hcju/isabelleps huggingface dataset:
fmdrop_lookup: fixes a :: "'d"   and m :: "('d, 'c) fmap" shows "fmdrop a m $$ a = None"
absorb1: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "a \<squnion> b = a"
is_SNF_def: fixes A :: "'a mat"   and R :: "'a mat \<times> 'a mat \<times> 'a mat" shows "is_SNF A R = (case R of (P, S, Q) \<Rightarrow> P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_col A) (dim_col A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> S = P * A * Q)"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "hom.ide a"   and "hom.ide a'"   and "hom'.in_hom g (e \<star> a' \<star> src f) (e \<star> a \<star> src f)" shows "\<exists>fa. hom.in_hom fa a' a \<and> e \<star> fa \<star> src f = g"
of_int_minus: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- z) = uminus (ring_1.of_int one plus zero minus z)"
sel_1: fixes x1 :: "'d"   and x2 :: "'e" shows "fst (x1, x2) = x1"
Sup_subset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
nth_enumerate_eq: fixes m :: "nat"   and xs :: "'b list"   and n :: "nat" assumes "m < length xs" shows "enumerate n xs ! m = (n + m, xs ! m)"
sub: fixes a :: "'a" assumes "a \<in> L" shows "a \<in> G"
lead_coeff_map_poly_nz: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "f (lead_coeff p) \<noteq> (0::'a)"   and "f (0::'b) = (0::'a)" shows "lead_coeff (map_poly f p) = f (lead_coeff p)"
nested_submodules: fixes W :: "'a vec set"   and X :: "'a vec set" assumes "LinearCombinations.submodule class_ring W V"   and "LinearCombinations.submodule class_ring X V"   and "X \<subseteq> W" shows "LinearCombinations.submodule class_ring X (vs W)"
gco_approx_def: fixes prog :: "'a program_scheme" shows "gco_approx prog \<equiv> insert None (Some ` gc_approx prog)"
Basis_nonneg: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "less_eq zero i"
is_closest_def: fixes v :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and x :: "'b"   and a :: "'a" shows "is_closest v s x a = (a \<in> s \<and> (\<forall>b. b \<in> s \<longrightarrow> \<bar>v a - x\<bar> \<le> \<bar>v b - x\<bar>))"
distinct_12: fixes x7 :: "CTypeId" shows "CClassT x7 \<noteq> BoolT"
r_def: shows "r = (if conv_radius f = \<infinity> then R' + 1 else (R' + real_of_ereal (conv_radius f)) / 2)"
fps_const_mult: fixes c :: "'a"   and d :: "'a" shows "fps_const c * fps_const d = fps_const (c * d)"
encode_negative_transition_frame_axiom_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and v :: "'b" shows "encode_negative_transition_frame_axiom \<Pi> t v \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O>; deleting_operators = filter (\<lambda>op. ListMem v (delete_effects_of op)) ops in \<^bold>\<not> (Atom (State t (index vs v))) \<^bold>\<or> (Atom (State (Suc t) (index vs v)) \<^bold>\<or> \<^bold>\<Or>map (\<lambda>op. Atom (Operator t (index ops op))) deleting_operators)"
fixp_induct: fixes P :: "(nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bdd \<Rightarrow> (nat \<times> bdd) option) \<Rightarrow> bool" assumes "option.admissible (\<lambda>ite_impl_lu. P (curry (curry (curry ite_impl_lu))))"   and "P (\<lambda>ite_impl_lu i t. Map.empty)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>a aa aaa aaaa. case dcl aaaa (a, aa, aaa) of None \<Rightarrow> mi.param_opt_impl a aa aaa aaaa \<bind> (\<lambda>aaaa. case aaaa of (None, sa) \<Rightarrow> mi.lowest_tops_impl [a, aa, aaa] sa \<bind> (\<lambda>aaaa. case aaaa of (None, uu_) \<Rightarrow> None | (Some ab, uu_) \<Rightarrow> mi.restrict_top_impl a ab True sa \<bind> (\<lambda>aaaa. case aaaa of (ti, uu_) \<Rightarrow> mi.restrict_top_impl aa ab True sa \<bind> (\<lambda>aaaa. case aaaa of (tt, uu_) \<Rightarrow> mi.restrict_top_impl aaa ab True sa \<bind> (\<lambda>aaaa. case aaaa of (te, uu_) \<Rightarrow> mi.restrict_top_impl a ab False sa \<bind> (\<lambda>aaaa. case aaaa of (fi, uu_) \<Rightarrow> mi.restrict_top_impl aa ab False sa \<bind> (\<lambda>aaaa. case aaaa of (ft, uu_) \<Rightarrow> mi.restrict_top_impl aaa ab False sa \<bind> (\<lambda>aaaa. case aaaa of (fe, uu_) \<Rightarrow> s ti tt te sa \<bind> (\<lambda>aaaa. case aaaa of (tb, sa) \<Rightarrow> s fi ft fe sa \<bind> (\<lambda>aaaa. case aaaa of (fb, s) \<Rightarrow> ifmi' ab tb fb s \<bind> (\<lambda>aaaa. case aaaa of (r, s) \<Rightarrow> let s = updS s (a, aa, aaa) r in Some (r, s))))))))))) | (Some b, sa) \<Rightarrow> Some (b, sa)) | Some b \<Rightarrow> Some (b, aaaa))" shows "P mi.ite_impl_lu"
key_compare_same: fixes ko :: "'a key_order"   and x :: "'a" shows "key_compare ko x x = Eq"
N_fresh_not_KeyCryptNonce: fixes DK :: "nat"   and N :: "nat"   and evs :: "event list" assumes "\<forall>C. DK \<noteq> priEK C" shows "Nonce N \<notin> used evs \<longrightarrow> \<not> KeyCryptNonce DK N evs"
well_valued0_r: fixes as :: "'b list"   and a :: "'b tree" assumes "well_valued0 as a"   and "a \<noteq> Leaf" shows "r a \<le> length as"
zero_le_power_abs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times (abs a) n)"
ideal_sum_iff_gcd: fixes a :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "d \<in> carrier R" shows "(PIdl d = PIdl a <+> PIdl b) = d gcdof a b"
card_of_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "|f ` A| \<le>o |A|"
comp_hd_eq: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie> v"   and "u \<noteq> \<epsilon>"   and "v \<noteq> \<epsilon>" shows "hd u = hd v"
Collect_mono_iff: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
HInv3_R_def: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" shows "HInv3_R s p q d = (\<lparr>block = dblock s q, proc = q\<rparr> \<in> blocksRead s p d \<or> \<lparr>block = dblock s p, proc = p\<rparr> \<in> blocksRead s q d)"
CFGExit_wf_axioms: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"
id_bnf_def: shows "BNF_Composition.id_bnf \<equiv> \<lambda>x. x"
degree_smult_eq: fixes a :: "'a"   and p :: "'a poly" shows "Polynomial.degree (Polynomial.smult a p) = (if a = (0::'a) then 0 else Polynomial.degree p)"
scale_cancel_right: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> S" shows "(a *\<^sub>R x = b *\<^sub>R x) = (a = b \<or> x = (0::'a))"
spoly_alt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "punit.spoly p q = fst (punit.crit_pair p q) - snd (punit.crit_pair p q)"
to_nat_plus_one_less_card': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "mod_type.to_nat Rep a + 1 < CARD('b)"
A92_NotSourceSet_level1: shows "\<forall>C\<in>AbstrLevel level1. sA92 \<notin> Sources level1 C"
rel_cfg_action: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and cfg1 :: "'a cfg"   and cfg2 :: "'b cfg" assumes "rel_cfg P cfg1 cfg2" shows "rel_pmf P (action cfg1) (action cfg2)"
hoare_triple_def: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" shows "<P> c <Q> \<equiv> \<forall>h as \<sigma> r. (h, as) \<Turnstile> P \<and> run c (Some h) \<sigma> r \<longrightarrow> (let h' = the_state \<sigma>; as' = new_addrs h as h' in \<not> is_exn \<sigma> \<and> (h', as') \<Turnstile> Q r \<and> relH {a. a < lim h \<and> a \<notin> as} h h' \<and> lim h \<le> lim h')"
safe_hd_eq: shows "safe_hd tss = (Some ts, tss')"
induced_automorphism_surj_on_simplices: shows "\
 \<turnstile> X = X"
measurable_fst'': fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and P :: "'f measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(\<lambda>x. f (fst x)) \<in> M \<Otimes>\<^sub>M P \<rightarrow>\<^sub>M N"
