hcju/isabelleps huggingface dataset:
lookup_powsum_mpoly_aux: fixes X :: "nat set"   and k :: "nat"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "lookup (powsum_mpoly_aux X k) mon = (if infinite X \<or> k = 0 \<and> mon \<noteq> 0 then 0::'b else if k = 0 \<and> mon = 0 then of_nat (card X) else if finite X \<and> (\<exists>x\<in>X. mon = Poly_Mapping.single x k) then 1::'b else (0::'b))"
zmult_pos_bignumTr0: fixes z :: "int"   and x :: "int" shows "\<exists>L. \<forall>m>L. z < x + int m"
current_restrict_current: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "wave \<Gamma> f"   and "current \<Gamma> g" shows "current (quotient_web \<Gamma> f) (g \<upharpoonleft> \<Gamma> / f)"
simps_2: fixes uw :: "linear_poly"   and ux :: "rat" shows "is_le (LEQ uw ux) = True"
naturality: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "\<alpha>' (CCC.cod f) \<cdot> T.ToCT f = T.ToTC f \<cdot> \<alpha>' (CCC.dom f)"
simps_20: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x41 :: "nat set"   and x42 :: "nat set"   and x43 :: "nat set"   and x44 :: "nat set" shows "(case Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f4 x41 x42 x43 x44"
ceiling_less_zero: fixes x :: "'b" shows "(\<lceil>x\<rceil> < 0) = (x \<le> - (1::'b))"
order_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a = a \<sqinter> b)"
plus_bit_def: fixes a :: "bit"   and b :: "bit" shows "a + b = Abs_bit (Rep_bit a \<noteq> Rep_bit b)"
dependent_single: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "vector_space scale" shows "module.dependent scale {x} = (x = (0::'b))"
Succ_Shift: fixes Kl :: "'b list set"   and k :: "'b"   and kl :: "'b list" shows "BNF_Greatest_Fixpoint.Succ (BNF_Greatest_Fixpoint.Shift Kl k) kl = BNF_Greatest_Fixpoint.Succ Kl (k # kl)"
valuation_for_state_variables_def: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<tau> :: "('b \<Rightarrow> bool option) list" shows "valuation_for_state_variables \<Pi> \<pi> \<tau> \<equiv> let t' = length \<tau>; \<tau>\<^sub>\<Omega> = \<tau> ! (t' - 1); vs = \<Pi>\<^sub>\<V>; V\<^sub>1 = {State k (index vs v) |k v. k \<in> {0..<t'} \<and> v \<in> set vs}; V\<^sub>2 = {State k (index vs v) |k v. k \<in> {t'..length \<pi> + 1} \<and> v \<in> set vs}; \<A>\<^sub>1 = foldr (\<lambda>(k, v). valuation_for_state (\<Pi>\<^sub>\<V>) (\<tau> ! k) k v) (List.product [0..<t'] vs) \<A>\<^sub>0; \<A>\<^sub>2 = foldr (\<lambda>(k, v). valuation_for_state (\<Pi>\<^sub>\<V>) \<tau>\<^sub>\<Omega> k v) (List.product [t'..<length \<pi> + 2] vs) \<A>\<^sub>0 in override_on (override_on \<A>\<^sub>0 \<A>\<^sub>1 V\<^sub>1) \<A>\<^sub>2 V\<^sub>2"
term_ok_betapply: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and u :: "Core.term"   and uty :: "typ"   and tty :: "typ" assumes "term_ok \<Theta> f"   and "term_ok \<Theta> u"   and "typ_of f = Some (uty \<rightarrow> tty)"   and "typ_of u = Some uty" shows "term_ok \<Theta> (f \<bullet> u)"
NI_indirect_sources_def: shows "NI_indirect_sources \<equiv> \<forall>execs a n. realistic_executions execs \<longrightarrow> (let s_f = run n s0 execs in output_f (run n s0 (int.ipurge_l execs (current s_f))) a = output_f (run n s0 (int.ipurge_r execs (current s_f))) a)"
compat_def: fixes I :: "('a \<times> 'b) list"   and I' :: "('a \<times> 'b) list" shows "PlaneGraphIso.compat I I' \<equiv> \<forall>(x, y)\<in>set I. \<forall>(x', y')\<in>set I'. (x = x') = (y = y')"
keys_delete: fixes k :: "'c"   and m :: "('c, 'd) mapping" shows "Mapping.keys (Mapping.delete k m) = Mapping.keys m - {k}"
domNMT: fixes X :: "'a \<Rightarrow> 'b option" assumes "dom X \<noteq> {}" shows "X \<noteq> \<emptyset>"
hoare_Sup: fixes P :: "'a set"   and S :: "'b \<Rightarrow> 'a"   and q :: "'b" shows "(\<forall>p\<in>P. \<Turnstile> p{| S |}q) = \<Turnstile> \<Squnion>P{| S |}q"
