hcju/isabelleps huggingface dataset:
hyps: shows "[a\<noteq>b]Q \<longmapsto> \<alpha> \<prec> Q'"
Collect_mono: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
add_diff_cancel_right': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
set_integrable_abs_iff': fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and A :: "'a set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
ascP_comp_append_Cons: fixes f :: "'d list \<Rightarrow> 'c list"   and x :: "'c" shows "ascP (\<lambda>xs. f [] @ x # xs)"
alw_sfilter: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream"   and Q :: "'a stream \<Rightarrow> bool" assumes "alw (ev (holds P)) s" shows "alw Q (sfilter P s) = alw (\<lambda>x. Q (sfilter P x)) s"
simps_1: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e" shows "pre_RDR.exec \<delta> s [] = s"
invariantE: fixes P :: "'b \<Rightarrow> bool"   and succs :: "'b \<Rightarrow> 'b list"   and g' :: "'b"   and g :: "'b" assumes "invariant P succs"   and "g' \<in> set (succs g)"   and "P g" shows "P g'"
un: shows "einterval A B = (\<Union>i. {g (l i)<..<g (u i)})"
Abs_FingerTree_inverse: fixes y :: "('c, 'd) FingerTreeStruc" assumes "y \<in> {t. FingerTreeStruc.ft_invar t}" shows "Rep_FingerTree (Abs_FingerTree y) = y"
hom_zero: shows "Ring_Hom_Poly.of_int_poly 0 = 0"
lub_below: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "class.cpo below"   and "po.chain below S"   and "\<And>i. below (S i) x" shows "below (po.lub below (range S)) x"
finite_ball_avoid: fixes S :: "'b set"   and X :: "'b set"   and p :: "'b" assumes "open S"   and "finite X"   and "p \<in> S" shows "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)"
fun_upd_upd: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d"   and z :: "'d" shows "f(x := y, x := z) = f(x := z)"
assms: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
case_prodD: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
Units_closed: fixes x :: "nat \<times> nat" assumes "x \<in> Units S" shows "x \<in> carrier S"
simps_2: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option" shows "none_MT_rules C [] = True"
empty_correct: shows "local.map_of empty \<subseteq>\<^sub>m Map.empty"
intro_elim_3_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]" shows "[\<phi> \<^bold>\<or> \<psi> in v]"
assms_4: shows "I \<cdot> J \<subseteq> P"
after_def: fixes P :: "'e process"   and xs :: "'e list" shows "P \<setminus> xs \<equiv> Abs_process (futures P xs, future_divergences P xs)"
dfa_of_pf'_induct: fixes P :: "nat \<Rightarrow> pf \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "pf" assumes "\<And>n ks l. P n (Eq ks l)"   and "\<And>n ks l. P n (Le ks l)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (And p q)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (Or p q)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (Imp p q)"   and "\<And>n p. P (Suc n) p \<Longrightarrow> P n (Exist p)"   and "\<And>n p. P n (Neg (Exist (Neg p))) \<Longrightarrow> P n (Forall p)"   and "\<And>n p. P n p \<Longrightarrow> P n (Neg p)" shows "P a0 a1"
no_id_collision: shows "I m \<notin> I ` insert_messages M"
induct: fixes x1 :: "nat"   and x2 :: "'a set"   and x3 :: "'a set"   and P :: "nat \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "dimension x1 x2 x3"   and "\<And>K. P 0 K {\<zero>}"   and "\<And>v E n K. \<lbrakk>v \<in> carrier R; v \<notin> E; dimension n K E; P n K E\<rbrakk> \<Longrightarrow> P (Suc n) K (line_extension K v E)" shows "P x1 x2 x3"
equivalent_objects_symmetric: fixes a :: "'c"   and b :: "'c" assumes "C.equivalent_objects a b" shows "C.equivalent_objects b a"
mdpt_lt: assumes "- b / (2 * a) < z" shows "a * z\<^sup>2 + b * z + c < 0"
is_none_map_option: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c option" shows "Option.is_none (map_option f x) = Option.is_none x"