check_poly_weak_mono_discrete_def: fixes p :: "('b monom \<times> 'c) list"   and v :: "'b" shows "check_poly_weak_mono_discrete p v \<equiv> check_poly_ge (poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'c), PVar v] else PVar w)) p) p"
hIm_add: fixes x :: "complex star"   and y :: "complex star" shows "hIm (x + y) = hIm x + hIm y"
prod_list_power: fixes xs :: "'a list"   and n :: "nat" shows "prod_list xs ^ n = (\<Prod>x\<leftarrow>xs. x ^ n)"
eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
sourcenodes_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and xs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.sourcenodes sourcenode xs \<equiv> map sourcenode xs"
parallel_eoracle_def: fixes eoracle1 :: "'v \<Rightarrow> 'p \<Rightarrow> (('r \<times> 't list) \<times> 'v) spmf"   and eoracle2 :: "'w \<Rightarrow> 'q \<Rightarrow> (('s \<times> 'u list) \<times> 'w) spmf"   and state :: "'v \<times> 'w" shows "parallel_eoracle eoracle1 eoracle2 state \<equiv> map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) \<circ> (eoracle1 \<ddagger>\<^sub>O eoracle2) state"
local_order_prefix_closed_last: fixes x :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat"   and y :: "('a \<times> ('a, 'b) operation) event"   and xs :: "('a \<times> ('a, 'b) operation) event list" assumes "x \<sqsubset>\<^sup>i y"   and "xs @ [y] prefix of i" shows "x \<in> set xs"
Cons_1: fixes q :: "('a monom \<times> 'b) list" assumes "poly_inv p"   and "poly_inv q" shows "poly_inv (poly_add p q)"
abs_eq: fixes x :: "(real, 'a) Finite_Cartesian_Product.vec" shows "to_st_vec x = Abs_st_vec (if stoch_vec x \<and> non_neg_vec x then x else \<chi>i. if i = undefined then 1 else 0)"
below_bottom_iff: fixes x :: "'c" shows "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas (\<phi> W\<^sub>n \<psi>) = restricted_subformulas_inner \<phi> \<union> restricted_subformulas \<psi>"
density_eq: fixes A :: "'c set" assumes "A \<in> sets M" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A f'"
simps_7: fixes i :: "int"   and a :: "Cooper.num" shows "numbound0 (Mul i a) = numbound0 a"
formulaEntailsResolvent: fixes formula :: "Literal list list"   and clause1 :: "Literal list"   and clause2 :: "Literal list"   and literal :: "Literal" assumes "formulaEntailsClause formula clause1"   and "formulaEntailsClause formula clause2" shows "formulaEntailsClause formula (resolve clause1 clause2 literal)"
Subobjs_nonempty: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs P C Cs" shows "Cs \<noteq> []"
qAFreshAbs_imp_qFreshAbs_preserves_alphaAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and z :: "'var"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs A"   and "qAFreshAbs zs z A"   and "A $= A'" shows "qFreshAbs zs z A'"
acquired_union_notin_first: fixes a :: "nat"   and pending_write :: "bool"   and sb :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "a \<in> acquired pending_write sb (A \<union> B)"   and "a \<notin> A" shows "a \<in> acquired pending_write sb B"
that: fixes M :: "'a" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<turnstile> the (subject \<alpha>) \<leftrightarrow> M"   and "B \<sharp>* M" shows "thesis"
w_W: shows "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
simp: fixes p :: "'c poly"   and k :: "nat" shows "decompose_prod_root_unity_main p k = (if k = 0 then ([], p) else let q = root_unity k in if q dvd p then if p = 0 then ([], 0) else map_prod ((#) k) id (decompose_prod_root_unity_main (p div q) k) else decompose_prod_root_unity_main p (k - 1))"
