hcju/isabelleps huggingface dataset:
SD_singleton: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier"   and "y \<in> carrier" shows "SD le (return_pmf x) (return_pmf y) = le x y"
false_Guard1: fixes g :: "'c \<Rightarrow> bool"   and a :: "('d, 'c) alpha_rp_scheme"   and P :: "('d, 'c) action"   and b :: "('d, 'c) alpha_rp_scheme" assumes "g (alpha_rp.more a) = False" shows "action.relation_of g `&` P (a, b) = action.relation_of Stop (a, b)"
Rats_no_bot_less: fixes x :: "real" shows "\<exists>q\<in>\<rat>. q < x"
"1_2": shows "feasible (init_conf prb) (trace es (labelling (black prb)))"
u\<^sub>f_def: shows "u\<^sub>f = g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0"
scaleRat_less1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less a b"   and "0 < k" shows "less (scaleRat k a) (scaleRat k b)"
max_absorb2: fixes x :: "'h"   and y :: "'h" assumes "x \<le> y" shows "max x y = y"
map_apply_def: shows "map_apply = (\<lambda>f x. the (f x))"
ccompare_uprod_def: shows "ccompare = map_option uprod_compare ccompare_comparator"
simps_4: fixes a :: "char"   and as :: "char list"   and b :: "char"   and bs :: "char list" shows "(Iface (a # as) < Iface (b # bs)) = (if a = b then Iface as < Iface bs else a < b)"
diff_left_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq b a" shows "less_eq (minus c a) (minus c b)"
filters_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.filters less_eq = {X. X \<noteq> {} \<and> X \<in> ord.upsets less_eq \<and> ord.filtered less_eq X}"
statement52_existence: fixes a3 :: "proj2"   and a :: "(proj2, 3) vec" assumes "proj2_no_3_Col (insert a3 (range (($) a)))" shows "\<exists>A. apply_cltn2 (proj2_abs (vector [1, 1, 1])) A = a3 \<and> (\<forall>j. apply_cltn2 (proj2_abs (axis j 1)) A = a $ j)"
pos_emb_step_at_opp: fixes t :: "('c, 'd) tm"   and p :: "dir list"   and d1 :: "dir"   and q :: "dir list"   and d2 :: "dir" shows "position_of t (p @ [d1]) = position_of (emb_step_at (p @ [opp d1] @ q) d2 t) (p @ [d1])"
length4D: fixes xs :: "'a list" assumes "|xs| = 4" shows "\<exists>a b c d. xs = [a, b, c, d]"
weight_pair_spmf: fixes p :: "'c spmf"   and q :: "'d spmf" shows "weight_spmf (pair_spmf p q) = weight_spmf p * weight_spmf q"
STRUCT_EQ_tag_def: fixes x :: "'g"   and y :: "'g" shows "STRUCT_EQ_tag x y \<equiv> x = y"
simps_1: fixes n :: "nat"   and a :: "'a"   and m :: "nat" shows "pre_wf_formula n (FQ a m) = (a \<in> set \<Sigma> \<and> m < n)"
diff_gt_0_iff_gt_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(0 < a - b) = (a = top \<and> b = top \<or> b < a)"
bit_exp_iff: 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 m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero \<and> m = n)"
assms_2: shows "c \<notin>\<^sub>\<circ> B"
deterministic_def: fixes e :: "((nat \<times> nat) \<times> transition) fset" shows "deterministic e = (\<forall>s r l i. size (possible_steps e s r l i) \<le> 1)"
fls_deriv_const: fixes x :: "'a" shows "fls_deriv (fls_const x) = 0"
aux: fixes k :: "nat"   and i :: "nat" assumes "round (rho (k + i)) p = round (rho k) p" shows "state (rho (k + i)) p = state (rho k) p"
inject: fixes x1 :: "'a literal multiset multiset"   and x2 :: "'a literal multiset"   and x3 :: "'a literal multiset"   and y1 :: "'a literal multiset multiset"   and y2 :: "'a literal multiset"   and y3 :: "'a literal multiset" shows "(Infer x1 x2 x3 = Infer y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
FAcc1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and F :: "String.literal"   and D :: "String.literal" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e\<bullet>F{D},s\<rangle> -ta\<rightarrow> \<langle>e'\<bullet>F{D},s'\<rangle>"
get_ancestors_si_locs_def: shows "get_ancestors_si_locs = get_parent_locs \<union> get_host_locs"
