hcju/isabelleps huggingface dataset:
intros_1: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
inf_absorb2: fixes y :: "'c multiset"   and x :: "'c multiset" assumes "y \<subseteq># x" shows "x \<inter># y = y"
is_ub_thelub: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a"   and x :: "nat" assumes "class.cpo below"   and "po.chain below S" shows "below (S x) (po.lub below (range S))"
sel_2: fixes x1 :: "'e"   and x2 :: "'d" shows "snd (x1, x2) = x2"
\<tau>Red: shows "s -\<triangleright>ttas\<rightarrow>* s'"
scale_ms_aux: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and c :: "real" assumes "is_expansion_aux xs f basis"   and "basis_wf basis" shows "is_expansion_aux (scale_ms_aux c xs) (\<lambda>x. c * f x) basis"
left_assoc: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "ac_operator f" shows "f a (f b c) = f (f a b) c"
eq: shows "lt ` set (comp_min_basis xs @ []) = lt ` set (comp_red_basis_aux (comp_min_basis xs) [])"
is_type_pTs: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and S :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b" assumes "wf_prog wf_md P"   and "(C, S, fs, ms) \<in> set P"   and "(M, b, Ts, T, m) \<in> set ms" shows "set Ts \<subseteq> types P"
unique_idxD1: fixes bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and f :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and g :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "unique_idx bs data"   and "f \<in> set bs"   and "g \<in> set bs"   and "fst (snd f) = fst (snd g)" shows "f = g"
ennreal_le_of_nat_iff: fixes r :: "real"   and i :: "nat" shows "(ennreal r \<le> of_nat i) = (r \<le> real i)"
Abs_supports_2: fixes x :: "'a"   and as :: "atom set" shows "(supp x - as) supports ([as]res. x)"
ProofStepInduct_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" shows "ProofStepInduct G v0 v1 paths P_new sep_size \<equiv> NoSmallSeparationsInduct G v0 v1 sep_size \<and> DisjointPathsPlusOne G v0 v1 paths P_new \<and> ProofStepInduct_axioms paths sep_size"
po_eq_conv: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below" shows "(x = y) = (below x y \<and> below y x)"
simps_1: shows "X5d X5.Xd1 = D1"
ortho: fixes w :: "'a"   and x :: "'a" assumes "w \<in> T" shows "orthogonal (p2 x) w"
inout_arcs_arc_simps_6: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d" assumes "e \<in> arcs G" shows "in_arcs G u \<inter> {} = {}"
Ninv: fixes x :: "int \<times> int" shows "INum (Ninv x) = (1::'a) / INum x"
wfP_iff_no_inf_chain: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wfP (\<lambda>x y. p y x) = (\<nexists>f. inf_chain p f)"
simps_199: fixes x121 :: "expr"   and x122 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "x121;; x122 \<noteq> Call x101 x102 x103 x104"
case_4: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x4 :: "nat" shows "(case \<not>unicast x4 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f4 x4"
simps_2: fixes f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "num_funs (Fun f ts) = Suc (sum_list (map num_funs ts))"
pmf_bind: fixes N :: "'b pmf"   and f :: "'b \<Rightarrow> 'a pmf"   and i :: "'a" shows "pmf (N \<bind> f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
IH_2: assumes "receive\<langle>t\<rangle> \<in> set S" shows "fv t \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
dlet1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and p :: "pat"   and env' :: "(char list \<times> v) list"   and mn :: "char list list"   and locs :: "locn \<times> locn" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s2) p v1 [] = Match env'" shows "evaluate_dec ck mn env s1 (Dlet locs p e) (s2, Rval (make_sem_env (alist_to_ns env') nsEmpty))"