NonMemLocInvariant: fixes l :: "Locs"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals" shows "\<turnstile> #l \<notin> #MemLoc \<longrightarrow> \<box>MemInv mm l"
mult_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "complex"   and x :: "'c"   and y :: "'c" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times (scaleC a x) y = scaleC a (times x y)"
asIMOD_igOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> 'a) option"   and binp :: "'bindex \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> ('varSort, 'a) sAbs) option" shows "igOp (asIMOD SEM) delta inp binp = (\<lambda>val. if sWlsVal SEM val then sOp SEM delta (lift (\<lambda>X. X val) inp) (lift (\<lambda>A. A val) binp) else undefined)"
assms: shows "\<C> \<turnstile> es @ [e] : t1s _> t4s"
is_processT5_S4: fixes s :: "'c event list"   and P :: "'c process"   and c :: "'c event" assumes "(s, {}) \<in> F P"   and "(s, {c}) \<notin> F P" shows "(s @ [c], {}) \<in> F P"
next_not_nil_left: fixes x :: "'node" shows "(nil, x) \<notin> next"
g2_def: shows "g2 = (\<lambda>j. if 1 \<le> j then f2 (j - 1) else b)"
prems_2: shows "c \<notin> normalize_sort cs (insert ic s)"
mem_simps_11: fixes A :: "'b"   and C :: "'b set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
surjective: fixes r :: "('c, 'd) carrier_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = carrier.more r\<rparr>"
card_of_UNIV: fixes A :: "'d set" shows "|A| \<le>o |UNIV|"
fbrelation_vintersection_vrange: fixes s :: "V" assumes "\<R>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> \<R>\<^sub>\<bullet> s = []\<^sub>\<circ>" shows "r \<inter>\<^sub>\<circ> s = []\<^sub>\<circ>"
assms_5: shows "A *\<^sub>v (mat_of_cols nc ss *\<^sub>v v) = 0\<^sub>v nr"
nop_v11_vrange: shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
fold_set_fold_remdups: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and y :: "'l"   and xs :: "'k list" assumes "comp_fun_commute f" shows "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
verit_comp_simplify1_3: fixes b' :: "'g"   and a' :: "'g" shows "(\<not> b' \<le> a') = (a' < b')"
jonsson2a: shows "(\<exists>g. \<forall>x y. x \<lhd> y = x \<cdot> g y) = (\<forall>x y. x \<lhd> y = x \<cdot> ((1::'a) \<lhd> y))"
simps_2: fixes x :: "char list"   and G :: "diagram" shows "top_ass (Exists_dia x G) = Exists_int x (top_dia G)"
ipath_def: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b" shows "ipath E r \<equiv> \<forall>i. (r i, r (Suc i)) \<in> E"
simps_10: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>n \<phi> \<or> (\<exists>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>n \<psi>))"
simps_3: fixes t1 :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq1 :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t2 :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq2 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.meld (t1 # bq1) (t2 # bq2) = (if BinomialHeapStruc.rank t1 < BinomialHeapStruc.rank t2 then t1 # BinomialHeapStruc.meld bq1 (t2 # bq2) else if BinomialHeapStruc.rank t2 < BinomialHeapStruc.rank t1 then t2 # BinomialHeapStruc.meld (t1 # bq1) bq2 else BinomialHeapStruc.ins (BinomialHeapStruc.link t1 t2) (BinomialHeapStruc.meld bq1 bq2))"
homeomorphic_aff_dim_le: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "rel_interior S \<noteq> {}" shows "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
qinv: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) Q"
sp_fundef_def: fixes F :: "'a \<Rightarrow> 'b fundef option"   and fd :: "'c fundef"   and xs :: "('d, 'e, 'a, 'op, 'opinl, 'opubx, 'f, 'g) Ubx.instr list" shows "sp_fundef F fd xs \<equiv> Subx.sp F xs (replicate (arity fd) None)"
