hcju/isabelleps huggingface dataset:
inj_on_image_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "(f ` A = f ` B) = (A = B)"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "enn2ereal (x * xa) = enn2ereal x * enn2ereal xa"
safeE: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> 'd) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "'c"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "'d" assumes "safe_free_flowing (ts, m, \<S>)"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, x, \<D>, \<O>, \<R>)" shows "map owned ts,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
assms_1: shows "\<forall>i<m. \<forall>j<n. P i \<and> Q j"
Quotient_uint16: shows "Quotient (=) Abs_uint16 Rep_uint16 cr_uint16"
iexE: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "\<diamond> t I. P t" obtains t :: "nat" where "t \<in> I" and "P t"
lfp_def: fixes f :: "'b \<Rightarrow> 'b" shows "lfp f = \<Sqinter> {u. f u \<le> u}"
False: shows "\<not> (\<alpha> = 1 \<or> \<beta> = 0)"
triangle_right': shows "\<r>[f\<^sup>*] \<cdot> (f\<^sup>* \<star> \<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<eta> \<star> f\<^sup>*) \<cdot> \<l>\<^sup>-\<^sup>1[f\<^sup>*] = f\<^sup>*"
Min_Un: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "linorder.Min less_eq (A \<union> B) = ord.min less_eq (linorder.Min less_eq A) (linorder.Min less_eq B)"
total: shows "\<forall>x\<in>T. \<forall>y\<in>T. x \<noteq> y \<longrightarrow> x \<prec> y \<or> y \<prec> x"
Dup_head: fixes p :: "('b, 'c) fm"   and i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and q :: "('b, 'c) fm" assumes "Dup p i ((ps, a) # branch) xs" shows "Dup p i ((q # ps, a) # branch) xs"
deg_of_strict_monoI: fixes x :: "real"   and y :: "real" assumes "x < y" shows "deg_of x < deg_of y"
atoms_nTimes: fixes r :: "'b rexp"   and s :: "'b rexp" shows "atoms (nTimes r s) \<subseteq> atoms r \<union> atoms s"
Acts_JN: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "Acts (JOIN I F) = insert Id (\<Union>i\<in>I. Acts (F i))"
distinct_7: fixes x1 :: "nat"   and x51 :: "expr"   and x52 :: "expr" shows "Var x1 \<noteq> <x51 ,  x52>"
nonEmpty_neq: fixes \<Gamma> :: "'a multiset"   and C :: "'a multiset" assumes "\<Gamma> \<noteq> \<Empt>" shows "\<Gamma> + C \<noteq> C"
less_eq_Quantity_ext_def: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "(x \<le> y) = (mag x \<le> mag y \<and> Quantity.dim x = Quantity.dim y \<and> Quantity.more x \<le> Quantity.more y)"
eq_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a = b) = (a \<subseteq># b \<and> b \<subseteq># a)"
open_greaterThanLessThan: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThanLessThan less a b)"
eval_fmla_cnj: fixes e :: "(name, nat) finfun"   and A :: "fmla"   and B :: "fmla" shows "eval_fmla e (A AND B) = (eval_fmla e A \<and> eval_fmla e B)"
Nats_mult: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_1 one times plus zero"   and "a \<in> semiring_1.Nats one plus zero"   and "b \<in> semiring_1.Nats one plus zero" shows "times a b \<in> semiring_1.Nats one plus zero"
r_s_mod_s: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "(r * s - Suc 0) mod s = s - Suc 0"
boundedE: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<preceq>\<^sub>t ord_term_lin.min b c" obtains "a \<preceq>\<^sub>t b" and "a \<preceq>\<^sub>t c"
map_simp: fixes f :: "'a" assumes "S.arr f" shows "S.map f = f"
hyps: fixes w :: "bool stream" assumes "n \<le> matur" shows "rn_rev_price N der matur n w = expl_cond_expect N (proj_stoch_proc geom_proc (matur - n)) (discounted_value r (\<lambda>m. der) n) w"
new_unalive_old_Store: fixes s :: "Store"   and t :: "New" shows "\<not> alive (new s t) s"
ordered_resolution_is_complete_model_resolution: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'a set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.Complete (propositional_atoms.ordered_model_resolvent atom_ordering Sel)"