cong: fixes ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "ordered_term.ord_p ord_term_strict \<equiv> ordered_term.ord_p ord_term_strict"
ereal_less_4: fixes r :: "real" shows "(1 < ereal r) = (1 < r)"
green_typeI_chain_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" shows "green_typeI_chain i j F two_chain s \<equiv> R2 i j \<and> green_typeI_chain_axioms i j F two_chain s"
plus_period: fixes x :: "'b" shows "cos (x + (2::'b) * of_real pi) = cos x"
type_less_x_Tuple_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> = Tuple \<pi>"   and "strict_subtuple (\<le>) \<pi> \<xi>" shows "\<tau> < Tuple \<xi>"
da_shunt2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> ad y + z" shows "x \<cdot> d y \<le> z"
simps: fixes "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var"   and r :: "('form, 'rule, 'subst, 'var) itnode"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" shows "globalize is f (RNode r ants) = RNode (globalize_node is f r) (mapWithIndex (\<lambda>i'. globalize (is @ [i']) (rerename (a_fresh (inPorts' (iNodeOf (RNode r ants)) ! i')) (iAnnot (RNode r ants)) (isidx is) f)) ants)"
interchange_rows_i: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "interchange_rows A i j $ i = A $ j"
SN_iff_wf: fixes A :: "('b \<times> 'b) set" shows "SN A = wf (A\<inverse>)"
fvi_less_nfv: fixes \<phi> :: "'a MFOTL.formula" shows "\<forall>i\<in>fv \<phi>. i < MFOTL.nfv \<phi>"
fresh_7: fixes a :: "name"   and x2 :: "pi"   and x1 :: "pi" shows "a \<sharp> x2 \<oplus> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
state_block_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "state_block f x = f x"
inj_on_iff_eq_0: fixes s :: "'b set" assumes "m1.subspace s" shows "inj_on f s = (\<forall>x\<in>s. f x = (0::'c) \<longrightarrow> x = (0::'b))"
PQ_imp: fixes x :: "'a" assumes "z \<le> x"   and "\<forall>y\<ge>x. P y" shows "Q x"
porder_onI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "refl_onP A r"   and "antisymp r"   and "transp r" shows "porder_on A r"
fresherI: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "nsqn\<^sub>r r < nsqn\<^sub>r r' \<or> nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r" shows "r \<sqsubseteq> r'"
c_1: shows "c \<in> carrier (Ring_Divisibility.mult_of R)"
nonneg_cexpr_shift_iff2: fixes V :: "nat set"   and t1 :: "pdf_type"   and t2 :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" shows "nonneg_cexpr (shift_var_set (shift_var_set V)) (case_nat t1 (case_nat t2 \<Gamma>)) (map_vars (case_nat 0 (\<lambda>x. Suc (Suc x))) e) = nonneg_cexpr (shift_var_set V) (case_nat t1 \<Gamma>) e"
proj_sum: fixes k :: "'k"   and f :: "'d \<Rightarrow> 't \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "proj_poly k (sum f A) = (\<Sum>a\<in>A. proj_poly k (f a))"
composition_series_length_one: shows "(length \<GG> = 1) = (\<GG> = [{\<one>}])"
polynomial_incl: fixes K :: "'a list set"   and p :: "'a list list" assumes "polynomial\<^bsub>poly_ring R\<^esub> K p" shows "set p \<subseteq> K"
Ds_def: shows "Ds = (ARG_MIN length Ds. gallery (C # Ds @ [D]))"
bw_diff_gbinomial: fixes k :: "nat"   and n :: "int" assumes "0 < k" shows "bw_diff (\<lambda>x. x + n gchoose k) = (\<lambda>x. x + n - 1 gchoose (k - 1))"
necess_cond_2_rhs: shows "int (size {#x \<in># mset_set \<V> \<times># \<B>. fst x \<in> snd x#}) = \<b> * \<k>"
card_length: fixes xs :: "'a list" shows "card (set xs) \<le> length xs"
facet_of_imp_subset: fixes F :: "'b set"   and S :: "'b set" assumes "F facet_of S" shows "F \<subseteq> S"
assms_4: fixes var :: "Vars.var"   and c :: "'a"   and exp :: "exp" assumes "{atom var} \<sharp>* c"   and "\<And>c. P1 c exp" shows "P1 c (Lam [var]. exp)"