fun_eq_zeroI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> supp_fun f \<Longrightarrow> f x = (0::'c)" shows "f = 0"
coinduct: fixes X :: "('test, 'atom) com \<Rightarrow> bool"   and x :: "('test, 'atom) com" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>c. x = c \<and> (\<forall>x xa xb. (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> X xa \<or> siso xa) \<and> (\<forall>x xa xb xc. x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>c (xb, xc) \<longrightarrow> (\<exists>t'. xc \<approx> t' \<and> (c, xa) \<rightarrow>c (xb, t'))) \<and> (\<forall>x xa xb. x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>t xb \<longrightarrow> (\<exists>t'. xb \<approx> t' \<and> (c, xa) \<rightarrow>t t'))" shows "siso x"
Par_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D \<or> B A Par C D \<or> A B Par D C \<or> B A Par D C \<or> C D Par A B \<or> C D Par B A \<or> D C Par A B \<or> D C Par B A" shows "A B Par C D"
take_bit_Suc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit (Suc n) a = plus (times (take_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))) (numeral.numeral one plus (num.Bit0 num.One))) (modulo a (numeral.numeral one plus (num.Bit0 num.One)))"
oldUses_inv: fixes n :: "'node" assumes "n \<in> set (\<alpha>n g)" shows "oldUses g n = var g ` u_g g n"
assms_7: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
span_scale: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c" assumes "x \<in> phull S" shows "c \<cdot> x \<in> phull S"
h2: shows "\<forall>s t. s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, A) \<longrightarrow> \<not> (t \<circ> B \<le>\<^sub>\<K> B + C \<and> t \<circ> C \<le>\<^sub>\<K> B + C)"
WHILET_rule: fixes R :: "('c \<times> 'c) set"   and I :: "'c \<Rightarrow> bool"   and s :: "'c"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE\<^sub>T b f s \<le> SPEC \<Phi>"
pt_var1: fixes x :: "'a" shows "[] \<bullet> x = x"
l_zero: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "\<zero>\<^bsub>poly_ring R\<^esub> \<oplus>\<^bsub>poly_ring R\<^esub> x = x"
elims_3: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b tree" assumes "\<not> bst_wrt x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "\<not> ((\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r)"
extremum_unique: fixes a :: "nat" shows "(0 dvd a) = (a = 0)"
assms_4: shows "val_Zp a = val_Zp b"
rs_def: shows "rs = (\<lambda>b. {t. P t \<and> 0 < t \<and> t < b})"
iso_point_in_block_img_iff: fixes p :: "'a"   and bl :: "'a set" assumes "p \<in> \<V>"   and "bl \<in># \<B>" shows "(p \<in> bl) = (\<pi> p \<in> \<pi> ` bl)"
USdB_def: fixes d :: "'d" shows "\<approx>\<^bsub>d\<^esub> \<equiv> \<Union> {r. Strong_d_Bisimulation d r}"
up'_fl_fr: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and p_r :: "(nat \<times> int) list"   and p_l :: "(nat \<times> int) list"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < length p"   and "p = child p_r right d"   and "p = child p_l left d" shows "fst (up' d lm p \<alpha>) = (\<Sum>p'\<in>lgrid p {d} (lm + level p). \<alpha> p' * l2_\<phi> (p' ! d) (p_r ! d), \<Sum>p'\<in>lgrid p {d} (lm + level p). \<alpha> p' * l2_\<phi> (p' ! d) (p_l ! d))"
a_ni_c': fixes A :: "'a" assumes "A \<in> set As" shows "A \<notin> atms_of (C_of A)"
monoid_axioms: shows "monoid Partition ([\<cdot>]) (Class \<one>)"
hulls_inter: fixes S :: "'b list set set" shows "\<langle>\<Inter> {\<langle>G\<rangle> |G. G \<in> S}\<rangle> = \<Inter> {\<langle>G\<rangle> |G. G \<in> S}"
that: fixes v\<^sub>1 :: "value" assumes "rs, \<Gamma>\<^sub>1 \<turnstile>\<^sub>v t \<down> v\<^sub>1"   and "\<turnstile>\<^sub>v v\<^sub>1 \<approx> v\<^sub>2" shows "thesis"