hcju/isabelleps huggingface dataset:
swapEnv_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "goodEnv (rho &[x \<and> y]_xs)"
find_None_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(List.find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
Sigma_empty1: fixes B :: "'d \<Rightarrow> 'e set" shows "Sigma {} B = {}"
pi_iteratei: fixes S :: "('b, 'c) RBT.rbt" shows "proper_it (rm.iteratei S) (rm.iteratei S)"
float_divl_pos_less1_bound: fixes x :: "float"   and prec :: "nat" assumes "0 < real_of_float x"   and "real_of_float x \<le> 1"   and "1 \<le> prec" shows "1 \<le> real_of_float (float_divl prec 1 x)"
row_space_def: fixes n :: "nat"   and B :: "'b mat" shows "vec_space.row_space n B = LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (rows B))"
cPar1F: fixes \<alpha> :: "freeRes"   and P' :: "pi"   and C :: "'a" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "F C (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"
rel_pmf_mono: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c pmf"   and g :: "'d pmf"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_pmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_pmf B f g"
continuous_map_of_snd: fixes X :: "'a topology"   and Y :: "'b topology"   and Z :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map (prod_topology X Y) Z (f \<circ> snd) = (topspace X = {} \<or> continuous_map Y Z f)"
pmf_bind: fixes N :: "'e pmf"   and f :: "'e \<Rightarrow> 'd pmf"   and i :: "'d" shows "pmf (N \<bind> f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
mv_box_inv_init: fixes m :: "nat"   and initlm :: "nat list"   and n :: "nat" assumes "m < length initlm"   and "n < length initlm"   and "m \<noteq> n" shows "mv_box_inv (0, initlm) m n initlm"
order_reflect_poly_ge: fixes c :: "'a"   and p :: "'a poly" assumes "c \<noteq> (0::'a)"   and "p \<noteq> 0" shows "order ((1::'a) / c) p \<le> order c (reflect_poly p)"
invL_FL_bisimilar_def: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" shows "invL_FL_bisimilar F P Q \<equiv> EF (F, P) \<sim>\<cdot>\<^sub>L EF (F, Q)"
eqvt_at_apply: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and p :: "perm" assumes "eqvt_at f x" shows "(p \<bullet> f) x = f x"
ustar_def: fixes P :: "'b hrel" shows "P\<^sup>\<star> = (\<Sqinter> i \<in> {0..} \<bullet> (power.power II (;;) P) i)"
Rice_rec_inseparable: fixes n :: "nat"   and m :: "nat" shows "rec_inseparable {k. nat_to_ce_set k = nat_to_ce_set n} {k. nat_to_ce_set k = nat_to_ce_set m}"
1: fixes x :: "real" assumes "- (pi / 2) < x"   and "x * 2 < pi" shows "(tan has_real_derivative 1 + (tan x)\<^sup>2) (at x)"
Mp_Mp: fixes f :: "int poly" shows "pn.eq_m (pn.Mp f) f"
f: shows "bij_betw f (set_pmf p) (set_pmf q)"
ff_def: shows "ff \<equiv> \<lambda>j. real_of_int (a j) powr (1 / real_of_int (2 ^ j))"
distinct_55: fixes x7 :: "('a, 'b, 'c) expr"   and x8 :: "('a, 'b, 'c) expr" shows "Rfork x7 \<noteq> Rjoin x8"
eq_cong_ax_holds: fixes e :: "(name, nat) finfun" shows "eval_fmla e eq_cong_ax"
bitlen_le_iff_floorlog: fixes x :: "int"   and w :: "int" shows "(bitlen x \<le> w) = (0 \<le> w \<and> floorlog 2 (nat x) \<le> nat w)"
eq_iff_18: fixes hash :: "hash"   and "term" :: "Syntax.term"   and hasha :: "hash"   and terma :: "Syntax.term" shows "(Hashed hash term = Hashed hasha terma) = (hash = hasha \<and> term = terma)"
drop_bit_half: 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 "drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One))) = divide (drop_bit n a) (numeral.numeral one plus (num.Bit0 num.One))"
pf_nonneg_mat_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "pf_nonneg_mat A \<equiv> non_neg_mat A"
finite_image_set: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "finite {x. P x}" shows "finite {f x |x. P x}"
chordal_circle_det_positive: fixes x :: "real"   and y :: "real" assumes "x * y < 0" shows "0 < x / (x - y)"
C: shows "x # xs \<cong> {ys, zs, P}"