inner_sum_left: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b" shows "sum f A \<bullet> y = (\<Sum>x\<in>A. f x \<bullet> y)"
cases: fixes x :: "('a \<times> 'b) tree" obtains l :: "('a \<times> 'b) tree" and a :: "'a" and uu_ :: "'b" and r :: "('a \<times> 'b) tree" where "x = \<langle>l, (a, uu_), r\<rangle>"   | "x = \<langle>\<rangle>"
last_conv_nth: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - 1)"
insert_absorb2: fixes x :: "'b"   and A :: "'b set" shows "insert x (insert x A) = insert x A"
half_gt_zero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
match_implicit_pres_def: fixes effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and s :: "nat \<Rightarrow> nat option" shows "match_implicit_pres effs s \<equiv> \<forall>(uu_, x, vp, uu_)\<in>set effs. case vp of None \<Rightarrow> True | Some v \<Rightarrow> s x = Some v"
bdd: fixes x :: "'a" shows "bdd_above (range (\<lambda>i. f i x))"
parts_sep: fixes s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and s :: "(agent \<times> msg) set"   and Z :: "msg" assumes "s' = insert (Spy, X) (insert (Spy, Y) s) \<and> (Spy, \<lbrace>X, Y\<rbrace>) \<in> s"   and "Z \<in> parts_msg X \<or> Z \<in> parts_msg Y" shows "Z \<in> parts (used s)"
not_ccw_eq: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "distinct3 p q r"   and "in3 S p q r" shows "(\<not> ccw p q r) = ccw p r q"
Map_preserves_objects: fixes a :: "('a E.term, 'a) arr" assumes "obj a" shows "B.obj (Map a)"
simps_3: fixes A :: "hfm"   and B :: "hfm" shows "closed_hfm (A HOR B) = (closed_hfm A \<and> closed_hfm B)"
substAbs_compose2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "goodAbs A"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "A $[Y / y]_ys $[Z / z]_zs = A $[Z / z]_zs $[Y #[Z / z]_zs / y]_ys"
P_D2_H2_h_substitutes: fixes h :: "H2" shows "substitutes (P_D2_H2_h h)"
AB: shows "A \<subseteq> carrier G \<and> B \<subseteq> carrier G"
assms_1: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
set_eq2: shows "{x \<in> carrier (mult_of R). G.ord x = d} = ([^]) a ` {n \<in> {1..d}. G.ord (a [^] n) = d}"
initState_CA: fixes i :: "nat" shows "initState (CA i) = (if i = 0 then Key ` ({priEK RCA, priSK RCA} \<union> pubEK ` range CA \<union> pubSK ` range CA) else {Key (priEK (CA i)), Key (priSK (CA i)), Key (pubEK (CA i)), Key (pubSK (CA i)), Key (pubEK RCA), Key (pubSK RCA)})"
greaterThan_subset_iff: fixes x :: "'a"   and y :: "'a" shows "({x<..} \<subseteq> {y<..}) = (y \<le> x)"
inL2: fixes F :: "'a formula"   and G :: "'a formula"   and H :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "F, G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "G, H, F, \<Gamma> \<Rightarrow> \<Delta>"
diff2_2: shows "local.rev_curr f m \<in> rs.FINFUNC"
minus_one_power_one_word: fixes x :: "1 word" shows "(- 1) ^ unat x = (if unat x = 0 then 1 else - 1)"
sfilter_const_FF: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "sfilter\<cdot>(\<Lambda> x. FF)\<cdot>xs = [::]"
set_of_map_def: fixes m :: "'a \<Rightarrow> 'b option" shows "set_of_map m = {(k, v) |k v. m k = Some v}"
all_bool_eq: fixes P :: "bool \<Rightarrow> bool" shows "(\<forall>b. P b) = (P True \<and> P False)"
obj_def': fixes a :: "'a" shows "B.obj a = (B.arr a \<and> trg\<^sub>B a = a)"
d_sub_identity: fixes x :: "'a" shows "d x \<le> top"
nonzero_inverse_mult_distrib: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<noteq> \<zero>"   and "b \<noteq> \<zero>" shows "inv (a \<otimes> b) = inv b \<otimes> inv a"
RootExSem_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "RootExSem (SAs (HA ST)) (CompFun (HA ST)) (StepConf (HA ST) (Conf ST) TS)"
subst_fv_dom_ground_if_ground_img: fixes t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<subseteq> subst_domain s"   and "ground (subst_range s)" shows "fv (t \<cdot> s) = {}"
