hcju/isabelleps huggingface dataset:
onll_ostep_invariant_sterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('i \<times> ('f, 'g, 'e, 'h) seqp, 'j) automaton"   and S :: "'i \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool"   and U :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and P :: "('i \<times> 'h) \<times> 'j \<times> 'i \<times> 'h \<Rightarrow> bool"   and \<sigma> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and a :: "'j"   and \<sigma>' :: "'i"   and q :: "('f, 'g, 'e, 'h) seqp"   and l' :: "'h"   and p' :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a"   and "l' \<in> labels \<Gamma> q"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P ((\<sigma>, l), a, \<sigma>', l')"
subst_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and f :: "'d \<Rightarrow> 'e option"   and mem :: "'d \<Rightarrow> 'e" assumes "sifum_security_init dma \<C>_vars eval" shows "subst f mem = (\<lambda>x. case f x of None \<Rightarrow> mem x | Some v \<Rightarrow> v)"
quadratic_quadratic_at_most_2_intersections': fixes b2 :: "complex"   and B2 :: "complex"   and b1 :: "complex"   and B1 :: "complex"   and qf1 :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and c :: "complex"   and qf2 :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and C :: "complex"   and x1 :: "complex"   and y1 :: "complex"   and x2 :: "complex"   and y2 :: "complex"   and x :: "complex"   and y :: "complex" assumes "b2 \<noteq> B2 \<or> b1 \<noteq> B1"   and "(b2 - B2)\<^sup>2 + (b1 - B1)\<^sup>2 \<noteq> 0"   and "qf1 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + b1 * x + b2 * y + c)"   and "qf2 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + B1 * x + B2 * y + C)"   and "qf1 x1 y1 = 0"   and "qf2 x1 y1 = 0"   and "qf1 x2 y2 = 0"   and "qf2 x2 y2 = 0"   and "(x1, y1) \<noteq> (x2, y2)"   and "qf1 x y = 0"   and "qf2 x y = 0" shows "(x, y) = (x1, y1) \<or> (x, y) = (x2, y2)"
simps_2: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.negative (Neg A) = True"
cod_cod: fixes f :: "'t" assumes "S'.arr f" shows "S'.cod (S'.cod f) = S'.cod f"
Gets_imp_knows: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> orb" shows "X \<in> knows B evs"
llbound_lbound: fixes A :: "'a set set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> I" shows "lbound A = llbound"
map_sel_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b stream" shows "stl (smap f a) = smap f (stl a)"
inj_on_to_nat_on: fixes A :: "'b set" assumes "countable A" shows "inj_on (to_nat_on A) A"
\<nu>_def: shows "\<nu> \<equiv> \<lambda>l. case H l of (n, M, j) \<Rightarrow> n"
\<omega>consistent_impliesStd1: assumes "B.\<omega>consistent" shows "B.\<omega>consistentStd1"
mk_iso_nomulti_def: fixes G :: "('c, 'd) pre_digraph"   and H :: "('e, 'f) pre_digraph"   and fv :: "'c \<Rightarrow> 'e" shows "mk_iso_nomulti G H fv \<equiv> \<lparr>iso_verts = fv, iso_arcs = the_inv_into (arcs H) (arc_to_ends H) \<circ> pairself fv \<circ> arc_to_ends G, iso_head = head H, iso_tail = tail H\<rparr>"
Ball_set_table: fixes l :: "('c \<times> 'd) list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<forall>(x, y)\<in>set l. P x y" shows "\<forall>x. ! y:table_of l x: P x y"
iterates_def: shows "iterates \<equiv> \<lambda>uu uua. corec_llist (\<lambda>(f, x). False) (\<lambda>(f, x). x) (\<lambda>(f, x). False) (\<lambda>(f, x). undefined) (\<lambda>(f, x). (f, f x)) (uu, uua)"
less_rank_noninjective: fixes A :: "((real, 'a) vec, 'b) vec" shows "(rank A < CARD('a)) = (\<not> inj ((*v) A))"
