseteq> A \<inter> B"
mult_not_zero: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
cl_insert_absorb: fixes X :: "'a set"   and x :: "'a" assumes "X \<subseteq> carrier"   and "x \<in> cl X" shows "cl (insert x X) = cl X"
inf2E: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "(A \<sqinter> B) x y" obtains "A x y" and "B x y"
semiring_simprules_4: fixes R :: "('c, 'd) ring_scheme" assumes "semiring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
diff_closed: fixes V :: "'c set"   and x :: "'c"   and y :: "'c" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x - y \<in> V"
Map_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Map (local.dom f) = Id (Dom f)"
act_unitl: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Dla :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and e :: "'c"   and p :: "'d" assumes "partial_mon_sg_laction act Dla"   and "e \<in> E" shows "Dla e p \<and> act e p = p"
inf_top_left: fixes x :: "'b" shows "\<top> \<sqinter> x = x"
ccProd_Union2: fixes S :: "Vars.var set"   and S' :: "Vars.var set set" shows "S G\<times> \<Union> S' = lub ((G\<times>) S ` S')"
1: fixes j :: "nat" assumes "j < length sl" shows "sl ! j = fst (\<pi> !! j)"
Scons_Scons_eq: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" shows "(Scons M N = Scons M' N') = (M = M' \<and> N = N')"
True: fixes F :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (%x. True)"
VarP_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm" assumes "H \<turnstile> x EQ x'" shows "H \<turnstile> VarP x IFF VarP x'"
cf_pos_poly_represents: fixes p :: "int poly"   and x :: "'a" shows "cf_pos_poly p represents x = p represents x"
a_minus_def: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
trail_connected: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "Digraph_Component.connected G"   and "trail u p v"   and "set p \<noteq> arcs G" shows "\<exists>e. e \<in> arcs G - set p \<and> (tail G e \<in> set (awalk_verts u p) \<or> head G e \<in> set (awalk_verts u p))"
zero_fcompare: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_zero x"   and "IEEE.is_zero y" shows "fcompare x y = ccode.Eq"
USUP_image_eq: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set"   and g :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Squnion> i \<in> f ` A \<bullet> g i) = (\<Squnion> i \<in> A \<bullet> g (f i))"
bst_joinR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "Tree2.bst l"   and "Tree2.bst r"   and "\<forall>x\<in>Tree2.set_tree l. x < a"   and "\<forall>y\<in>Tree2.set_tree r. a < y" shows "Tree2.bst (joinR l a r)"
right_unit: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> G" shows "compose S a (identity S) = a"
insert: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'e" assumes "semilattice_set f"   and "finite A"   and "A \<noteq> {}" shows "semilattice_set.F f (insert x A) = f x (semilattice_set.F f A)"
preserves_arr: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "B.arr (F f)"
mono_fun_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "mono_fun R = (\<forall>i. mono (R i))"
dom_simp: fixes f :: "'a" assumes "arr f" shows "local.dom f = C.dom f"
inject_5: fixes x5 :: "'d val"   and y5 :: "'d val" shows "(Val x5 = Val y5) = (x5 = y5)"
validCI_act: fixes n :: "nat"   and cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'cmp" assumes "\<exists>i\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (the_enat \<langle>cid #\<^bsub>enat n\<^esub>inf_llist t\<rangle>)" shows "eval cid t t' n \<gamma>"
normal_trans: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>* c'" shows "c \<Rightarrow>\<^sub>G\<^sup>* c'"
nonneg_int_cases: fixes k :: "int" assumes "0 \<le> k" obtains n :: "nat" where "k = int n"
less_n: fixes i :: "nat" shows "odd_of_distinct p i * i \<le> n"
add_mset_eq_single: fixes b :: "'b"   and M :: "'b multiset"   and a :: "'b" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
norm\<psi>: shows "norm \<psi> = sqrt (infsetsum' f UNIV)"
inducts: fixes p :: "'a \<times> 'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "sndsp p x"   and "P (snd p)" shows "P x"
foldri_def: fixes l :: "'e list" shows "foldri l = foldli (rev l)"
path_suc: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "path \<sigma> (Suc k) = fst (local.step (path \<sigma> k, \<sigma>\<^bsup>k\<^esup>))"
varsRemoveAllClauseSuperset: fixes formula :: "Literal list list"   and clause :: "Literal list" shows "vars formula - vars clause \<subseteq> vars (removeAll clause formula)"