geodesic_segment_param_6: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "dist x (geodesic_segment_param G x t) = t"
correctCompositionDiffLevelsA6: shows "correctCompositionDiffLevels sA6"
arbitrage_val_process: fixes Mkt :: "('a, 'c) discrete_market"   and pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "arbitrage_process Mkt pf1"   and "self_financing Mkt pf2"   and "trading_strategy pf2"   and "\<forall>n w. cls_val_process Mkt pf1 n w = cls_val_process Mkt pf2 n w" shows "arbitrage_process Mkt pf2"
mask_shiftl_decompose: fixes m :: "nat"   and n :: "nat" shows "mask m << n = mask (m + n) AND NOT (mask n)"
reach: fixes x :: "CValue" shows "(\<Squnion>n. CValue_take n\<cdot>x) = x"
iterates_above_successor: fixes y :: "'a"   and a :: "'a" assumes "y \<in> iterates_above a"   and "a \<in> Field leq" shows "y = a \<or> y \<in> iterates_above (f a)"
edka4_refine: shows "edka4 \<le> \<Down> Id edka3"
continuous_attains_inf: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "compact s"   and "s \<noteq> {}"   and "continuous_on s f" shows "\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y"
bezout_matrix_JNF_def: fixes A :: "'b mat"   and a :: "nat"   and b :: "nat"   and j :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "bezout_matrix_JNF A a b j bezout = Matrix.mat (dim_row A) (dim_row A) (\<lambda>(x, y). let (p, q, u, v, d) = bezout (A $$ (a, j)) (A $$ (b, j)) in if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then 1::'b else (0::'b))"
recurrent_subset_infty_def': fixes A :: "'a set" shows "recurrent_subset_infty A = (\<Inter>m. \<Union>n\<in>{m..}. A \<inter> (T ^^ n) -` A)"
find_None_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
ldrop_eq_LNil: fixes n :: "enat"   and xs :: "'b llist" shows "(ldrop n xs = LNil) = (llength xs \<le> n)"
LT_def: shows "LT \<equiv> Ordering_abs\<cdot>(sinl\<cdot>ONE)"
simps_2: fixes C :: "'a set" shows "wall_crossings [C] = []"
\<gamma>_in_vhom: shows "\<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright>"
filtrated_prob_space_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" shows "filtrated_prob_space M F \<equiv> prob_space M \<and> filtrated_prob_space_axioms M F"
not_le: fixes y :: "'a list"   and x :: "'a list" shows "(\<not> y \<le>lex x) = (x <lex y)"
listpr_def: shows "listpr = prod_list (nths (dims Aw) (Collect even))"
Class_widen: fixes P :: "'a prog"   and C :: "String.literal"   and T :: "ty" assumes "P \<turnstile> Class C \<le> T" shows "\<exists>D. T = Class D"
operator_effect__strips_2: fixes s' :: "'a \<Rightarrow> bool option"   and s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and v :: "'a" assumes "s' = s \<then> op"   and "v \<notin> set (add_effects_of op) \<and> v \<in> set (delete_effects_of op)" shows "s' v = Some False"
less_fun: fixes v :: "val"   and f :: "(val \<times> val) fset" assumes "v \<sqsubseteq> val.Fun f" obtains f' :: "(val \<times> val) fset" where "v = val.Fun f'" and "fset f' \<subseteq> fset f"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> ns_shared" shows "K \<notin> keysFor (parts (knows Spy evs))"
continuous_iff_lsc_usc: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "isCont f x0 = (lsc_at x0 f \<and> usc_at x0 f)"
Int_Un_eq_2: fixes S :: "'c set"   and T :: "'c set" shows "S \<inter> T \<union> T = T"
le_imp_diff_is_add: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "a \<le> b" shows "(b - a = c) = (b = c + a)"
Pdrop_conforms: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "vmc_path G P v0 p \<sigma>" shows "ParityGame.path_conforms_with_strategy G p (ldropn n P) \<sigma>"
lift_const: fixes f :: "nat \<Rightarrow> real"   and z :: "nat"   and C :: "real" shows "insertion f (liftPoly 0 z (Const C)) = insertion f (Const C)"