assms_3: shows "\<forall>x\<in>s. \<forall>y\<in>s. f x \<le> f y \<or> f y \<le> f x"
c_eval: shows "c_sinvar m = (\<lambda>G. sinvar G nP)"
sgn_real_alg: fixes x :: "real_alg" shows "sgn (real_of x) = real_of (sgn x)"
CconcNone: fixes a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "dom (C a) = {}"   and "p \<noteq> []" shows "C (list2FWpolicy (a # p)) x = C (list2FWpolicy p) x"
open_dist: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.metric_space dist uniformity open" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
ThinishChamberComplex_def: fixes X :: "'b set set" shows "ThinishChamberComplex X \<equiv> ChamberComplex X \<and> ThinishChamberComplex_axioms X"
red_wf_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> punit.dgrad_p_set d m" shows "wfP (punit.red F)\<inverse>\<inverse>"
1: shows "\<exists>x<Suc n. \<phi> i x \<noteq> f x"
lprefix_refl: fixes xs :: "'c llist" shows "lprefix xs xs"
echelon_form_implies_echelon_form_upt: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "echelon_form A" shows "echelon_form_upt_k A k"
add_left_imp_eq: fixes a :: "'k"   and b :: "'k"   and c :: "'k" assumes "a + b = a + c" shows "b = c"
convergent_LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'd" shows "convergent X = X \<longlonglongrightarrow> lim X"
rep_eq: fixes x :: "uint32"   and xa :: "uint32" shows "Rep_uint32 (x * xa) = Rep_uint32 x * Rep_uint32 xa"
distinct_33: fixes x31 :: "POPLmarkRecord.type"   and x32 :: "trm"   and x71 :: "trm"   and x72 :: "char list" shows "(\<lambda><:x31. x32) \<noteq> x71..x72"
inject_2: fixes x41 :: "'a fm"   and x42 :: "'a fm"   and y41 :: "'a fm"   and y42 :: "'a fm" shows "(And x41 x42 = And y41 y42) = (x41 = y41 \<and> x42 = y42)"
X: fixes i :: "'b" assumes "i \<in> I" shows "random_variable (M' i) (X i)"
substitutes_appE': fixes A :: "'a trm"   and B :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and X :: "'a trm" assumes "substitutes (App A B) x M X" obtains "App Aa__ Ba__ = Unit" and "X = Unit"   | "App Aa__ Ba__ = Var xa__" and "X = M"   | x :: "'a" where "App Aa__ Ba__ = Var x" and "X = Var x" and "x \<noteq> xa__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "App Aa__ Ba__ = App Ab__ Bb__" and "X = App A'__ B'__" and "substitutes Ab__ xa__ M A'__" and "substitutes Bb__ xa__ M B'__"   | y :: "'a" and A :: "'a trm" and A' :: "'a trm" and T :: "PreSimplyTyped.type" where "App Aa__ Ba__ = Fn y T Ac__" and "X = Fn y T A'a__" and "xa__ \<noteq> y" and "y \<notin> fvs M" and "substitutes Ac__ xa__ M A'a__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "App Aa__ Ba__ = SimplyTyped.Pair A B" and "X = SimplyTyped.Pair A' B'" and "substitutes A xa__ M A'" and "substitutes B xa__ M B'"   | P :: "'a trm" and P' :: "'a trm" where "App Aa__ Ba__ = Fst P__" and "X = Fst P'__" and "substitutes P__ xa__ M P'__"   | P :: "'a trm" and P' :: "'a trm" where "App Aa__ Ba__ = Snd P" and "X = Snd P'" and "substitutes P xa__ M P'"
iso_digraphI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
Ln_series': fixes z :: "complex" assumes "cmod z < 1" shows "(\<lambda>n. - ((- z) ^ n) / of_nat n) sums Ln (1 + z)"
x_to_bpA: shows "from x to bpA = (- \<beta> * kab) ** dirB"
Lifted_def: shows "Lifted \<equiv> \<Lambda> a. lifted_abs\<cdot>(up\<cdot>a)"
expand: fixes operation :: "'a operation"   and operation' :: "'a operation" assumes "is_INS operation = is_INS operation'"   and "\<lbrakk>is_INS operation; is_INS operation'\<rbrakk> \<Longrightarrow> pos operation = pos operation' \<and> op_idx operation = op_idx operation' \<and> op_int operation = op_int operation'"   and "\<lbrakk>\<not> is_INS operation; \<not> is_INS operation'\<rbrakk> \<Longrightarrow> pos operation = pos operation' \<and> op_idx operation = op_idx operation' \<and> op_int operation = op_int operation'" shows "operation = operation'"