v2: fixes h :: "'a" shows "Proj M *\<^sub>V h \<in> space_as_set M"
sphere_1D_doubleton_zero: fixes r :: "real" assumes "DIM('a) = 1"   and "0 < r" obtains x :: "'a" and y :: "'a" where "sphere (0::'a) r = {x, y} \<and> dist x y = 2 * r"
simps_16: fixes x31 :: "nat"   and x32 :: "nat" shows "recf.Id x31 x32 \<noteq> S"
master_theorem: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> = (\<exists>X\<subseteq>subformulas\<^sub>\<mu> \<phi>. \<exists>Y\<subseteq>subformulas\<^sub>\<nu> \<phi>. (\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>) \<and> (\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)) \<and> (\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)))"
m_weakly_prefers_def: fixes m :: "'m"   and \<mu> :: "('m \<times> 'w) set" shows "m_weakly_prefers m \<mu> = {w' \<in> Field (Pm m). \<forall>w\<in>{w. (m, w) \<in> \<mu>}. (w, w') \<in> Pm m}"
prems_1: shows "set (x # xs) \<subseteq> inputs M"
zip_eq_Nil_iff: fixes xs :: "'c list"   and ys :: "'d list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
assms_2: shows "f \<approx>>\<^bsub>CatExp A B\<^esub> g"
widen_is_class_type_of: fixes T :: "ty"   and C :: "String.literal"   and P :: "'a prog"   and T' :: "ty" assumes "class_type_of' T = \<lfloor>C\<rfloor>"   and "P \<turnstile> T' \<le> T"   and "T' \<noteq> NT" obtains C' :: "String.literal" where "class_type_of' T' = \<lfloor>C'\<rfloor>" and "P \<turnstile> C' \<preceq>\<^sup>* C"
mult_locally_cancel: fixes s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Z :: "'a multiset"   and Y :: "'a multiset" assumes "trans s"   and "locally_irrefl s (X + Z)"   and "locally_irrefl s (Y + Z)" shows "((X + Z, Y + Z) \<in> mult s) = ((X, Y) \<in> mult s)"
SKIP_in_constrains_iff: fixes A :: "'c set"   and B :: "'c set" shows "(\<bottom> \<in> A co B) = (A \<subseteq> B)"
that: shows "0 \<in>\<^sub>r A \<or> 0 \<in>\<^sub>r B"
uv_5: shows "z = u\<^sup>2 + v\<^sup>2"
lunit_naturality: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "\<mu> \<cdot>\<^sub>B B.lunit (B.dom \<mu>) = B.lunit (B.cod \<mu>) \<cdot>\<^sub>B trg\<^sub>B \<mu> \<star>\<^sub>B \<mu>"
compet_rand_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and c :: "real"   and S0 :: "'c list set" shows "Partial_Cost_Model.compet_rand A c S0 = (\<forall>s\<in>S0. \<exists>b\<ge>0. \<forall>rs. static s rs \<longrightarrow> T\<^sub>p_on_rand' A (fst A s \<bind> (\<lambda>is. return_pmf (s, is))) rs \<le> c * real (T\<^sub>p_opt s rs) + b)"
sethi_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "sethi_instr instr \<equiv> let op_list = snd instr; imm22 = get_operand_w22 (op_list ! 0); rd = get_operand_w5 (op_list ! 1) in if rd \<noteq> 0 then do curr_win \<leftarrow> get_curr_win (); write_reg (ucast imm22 << 10) curr_win rd; return () od else return ()"
zcount_ne_zero_iff: fixes M :: "'a zmultiset"   and x :: "'a" shows "(zcount M x \<noteq> 0) = (x \<in>#\<^sub>z M)"
GuardStrip: fixes P :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f"   and g :: "'d set" assumes "P \<subseteq> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
br_iq_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "br_iq \<delta> \<equiv> {lhs r |r. r \<in> \<delta> \<and> rhsq r = []}"
open_Union: fixes "open" :: "'b set \<Rightarrow> bool"   and K :: "'b set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. open S" shows "open (\<Union> K)"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a = a \<sqinter> b)"
coeff_add: fixes p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int"   and n :: "nat" assumes "p \<in> carrier (UP INTEG)"   and "q \<in> carrier (UP INTEG)" shows "up_ring.coeff (UP INTEG) (p \<oplus>\<^bsub>UP INTEG\<^esub> q) n = up_ring.coeff (UP INTEG) p n \<oplus>\<^bsub>INTEG\<^esub> up_ring.coeff (UP INTEG) q n"
OclAllInstances_at_post_def: shows "OclAllInstances_at_post = OclAllInstances_generic snd"
infdist_closed: fixes e :: "real"   and S :: "'b set" shows "closed {z. e \<le> infdist z S}"
drop: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and p :: "'c" assumes "matches \<gamma> m Drop p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny"
"4_2": shows "length (y # ys) \<le> length (x # xs)"
WTrt_binop_simps_1: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Eq\<guillemotright>T2 : T) = (T = Boolean)"
vderiv_on_continuous_on: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "(f has_vderiv_on f') S" shows "continuous_on S f"