unmatching_insert_agnostic: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "('a, 'c) flow_entry_match"   and p :: "'b"   and ft :: "('a, 'c) flow_entry_match list" assumes "\<not> \<gamma> (ofe_fields a) p" shows "OF_same_priority_match2 \<gamma> (a # ft) p = OF_same_priority_match2 \<gamma> ft p"
rep_eq: fixes x :: "'a dual set" shows "undual (Inf x) = Sup (undual ` x)"
simps_3: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and uu :: "nat \<times> nat" assumes "Network c s t" shows "project_operation (SAT_PUSH' uu) = PUSH"
strict_order_iff: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "(a \<prec> b) = (a = linordered_monoid.min a b \<and> a \<noteq> b)"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (*s) cart_basis"
nrcons_list: fixes t :: "('c, 'd) FingerTreeStruc"   and a :: "('c, 'd) Node" shows "toList (nrcons t a) = toList t @ nodeToList a"
C_eq_All_untilSortedQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = Cp p"
REFL_TC_CONJ: fixes R' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "reflexive R'"   and "\<forall>x y. P x \<and> P y \<longrightarrow> R' x y \<longrightarrow> R x y"   and "\<not> R\<^sup>+\<^sup>+ x y" shows "\<not> R'\<^sup>+\<^sup>+ x y \<or> (\<exists>z. \<not> P z \<and> R'\<^sup>+\<^sup>+ x z \<and> R'\<^sup>+\<^sup>+ z y)"
power_increasing: fixes n :: "nat"   and N :: "nat"   and a :: "'a" assumes "n \<le> N"   and "(1::'a) \<le> a" shows "a ^ n \<le> a ^ N"
singleton_insert_inj_eq': fixes a :: "'b"   and A :: "'b set"   and b :: "'b" shows "(insert a A = {b}) = (a = b \<and> order.greater_eq {b} A)"
finite_has_maximal: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq m b \<longrightarrow> m = b"
system_automaton_alt_def: shows "s2.system_automaton = stutter_extend s2.system_automaton'"
ord_gpv_compp: shows "ord_gpv OO ord_gpv = ord_gpv"
leg1_is_map: shows "is_left_adjoint r\<^sub>0s\<^sub>1.p\<^sub>1"
star_slide_var: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x = x \<cdot> x\<^sup>\<star>"
list_all2_refl: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>x. P x x" shows "list_all2 P xs xs"
subset_eq: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
cmf: shows "continuous_map (subtopology (Euclidean_space m) U) (Euclidean_space n) f"
object_ptr_kinds_eq: fixes ptrs :: "(_) object_ptr list" shows "h \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs = h' \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs"
ttree_contI2: fixes f :: "'b ttree \<Rightarrow> 'c ttree"   and t :: "'b list \<Rightarrow> 'c list set" assumes "\<And>x. paths (f x) = \<Union> (t ` paths x)"   and "[] \<in> t []" shows "cont f"
if_distrib_fun: fixes c :: "bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(if c then f else g) x = (if c then f x else g x)"
rep_eq: fixes x :: "'a"   and xa :: "'a fset" shows "fset (finsert x xa) = insert x (fset xa)"
feasible_def: fixes c :: "('d, 'e) conf"   and ls :: "('d, 'e) label list" shows "feasible c ls \<equiv> \<exists>c'. se_star c ls c' \<and> Conf.sat c'"
x_as_monom: shows "[:0::'a, 1::'a:] = Polynomial.monom (1::'a) 1"
axioms_2: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_csgrp \<alpha> \<SS>" shows "\<Z>_csgrp_axioms \<SS>"
ltf: shows "lt f = map_component (\<lambda>k. k - length bs) (lt h)"
isNSCont_NSLIM_iff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "isNSCont f a = f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a"
netmask_def': fixes I :: "nat set"   and \<sigma> :: "nat \<Rightarrow> 'c"   and \<zeta> :: "'d" shows "netmask I (\<sigma>, \<zeta>) = (\<lambda>i. if i \<in> I then Some (\<sigma> i) else None, \<zeta>)"
OP_def: fixes x :: "'d" shows "OP x \<equiv> x"
mono: fixes a :: "'d upred"   and c :: "'d upred"   and b :: "'d upred"   and d :: "'d upred" assumes "c \<sqsubseteq> a"   and "d \<sqsubseteq> b" shows "(c \<and> d) \<sqsubseteq> (a \<and> b)"
Inl_Inr_False: fixes x :: "'b"   and y :: "'c" shows "(Inl x = Inr y) = False"
hh2: shows "(A2, B2, C2, D2) = H2"
in_notinI: fixes j :: "'c"   and i :: "'c" assumes "(j, i) \<notin> t \<or> j = i"   and "i \<in> Field t"   and "j \<in> Field t" shows "(i, j) \<in> t"