length_Cons: fixes x :: "'m"   and xs :: "'m list" shows "length (x # xs) = Suc (length xs)"
IH_1: fixes d :: "nat"   and t' :: "'a tree" assumes "ins3 n d x l = Bal2 t'"   and "bal_i n (height l + d)" shows "real (T_ins3 n d x l) + \<Phi> t' - \<Phi> l \<le> (6 * e + 2) * real (height l + 1)"
prime_factorization_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize one = {#}"
terminates_takeWhile_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and P :: "'f \<Rightarrow> bool" assumes "terminates g" shows "terminates (takeWhile_raw P g)"
that: fixes K :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite K"   and "K \<subseteq> T"   and "disjnt K S"   and "continuous_on (T - K) g"   and "g ` (T - K) \<subseteq> rel_frontier U"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
trans: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s1 :: "'e"   and s2 :: "'e"   and s3 :: "'e" assumes "pre_RDR.less_eq \<delta> s1 s2"   and "pre_RDR.less_eq \<delta> s2 s3" shows "pre_RDR.less_eq \<delta> s1 s3"
Iff_Ann2: fixes p :: "'a pfm"   and p' :: "'a pfm"   and r :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')" shows "\<turnstile>\<^sub>! ([r]\<^sub>! p \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p')"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (le b a \<and> le a b)"
lenlex_def: fixes r :: "('b \<times> 'b) set" shows "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
triangular_to_jnf_steps_dims_3: fixes A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_2 A) = dim_row A"
sub_mult_one_circ: fixes x :: "'a" shows "x * (1::'a)\<^sup>\<circ> \<le> (1::'a)\<^sup>\<circ> * x"
compactI: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set" assumes "class.topological_space open"   and "\<And>C. \<lbrakk>\<forall>t\<in>C. open t; s \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'" shows "topological_space.compact open s"
AssumeH_2: fixes A1 :: "fmla"   and A2 :: "fmla"   and H :: "fmla set" shows "insert A1 (insert A2 H) \<turnstile> A2"
s1_1_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "s1_1 x1 y1 = s1_1 x2 y2" shows "x1 = x2 \<and> y1 = y2"
v_not_in_vars_extract_var: fixes v :: "nat"   and p :: "'a mpoly" shows "v \<notin> vars (extract_var p v)"
dom_dom: fixes f :: "'a" assumes "A.arr f" shows "A.dom (A.dom f) = A.dom f"
sfis_char: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and A :: "'b set" assumes "measure_space M"   and "A \<in> measurable_sets M" shows "Measure.measure M A \<in> sfis \<chi> A M"
assms_2: shows "Q \<longmapsto> a<\<nu>x> \<prec> Q'"
round_robin_\<alpha>_def: shows "round_robin_\<alpha> = apfst queue_\<alpha>"
fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_is_fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s: fixes F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = set (fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F)"
snd_echelon_form_of_column_k_det_eq: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and n :: "'d"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and k :: "nat" shows "snd (echelon_form_of_column_k_det bezout (n, A, i) k) = echelon_form_of_column_k bezout (A, i) k"
map_of_defined_if_constructed_from_list_of_constant_assignments: fixes l :: "('a \<times> 'b) list"   and a :: "'b"   and xs :: "'a list" assumes "l = map (\<lambda>x. (x, a)) xs" shows "\<forall>x\<in>set xs. map_of l x = Some a"
chambersubcomplex_image: shows "codomain.ChamberSubcomplex (f \<turnstile> X)"
blinding_hash_eq: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "blinding_respects_hashes h bo"   and "bo x y" shows "h x = h y"
hoaret_complete': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
sscan_scons_snth: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and i :: "nat" shows "(a ## sscan f xs a) !! i = fold f (stake i xs) a"