cases: fixes t :: "nat"   and a1 :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "clstep t a1 a2" obtains c :: "bool list" and n :: "nat" and ts :: "nat \<Rightarrow> thread" and c' :: "bool list" and n' :: "nat" and s' :: "thread" where "a1 = (c, n, ts)" and "a2 = (c', n', ts(t := s'))" and "t < N" and "cstep_sng (c, n, ts t) (c', n', s')"
analytic_imp_holomorphic: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f analytic_on S" shows "f holomorphic_on S"
simps_1: fixes uu :: "nat"   and uv :: "nat"   and I :: "float interval list"   and uw :: "float interval list"   and c :: "float"   and env :: "taylor_model list" shows "approx_tm uu uv I uw (Num c) env = Some (tm_const c)"
numeral_eq_Suc: fixes k :: "Num.num" shows "numeral k = Suc (pred_numeral k)"
set_encode_insert: fixes A :: "nat set"   and n :: "nat" assumes "finite A"   and "n \<notin> A" shows "set_encode (insert n A) = 2 ^ n + set_encode A"
subnetOf_lemma3: fixes a :: "int"   and b :: "int" shows "\<exists>x. x \<in> subnet_of (a, b)"
independent_explicit_module: fixes s :: "'b set" shows "cindependent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *\<^sub>C v) = (0::'b) \<longrightarrow> v \<in> t \<longrightarrow> u v = 0)"
succsr_is_node: fixes x :: "int"   and y :: "int"   and n :: "nat"   and ks :: "int list"   and l :: "int" assumes "(x, y) \<in> (succsr (dioph_succs n ks))\<^sup>*"   and "dioph_is_node ks l x" shows "dioph_is_node ks l y"
safe_get_and: fixes \<phi> :: "formula" assumes "safe_formula \<phi>" shows "list_all safe_neg (get_and_list \<phi>)"
assms_3: shows "V \<subseteq> V\<^bsub>G'\<^esub>"
inext_less_mono_infin: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "infinite I" shows "inext a I < inext b I"
assms: shows "v \<in> type_universe (dist_param_type dst)"
xsys': shows "bv_msb xs = bv_msb ys"
wf_until_aux_Cons3: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and pos :: "bool"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and t :: "nat"   and a1 :: "event_data option list set"   and a2 :: "event_data option list set"   and aux :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and ne :: "nat" assumes "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> ((t, a1, a2) # aux) ne" shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. \<exists>j\<ge>ne. j < Suc (ne + length aux) \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> ne) I \<and> Formula.sat \<sigma> V (map the v) j \<psi> \<and> (\<forall>k\<in>{ne..<j}. if pos then Formula.sat \<sigma> V (map the v) k \<phi> else \<not> Formula.sat \<sigma> V (map the v) k \<phi>)) a2"
isomorphic_unit_right: fixes f :: "'a" assumes "B.ide f" shows "f \<star> src f \<cong>\<^sub>B f"
compute_floor_fl: fixes m :: "int"   and e :: "int" shows "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div 2 ^ nat (- e)) 0)"
indexed_fmember_is_fmember: fixes x :: "'b"   and i :: "nat"   and s :: "'b fset" assumes "x |\<in>|\<^bsub>i\<^esub> s" shows "x |\<in>| s"
pairwise_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
distinct_21: fixes x3 :: "'a"   and x51 :: "'a rexp"   and x52 :: "'a rexp" shows "Atom x3 \<noteq> Times x51 x52"
aligned_at_least_t2n_diff: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "is_aligned x n"   and "is_aligned y n"   and "x < y" shows "x \<le> y - 2 ^ n"
DiffD2: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
dvd_imp_mod_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times a b" shows "modulo b a = zero"
coprime: shows "pairwise coprime ((\<lambda>(c, n). [:1::'a, - c:] ^ Suc n) ` set cs)"
prems: shows "i < \<^bold>|r \<^sup>@ Suc k\<^bold>|"
no_step: fixes A :: "('b \<times> 'b) set"   and a :: "'b" assumes "A `` {a} = {}" shows "a \<in> NF A"