L: shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) (return_pmf ([x, y], h)) \<sigma>') (last \<sigma>') [x, y] = TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) \<sigma>') (last \<sigma>') [x, y]"
eq_qe_backwards: fixes eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list" assumes "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
Ker_equality: shows "induced.Ker = {a |a. a \<in> contained.Partition \<and> induced a = \<one>'}"
tuple_since'_keys: fixes as :: "'a option list" assumes "as \<in> Mapping.keys tuple_since'" shows "as \<in> Mapping.keys tuple_since"
assms_2: shows "n < \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n'"
span_Un: fixes S :: "'d set"   and T :: "'d set" shows "cspan (S \<union> T) = {x + y |x y. x \<in> cspan S \<and> y \<in> cspan T}"
distinct_1: fixes x1 :: "('e, 'f, 'g) Tree set['e]"   and x2 :: "('e, 'f, 'g) Tree" shows "tConj x1 \<noteq> tNot x2"
subset_pairs_def: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" shows "(A \<subseteq>\<^sub>p B) = (\<forall>x. x \<in>\<^sub>p A \<longrightarrow> x \<in>\<^sub>p B)"
small_Times: fixes A :: "'c set"   and B :: "'d set" assumes "small A"   and "small B" shows "small (A \<times> B)"
OrdInd_lemma: fixes j :: "name"   and i :: "name"   and A :: "fm" assumes "atom j \<sharp> (i, A)" shows "{OrdP (Var i)} \<turnstile> SyntaxN.All i (OrdP (Var i) IMP All2 j (Var i) (A(i::=Var j)) IMP A) IMP A"
univ_poly_units_incl: fixes K :: "'a set" assumes "subring K R" shows "Units (K [X]) \<subseteq> {[k] |k. k \<in> K - {\<zero>}}"
D: shows "\<forall>cfs. P' x \<Turnstile> cfs\<box> \<longrightarrow> Q x (cfs ! 0) (cfs ! (length cfs - 1))"
hyps_3: shows "n'' -as''\<rightarrow>\<^sub>d* n'"
list_all2_length: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
lfp_to_gfp_var: fixes f :: "'a \<Rightarrow> 'a" shows "lfp f = \<partial> (gfp (\<partial>\<^sub>F f))"
hm_size_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable" shows "<is_hashmap m ht> hm_size ht <\<lambda>r. is_hashmap m ht * \<up> (r = card (dom m))>"
AX_def: fixes f :: "'a set" shows "AX f \<equiv> {s. {f0. s \<rightarrow>\<^sub>i f0} \<subseteq> f}"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus_iii: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "is_serial_solution_for_problem \<Psi> \<psi>"   and "op \<in> set \<psi>" shows "are_operator_effects_consistent op op"
combining_sys_satisfies_properties_helper: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and subsets1 :: "nat list list"   and signs1 :: "rat list list"   and matrix1 :: "rat mat"   and subsets2 :: "nat list list"   and signs2 :: "rat list list"   and matrix2 :: "rat mat" assumes "p \<noteq> 0"   and "0 < length qs1"   and "\<forall>q. List.member qs1 q \<longrightarrow> coprime p q"   and "0 < length qs2"   and "\<forall>q. List.member qs2 q \<longrightarrow> coprime p q"   and "satisfies_properties p qs1 subsets1 signs1 matrix1"   and "satisfies_properties p qs2 subsets2 signs2 matrix2" shows "satisfies_properties p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_matrix (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
preserves_ide: fixes a :: "'a" assumes "S.ide a" shows "S.ide (L a)"
hyp1_unfolded: shows "matrix_to_iarray (snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])) = snd (foldl Gauss_Jordan_column_k_iarrays (0, matrix_to_iarray A) [0..<Suc k])"
interpret_floatarith_fresh_cong: fixes f :: "floatarith"   and xs :: "real list"   and ys :: "real list" assumes "\<And>i. \<not> fresh_floatarith f i \<Longrightarrow> xs ! i = ys ! i" shows "interpret_floatarith f ys = interpret_floatarith f xs"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "USW1 x1 \<noteq> USW2 x2"
