hcju/isabelleps huggingface dataset:
i_def_4: shows "\<forall>j<i. \<not> cond ((v @ [s]) ! j)"
ExtCategoryMakeCat: fixes C :: "('d, 'e, 'f) Category_scheme" shows "ExtCategory (MakeCat C)"
atLeast_subset_iff: fixes x :: "'d"   and y :: "'d" shows "({x..} \<subseteq> {y..}) = (y \<le> x)"
append_eq_appendI: fixes xs :: "'b list"   and xs1 :: "'b list"   and zs :: "'b list"   and ys :: "'b list"   and us :: "'b list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
1: fixes x :: "'b"   and d :: "real" assumes "x \<in> S"   and "0 < d" shows "\<exists>i. i \<in> {i \<in> K. 0 < r i} \<and> x \<in> cball (a i) (r i) \<and> r i < d"
suffix_length_suffix: fixes ps :: "'a list"   and xs :: "'a list"   and qs :: "'a list" assumes "suffix ps xs"   and "suffix qs xs"   and "length ps \<le> length qs" shows "suffix ps qs"
expl_cond_exp_integrable: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "integrable M X"   and "disc_fct Y"   and "point_measurable M (space N) Y" shows "integrable M (expl_cond_expect M Y X)"
Zorns_po_lemma: fixes r :: "('a \<times> 'a) set" assumes "Partial_order r"   and "\<And>C. C \<in> Chains r \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
simps: fixes x :: "'a ext0"   and y :: "'a ext0" shows "x * y = x \<sqinter> y"
8: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<squnion> y"
td_ext_uint: shows "td_ext uint word_of_int (uints LENGTH('a)) (\<lambda>w. w mod 2 ^ LENGTH('a))"
characteristic_weak_formula_card_aux: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "|{indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q |Q. \<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q}| <o natLeq +c |UNIV|"
inf_sup_ord_4: fixes y :: "'c \<times> 'd"   and x :: "'c \<times> 'd" shows "less_eq_prod y (sup_prod x y)"
finite_dimensionE': fixes K :: "'a set"   and E :: "'a set" assumes "finite_dimension K E" obtains n :: "nat" where "dimension n K E"
irreducible\<^sub>dI: fixes p :: "'b poly" assumes "0 < degree p"   and "\<And>q r. \<lbrakk>0 < degree q; degree q < degree p; 0 < degree r; degree r < degree p; p = q * r\<rbrakk> \<Longrightarrow> False" shows "irreducible\<^sub>d p"
derives_implies_leftderives_gen: fixes a :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "derives a (u @ v)"   and "is_word u" shows "\<exists>w. leftderives a (u @ w) \<and> (v = [] \<longrightarrow> w = []) \<and> (\<forall>X. is_first X v \<longrightarrow> is_first X w)"
compactI: fixes "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set" assumes "class.topological_space open"   and "\<And>C. \<lbrakk>\<forall>t\<in>C. open t; s \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'" shows "topological_space.compact open s"
ord_rec_def: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf"   and x :: "hf" shows "ord_rec T G H x = (THE y. ord_recp T G H x y)"
streett_leadsto: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "\<turnstile> (\<box>\<diamond>Init F \<longrightarrow> \<box>\<diamond>G) = (\<diamond>(F \<leadsto> G))"
collection_binop_type_det: fixes op :: "collection_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "collection_binop_type op \<tau> \<sigma> \<rho>\<^sub>1"   and "collection_binop_type op \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
path_components_of_maximal: fixes C :: "'a set"   and X :: "'a topology"   and S :: "'a set" assumes "C \<in> path_components_of X"   and "path_connectedin X S"   and "\<not> disjnt C S" shows "S \<subseteq> C"
hyps_4: fixes \<rho> :: "Vars.var \<Rightarrow> Value" assumes "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "\<lbrakk> body \<rbrakk>\<^bsub>\<lbrace>as @ \<Gamma>\<rbrace>\<rho>\<^esub> = \<lbrakk> v \<rbrakk>\<^bsub>\<lbrace>\<Delta>\<rbrace>\<rho>\<^esub>"