xor_compl_right: fixes x :: "'a"   and y :: "'a" shows "x XOR NOT y = NOT (x XOR y)"
length_rewrite: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" shows "length (rewrite xs i x) = length xs"
mem_cball: fixes y :: "'c"   and x :: "'c"   and e :: "real" shows "(y \<in> cball x e) = (dist x y \<le> e)"
commute: fixes a :: "'c"   and b :: "'c" shows "a XOR b = b XOR a"
SigTy: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" assumes "Signature S"   and "Sig S \<triangleright> Vx : A \<turnstile> e : B" shows "A \<in> Ty\<^bsub>S\<^esub> \<and> B \<in> Ty\<^bsub>S\<^esub>"
simps: fixes x :: "nat"   and y :: "nat" shows "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
NextRecvUpdD_4: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q :: "'b" assumes "NextRecvUpd' c0 c1 p q" shows "\<forall>p'. prop_config c1 p' = (if p' = q then cm_all (prop_config c0 q) (hd (c_msg (exchange_config c0) p q)) else prop_config c0 p')"
assms_1: shows "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q"
funionM_comp_fun_commute: shows "comp_fun_commute funionM"
modulo_mod_ring_def: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "x mod y = (if y = 0 then x else 0)"
at_res1: fixes c :: "cars" shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1 \<longrightarrow> (\<forall>x v'. v = x > v' \<longrightarrow> 0 < \<parallel>ext v'\<parallel> \<and> len v' ts c = ext v' \<and> restrict v' (res ts) c = lan v' \<and> |lan v'| = 1)"
pi_def: shows "pi = (\<lambda>x i. x (to_nat i))"
\<T>_def: fixes state :: "('a, 'b) state" shows "\<T> state = (case state of State x1 x2 x3 x4 x5 x6 \<Rightarrow> x1)"
less_square_imp_div_less: fixes i :: "nat"   and E :: "nat"   and D :: "nat" assumes "i < E * D" shows "i div E < D"
nn_integral_0_1_power: fixes n :: "nat" shows "\<integral>\<^sup>+x\<in>{0<..<1}. ennreal (x ^ n)\<partial>lborel = ennreal (1 / (real n + 1))"
hcomp_simps_1: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.hseq \<nu> \<mu>" shows "Span.src (\<nu> \<circ> \<mu>) = Span.src \<mu>"
Polys_closed_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a \<in> P[X]" shows "sum f A \<in> P[X]"
neutral_const: fixes A :: "'d set" shows "sup_monoid.sum (\<lambda>_. bot) A = bot"
wf_fmla_atm_is_atom: fixes f :: "object atom formula" assumes "wf_fmla_atom objT f" shows "is_predAtom f"
I_tndsto: shows "(I \<longlongrightarrow> 0) (at_right 0)"
simps_1: fixes c :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and c' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "conc.meval_sched_abv c [] c' = (c = c')"
multiset: fixes l :: "'b" shows "multiset l = Heap.multiset (as_tree l)"
underS_supr: fixes b :: "'a"   and A :: "'a set" assumes "b \<in> local.underS (local.supr A)"   and "A \<subseteq> Field r" shows "\<exists>a\<in>A. b \<in> local.underS a"
rel2p_def: fixes R :: "('d \<times> 'e) set"   and x :: "'d"   and y :: "'e" shows "rel2p R x y \<equiv> (x, y) \<in> R"
hyps: fixes c :: "('p, 's, 'm) configuration" assumes "trace c t'' c'"   and "suffix t' t''" shows "\<exists>c''. trace c'' t' c'"
eq: fixes x :: "'a" assumes "x \<in> T" shows "f' x = (g \<circ> f) x"
\<R>\<^sub>1p_\<R>\<^sub>1_eq: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "\<R>\<^sub>1p \<Gamma>' \<S>' P' = (\<lambda>x xa xb xc xd xe. \<langle>x, xa, xb\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>xc, xd, xe\<rangle>)"
add_divide_eq_if_simps_6: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and z :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus (uminus (divide a z)) b = (if z = zero then uminus b else divide (minus (uminus a) (times b z)) z)"
t_wcode_adjust_def: shows "t_wcode_adjust = [(W1, 1), (R, 2), (Nop, 2), (R, 3), (R, 3), (R, 4), (L, 8), (L, 5), (L, 6), (W0, 5), (L, 6), (R, 7), (W1, 2), (Nop, 7), (L, 9), (W0, 8), (L, 9), (L, 10), (L, 11), (L, 10), (R, 0), (L, 11)]"