B: fixes x :: "nat \<Rightarrow> val"   and y :: "val" assumes "x \<in> space (state_measure V \<Gamma>)"   and "y \<in> space (stock_measure t)" shows "case_nat y (merge V V' (x, \<rho>)) = merge (shift_var_set V) (Suc ` V') (case_nat y x, case_nat undefined \<rho>)"
open_Collect_less: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "open {x. f x < g x}"
DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" shows "\<Gamma>\<turnstile> (DynCom c, Normal s) \<rightarrow> (c s, Normal s)"
phi_def: fixes g :: "'g"   and v :: "'a" shows "other.phi g v = phis' g (other.defNode g v, v)"
PiE_restrict: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
pfinite_converter_3: shows "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"
is_closed_eq: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "is_closed A"   and "x \<in> A"   and "x .= x'"   and "x' \<in> carrier S" shows "x' \<in> A"
add_is_0: fixes m :: "nat"   and n :: "nat" shows "(m + n = 0) = (m = 0 \<and> n = 0)"
max_degree_adjugate: fixes A :: "('b, 'c) sq_matrix" shows "max_degree (adjugate (\<^bold>X - \<^bold>C A)) = CARD('c) - 1"
simps_4: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and \<rho> :: "complex mat" shows "denote (Measure n M S) \<rho> = denote_measure n M (map denote S) \<rho>"
states_def: fixes P :: "'a prog"   and mxs :: "nat"   and mxl :: "nat" shows "states P mxs mxl \<equiv> fst (JVM_SemiType.sl P mxs mxl)"
success_tapI: fixes f :: "heap \<Rightarrow> 'a"   and h :: "heap" shows "success (tap f) h"
ideal_condition: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "I \<subseteq> carrier R"   and "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> y \<in> I"   and "\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I" shows "ideal R I"
tj_stack_reach_stack: fixes u :: "'a" assumes "u \<in> set (tj_stack s)" shows "\<exists>v\<in>set (stack s). (u, v) \<in> E\<^sup>* \<and> \<delta> s v \<le> \<delta> s u"
eval_poly_def: fixes h :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "eval_poly h p = poly (map_poly h p)"
rtrancl_reflcl: fixes R :: "('a \<times> 'a) set" shows "(R\<^sup>=)\<^sup>* = R\<^sup>*"
s_prod_isor: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" assumes "x \<subseteq> y" shows "x \<cdot> z \<subseteq> y \<cdot> z"
lunit_eqI: fixes f :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "ide f"   and "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>"   and "trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<bullet> \<a>\<^sup>-\<^sup>1[trg f, trg f, f]" shows "\<mu> = \<l>[f]"
prj_simps_4: shows "C.dom \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1 = HH\<mu>\<nu>H\<pi>\<rho>.chine"
hseqI': fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "C.hseq \<nu> \<mu>"
rep_eq: shows "semilattice_set_apply max_sls = max"
Exists: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula" assumes "wf_mformula \<sigma> j P V (Suc n) (lift_envs R) \<phi> \<phi>'" shows "wf_mformula \<sigma> j P V n R (MExists \<phi>) (formula.Exists \<phi>')"
hyps: fixes ys :: "'a list" assumes "distinct ys" shows "distinct (local.filter_min_aux xs ys)"
lower_eq_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "(punit.lower p v = punit.lower q v) = (\<forall>u. u \<prec> v \<longrightarrow> lookup p u = lookup q u)"
unfold_while: fixes P :: "(char list list \<times> char list exp) prog"   and b :: "char list exp"   and c :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" shows "P \<turnstile> \<langle>while (b) c,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = P \<turnstile> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
Max_less_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Max less_eq A) x = (\<forall>a\<in>A. less a x)"