kD_quality_increases: fixes i :: "nat"   and \<xi> :: "state"   and \<xi>' :: "state" assumes "i \<in> kD (rt \<xi>)"   and "quality_increases \<xi> \<xi>'" shows "i \<in> kD (rt \<xi>')"
ineq_apply_subst: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "subst_domain \<delta> \<inter> set X = {}" shows "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)\<rangle>\<^sub>s\<^sub>t"
valid_11: shows "valid ts\<^sub>s\<^sub>b'"
leg0_composite_1: shows "H\<mu>H\<nu>\<pi>.leg0 = \<pi>.leg0 \<cdot> \<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>0"
isTable_deleteAround: fixes E :: "nat \<Rightarrow> nat"   and vs :: "nat list"   and a :: "nat"   and b :: "nat"   and ps :: "(nat \<times> nat) list"   and g :: "graph" assumes "isTable E vs ((a, b) # ps)" shows "isTable E vs (deleteAround g a ps)"
chine_simps_2: shows "C.dom \<mu>'.chine = \<mu>'.dom.apex"
swap_child_eq_on: fixes arr1 :: "int \<Rightarrow> 'a"   and arr2 :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and c :: "int"   and p :: "int" assumes "arr1 = arr2 on - {l..<r}"   and "c = l_child l p \<or> c = r_child l p"   and "l \<le> p"   and "c < r" shows "arr1 = swap arr2 p c on - {l..<r}"
zip_fun_simps_5: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b" shows "fst ` range (f \<parallel> g) = range f"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A"
v11_vcomp_vconverse: shows "(\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ> \<circ>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr> = vid_on (\<AA>\<lparr>Arr\<rparr>)"
le_minus_divide_eq: 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"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq a (uminus (divide b c)) = (if less zero c then less_eq (times a c) (uminus b) else if less c zero then less_eq (uminus b) (times a c) else less_eq a zero)"
Allowed_rename: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program" assumes "bij h" shows "Allowed (rename h F) = rename h ` Allowed F"
Rep_filter_iff_eventually: fixes F :: "'a filter"   and P :: "'a \<Rightarrow> bool" shows "Rep_filter F P = eventually P F"
mid_eq_a: shows "midpoint (a - complex_of_real e) (a + complex_of_real e) = a"
order_le_less: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (x < y \<or> x = y)"
check_extension_def: fixes A :: "(nat, int mpoly) fmap"   and \<V> :: "nat set"   and i :: "nat"   and v :: "nat"   and p :: "int mpoly" shows "check_extension A \<V> i v p = SPEC (\<lambda>b. b \<longrightarrow> i \<notin># dom_m A \<and> v \<notin> \<V> \<and> (p + Var v)\<^sup>2 - (p + Var v) \<in> More_Modules.ideal polynomial_bool \<and> vars (p + Var v) \<subseteq> \<V>)"
preserves_other: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and t'' :: "'t"   and x'' :: "'x" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P t x m"   and "P t'' x'' m" shows "P t'' x'' m'"
A: shows "\<forall>r<c. \<forall>\<^sub>F x in F. f x < ereal r"
set_10: fixes x21 :: "('b, 'a, 'c) expr"   and x22 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Apply x21 x22) = LID\<^sub>E x21 \<union> LID\<^sub>E x22"
generate_consistent: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H (G\<lparr>carrier := generate G {a}\<rparr>)" shows "generate (G\<lparr>carrier := generate G {a}, carrier := H\<rparr>) K = generate (G\<lparr>carrier := generate G {a}\<rparr>) K"
routing_port_ranges_subsets: fixes a1 :: "char list"   and b1 :: "'b wordinterval"   and tbl :: "'b routing_rule list"   and s :: "'b wordinterval" assumes "(a1, b1) \<in> set (routing_port_ranges tbl s)" shows "wordinterval_to_set b1 \<subseteq> wordinterval_to_set s"