ctr_1: fixes xs :: "'a llist"   and P :: "'a \<Rightarrow> bool" assumes "lnull xs \<or> \<not> P (lhd xs)" shows "ltakeWhile P xs = LNil"
exhaust_disc: fixes val :: "val" obtains "val = UnitVal"   | "is_BoolVal val"   | "is_IntVal val"   | "is_RealVal val"   | "is_PairVal val"
Rep_lam: fixes x :: "lam" shows "Rep_lam x \<in> Collect lam_Rep_set"
compact_Union: fixes S :: "'c set set" assumes "finite S"   and "\<And>T. T \<in> S \<Longrightarrow> compact T" shows "compact (\<Union> S)"
rec_newconf_def: shows "rec_newconf = Cn 2 rec_trpl [Cn 2 rec_newleft [Cn 2 rec_left [recf.id 2 1], Cn 2 rec_right [recf.id 2 1], Cn 2 rec_actn [recf.id 2 0, Cn 2 rec_stat [recf.id 2 1], Cn 2 rec_right [recf.id 2 1]]], Cn 2 rec_newstat [recf.id 2 0, Cn 2 rec_stat [recf.id 2 1], Cn 2 rec_right [recf.id 2 1]], Cn 2 rec_newrght [Cn 2 rec_left [recf.id 2 1], Cn 2 rec_right [recf.id 2 1], Cn 2 rec_actn [recf.id 2 0, Cn 2 rec_stat [recf.id 2 1], Cn 2 rec_right [recf.id 2 1]]]]"
dsum_iOp_func: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_mOp I A \<in> carr_dsumag I A \<rightarrow> carr_dsumag I A"
simps: fixes a :: "binding list" shows "\<turnstile> a ok = (a = [] \<or> (\<exists>\<Gamma> X T. a = TVarB X T # \<Gamma> \<and> \<turnstile> \<Gamma> ok \<and> X \<sharp> ty_dom \<Gamma> \<and> T closed_in \<Gamma>) \<or> (\<exists>\<Gamma> x T. a = VarB x T # \<Gamma> \<and> \<turnstile> \<Gamma> ok \<and> x \<sharp> trm_dom \<Gamma> \<and> T closed_in \<Gamma>))"
strict_trans2: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a <s b"   and "b \<le>s c" shows "a <s c"
setbydecision_def: fixes iface :: "iface"   and rs :: "'a common_primitive rule list"   and dec :: "final_decision" shows "setbydecision TYPE('b) iface rs dec = {ip. \<exists>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs Undecided = Decision dec}"
subdiv_H: shows "subdivision_pair (gen_contr_graph (with_proj K) (verts3 (with_proj K))) H"
MkArr_Map: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" shows "Cop_S.MkArr (Cop_S.Dom f) (Cop_S.Cod f) (Cop_S.Map f) = f"
SourcesA42_L1: shows "Sources level1 sA42 = {}"
m_1: fixes m' :: "nat" assumes "m' \<in> set ms" shows "m < m'"
walk_takeWhile: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> bool" assumes "walk xs" shows "walk (takeWhile f xs)"
chamber_vertex_outside_facet_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and v :: "'b"   and z :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "v \<notin> z"   and "SimplicialComplex.maxsimp X (insert v z)" shows "f v \<notin> f ` z"
length_filter_less: fixes x :: "'c"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "|filter P xs| < |xs|"
dres_internal_simps_1: shows "dSUCCEEDi = dSUCCEED"
hlog_def: fixes a :: "real star"   and x :: "real star" shows "hlog a x = (*f2* log) a x"
assms_1: fixes x :: "'r"   and y :: "'r" assumes "x \<in> R"   and "y \<in> R" shows "f (x + y) = f x + f y"
simps_2: fixes r :: "real"   and n :: "nat"   and w :: "'b" shows "disc_rfr_proc r (Suc n) w = (1 + r) * disc_rfr_proc r n w"
sgn_finite_3_def: shows "sgn = (\<lambda>x. x)"
inject: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and y1 :: "real"   and y2 :: "real"   and y3 :: "real"   and y4 :: "real" shows "(Quat x1 x2 x3 x4 = Quat y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
assms_2: shows "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"
distinct_1: fixes x21 :: "Unary_PCF.type"   and x22 :: "Unary_PCF.type" shows "\<B> \<noteq> x21 \<rightarrow> x22"
simps_1: fixes uu :: "'d \<Rightarrow> bool" shows "find uu [] = None"
invertible_right_inverse: fixes u :: "'a set" assumes "quotient.invertible u"   and "u \<in> Partition" shows "u [\<cdot>] quotient.inverse u = Class \<one>"
Ln'_of_real_neg: fixes x :: "real" assumes "x < 0" shows "Ln' (complex_of_real x) = complex_of_real (ln (- x)) + \<i> * complex_of_real pi"