trivial_RSubmodule: shows "RModule FG (\<cdot>) 0 \<and> 0 \<subseteq> V"
associativity: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
between_def: fixes vs :: "'c list"   and ram\<^sub>1 :: "'c"   and ram\<^sub>2 :: "'c" shows "between vs ram\<^sub>1 ram\<^sub>2 \<equiv> let (pre\<^sub>1, post\<^sub>1) = splitAt ram\<^sub>1 vs in if ram\<^sub>2 \<in> set post\<^sub>1 then let (pre\<^sub>2, post\<^sub>2) = splitAt ram\<^sub>2 post\<^sub>1 in pre\<^sub>2 else let (pre\<^sub>2, post\<^sub>2) = splitAt ram\<^sub>2 pre\<^sub>1 in post\<^sub>1 @ pre\<^sub>2"
gsfw_join_valid: fixes f1 :: "('b simple_match \<times> 'c) list"   and f2 :: "('b simple_match \<times> 'd) list" assumes "gsfw_valid f1"   and "gsfw_valid f2" shows "gsfw_valid (generalized_fw_join f1 f2)"
hyps: shows "[a\<frown>b]Q \<longmapsto> \<alpha> \<prec> Q'"
coinduct_strong: fixes R :: "'a tree \<Rightarrow> 'a tree \<Rightarrow> bool"   and tree :: "'a tree"   and tree' :: "'a tree" assumes "R tree tree'"   and "\<And>tree tree'. R tree tree' \<Longrightarrow> root tree = root tree' \<and> (R (left tree) (left tree') \<or> left tree = left tree') \<and> (R (right tree) (right tree') \<or> right tree = right tree')" shows "tree = tree'"
dim_mono: fixes V :: "'b set"   and W :: "'b set" assumes "V \<subseteq> span W" shows "dim V \<le> dim W"
table_Un: fixes n :: "nat"   and V :: "nat set"   and X :: "'a option list set"   and Y :: "'a option list set" assumes "table n V X"   and "table n V Y" shows "table n V (X \<union> Y)"
recurrence_time_multiples: fixes T :: "real"   and x :: "'a"   and t :: "real"   and n :: "nat" assumes "T \<in> existence_ivl0 x"   and "T \<noteq> 0"   and "flow0 x T = x" shows "flow0 x (t + T * real n) = flow0 x t"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool" assumes "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2"   and "FWdelay_bisimulation_lift_axioms r1 r2 bisim \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_lift final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"
modular_2_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> (x \<cdot> z ; y\<^sup>T) ; y"
simps_4: fixes v :: "'d"   and va :: "('d, 'c) Term.term list"   and x :: "'c"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and bs :: "('c \<times> ('d, 'c) Term.term) list" shows "unify ((Fun v va, Var x) # E) bs = (if x \<in> vars_term (Fun v va) then None else unify (subst_list (subst x (Fun v va)) E) ((x, Fun v va) # bs))"
in_rtrancl_def: fixes TI :: "('b \<times> 'b) list"   and a :: "'b"   and b :: "'b" shows "in_rtrancl TI a b \<equiv> a = b \<or> in_trancl TI a b"
scale_minus_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c" assumes "module scale" shows "scale a (- x) = - scale a x"
rv_times_rv: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "(\<lambda>w. f w * g w) \<in> rv M"
p_e_non_empty: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "r \<in> p_e R M N" shows "fst r \<noteq> []"
ln_inj_iff: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "(ln x = ln y) = (x = y)"
summable_LIMSEQ': fixes f :: "nat \<Rightarrow> 'e" assumes "summable f" shows "(\<lambda>n. sum f {..n}) \<longlonglongrightarrow> suminf f"
norm_le_in_cubeI: fixes x :: "'b"   and y :: "'b" assumes "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i\<bar> \<le> \<bar>y \<bullet> i\<bar>" shows "norm x \<le> norm y"
raise_bind: fixes e :: "String.literal"   and f :: "'b \<Rightarrow> 'a Heap" shows "raise e \<bind> f = raise e"
spies_evs_rev: fixes evs :: "event list" shows "knows Spy evs = knows Spy (rev evs)"
pruned: fixes v :: "'d" assumes "v |\<in>| vertices" shows "\<exists>pth v'. path v v' pth \<and> terminal_vertex v'"
ortho: shows "orthogonal_current (web_of_network \<Delta>) f S"