sames: shows "{x. f x \<noteq> c} - {i, b} = {x. f (swap b i x) \<noteq> c} - {i, b}"
helper_masked_ucast_generic: fixes n :: "nat"   and m :: "nat"   and b :: "16 word" assumes "n + 16 \<le> m"   and "m < 128" shows "(UCAST(16 \<rightarrow> 128) b << n) && (mask 16 << m) = 0"
are_all_operator_effects_consistent_def: fixes ops :: "('a, 'b) sas_plus_operator list" shows "are_all_operator_effects_consistent ops \<equiv> list_all (\<lambda>op. list_all (are_operator_effects_consistent op) ops) ops"
set_step: fixes s :: "'c list"   and r :: "'c"   and act :: "nat \<times> nat list" shows "set (step s r act) = set s"
wfP_trancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfP r" shows "wfP r\<^sup>+\<^sup>+"
tendsto_compose_at: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and F :: "'b filter"   and g :: "'c \<Rightarrow> 'd"   and z :: "'d" assumes "(f \<longlongrightarrow> y) F"   and "g \<midarrow>y\<rightarrow> z"   and "\<forall>\<^sub>F w in F. f w = y \<longrightarrow> g y = z" shows "((g \<circ> f) \<longlongrightarrow> z) F"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Arr (t \<^bold>\<cdot> u) = Seq t u"
incidA_lAB: fixes A :: "Points"   and B :: "Points" shows "incid A (line A B)"
inverse_unique: fixes a :: "'a"   and b :: "'a" assumes "a * b = (1::'a)" shows "inverse a = b"
contained_less_eq: fixes xq :: "'a seq"   and Q :: "'a pred" shows "contained xq Q = (pred_of_seq xq \<le> Q)"
execn_callAbrupt: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and n :: "nat"   and t :: "'d"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Abrupt t" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =Suc n\<Rightarrow> Abrupt (return s t)"
bounded_linear_inner_left: fixes b :: "'e" shows "bounded_linear (\<lambda>a. a \<bullet> b)"
zip_upt_self_aux: fixes xs :: "'d list" shows "zip [0..<length xs] xs = map (\<lambda>i. (i, xs ! i)) [0..<length xs]"
prefix_conc_length: fixes w :: "'b list"   and w' :: "nat \<Rightarrow> 'b" shows "prefix (length w) (w \<frown> w') = w"
finite_measure_mono: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c pmf" assumes "A \<subseteq> B"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A \<le> measure_pmf.prob M B"
bCollect_param: fixes R :: "('c \<times> 'd) set" shows "(\<lambda>y a. {x \<in> y. a x}, \<lambda>z a'. {x \<in> z. a' x}) \<in> \<langle>R\<rangle>set_rel \<rightarrow> (R \<rightarrow> bool_rel) \<rightarrow> \<langle>R\<rangle>set_rel"
is_ground_clss_mono: fixes CC :: "'a literal multiset set"   and DD :: "'a literal multiset set" assumes "CC \<subseteq> DD"   and "is_ground_clss DD" shows "is_ground_clss CC"
word_lsb_alt: fixes w :: "'b word" shows "lsb w = w !! 0"
sup_aci_2: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
bound_of_def: fixes P :: "'b \<Rightarrow> real" shows "bound_of P \<equiv> Sup (range P)"
insert_invar: fixes q :: "('c, 'd) SkewBinomialTree list"   and e :: "'c"   and a :: "'d" assumes "invar q" shows "invar (local.insert e a q)"
sorter: fixes n :: "nat"   and R :: "(nat \<times> nat) set" assumes "linorder_on {..<n} R" shows "is_sorting R [0..<n] (eval_sorter R (sorter n))"
almost_full_onI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>f. \<forall>i. f i \<in> A \<Longrightarrow> good P f" shows "almost_full_on P A"
get_component_to_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and to :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr' \<rightarrow>\<^sub>r to"   and "ptr \<in> set to" shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c"
simps_5: fixes vc :: "16 word"   and vd :: "of_match_field set" shows "prerequisites (VlanId vc) vd = True"
bv_msb_extend_same: fixes w :: "bit list"   and b :: "bit"   and n :: "nat" assumes "bv_msb w = b" shows "bv_msb (bv_extend n b w) = b"
Ex: fixes A :: "dbfm"   and name :: "name" assumes "wf_dbfm A" shows "wf_dbfm (DBEx (abst_dbfm name 0 A))"
