hcju/isabelleps huggingface dataset:
step: fixes x :: "'a node \<times> 'a node set" assumes "((\<mu> U\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> \<eta> \<in> old (fst x) \<union> new (fst x) \<or> \<mu> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> U\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> \<eta> \<in> old nd \<or> \<mu> \<in> old nd \<and> \<mu> U\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>n\<in>snd x. \<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd) \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>r. \<forall>n\<in>snd r. \<forall>nd\<in>snd r. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd)"
E5: shows "vpeq_subj_obj u (set_object_value x y s1) (set_object_value x y s2)"
y_def: shows "y = (SOME y. y \<in> r' / s')"
assms_2: shows "\<forall>i\<in>I. \<exists>p. p tagged_division_of i \<and> d fine p"
foldl_prs_aux: fixes R1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs1 :: "'c \<Rightarrow> 'd"   and rep1 :: "'d \<Rightarrow> 'c"   and R2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs2 :: "'e \<Rightarrow> 'f"   and rep2 :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'd"   and e :: "'d"   and l :: "'f list" assumes "Quotient3 R1 abs1 rep1"   and "Quotient3 R2 abs2 rep2" shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
quotient_ring_axioms: shows "Comm_Ring.quotient_ring (R\<setminus>I) R (+) (\<cdot>) \<zero> \<one>"
map_add_left_dom_eq: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>1' :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"   and "h\<^sub>0 \<bottom> h\<^sub>1"   and "h\<^sub>0' \<bottom> h\<^sub>1'"   and "dom h\<^sub>0 = dom h\<^sub>0'" shows "h\<^sub>0 = h\<^sub>0'"
bounded_clinear_eq_on: fixes A :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and t :: "'c" assumes "bounded_clinear A"   and "bounded_clinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
ext_csrp_fn_nonzero: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and g' :: "'a set" assumes "Idomain S"   and "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "g' \<in> carrier R'"   and "g' \<noteq> \<zero>\<^bsub>R'\<^esub>" shows "polyn_expr R X (deg_n R' (S /\<^sub>r P) Y g') (deg_n R' (S /\<^sub>r P) Y g', cmp (csrp_fn S P) (snd (s_cf R' (S /\<^sub>r P) Y g'))) \<noteq> \<zero>"
disj_is_nnf: fixes F :: "'b formula" assumes "is_disj F" shows "is_nnf F"
distinct_4: fixes x3 :: "'b"   and x1 :: "'b" shows "mid x3 \<noteq> left x1"
R1: shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
out213_suma__conga: 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 C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.Out Bet B A C" shows "Tarski_neutral_dimensionless.CongA Bet Cong D E F G H I"
unrest_true: fixes x :: "'a \<Longrightarrow> 'b" shows "x \<sharp> true"
remdups_eq_nil_right_iff: fixes x :: "'b list" shows "([] = remdups x) = (x = [])"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" shows "\<guillemotleft>Y a' f : Y a' a \<rightarrow>\<^sub>S Y a' b\<guillemotright>"
alg_mult_eq_count_roots: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "domain R"   and "p \<in> carrier (poly_ring R)" shows "ring.alg_mult R p = count (ring.roots R p)"
TS_xr'_3: fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list"   and r :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (question (Atom x))"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)" shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) (qs @ r) = Partial_Cost_Model.config' (rTS h0) ([x, y], rev qs @ h) r"
fix_eq: fixes F :: "'c \<rightarrow> 'c" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