comm_ring_axioms_def: fixes R :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_ring_axioms R multiplication \<equiv> \<forall>a b. a \<in> R \<longrightarrow> b \<in> R \<longrightarrow> multiplication a b = multiplication b a"
param_and_cong1: fixes a :: "bool"   and a' :: "bool"   and b :: "bool"   and b' :: "bool" assumes "(a, a') \<in> bool_rel"   and "\<lbrakk>a; a'\<rbrakk> \<Longrightarrow> (b, b') \<in> bool_rel" shows "(a \<and> b, a' \<and> b') \<in> bool_rel"
rb_aux_inv_D8: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv (bs, ss, ps)"   and "a \<in> set bs"   and "b \<in> set bs"   and "is_regular_spair a b"   and "Inl (a, b) \<notin> set ps"   and "Inl (b, a) \<notin> set ps"   and "\<not> is_RB_in dgrad rword (set bs) (lt (spair a b))" obtains p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" where "p \<in> set ps" and "sig_of_pair p = lt (spair a b)" and "\<not> sig_crit' bs p"
finsum_closed: fixes G :: "('h, 'i) ring_scheme"   and f :: "'j \<Rightarrow> 'h"   and A :: "'j set" assumes "abelian_monoid G"   and "f \<in> A \<rightarrow> carrier G" shows "finsum G f A \<in> carrier G"
all_aligned_on_semifin_chain: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and X :: "'b set"   and y :: "'b"   and z :: "'b" assumes "MinkowskiChain \<E> \<P> betw"   and "MinkowskiBetweenness.semifin_chain betw f x X"   and "y \<in> X"   and "z \<in> X"   and "x \<noteq> y"   and "x \<noteq> z"   and "y \<noteq> z" shows "betw x y z \<or> betw x z y"
homotopic_nearby_paths: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "open S"   and "path_image g \<subseteq> S" shows "\<exists>e>0. \<forall>h. path h \<and> pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and> (\<forall>t\<in>{0..1}. norm (h t - g t) < e) \<longrightarrow> homotopic_paths S g h"
smap_ctr: fixes f :: "'e \<Rightarrow> 'd"   and s :: "'e stream"   and x :: "'d"   and s' :: "'d stream" shows "(smap f s = x ## s') = (f (shd s) = x \<and> smap f (stl s) = s')"
constant_rules_mono: fixes C\<^sub>1 :: "'b set"   and C\<^sub>2 :: "'b set" assumes "C\<^sub>1 \<subseteq> C\<^sub>2" shows "constant_rules C\<^sub>1 \<subseteq> constant_rules C\<^sub>2"
has_vderiv_on_subset: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and S :: "real set"   and T :: "real set" assumes "(f has_vderiv_on f') S"   and "T \<subseteq> S" shows "(f has_vderiv_on f') T"
wf_cdecl_def: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (D, C) \<notin> (subcls1 P)\<^sup>* \<and> (\<forall>(M, b, Ts, T, m)\<in>set ms. \<forall>D' b' Ts' T' m'. P \<turnstile> D sees M, b' :  Ts'\<rightarrow>T' = m' in D' \<longrightarrow> b = b' \<and> P \<turnstile> Ts' [\<le>] Ts \<and> P \<turnstile> T \<le> T')) \<and> wf_clinit ms"
atoms_list_disjE: fixes a :: "'a"   and fs :: "'a fm list" assumes "a \<in> atoms (list_disj fs)" shows "a \<in> \<Union> (atoms ` set fs)"
remainder_recovery_imp: fixes K :: "'a set"   and A :: "'a set"   and \<psi> :: "'a"   and \<phi> :: "'a"   and B :: "'a set" assumes "K = Cn A"   and "K \<turnstile> \<psi>"   and "\<tturnstile> (\<psi> .\<longrightarrow>. \<phi>)"   and "B \<in> K .\<bottom>. \<phi>" shows "B \<in> K .\<bottom>. \<psi>"
cbelow_on_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and t :: "'a"   and c :: "'c"   and z :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "d t \<le> m"   and "c \<noteq> (0::'c)"   and "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) z (\<lambda>a b. red F a b \<or> red F b a) p q" shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (monom_mult c t z) (\<lambda>a b. red F a b \<or> red F b a) (monom_mult c t p) (monom_mult c t q)"
hcomp_simps_4: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.hseq \<nu> \<mu>" shows "C.cod (\<nu> \<star>\<^sub>C \<mu>) = C.cod \<nu> \<star>\<^sub>C C.cod \<mu>"
assms: shows "\<forall>r. \<not> evaluate_dec False mn env s d r"