Inf_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and a :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (insert a A) = inf a (Inf A)"
add_pos: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq zero a" shows "less_eq b (plus b a)"
xf01: fixes j :: "nat" shows "xf j ` S \<subseteq> {0..1}"
pasting_lemma: fixes I :: "'b set"   and X :: "'c topology"   and T :: "'b \<Rightarrow> 'c set"   and Y :: "'d topology"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x" shows "continuous_map X Y g"
spick_red_geom_filt: fixes n :: "nat"   and x :: "bool" shows "(\<lambda>w. spick w n x) \<in> G n \<rightarrow>\<^sub>M G (Suc n)"
subgroup_mult_id: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "H <#>\<^bsub>G\<^esub> H = H"
leg0_simps_4: shows "local.dom p\<^sub>0 = p\<^sub>0"
between_exchange2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Bet B C D" shows "Bet A C D"
map_component_comp: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and x :: "'t" shows "map_component f (map_component g x) = map_component (\<lambda>k. f (g k)) x"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod p ^ n \<noteq> 0"   and "1 < p ^ n" shows "q.degree_m f = degree f"
prod_nonneg: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less_eq zero (f a)" shows "less_eq zero (comm_monoid_mult.prod times one f A)"
card_Mi_eq_card_edge_as_set_Mi: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and i :: "nat"   and L :: "'b \<Rightarrow> nat" assumes "matching V E M" shows "card (matching_i i V E M L) = card (edge_as_set ` matching_i i V E M L)"
Z_def: shows "Z = {{(False, False), (False, True)}, {(False, False), (True, False)}, {(False, False), (False, True), (True, False), (True, True)}}"
underS_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "Order_Relation.underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
eq_m_irreducible_m: fixes f :: "int poly"   and g :: "int poly" assumes "f =m g" shows "irreducible_m f = irreducible_m g"
exhaust: fixes y :: "(_) shadow_root_ptr" obtains x1 :: "nat" where "y = shadow_root_ptr.Ref x1"   | x2 :: "'m" where "y = shadow_root_ptr.Ext x2"
Subspace_Span: fixes ms :: "'v list" assumes "set ms \<subseteq> V" shows "Subspace (RSpan ms)"
order_less_imp_not_less: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "(\<not> y < x) = True"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "degree (MPoly x) = (\<lambda>v. Max (insert 0 ((\<lambda>m. poly_mapping.lookup m v) ` Poly_Mapping.keys x)))"
LT16: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> H) \<longrightarrow> (G \<leadsto> H) \<longrightarrow> F \<or> G \<leadsto> H"
simps_1: fixes uu :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and uv :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" shows "step_score [] uu uv = 0"
WT2: shows "\<I>' \<turnstile>g gpv' \<surd>"
le_TC_Ord: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "\<beta> \<sqsubseteq> \<alpha>"   and "Ord \<alpha>" shows "Ord \<beta>"
dyad_rec_of_nat: fixes b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and m :: "nat" shows "dyad_rec b l r (real m) = b m"
LFP_lemma2: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "f (LFP f) \<sqsubseteq> LFP f"
Dynkin_subset: fixes N :: "'a set set" assumes "N \<subseteq> M" shows "Dynkin \<Omega> N \<subseteq> M"
one_in_Polys: fixes X :: "'b set" shows "1 \<in> P[X]"
insert_code_1: fixes x :: "'c"   and xs :: "'c list" shows "insert x (set xs) = set (List.insert x xs)"
simps_2: fixes i :: "'b memref"   and "is" :: "'b memref list" shows "prog_instrs (i # is) = (case i of Prog\<^sub>s\<^sub>b x xa is' \<Rightarrow> is' @ prog_instrs is | _ \<Rightarrow> prog_instrs is)"