drop_all_iff: fixes n :: "nat"   and xs :: "'d list" shows "(drop n xs = []) = (length xs \<le> n)"
linear_def: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" shows "Vector_Spaces.linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"
a: shows " Lam [a].t \<longrightarrow>\<^sub>\<beta> t'"
unit_simps_4: fixes a :: "'a" assumes "B.obj a" shows "B.dom \<i>\<^sub>B[a] = a \<star>\<^sub>B a"
n_even: fixes n :: "nat" assumes "even n" shows "2 * (n div 2) = n"
lookup_HSem_other: fixes y :: "'d"   and h :: "('d \<times> 'e) list"   and ESem :: "'e \<Rightarrow> ('d \<Rightarrow> 'f) \<rightarrow> 'f"   and \<rho> :: "'d \<Rightarrow> 'f" assumes "y \<notin> domA h" shows "(has_ESem.HSem ESem h\<cdot>\<rho>) y = \<rho> y"
ValidD: fixes P :: "'b set"   and c :: "'b com"   and Q :: "'b set"   and s :: "'b"   and s' :: "'b" assumes "\<turnstile> P c Q"   and "Sem c s s'"   and "s \<in> P" shows "s' \<in> Q"
simps_3: fixes E :: "nat \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm" shows "\<lbrakk>E, F, G\<rbrakk> (p \<^bold>\<longrightarrow> q) = (\<lbrakk>E, F, G\<rbrakk> p \<longrightarrow> \<lbrakk>E, F, G\<rbrakk> q)"
prems_1: shows "mbufn_take f z buf = (z', buf')"
balR_def: fixes A :: "('b \<times> nat) tree"   and a :: "'b"   and BC :: "('b \<times> nat) tree" shows "balR A a BC = (if ht BC = ht A + 2 then case BC of \<langle>B, (c, x), C\<rangle> \<Rightarrow> if ht B \<le> ht C then node (node A a B) c C else case B of \<langle>B\<^sub>1, (b, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) b (node B\<^sub>2 c C) else node A a BC)"
less_restrictive_saturated: fixes R1 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and R2 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" assumes "less_restrictive R1 R2"   and "saturated_binary_rule R1 S" shows "saturated_binary_rule R2 S"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "mset_set A = add_mset x (mset_set (A - {x}))"
to_list: shows "SIDE_GEN_ALGO (is_set_to_list A Rs1 tol)"
retraction_strict: fixes f :: "'d \<rightarrow> 'c"   and g :: "'c \<rightarrow> 'd" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "f\<cdot>\<bottom> = \<bottom>"
simps: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set" shows "ASC_Example.well_formed_rel rel = (finite rel \<and> (\<forall>s1 x y. x \<notin> (fst \<circ> fst \<circ> snd) ` rel \<or> y \<notin> (snd \<circ> fst \<circ> snd) ` rel \<longrightarrow> (\<nexists>s2. (s1, (x, y), s2) \<in> rel)) \<and> rel \<noteq> {})"
adds_termE: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "term_powerprod.adds_term pair_of_term u v" obtains t :: "'b" where "v = term_powerprod.splus pair_of_term term_of_pair t u"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and net :: "real filter" shows "(f has_vector_derivative f') net = D f \<mapsto> (\<lambda>x. x *\<^sub>R f') net"
simps_1: fixes w :: "nat"   and a :: "'a" shows "height (Leaf w a) = 0"
True_True_steps_concI: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> NAe.steps L w" shows "(True # p, True # q) \<in> NAe.steps (RegExp2NAe.conc L R) w"
wfp_on_restrict_to_tranclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "wfp_on (restrict_to P A)\<^sup>+\<^sup>+ A"
elims: fixes x :: "'a"   and xa :: "'a tree \<times> nat"   and y :: "'a tree \<times> nat" assumes "nxt x xa = y" obtains x :: "'a" and tn :: "'a tree \<times> nat" where "xa__ = x" and "xa = tn" and "y = insert3 x tn"
reflect_poly_dvdI: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "reflect_poly p dvd reflect_poly q"
p_2: shows "defNode g p \<notin> set (tl ns)"
simps_2: fixes v :: "'v"   and v1 :: "'v"   and w :: "'w"   and v2 :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_path v ((v1, w, v2) # p) v' = (v = v1 \<and> (v1, w, v2) \<in> E \<and> is_path v2 p v')"