set_8: fixes x3 :: "'a" shows "LID\<^sub>V (Loc x3) = {x3}"
cases: fixes a1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and a2 :: "('e \<Rightarrow> real) set set"   and a3 :: "'d"   and a4 :: "('e \<Rightarrow> real) set"   and a5 :: "'d"   and a6 :: "('e \<Rightarrow> real) set" assumes "a1,a2 \<turnstile> \<langle>a3, a4\<rangle> \<leadsto>* \<langle>a5, a6\<rangle>" obtains A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" and \<R> :: "('e \<Rightarrow> real) set set" and l :: "'d" and R :: "('e \<Rightarrow> real) set" where "a1 = Ab__" and "a2 = \<R>'__" and "a3 = lb__" and "a4 = R__" and "a5 = lb__" and "a6 = R__"   | A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" and \<R> :: "('e \<Rightarrow> real) set set" and l :: "'d" and R :: "('e \<Rightarrow> real) set" and l' :: "'d" and R' :: "('e \<Rightarrow> real) set" and l'' :: "'d" and R'' :: "('e \<Rightarrow> real) set" where "a1 = A" and "a2 = \<R>" and "a3 = l" and "a4 = R" and "a5 = l''" and "a6 = R''" and "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>" and "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto> \<langle>l'', R''\<rangle>"
inject: fixes x21 :: "'c"   and x22 :: "'c"   and y21 :: "'c"   and y22 :: "'c" shows "(Point x21 x22 = Point y21 y22) = (x21 = y21 \<and> x22 = y22)"
singleton_Un_iff: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
simps_2: fixes b1 :: "bool"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (Bc b1) (Eq v va) = True"
n_def: shows "n \<equiv> LEAST n. (deriv ^^ n) f \<xi> \<noteq> 0"
closed_sum_closure_right: fixes A :: "'b set"   and B :: "'b set" shows "A +\<^sub>M closure B = A +\<^sub>M B"
monoid_homomorphism_axioms: shows "monoid_homomorphism \<eta> G (\<cdot>) \<one> G' (\<cdot>') \<one>'"
unity_div: fixes d :: "nat"   and k :: "nat"   and a :: "int" assumes "d dvd k" shows "unity_root k (a * int d) = unity_root (k div d) a"
extremum: fixes a :: "'a word" shows "a \<le> - 1"
extensional_arb: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a" assumes "f \<in> extensional A"   and "x \<notin> A" shows "f x = undefined"
zero_and_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and zero a = zero"
vcompD: fixes a :: "V"   and c :: "V"   and r :: "V"   and s :: "V" assumes "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s" shows "\<exists>b. \<langle>b, c\<rangle> \<in>\<^sub>\<circ> r \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> s"
rimg_carrier: fixes f :: "'c \<Rightarrow> 'd"   and A :: "('c, 'e) Ring_scheme"   and R :: "('d, 'f) Ring_scheme" assumes "f \<in> rHom A R" shows "carrier (rimg A R f) = f ` carrier A"
checker3_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "checker3 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o; s_stop_e = s\<^sub>e + rel_dist_to_stop v\<^sub>e a\<^sub>e; s_stop_o = s\<^sub>o + rel_dist_to_stop v\<^sub>o a\<^sub>o in precond \<and> (s_stop_e < s\<^sub>o \<or> s\<^sub>o \<le> s_stop_e \<and> s_stop_e < s_stop_o \<and> \<not> (a\<^sub>e < a\<^sub>o \<and> v\<^sub>o < v\<^sub>e \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o < 0 \<and> distance * (a\<^sub>o - a\<^sub>e) \<le> (v\<^sub>o - v\<^sub>e)\<^sup>2 / 2))"