span_subspace: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> local.span A"   and "local.subspace B" shows "local.span A = B"
sorted_repeat_poly_list_rel_with0_wrtl_Cons_iff: fixes ys :: "char list list"   and n :: "int"   and p :: "(char list list \<times> int) list"   and a :: "(char list multiset \<times> int) multiset"   and S :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" shows "(((ys, n) # p, a) \<in> sorted_repeat_poly_list_rel_with0 S) = ((p, remove1_mset (mset ys, n) a) \<in> sorted_repeat_poly_list_rel_with0 S \<and> (mset ys, n) \<in># a \<and> (\<forall>x\<in>set p. S ys (fst x)) \<and> sorted_wrt var_order ys \<and> distinct ys)"
fds_mangoldt_times_zeta: shows "fds mangoldt * fds_zeta = (\<chi>x. of_real (ln (real x)))"
proper_interval_simps_3: fixes x :: "'a" shows "proper_interval (Some x) None = (\<exists>z. x < z)"
zero_in_full_pmdl: fixes K :: "unit set" shows "0 \<in> local.punit.full_pmdl K"
min_height_bal: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, zs)" shows "min_height t = nat \<lfloor>log 2 (real (n + 1))\<rfloor>"
filter_def: shows "filter \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. []) (\<lambda>x xs xsa P. if P x then x # xsa P else xsa P) uua uu"
all_security_requirements_fulfilled_mono: fixes M :: "'b SecurityInvariant_configured list"   and E' :: "('b \<times> 'b) set"   and E :: "('b \<times> 'b) set"   and V :: "'b set" assumes "valid_reqs M"   and "E' \<subseteq> E"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E\<rparr>" shows "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E'\<rparr>"
MBC_def: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "MBC t n = (SOME b. mbc_cond t n b)"
ivl_cong: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "sum g {a..<b} = sum h {c..<d}"
contI2: fixes f :: "'c \<Rightarrow> 'd" assumes "monofun f"   and "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))" shows "cont f"
iprev_predicate_change_exists2_all: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P b"   and "\<forall>k\<in>I \<down>\<le> a. P k" shows "\<exists>n\<in>I. a < n \<and> n \<le> b \<and> \<not> P n \<and> (\<forall>k\<in>I \<down>< n. P k)"
inner_scaleR_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and r :: "real"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (scaleR r y) = r * inner x y"
bn_finite: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and bn :: "'e \<Rightarrow> atom set"   and \<alpha> :: "'e" assumes "class.bn permute bn" shows "finite (bn \<alpha>)"
IH_2: fixes i :: "int"   and j :: "int" assumes "inorder r = [i..j]"   and "i \<le> j + 1" shows "(\<Sum>x\<in>leaves i r. f (fst x)) = sum f {i..j + 1}"
approx_f: fixes j :: "nat"   and x :: "'a" assumes "t j \<le> t0 + e1'" shows "dist (f (t j) x) (g (t j) x) \<le> euler_C TYPE('a) / 2 * stepsize j"
wf_fold_rderiv: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and k :: "nat"   and x :: "'x" assumes "local.wf idx \<phi>" shows "local.wf idx (fold rderiv (replicate k x) \<phi>)"
project_coord_inner: fixes i :: "'a"   and x :: "'a"   and b :: "'a"   and n :: "real" assumes "\<bar>i\<bar> \<in> Basis" shows "project_coord x b n \<bullet> i = (if i = b then n else if i = - b then - n else x \<bullet> i)"
case_cong: fixes option :: "'b option"   and option' :: "'b option"   and f1 :: "'c"   and g1 :: "'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "option = option'"   and "option' = None \<Longrightarrow> f1 = g1"   and "\<And>x2. option' = Some x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case option of None \<Rightarrow> f1 | Some x2 \<Rightarrow> f2 x2) = (case option' of None \<Rightarrow> g1 | Some x2 \<Rightarrow> g2 x2)"
fromIMor_termFSwMorph: 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" assumes "termFSwImorph h hA (fromMOD MOD)" shows "termFSwMorph (fromIMor h) (fromIMorAbs hA) MOD"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "TE.int \<xi> (Var x) = \<xi> x"
inf_eq_top_iff: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "(inf x y = top) = (x = top \<and> y = top)"