cfg_on_inv: fixes cfg :: "'s cfg"   and \<omega> :: "'s cfg stream"   and s :: "'s" assumes "MC.enabled cfg \<omega>"   and "cfg \<in> cfg_on s" shows "pred_stream (\<lambda>cfg. cfg \<in> cfg_on (state cfg)) \<omega>"
IH: fixes y :: "nat" assumes "y < n" shows "set_encode (set_decode y) = y"
normalize_monom: fixes a :: "'b"   and n :: "nat" shows "normalize (monom a n) = monom (normalize a) n"
lfinite_lset: fixes xs :: "'a llist" assumes "lfinite xs"   and "\<not> lnull xs" shows "llast xs \<in> lset xs"
in_enum: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'b" assumes "class.enum enum enum_all enum_ex" shows "x \<in> set enum"
RETURN_SPEC_conv: fixes r :: "'a" shows "RETURN r = SPEC (\<lambda>x. x = r)"
fresh_10: fixes a :: "coname"   and x4 :: "coname"   and x5 :: "trm"   and x2 :: "coname"   and x3 :: "trm"   and x1 :: "coname" shows "a \<sharp> AndR <x4>.x5 <x2>.x3 x1 = (a \<sharp> [x4].x5 \<and> a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
K_concrete_finite_consistency: fixes G :: "'b fm list" assumes "set G \<in> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})" shows "\<not> \<turnstile> imply G \<^bold>\<bottom>"
reset_set2: fixes cs :: "'c list"   and ts :: "'d list"   and u :: "'c \<Rightarrow> 'd" shows "\<forall>c. c \<notin> set cs \<longrightarrow> set_clocks cs ts u c = u c"
hcomp_in_hhom: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and a :: "'c"   and b :: "'c"   and \<nu> :: "'c"   and c :: "'c" assumes "horizontal_composition V H src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b"   and "horizontal_homs.in_hhom V src trg \<nu> b c" shows "horizontal_homs.in_hhom V src trg (H \<nu> \<mu>) a c"
JOIN_def: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "JOIN I F = mk_program (\<Inter>i\<in>I. Init (F i), \<Union>i\<in>I. Acts (F i), \<Inter>i\<in>I. AllowedActs (F i))"
streams_iff_sset: fixes s :: "'b stream"   and A :: "'b set" shows "(s \<in> streams A) = (sset s \<subseteq> A)"
ipresCons_imp_ipresWlsAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igConsIPresIGWls Delta MOD" shows "FixSyn.ipresWlsAll Delta h hA MOD"
CC'_def: shows "CC' = (\<Union>E\<in>EE. DD_of E - {{#- L#} |L. L \<in># E})"
Vfbasis_def: shows "Vfbasis \<equiv> SOME vs. set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) vs"
mset_single_cases': fixes s :: "'a"   and c :: "'a multiset"   and r' :: "'a"   and c' :: "'a multiset" assumes "add_mset s c = add_mset r' c'" obtains "s = r'" and "c = c'"   | cc :: "'a multiset" where "c' = {#s#} + cc" and "c = {#r'#} + cc" and "c' - {#s#} = cc" and "c - {#r'#} = cc"
absolutely_integrable_on_iff_nonneg: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "(f absolutely_integrable_on S) = (f integrable_on S)"
axioms_3: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "functor B A F"
below_val_zero: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "prime p"   and "x \<in> padic_set p"   and "x (Suc n) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc n)\<^esub>" shows "padic_val p x \<le> int n"
decomposeLS_def: fixes s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" shows "decomposeLS s = (case cPGM s of [] \<Rightarrow> {} | c # x \<Rightarrow> decompose_com c)"
Diff_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(c \<in> A\<setminus>B) = (c \<in> A \<and> c \<notin> B)"
ins2_iff_ins_3: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree"   and n' :: "nat"   and h' :: "nat" assumes "RBTi2 TYPE('b) bal_i e" shows "(RBTi2.ins2 bal_i n d x t = Unbal2 t' n' h') = (RBTi1.ins bal_i n d x t = Unbal t' \<and> n' = size t' \<and> h' = height t')"
interval_of_add: fixes a :: "'a"   and b :: "'a" shows "interval_of (a + b) = interval_of a + interval_of b"
inject_2: fixes x21 :: "brick list"   and x22 :: "wall"   and y21 :: "brick list"   and y22 :: "wall" shows "(x21 * x22 = y21 * y22) = (x21 = y21 \<and> x22 = y22)"