valuate_depend: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'b"   and v' :: "nat \<Rightarrow> 'b" assumes "\<forall>x\<in>vars p. v x = v' x" shows "p \<lbrace> v \<rbrace> = p \<lbrace> v' \<rbrace>"
exhaust: fixes y :: "rule" obtains "y = Idle"   | x2 :: "nat" where "y = Ax x2"   | x3 :: "fmla" where "y = NegL x3"   | x4 :: "fmla" where "y = NegR x4"   | x51 :: "fmla" and x52 :: "fmla" where "y = ConjL x51 x52"   | x61 :: "fmla" and x62 :: "fmla" where "y = ConjR x61 x62"
radical_sqrt_line_line_intersection: fixes A :: "point"   and B :: "point"   and C :: "point"   and D :: "point"   and X :: "point" assumes "abscissa A \<in> radical_sqrt"   and "ordinate A \<in> radical_sqrt"   and "abscissa B \<in> radical_sqrt"   and "ordinate B \<in> radical_sqrt"   and "abscissa C \<in> radical_sqrt"   and "ordinate C \<in> radical_sqrt"   and "abscissa D \<in> radical_sqrt"   and "ordinate D \<in> radical_sqrt"   and "\<not> parallel A B C D"   and "is_intersection X A B C D" shows "abscissa X \<in> radical_sqrt \<and> ordinate X \<in> radical_sqrt"
midpoints_in_convex_hull: fixes x :: "'a"   and s :: "'a set"   and y :: "'a" assumes "x \<in> convex hull s"   and "y \<in> convex hull s" shows "midpoint x y \<in> convex hull s"
lfp_omega_greatest: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b \<Rightarrow> 'c" assumes "\<And>u. Omega_fun f g u \<le> u \<Longrightarrow> A \<le> u" shows "A \<le> omega_fun f \<circ> g"
invariantE: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "invariant rel I"   and "rel x y"   and "I x" shows "I y"
B: shows "\<forall>\<^sub>F n in sequentially. Sigma_Algebra.measure (distr M borel (\<lambda>x. birkhoff_sum f n x / B n)) {x. C0 \<le> \<bar>x\<bar>} < alpha"
Sup_insert: fixes a :: "'b"   and A :: "'b set" shows "\<Sqinter> insert a A = a \<sqinter> \<Sqinter> A"
oinvariant_def: fixes A :: "('c \<times> 'd, 'e) automaton"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<times> 'd \<Rightarrow> bool" shows "(A \<Turnstile> (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. P s)"
rep_eq: fixes x :: "real_alg" shows "\<lfloor>x\<rfloor> = floor_3 (rep_real_alg x)"
cobounded2: fixes b :: "'d"   and a :: "'d" shows "b \<le> max a b"
m_lcomm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> (y \<otimes>\<^bsub>P\<^esub> z) = y \<otimes>\<^bsub>P\<^esub> (x \<otimes>\<^bsub>P\<^esub> z)"
sum_comp_morphism: fixes h :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "h (0::'d) = (0::'c)"   and "\<And>x y. h (x + y) = h x + h y" shows "sum (h \<circ> g) A = h (sum g A)"
eq: shows "of_int x = of_int (x div d) * of_int d + of_int (x mod d)"
permute_list_map: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'a list"   and g :: "'a \<Rightarrow> 'b" assumes "f permutes {..<length xs}" shows "permute_list f (map g xs) = map g (permute_list f xs)"
top_edge_explicit: shows "(\<lambda>x. twoC (x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
comp_red_basis_aux_pmdl: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_minimal_basis (set (xs @ ys))"   and "distinct (xs @ ys)" shows "pmdl (set (comp_red_basis_aux xs ys)) \<subseteq> pmdl (set (xs @ ys))"
Lcm_nat_insert: fixes n :: "nat"   and M :: "nat set" shows "Lcm (insert n M) = lcm n (Lcm M)"
field_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "field_axioms R \<equiv> Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}"
HMA_T_def: fixes n :: "nat"   and uu :: "'b itself" shows "HMA_T n uu = (n = CARD('b))"
negate_def: fixes P :: "'b \<Rightarrow> bool" shows "\<N> P = (\<lambda>s. \<not> P s)"
continuous_imp_measurable_on: fixes f :: "'a \<Rightarrow> 'b" assumes "continuous_on UNIV f" shows "f measurable_on UNIV"