le_bot: fixes bot :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.order_bot bot less_eq less"   and "less_eq a bot" shows "a = bot"
assms: shows "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}"
Assign: fixes a :: "aexp"   and x :: "char list"   and ps :: "char list \<Rightarrow> int option"   and v :: "int"   and ps' :: "char list \<Rightarrow> int option" assumes "vars a \<union> {x} \<subseteq> dom ps"   and "paval a ps = v"   and "ps' = ps(x \<mapsto> v)" shows "(x ::= a, ps) \<Rightarrow>\<^sub>A Suc 0 \<Down> ps'"
bij_project: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "bij (project (\<lambda>(x, u). h x) UNIV)"
ba: shows "[b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<D>\<^sub>\<circ> f)\<inverse>\<^sub>\<bullet>"
nset_Tr51: fixes j :: "nat"   and n :: "nat" assumes "j \<in> nset (Suc 0) (Suc (Suc n))"   and "j \<noteq> Suc 0" shows "j - Suc 0 \<in> nset (Suc 0) (Suc n)"
convex_rel_interior_iff: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "S \<noteq> {}" shows "(z \<in> rel_interior S) = (\<forall>x\<in>S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
UN_simps_7: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>x\<in>C. A - B x) = A - \<Inter> (B ` C)"
one_dim_iso_of_zero: shows "one_dim_iso (0::'b) = (0::'a)"
le_int_of_integer: fixes x :: "integer"   and y :: "integer" shows "(x \<le> y) = (int_of_integer x \<le> int_of_integer y)"
scale_right_diff_distrib: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d"   and y :: "'d" assumes "module scale" shows "scale a (x - y) = scale a x - scale a y"
arg_min_SOME_Min: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S" shows "arg_min_on f S = (SOME y. y \<in> S \<and> f y = Min (f ` S))"
simps_1: fixes A :: "'o"   and uu :: "'o"   and uv :: "'a" shows "Dom (MkArr A uu uv) = A"
solutions_le_iff_pell_valuation_le: fixes a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "solution (a, b)"   and "solution (x, y)" shows "(a \<le> x \<and> b \<le> y) = (pell_valuation (case (a, b) of (x, y) \<Rightarrow> (int x, int y)) \<le> pell_valuation (case (x, y) of (x, y) \<Rightarrow> (int x, int y)))"
simps_3: fixes v :: "fml" shows "Ando (Afml v) undeff = undeff"
weak_le_iff_meet: fixes x :: "int"   and y :: "int" assumes "x \<in> carrier \<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>"   and "y \<in> carrier \<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>" shows "(x \<sqsubseteq>\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> y) = (x \<squnion>\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> y .=\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> y)"
a: shows "\<Delta> @ [(x, T')] @ \<Gamma> \<turnstile> e : T"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
jordan_hoelder_multisets: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list"   and \<HH> :: "'c set list" assumes "Group.group G"   and "finite (carrier G)"   and "composition_series G \<GG>"   and "composition_series G \<HH>" shows "mset (map group.iso_class (normal_series.quotients G \<GG>)) = mset (map group.iso_class (normal_series.quotients G \<HH>))"
eq: fixes x :: "real" shows "indicat_real {a..b} (x + c) = indicat_real {a - c..b - c} x"
method_declared_inI: fixes c :: "class"   and sig :: "sig"   and m :: "methd"   and G :: "prog"   and C :: "qtname" assumes "table_of (methods c) sig = Some m"   and "class G C = Some c" shows "G\<turnstile> mdecl (sig, m) declared_in C"
map_add_le_mapE: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option"   and h :: "'b \<Rightarrow> 'c option" assumes "g \<Oplus> f \<subseteq>\<^sub>m h" shows "g \<subseteq>\<^sub>m h"
assoc_mult_mat: fixes A :: "'a Matrix.mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'a Matrix.mat"   and n\<^sub>3 :: "nat"   and C :: "'a Matrix.mat"   and n\<^sub>4 :: "nat" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "C \<in> carrier_mat n\<^sub>3 n\<^sub>4" shows "A * B * C = A * (B * C)"