times_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder_stone_relation_algebra_expansion one times uminus less_eq less bot top inf sup conv" shows "times x y = ord.min less_eq x y"
stutter_extend_edgesI_edge: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set"   and V :: "'a set" assumes "(u, v) \<in> E" shows "(u, v) \<in> stutter_extend_edges V E"
Top_provable: fixes \<Gamma> :: "'b formula set" shows "\<Gamma> \<turnstile> \<top>"
sign_simps_10: fixes ab :: "'b"   and cb :: "'b"   and bb :: "'b" shows "(ab = cb - bb) = (ab + bb = cb)"
map_def: fixes f :: "'a" shows "local.map f = (if arr f then f else null)"
empty: fixes r :: "('a \<times> 'a) set" assumes "MaxR r" shows "MaxR.MaxR_opt r {} = None"
count_greater_eq_Suc_zero_iff: fixes M :: "'k multiset"   and x :: "'k" shows "(Suc 0 \<le> count M x) = (x \<in># M)"
Can_and_Diag_implies_Ide: fixes t :: "'a local.term" assumes "Can t"   and "Diag t" shows "Ide t"
exhaust: fixes y :: "'c tree" obtains "y = \<langle>\<rangle>"   | x21 :: "'c tree" and x22 :: "'c" and x23 :: "'c tree" where "y = \<langle>x21, x22, x23\<rangle>"
simps_16: fixes b :: "'a exp"   and e :: "'a exp" shows "sub_RI (while (b) e) = (sub_RI b \<or> sub_RI e)"
range_tm_tm_pi: fixes e :: "nat \<Rightarrow> real"   and prec :: "nat" shows "pi \<in>\<^sub>i range_tm e (tm_pi prec)"
H_fcs_full: fixes SS :: "'b set set" assumes "SS \<subseteq> H.face_cycle_sets" shows "H.face_cycle_set ` \<Union> SS = SS"
length_drop: fixes n :: "nat"   and xs :: "'g list" shows "length (drop n xs) = length xs - n"
nsg_in_Gp: fixes N :: "'a set"   and H :: "'a set" assumes "G \<triangleright> N"   and "G \<guillemotright> H "   and "N \<subseteq> H" shows "\<natural>H \<triangleright> N"
bv_to_nat_emsapss_encode_le: fixes m :: "bit list"   and x :: "nat" assumes "emsapss_encode m x \<noteq> []" shows "bv_to_nat (emsapss_encode m x) < 2 ^ (roundup x 8 * 8)"
aexec: shows "abc_steps_l (0, lm) ap stp = (as, am)"
Cons_1: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "g \<in> set gs" shows "monom_mult (1::'b) (pp_of_term v - lp g) g \<in> set (snd (sym_preproc_addnew gs vs fs v))"
inj_on_hom_induced_inclusion: fixes S :: "'b set"   and X :: "'b topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" shows "inj_on (hom_induced p (subtopology X S) {} X {} id) (carrier (homology_group p (subtopology X S)))"
simps: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping"   and cs :: "('a \<times> constraint) list"   and J :: "'a set"   and ds :: "'b ns_constraint list"   and s :: "'c" assumes "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns" shows "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.invariant_cs invariant_nsc to_ns cs J (ds, s) = (ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s)"
f'_def: fixes n :: "V" shows "f' n = f `\<^sub>\<circ> n"
gcd_zero_iff: fixes a :: "'b"   and b :: "'b" shows "(some_gcd a b = (0::'b)) = (a = (0::'b) \<and> b = (0::'b))"
cases: fixes x :: "('a, 'b) form" obtains p :: "('a, 'b) form" where "x = Neg p"   | "x = \<bottom>"   | "x = \<top>"   | v :: "'b" and va :: "'a FOL_Fitting.term list" where "x = Pred v__ va__"   | v :: "('a, 'b) form" and va :: "('a, 'b) form" where "x = And vb__ vaa__"   | v :: "('a, 'b) form" and va :: "('a, 'b) form" where "x = Or vc__ vab__"   | v :: "('a, 'b) form" and va :: "('a, 'b) form" where "x = Impl vd__ va"   | v :: "('a, 'b) form" where "x = Forall ve__"   | v :: "('a, 'b) form" where "x = Exists v"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> synth H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>A. P (Messages.Agent A)"   and "\<And>n. P (Number n)"   and "\<And>n. P (NonceA n)"   and "\<And>n. P (epubKA n)"   and "\<And>n. P (epriKA n)"   and "\<And>X. \<lbrakk>X \<in> synth H; P X\<rbrakk> \<Longrightarrow> P (Hash X)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P \<langle>X, Y\<rangle>"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P (Enc X Y)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P (Aenc X Y)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P (Sign X Y)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P (Exp X Y)" shows "P x"