impl_red_dfs_ahs_def: shows "impl_red_dfs_ahs \<equiv> \<lambda>x xa xb xc. REC\<^sub>T (\<lambda>xd (xe, xf). let y' = ahs.ins xf xe in NDFS_SI_Statistics.vis_red_nres \<bind> (\<lambda>xg. LIST_FOREACH' (RETURN (id (x xf))) is_None (\<lambda>xh xi. if ahs.memb xh xa then RETURN (red_init_witness xf xh) else RETURN None) None \<bind> (\<lambda>xh. case xh of None \<Rightarrow> LIST_FOREACH' (RETURN (id (x xf))) (\<lambda>(xi, y). is_None y) (\<lambda>xi (xj, xk). if \<not> ahs.memb xi xj then xd (xj, xi) \<bind> (\<lambda>(xl, xm). RETURN (xl, prep_wit_red xf xm)) else RETURN (xj, None)) (y', None) | Some xi \<Rightarrow> RETURN (y', xh)))) (xb, xc)"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c1 :: "com"   and x :: "nat"   and t :: "char list \<Rightarrow> int"   and y :: "nat"   and c2 :: "com" assumes "bval b s"   and "(c1, s) \<Rightarrow> x \<Down> t"   and "y = x + 1" shows "(IF b THEN c1 ELSE c2, s) \<Rightarrow> y \<Down> t"
adm_woL_def: fixes L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'd" shows "adm_woL L \<equiv> \<forall>f g i. isLim i \<and> (\<forall>j\<in>local.underS i. f j = g j) \<longrightarrow> L f i = L g i"
NextPropagate'_next': fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q :: "'b" assumes "NextPropagate' c0 c1 p" shows "local.next'\<^sup>+\<^sup>+ (prop_config c0 q) (prop_config c1 q)"
TC_induct_down: fixes a :: "V"   and b :: "V"   and P :: "V \<Rightarrow> bool" assumes "a \<sqsubset> b"   and "\<And>y. y \<in> elts b \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>y \<sqsubset> b; P y; z \<in> elts y\<rbrakk> \<Longrightarrow> P z"   and "small (Collect P)" shows "P a"
comp_converter_def: shows "comp_converter \<equiv> \<lambda>uu uua. corec_converter (\<lambda>(conv1, conv2) a. map_gpv (\<lambda>uu. (id (fst (fst uu)), Inr (snd (fst uu), snd uu))) id (inline run_converter (run_converter conv1 a) conv2)) (uu, uua)"
emBits_roundup_cancel: fixes emBits :: "nat" assumes "emBits mod 8 \<noteq> 0" shows "roundup emBits 8 * 8 - emBits = 8 - emBits mod 8"
comparator_list_simps_1: fixes comp\<^sub>'\<^sub>a :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "comparator_list comp\<^sub>'\<^sub>a [] [] = Eq"
simps_4: fixes a :: "'c"   and b :: "'c" shows "dbm_add (Le a) (Le b) = Le (a + b)"
t_\<theta>_id: shows "t \<cdot> Var(v := t) = t"
lens_create_def: fixes X :: "'c \<Longrightarrow> 'b"   and v :: "'c" shows "create\<^bsub>X\<^esub> v = put\<^bsub>X\<^esub> src\<^bsub>X\<^esub> v"
assms_3: shows "C \<in> cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) (length \<pi>))"
type_substitutivity_aux: fixes \<Delta> :: "(name \<times> ty) list"   and x :: "name"   and T' :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and e :: "trm"   and T :: "ty"   and e' :: "trm" assumes "\<Delta> @ [(x, T')] @ \<Gamma> \<turnstile> e : T"   and "\<Gamma> \<turnstile> e' : T'" shows "\<Delta> @ \<Gamma> \<turnstile> e[x::=e'] : T"
JFcol_bd: fixes n :: "nat" shows "\<forall>j1 j2. |JF1col n j1| \<le>o bd_F \<and> |JF2col n j2| \<le>o bd_F"
add_assoc: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3" shows "local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3) = local.add a (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
\<R>_def: shows "\<R> \<equiv> {Regions.region \<X> I r |I r. Regions.valid_region \<X> k I r}"
simps_2: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
B_def: shows "B = f 0 + 1 / c \<cdot>\<^sub>m M"
intros_34: fixes n :: "i32"   and i :: "nat" assumes "int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $Br_if i]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
repeat_add: fixes n :: "nat"   and m :: "nat"   and mx :: "'m" shows "repeat (n + m) mx = bind (repeat n mx) (\<lambda>_. repeat m mx)"
cqt_further_3: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