order_ow_axioms: shows "order_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"
mult_right_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "times x bot = bot"
p_coprime_right: fixes p :: "nat"   and a :: "nat" assumes "residues_prime p" shows "coprime a p = (\<not> p dvd a)"
inject_6: fixes x6 :: "arith_type"   and y6 :: "arith_type" shows "(arith_type x6 = arith_type y6) = (x6 = y6)"
right_scalingD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and c :: "real"   and s :: "'a" assumes "scaling t"   and "sound P"   and "0 \<le> c" shows "t P s * c = t (\<lambda>s. P s * c) s"
invertible_mat_zero: fixes A :: "'a mat" assumes "A \<in> carrier_mat 0 0" shows "invertible_mat A"
col_swaprows_4: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n" shows "col (swaprows_mat n k l) k = unit_vec n l"
measure_preserving_preserves_nn_integral: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "T \<in> measure_preserving M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>N N f = \<integral>\<^sup>+ x. f (T x) \<partial>M"
mono_exp_of_derived: fixes K :: "'c set"   and H :: "'c set"   and n :: "nat" assumes "K \<subseteq> H" shows "(derived H ^^ n) K \<subseteq> (derived H ^^ n) H"
dg_Par_components_2: fixes \<alpha> :: "V" shows "dg_Par \<alpha>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {T. arr_Par \<alpha> T}"
vrestr_meas: fixes g :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "g \<in> M \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "g --` A \<in> sets M"
simps_6: fixes x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "fos (ws1s.Ex1 x \<phi>) = remove1 x (fos \<phi>)"
Store_def: shows "Store = {s. (\<forall>l. aliveImpl (vals s l) s) \<and> (\<forall>l. \<not> aliveImpl (ref l) s \<longrightarrow> vals s l = init (ltype l)) \<and> (\<forall>l. \<tau> vals s l \<le> ltype l)}"
sfirst_eq_0: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "(sfirst P \<omega> = 0) = P \<omega>"
cospan: shows "trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)"
power_polys_i: fixes i :: "nat"   and n :: "nat"   and f :: "'i list"   and f' :: "'a mod_ring poly"   and g :: "'i list"   and g' :: "'a mod_ring poly"   and h :: "'i list"   and h' :: "'a mod_ring poly" assumes "i < n"   and "poly_rel f f'"   and "poly_rel g g'"   and "poly_rel h h'" shows "poly_rel (power_polys_i ff_ops g f h n ! i) (power_polys g' f' h' n ! i)"
map_prod_simp: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "map_prod f g (a, b) = (f a, g b)"
preserves_subset_stable: fixes v :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" shows "preserves v \<subseteq> stable {s. P (v s)}"
simps: fixes P :: "nat exp prog"   and a1 :: "nat exp list"   and a2 :: "heap \<times> val list \<times> sheap"   and a3 :: "nat exp list"   and a4 :: "heap \<times> val list \<times> sheap" shows "P \<turnstile>\<^sub>1 \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle> = ((\<exists>s. a1 = [] \<and> a2 = s \<and> a3 = [] \<and> a4 = s) \<or> (\<exists>e s\<^sub>0 v s\<^sub>1 es es' s\<^sub>2. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = Val v # es' \<and> a4 = s\<^sub>2 \<and> P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle> \<and> P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>) \<or> (\<exists>e s\<^sub>0 e' s\<^sub>1 es. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = throw e' # es \<and> a4 = s\<^sub>1 \<and> P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>))"
mult_add_distrib_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "C \<in> carrier_mat n nc" shows "A * (B + C) = A * B + A * C"
compl: fixes s :: "'a set" assumes "s \<in> C \<inter> sigma_sets \<Omega> M" shows "\<Omega> - s \<in> C"