dom_comp: fixes m :: "'f \<Rightarrow> 'e option"   and n :: "'d \<Rightarrow> 'f option" shows "dom (m \<circ>\<^sub>m n) \<subseteq> dom n"
is_zero_uminus_double: fixes x :: "double" shows "Code_Double.is_zero (- x) = Code_Double.is_zero x"
simps: fixes xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "lookup_raw (xs, ko) = lookup_pair ko xs"
dual_max_set: shows "ord.max_set (\<lambda>x y. y \<le> x) = min_set"
length_append: fixes xs :: "'c list"   and ys :: "'c list" shows "length (xs \<cdot> ys) = length xs + length ys"
find_consistent_signs_at_roots_len1: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "length qs = 1"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "set (find_consistent_signs_at_roots p qs) = set (characterize_consistent_signs_at_roots_copr p qs)"
COMPARE_intros_2: fixes y :: "real"   and x :: "real" assumes "y < x" shows "COMPARE x y \<equiv> GT"
inv_fg_cancel: fixes l :: "(bool \<times> 'd) list" shows "Cancelation.normalize (l @ inv_fg l) = []"
ludwig_waldmann_less: fixes \<alpha>2 :: "hmultiset"   and \<beta>2 :: "hmultiset"   and \<gamma> :: "hmultiset"   and \<alpha>1 :: "hmultiset"   and \<beta>1 :: "hmultiset"   and \<delta> :: "hmultiset" assumes "\<alpha>2 + \<beta>2 * \<gamma> < \<alpha>1 + \<beta>1 * \<gamma>"   and "\<beta>2 \<le> \<beta>1"   and "\<gamma> < \<delta>" shows "\<alpha>2 + \<beta>2 * \<delta> < \<alpha>1 + \<beta>1 * \<delta>"
integral_mono: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
R: shows "\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T)"
simps_2: fixes n :: "nat"   and k :: "nat" shows "fold_reduce n (Suc k) = fold_reduce n k \<circ> reduce (seqseq (n + k)) (n + k)"
eff_def: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and et :: "ex_table"   and t :: "(ty list \<times> ty err list) option" shows "eff i P pc et t = (case t of None \<Rightarrow> [] | \<lfloor>\<tau>\<rfloor> \<Rightarrow> norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> et)"
intTF_def: fixes \<sigma> :: "'b"   and a :: "'c" shows "intTF \<sigma> a \<equiv> True"
meet_idem: fixes x :: "'b" shows "x \<sqinter> x = x"
neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" obtains "x < y"   | "y < x"
seqr_mono: fixes P\<^sub>2 :: "('a, 'b) urel"   and P\<^sub>1 :: "('a, 'b) urel"   and Q\<^sub>2 :: "('b, 'c) urel"   and Q\<^sub>1 :: "('b, 'c) urel" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 ;; Q\<^sub>1 \<sqsubseteq> P\<^sub>2 ;; Q\<^sub>2"
assms_1: fixes r :: "'rule"   and s :: "'sequent"   and sl :: "'sequent fset"   and a :: "'a" assumes "r \<in> R"   and "eff r s sl"   and "inv (s, a)" shows "\<exists>s' a'. s' |\<in>| sl \<and> inv (s', a') \<and> pred (s, a) r (s', a')"
exhaust: fixes y :: "'b llist" obtains "y = <>"   | x21 :: "'b" and x22 :: "'b llist" where "y = x21 % x22"
IH: fixes M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" assumes "slp_of_fas' fas M slp = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs" shows "\<Union> (subterms ` set fas) \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
degree_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "deg R (a \<odot>\<^bsub>UP R\<^esub> X_poly R [^]\<^bsub>UP R\<^esub> n) = (if a = \<zero> then 0 else n)"
\<tau>move2_BlockNoneD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and T :: "ty"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk {V:T=None; e} pc xcp" shows "\<tau>move2 P h stk e pc xcp"
neg_less_eq_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) a = less_eq zero a"
invocationTypeExpr_noClassD: fixes E :: "env"   and e :: "expr"   and statT :: "ref_ty"   and m :: "'a member_scheme" assumes "E\<turnstile>e\<Colon>-RefT statT" shows "(\<forall>statC. statT \<noteq> ClassT statC) \<longrightarrow> invmode m e \<noteq> SuperM"