in_out_indep: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'d \<Longrightarrow> 'c" shows "$x \<bowtie> $y\<acute>"
fimageI1: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)" shows "x \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A"
ltrm_smult: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "monom P ((a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p))) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = a \<odot>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)"
ws_def: shows "ws = [countVertices g..<countVertices g + n]"
epred_llcp: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "\<not> lnull xs"   and "\<not> lnull ys"   and "lhd xs = lhd ys" shows "epred (llcp xs ys) = llcp (ltl xs) (ltl ys)"
rep_eq: fixes x :: "'c" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e = (\<lambda>b. x)"
distinct_delete_aux: fixes ps :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst ps)" shows "distinct (map fst (delete_aux k ps))"
nat_mult_2: fixes a :: "nat" shows "a + a = 2 * a"
assms_1: shows "(name, rhs') # rest = Rewriting_Sterm.compile rs"
lr_fps_denominator_nz: fixes cs :: "'a list" assumes "last cs \<noteq> (0::'a)"   and "cs \<noteq> []" shows "lr_fps_denominator cs \<noteq> 0"
LIMSEQ_le_const2: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a"   and a :: "'a" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> a" shows "x \<le> a"
kD_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and nip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "kD (rt(nip \<mapsto> v)) = insert nip (kD rt)"
poly_bigo: fixes F :: "'a filter"   and p :: "'a poly"   and k :: "nat" assumes "F \<le> at_infinity"   and "degree p \<le> k" shows "poly p \<in> O[F](\<lambda>x. x ^ k)"
distinct_1: fixes name' :: "name"   and trm' :: "trm"   and stack' :: "stack" shows "stack.Id \<noteq> [name']trm'\<ggreater>stack'"
Span_append: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and ms :: "'a list"   and ns :: "'a list" shows "R_scalar_mult.RSpan UNIV smult (ms @ ns) = R_scalar_mult.RSpan UNIV smult ms + R_scalar_mult.RSpan UNIV smult ns"
acomplete_def: fixes t :: "'c tree" shows "acomplete t = (height t - min_height t \<le> 1)"
substs_preserves_decompose_literal: fixes L :: "'b literal"   and t :: "'b trm"   and s :: "'b trm"   and polarity :: "sign"   and \<eta> :: "('b \<times> 'b trm) list" assumes "decompose_literal L t s polarity" shows "decompose_literal (subst_lit L \<eta>) (t \<lhd> \<eta>) (s \<lhd> \<eta>) polarity"
degen_outcome_cardinalities: shows "card local.degenerate_lotteries = card outcomes"
box_ne_empty_2: fixes a :: "'c"   and b :: "'c" shows "(box a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
top_greatest: shows "greatest L \<top>\<^bsub>L\<^esub> (carrier L)"
set_insert: fixes x :: "'b"   and xs :: "'b list" shows "set (List.insert x xs) = x \<triangleright> set xs"
assms_7: fixes a :: "floatarith"   and b :: "floatarith" shows "subterms (Binop a b) = insert (Binop a b) (subterms a \<union> subterms b)"
rHom_tOp: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('f, 'g) Ring_scheme"   and x :: "'d"   and y :: "'d"   and f :: "'d \<Rightarrow> 'f" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "f \<in> rHom A R" shows "f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y"