reuse_matches: fixes j :: "nat"   and s :: "'a list"   and t :: "'a list"   and i :: "nat" assumes "j \<le> length s"   and "\<forall>jj<j. t ! (i + jj) = s ! jj" shows "\<forall>jj<\<ff> s j - 1. t ! (i + (j - \<ff> s j + 1) + jj) = s ! jj"
p: shows "\<not> real_of_int j rdvd real_of_int (c * i) + Inum (real_of_int i # bs) e"
simps_3: fixes I :: "\<I>"   and v :: "event_data option list set"   and va :: "event_data option list set list"   and t :: "nat" shows "mprev_next I (v # va) [t] = ([], v # va, [t])"
le_transD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "le_trans t u"   and "sound P" shows "t P \<tturnstile> u P"
homotopic_paths_reversepath: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_paths s (reversepath p) (reversepath q) = homotopic_paths s p q"
e_app_elim: fixes v3 :: "val"   and e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v3 \<in> E (EApp e1 e2) \<rho>" obtains f :: "(val \<times> val) list" and v2 :: "val" and v2' :: "val" and v3' :: "val" where "VFun f \<in> E e1 \<rho>" and "v2 \<in> E e2 \<rho>" and "(v2', v3') \<in> set f" and "v2' \<sqsubseteq> v2" and "v3 \<sqsubseteq> v3'"
tendsto_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c"   and g :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
ring_axioms: shows "ring additive.Partition ([+]) ([\<cdot>]) (additive.Class \<zero>) (additive.Class \<one>)"
sat\<^sub>b_RESTRICT: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "sat\<^sub>b \<AA> \<phi> = (\<AA> \<Turnstile>\<^sub>b RESTRICT \<phi>)"
induct: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "('idx, 'pred, 'act) formula \<Rightarrow> bool" assumes "weak_formula x"   and "\<And>xset. \<lbrakk>finite (supp xset); \<And>x. x \<in> set_bset xset \<Longrightarrow> weak_formula x; \<And>x. x \<in> set_bset xset \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P (Conj xset)"   and "\<And>x. \<lbrakk>weak_formula x; P x\<rbrakk> \<Longrightarrow> P (Formula.Not x)"   and "\<And>x \<alpha>. \<lbrakk>weak_formula x; P x\<rbrakk> \<Longrightarrow> P \<langle>\<langle>\<alpha>\<rangle>\<rangle>x"   and "\<And>x \<phi>. \<lbrakk>weak_formula x; P x\<rbrakk> \<Longrightarrow> P \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x))" shows "P x"
mat_rearrange: fixes F :: "'d \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'c"   and i :: "'d"   and j :: "'e" assumes "finite UNIV"   and "finite UNIV" shows "\<Sum> {\<Sum> {F i k1 k2 j |k2. k2 \<in> UNIV} |k1. k1 \<in> UNIV} = \<Sum> {\<Sum> {F i k1 k2 j |k1. k1 \<in> UNIV} |k2. k2 \<in> UNIV}"
interaction_bounded_by_start: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
is_unit_div_mult_cancel_left: 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 a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times b one" shows "divide a (times a b) = divide one b"
cp_6: fixes p' :: "complex" assumes "p' \<notin> s - pts" shows "winding_number cp p' = 0 \<and> p' \<notin> path_image cp"
convergent_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "convergent (\<lambda>n. f (Suc n)) = convergent f"
mono_Sup: 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 \<Rightarrow> 'c) set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> {f. order.mono less_eq f}" shows "order.mono less_eq (\<Squnion>A)"
exec: shows "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
dual_equivalence: assumes "adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F \<epsilon>'.map \<eta>'.map" shows "adjoint_equivalence (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F G (inverse_transformation.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map \<eta>'.map) (inverse_transformation.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) Adj.FG.map \<epsilon>'.map)"