inj_on_map_of': fixes xs :: "('c \<times> 'b) list" assumes "distinct (map snd xs)" shows "inj_on (the \<circ> map_of xs) (set (map fst xs))"
mod_dist_3: fixes P :: "'a \<Rightarrow> assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> (\<exists>\<^sub>Ax. P x)) = (\<exists>x. h \<Turnstile> P x)"
intros_20: fixes t1 :: "t"   and v :: "v"   and t2 :: "t"   and sx :: "sx option" assumes "types_agree t1 v"   and "cvt t2 sx v = None" shows "\<lparr>[$C v, $Cvtop t2 Convert t1 sx]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
eventually_at: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
simps_3: fixes n :: "'b"   and v :: "'b"   and va :: "'b tree"   and vb :: "'b tree"   and vc :: "nat"   and vd :: "'b"   and ve :: "'b tree"   and vf :: "'b tree"   and vg :: "nat"   and h :: "nat" shows "delete_root (MKT n (MKT v va vb vc) (MKT vd ve vf vg) h) = (let (new_n, l') = delete_max (MKT v va vb vc) in mkt_bal_r new_n l' (MKT vd ve vf vg))"
incl: fixes A :: "'e set"   and B :: "'e set set"   and b :: "'e set" assumes "Congruence.partition A B"   and "b \<in> B" shows "b \<subseteq> A"
simps: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "Bochner_Integration.simple_bochner_integrable M f = (simple_function M f \<and> emeasure M {y \<in> space M. f y \<noteq> (0::'c)} \<noteq> \<infinity>)"
aux: fixes j' :: "nat" assumes "i \<le> j"   and "Suc j \<le> j'" shows "i \<le> j'"
converse_inject: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(r\<inverse> = s\<inverse>) = (r = s)"
distinct_delete: fixes al :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (list.map fst al)" shows "distinct (list.map fst (AList.delete k al))"
iteration_induct: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<infinity> \<le> x\<^sup>\<infinity> \<cdot> x"
ustar_inductl: fixes R :: "('b, 'c) urel"   and Q :: "('b, 'c) urel"   and P :: "'b hrel" assumes "Q \<sqsubseteq> R"   and "Q \<sqsubseteq> P ;; Q" shows "Q \<sqsubseteq> P\<^sup>\<star> ;; R"
LLL_invD_2: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "length (map of_int_hom.vec_hom fs) = m"
distinct_the_index_is_index: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "the_index xs (xs ! n) = n"
iTILL_add_neg: fixes n :: "nat"   and k :: "nat" shows "[\<dots>n] \<oplus>- k = (if k \<le> n then [\<dots>n - k] else {})"
fps_sin_deriv: fixes c :: "'a" shows "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
distinct_remove1: fixes xs :: "'g list"   and x :: "'g" assumes "distinct xs" shows "distinct (remove1 x xs)"
hom_one: shows "\<phi> \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
leadsTo_weaken_R: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A \<longmapsto> A'"   and "A' \<subseteq> B'" shows "F \<in> A \<longmapsto> B'"
poly_lift_hom_monom: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and r :: "'a"   and n :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "r \<in> carrier R" shows "poly_lift_hom R S \<phi> (monom (UP R) r n) = monom (UP S) (\<phi> r) n"
h1_is_pr: shows "(\<lambda>x y z. h z y x) \<in> PrimRec3"
l_to_tree_order_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and to_tree_order_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_to_tree_order_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptrs get_parent to_tree_order_si \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptrs get_parent \<and> l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order_si get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs type_wf known_ptr"