fi_scalar_prod_gso: fixes i :: "nat"   and j :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)"   and "j < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! i \<bullet> fs.gs.gso j = fs.gs.\<mu> i j * \<parallel>fs.gs.gso j\<parallel>\<^sup>2"
memo_def: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" shows "memo xs f \<equiv> let m = Mapping.tabulate xs f in (\<lambda>x. case Mapping.lookup m x of None \<Rightarrow> f x | Some y \<Rightarrow> y)"
that: fixes \<T> :: "'a set set" assumes "triangulation \<T>"   and "\<And>k. k \<in> \<T> \<Longrightarrow> diameter k < e"   and "\<And>k. k \<in> \<T> \<Longrightarrow> aff_dim k = d"   and "\<Union> \<T> = \<Union> \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>f. finite f \<and> f \<subseteq> \<T> \<and> C = \<Union> f"   and "\<And>k. k \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> k \<subseteq> C" shows "thesis"
fbox_loopi: fixes p :: "'a"   and i :: "'a"   and x :: "'a"   and q :: "'a" assumes "p \<le> d i"   and "d i \<le>  |x] i"   and "d i \<le> d q" shows "p \<le>  |loop x inv i ] q"
notin_range_Inner: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x :: "'m bi.vertex" assumes "countable_bipartite_web \<Gamma>" shows "(x \<notin> range bi.Inner) = (x = bi.SOURCE \<or> x = bi.SINK)"
lex_pp_of_exps: fixes a1 :: "nat"   and b1 :: "nat"   and i :: "nat"   and a2 :: "nat"   and b2 :: "nat"   and j :: "nat" shows "lex_pp (fst (rep_nat_term (Term_Order.of_exps a1 b1 i))) (fst (rep_nat_term (Term_Order.of_exps a2 b2 j))) = (a1 < a2 \<or> a1 = a2 \<and> b1 \<le> b2)"
B_def: shows "B \<equiv> \<Union>s\<in>A\<^sup>\<clubsuit>. {suff A s}"
intros: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and a :: "Arity"   and e :: "exp"   and as :: "Arity list" assumes "edom ae \<subseteq> domA \<Gamma> \<union> upds S"   and "Astack S \<sqsubseteq> a"   and "(ABinds \<Gamma>\<cdot>ae \<squnion> \<A>\<^bsub>a\<^esub> e \<squnion> AEstack as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae"   and "stack_consistent as S" shows "a_consistent (ae, a, as) (\<Gamma>, e, S)"
cod_eqI: fixes b :: "'a"   and f :: "'a" assumes "ide b"   and "seq b f" shows "cod f = b"
fnz: fixes z :: "complex" assumes "z \<in> S" shows "f z \<noteq> 0"
lt_of_nat_term_order_def: fixes to :: "'a nat_term_order" shows "lt_of_nat_term_order to = lt_of_key_order (key_order_of_nat_term_order to)"
starfun_exp_ln_iff: fixes x :: "real star" shows "((*f* exp) ((*f* real_ln) x) = x) = (0 < x)"
isomorphic_implies_hpar_1: fixes f :: "'d"   and f' :: "'d" assumes "D.isomorphic f f'" shows "D.ide f"
Cardano_1: shows "(root 3 u - e / (3 * root 3 u)) ^ 3 + e * (root 3 u - e / (3 * root 3 u)) + f = 0"
path_append_conv: fixes E :: "('a \<times> 'a) set"   and u :: "'a"   and p :: "'a list"   and v :: "'a"   and w :: "'a" shows "path E u (p @ [v]) w = (path E u p v \<and> (v, w) \<in> E)"
minus_fract: fixes a :: "'b"   and b :: "'b" shows "- Fract a b = Fract (- a) b"
m_inv_closed: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<^esub> x \<in> H"
not_\<omega>\<^sub>z_le_numeral: fixes n :: "num" shows "\<not> \<omega>\<^sub>z \<le> numeral n"
pr_trans: fixes A :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and C :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "A \<sqsubseteq> B"   and "B \<sqsubseteq> C" shows "A \<sqsubseteq> C"
rconjby_add: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (plus (uminus (plus x y)) z) (plus x y) = plus (plus (uminus y) (plus (plus (uminus x) z) x)) y"
rel_refl: fixes Ra :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l list" assumes "\<And>x. Ra x x" shows "list_all2 Ra x x"
prems_2: shows "char_poly A = (\<Prod>e\<leftarrow>e # es. [:- e, 1:])"