subspace_scale: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c"   and c :: "'b" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S" shows "scale c x \<in> S"
exec_move_SyncI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and V :: "nat"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (sync\<^bsub>V\<^esub> (e) e') h s ta h' s'"
decode_DIMACS_model'_def: fixes dimacs_M :: "int list"   and h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "decode_DIMACS_model' dimacs_M h prob \<equiv> decode_DIMACS_model dimacs_M h (rem_implicit_pres_ops prob)"
t_ins_form_aux: fixes x :: "'b"   and xt :: "'b bintree" shows "t_ins_form (t_ins_aux \<lparr>t_type.folding = False, item = x, subtrees = [xt]\<rparr>)"
s1: shows "\<forall>y. (- 1 * a * y\<^sup>2 + - 1 * b * y + - 1 * c < 0) = (0 < a * y\<^sup>2 + b * y + c)"
final2_simulation: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2"   and "final2 s2" shows "\<exists>s1'. s1 -\<tau>1\<rightarrow>* s1' \<and> s1' \<approx> s2 \<and> final1 s1'"
RELATES_def: fixes R :: "('c \<times> 'd) set" shows "RELATES R \<equiv> True"
confc': shows "conf_clinit P sh ((stk, loc, C, M, pc, Throwing Cs a) # frs)"
homotopy_equivalent_space_sym: fixes X :: "'c topology"   and Y :: "'d topology" shows "(X homotopy_equivalent_space Y) = (Y homotopy_equivalent_space X)"
cross3_simps_1: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" shows "cross3 a b \<equiv> vector [a $ 2 * b $ 3 - a $ 3 * b $ 2, a $ 3 * b $ 1 - a $ 1 * b $ 3, a $ 1 * b $ 2 - a $ 2 * b $ 1]"
spectral_radius_def: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "HMA_Connect.spectral_radius A = Max {uu_. \<exists>v ev. uu_ = cmod ev \<and> eigen_vector A v ev}"
n_def: shows "n = prefix_length ((=) (0::'a)) (coeffs p)"
fps_deriv_mult: fixes f :: "'b fps"   and g :: "'b fps" shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
acc_subset: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set" assumes "R1 \<subseteq> R2" shows "Wellfounded.acc R2 \<subseteq> Wellfounded.acc R1"
conjugate_int_def: fixes x :: "int" shows "conjugate x \<equiv> x"
assms_1: shows "g \<in> \<R>\<^sup>2"
diff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "Modules.additive f" shows "f (x - y) = f x - f y"
sccs_conv_sccs_verts: shows "H.sccs = (\<restriction>) H ` H.sccs_verts"
iwf_it: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c \<in> set conclusions" shows "Abstract_Task.iwf freshenLC renameLCs subst anyP antecedent consequent rules assumptions conclusions Abstract_Task.no_fresh_check (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c))) (fst (tree.root (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))"
simps_1: fixes tset :: "('a, 'c, 'b) Tree set['a]" shows "S_transform_Tree (tConj tset) = Conj\<^sub>\<alpha> (map_bset S_transform_Tree tset)"
less_eq_interval_def: fixes a :: "'b interval"   and b :: "'b interval" shows "(a \<le> b) = (lower b \<le> lower a \<and> upper a \<le> upper b)"
hensel_5: shows "\<forall>n>0. degree (hensel_fpxs2 $ n) < degree h"
numeral_code_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = (let m = numeral.numeral one plus n in plus m m)"
N_def: fixes V :: "'b set"   and L :: "'b \<Rightarrow> nat"   and i :: "nat" shows "N V L i = card {v \<in> V. L v = i}"
b'_insertion2: shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 1) = b'"
simps_2: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and e :: "('d, 'e, 'b) exp"   and es :: "('d, 'e, 'b) exp list" shows "J1_heap_base.\<tau>moves1 typeof_addr P h (e # es) = (J1_heap_base.\<tau>move1 typeof_addr P h e \<or> (\<exists>v. e = Val v \<and> J1_heap_base.\<tau>moves1 typeof_addr P h es))"
le_acc_def: fixes a :: "acc_modi"   and b :: "acc_modi" shows "(a \<le> b) = (a < b \<or> a = b)"