bintrunc_sbintrunc': fixes n :: "nat"   and w :: "int" assumes "0 < n" shows "take_bit n (signed_take_bit (n - 1) w) = take_bit n w"
respects_labels: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and v :: "'a" assumes "folding_g.label_wrt B \<phi>"   and "v \<in> \<Union> X" shows "\<phi> (g v) = \<phi> v"
le_fin_lproject_visible: fixes u :: "'action list"   and v :: "'action list" assumes "u \<preceq>\<^sub>F v" shows "lproject visible (llist_of u) \<le> lproject visible (llist_of v)"
IH_6: assumes "\<langle>c: t \<in> s\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t' s'. t = t' \<cdot> \<sigma> \<and> s = s' \<cdot> \<sigma> \<and> \<langle>c: t' \<in> s'\<rangle> \<in> set S"
termFSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Iteration.model_scheme" shows "termFSbImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSubstAll h hA MOD"
\<epsilon>_def: shows "\<epsilon> = sqrt (1 + \<delta>\<^sup>2) - 1"
step_def: fixes D :: "'b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and Q :: "'c \<Rightarrow> 'd"   and i :: "'b" shows "step D Q i = (INF j. D (i, j) (Q j))"
divide_by_norm_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "divide_by_norm A = (\<chi>a b. Projections.normalize (column b A) $ a)"
fags_carr_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "fags_carr f z i a = {x. \<exists>n. x = n\<Odot>a\<^bsub>f,i,z\<^esub>}"
prod_pos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less zero (f a)" shows "less zero (comm_monoid_mult.prod times one f A)"
atoms_pltl_sugar_2: shows "atoms_pltl true\<^sub>p = {}"
ChiPlus_ChiRel_Ex2: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiPlus A"   and "(S, T) \<notin> ChiRel A" shows "\<exists>U. (S, U) \<in> ChiRel A \<and> (U, T) \<in> ChiPlus A"
tends_f'n_g': fixes w :: "complex" assumes "w \<in> ball z r" shows "((\<lambda>n. f' n w) \<longlongrightarrow> g'__ w) F"
transition_system_traces_def: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transition_system_traces ex en ind \<equiv> traces ind \<and> transition_system_traces_axioms ex en ind"
between_switch: fixes p :: "hyp2"   and v :: "(real, 2) vec"   and q :: "hyp2" assumes "B\<^sub>\<real> hyp2_rep p v hyp2_rep q" shows "B\<^sub>K p hyp2_abs v q"
HyperA_def: shows "HyperA \<equiv> {b \<in> carrier_vec n. \<nexists>c. c \<in> carrier_vec n \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)}"
closure_mono: fixes a :: "'a set"   and b :: "'a set" assumes "a \<subseteq> b" shows "closure a \<subseteq> closure b"
discI: fixes tree :: "'a tree"   and x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" assumes "tree = \<langle>x21, x22, x23\<rangle>" shows "tree \<noteq> \<langle>\<rangle>"
coeff_monom_mult: fixes c :: "'b"   and n :: "nat"   and p :: "'b poly"   and k :: "nat" shows "coeff (monom c n * p) k = (if k < n then 0::'b else c * coeff p (k - n))"
assms: shows "(a\<^sub>p, b\<^sub>p) \<in> Tree_wf\<^sub>p"
foldSetD_backwards: fixes A :: "'e set"   and z :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "A \<noteq> {}"   and "(A, z) \<in> foldSetD D f e" shows "\<exists>x y. x \<in> A \<and> (A - {x}, y) \<in> foldSetD D f e \<and> z = f x y"
find_first_unique: fixes xs :: "'a list"   and i :: "nat" assumes "distinct xs"   and "i < length xs" shows "find_first (xs ! i) xs = i"
lunit_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "src (bicategory.lunit V H \<a> \<i> src trg f) = src f"
C'': fixes x :: "'a formula set" assumes "x \<in> C" shows "\<bottom> \<notin> x \<and> (\<forall>k. Atom k \<in> x \<longrightarrow> \<^bold>\<not> (Atom k) \<in> x \<longrightarrow> False) \<and> (\<forall>F G H. Con F G H \<longrightarrow> F \<in> x \<longrightarrow> G \<triangleright> H \<triangleright> x \<in> C) \<and> (\<forall>F G H. Dis F G H \<longrightarrow> F \<in> x \<longrightarrow> G \<triangleright> x \<in> C \<or> H \<triangleright> x \<in> C)"
propnot\<^sub>3_def: shows "propnot \<equiv> \<lambda>F. \<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<^bold>\<not>\<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>)"
zmult_zless_mono2_lemma: fixes i :: "int"   and j :: "int"   and k :: "nat" assumes "i < j"   and "0 < k" shows "int k * i < int k * j"
collect_pair_finite'': fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "finite {(x, y). P x \<and> Q y}" shows "finite {(x, y). P x \<and> Q y \<and> R x y}"