prime_elem_mono: fixes p :: "'a"   and q :: "'a" assumes "prime_elem p"   and "\<not> is_unit q"   and "q dvd p" shows "prime_elem q"
sel_2: fixes x1 :: "object atom formula"   and x2 :: "object ast_effect" shows "ground_action.effect (Ground_Action x1 x2) = x2"
s_ex: fixes s :: "real" assumes "0 \<le> s"   and "s \<le> h" shows "s \<in> existence_ivl0 x0"
ide_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "partial_magma.ide (span_vertical_category.vcomp C) \<mu> = (arrow_of_spans C \<mu> \<and> partial_magma.ide C (Chn \<mu>))"
NS_all_least: fixes f :: "'f"   and t :: "('f, 'b) Term.term" assumes "least f" shows "NS t (Fun f [])"
p_zero: shows "other.p 0 = s\<^sub>o"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod pl \<noteq> 0"   and "1 < pl" shows "pl.degree_m f = degree f"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Dom (t \<^bold>\<star> u) = Dom t \<^bold>\<star> Dom u"
indep_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>.indep_in \<E> X = (X \<subseteq> \<E> \<and> indep_in \<E> X)"
of_rat_to_rat: fixes x :: "'a" assumes "x \<in> \<rat>" shows "of_rat (to_rat x) = x"
assms_3: shows "isUnitClause bClause bLiteral (elements p)"
Sup_apply: fixes A :: "('c \<Rightarrow> 'b) set"   and x :: "'c" shows "Sup A x = (SUP f\<in>A. f x)"
lin_add: fixes x :: "'a"   and y :: "'a" shows "p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat \<Rightarrow> nat option"   and a3 :: "exp list"   and a4 :: "nat list" assumes "a1;a2 \<turnstile>+ a3 : a4" obtains CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" where "a1 = CT__" and "a2 = \<Gamma>__" and "a3 = []" and "a4 = []"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and es :: "exp list" and Cs :: "nat list" where "a1 = CT" and "a2 = \<Gamma>" and "a3 = e0 # es" and "a4 = C0 # Cs" and "CT;\<Gamma> \<turnstile> e0 : C0" and "CT;\<Gamma> \<turnstile>+ es : Cs"
rotation_rightward_line: fixes a :: "(real, 'a) vec"   and k :: "'a" obtains f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" where "orthogonal_transformation f" and "2 \<le> CARD('a) \<Longrightarrow> det (matrix f) = 1" and "f (norm a *\<^sub>R axis k 1) = a"
star_loop_least_fixpoint: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y * x \<squnion> z = x" shows "y\<^sup>\<star> * z \<le> x"
PiE_fun_upd: fixes y :: "'c"   and T :: "'d \<Rightarrow> 'c set"   and x :: "'d"   and f :: "'d \<Rightarrow> 'c"   and S :: "'d set" assumes "y \<in> T x"   and "f \<in> Pi\<^sub>E S T" shows "f(x := y) \<in> Pi\<^sub>E (insert x S) T"
\<tau>Execr_induct: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option"   and Q :: "'addr val list \<Rightarrow> 'addr val list \<Rightarrow> nat \<Rightarrow> 'addr option \<Rightarrow> bool" assumes "\<tau>Exec_mover ci P t e h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"   and "Q stk loc pc xcp"   and "\<And>stk' loc' pc' xcp' stk'' loc'' pc'' xcp''. \<lbrakk>\<tau>Exec_mover ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp'); \<tau>exec_move ci P t e h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp''); Q stk' loc' pc' xcp'\<rbrakk> \<Longrightarrow> Q stk'' loc'' pc'' xcp''" shows "Q stk'' loc'' pc'' xcp''"
part_equivp_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "part_equivp R"   and "R x y"   and "R y z" shows "R x z"
Collect_const: fixes P :: "bool" shows "{s. P} = (if P then UNIV else {})"
rtrancl_refl: fixes a :: "'c"   and r :: "('c \<times> 'c) set" shows "(a, a) \<in> r\<^sup>*"
If: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile> {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q}"   and "\<turnstile> {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q}" shows "\<turnstile> {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"