card_flipped_reflections: fixes w :: "'w" assumes "w \<in> W" shows "card (flipped_reflections w) = S_length w"
elims: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list"   and y :: "'c list" assumes "class.linorder less_eq less"   and "linorder.msort less x = y" obtains "xa__ = []" and "y = []"   | x :: "'c" where "xa__ = [x]" and "y = [x]"   | v :: "'c" and vb :: "'c" and vc :: "'c list" where "xa__ = v # vb # vc" and "y = linorder.merge less (linorder.msort less (take (length (v # vb # vc) div 2) (v # vb # vc))) (linorder.msort less (drop (length (v # vb # vc) div 2) (v # vb # vc)))"
runiq_def: fixes R :: "('b \<times> 'c) set" shows "runiq R = (\<forall>X. trivial X \<longrightarrow> trivial (R `` X))"
intros_1: fixes \<S> :: "s_context" assumes "\<C> \<turnstile> [e1, e2] : t1s _> t2s" shows "\<S>\<bullet>\<C> \<turnstile> $* [e1, e2] : t1s _> t2s"
times_left_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and z :: "'d"   and y :: "'d"   and x :: "'d" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero z"   and "less_eq y x" shows "less_eq (times y z) (times x z)"
neutral_const: fixes A :: "'c multiset" shows "(\<Sum>_\<in>#A. (0::'a)) = (0::'a)"
log_base_change: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "log b x = log a x / log a b"
has_overlap_def: fixes S :: "'b ivl set"   and y :: "'b ivl" shows "has_overlap S y = (\<exists>x\<in>S. overlap x y)"
ce_rel_lm_21: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "\<exists>p\<in>PrimRec3. \<forall>x y. ((x, y) \<in> r) = (\<exists>u. p x y u = 0)"
topological_basis_imp_subbasis: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "class.topological_space open"   and "topological_space.topological_basis open B" shows "open = generate_topology B"
lfp_lowerbound: fixes a :: "'b set acom"   and i :: "com"   and f :: "'b set acom \<Rightarrow> 'b set acom" assumes "a \<in> {c'. strip c' = i}"   and "f a \<le> a" shows "Collecting.lfp f i \<le> a"
i_append_Cons: fixes x :: "'d"   and xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" shows "(x # xs) \<frown> f = [x] \<frown> xs \<frown> f"
WTrt_elim_cases_14: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'c expr"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'c expr"   and e3 :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (e\<bullet>compareAndSwap(D\<bullet>F, e2, e3)) T" obtains U :: "ty" and C :: "String.literal" and T :: "ty" and fm :: "fmod" and T2 :: "ty" and T3 :: "ty" where "Ta__ = Boolean" and "J_heap_base.WTrt typeof_addr P h E e U" and "class_type_of' U = \<lfloor>C\<rfloor>" and "P \<turnstile> C has F:T (fm) in D" and "volatile fm" and "J_heap_base.WTrt typeof_addr P h E e2 T2__" and "P \<turnstile> T2__ \<le> T" and "J_heap_base.WTrt typeof_addr P h E e3 T3__" and "P \<turnstile> T3__ \<le> T"   | T2 :: "ty" and T3 :: "ty" where "Ta__ = Boolean" and "J_heap_base.WTrt typeof_addr P h E e NT" and "J_heap_base.WTrt typeof_addr P h E e2 T2" and "J_heap_base.WTrt typeof_addr P h E e3 T3"
iso_char: fixes f :: "'a \<times> 'a" shows "VoV.iso f = (VxV.iso f \<and> VoV.arr f \<and> VoV.arr (VxV.inv f))"
rm_vars_dom_subset: fixes V :: "'a set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain (rm_vars V s) \<subseteq> subst_domain s"
ordLess_ordIso_trans: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set"   and r'' :: "('h \<times> 'h) set" assumes "r <o r'"   and "r' =o r''" shows "r <o r''"
dord_fun_trans: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "\<lbrakk>ord s t; ord t u\<rbrakk> \<Longrightarrow> ord s u"   and "dord_fun ord s t"   and "dord_fun ord t u" shows "dord_fun ord s u"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'var option"   and x3 :: "('pname \<times> bool) option"   and x4 :: "bool"   and x5 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'var option \<Rightarrow> ('pname \<times> bool) option \<Rightarrow> bool \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "sum_SDG_edge x1 x2 x3 x4 x5"   and "\<And>n m n' m'. \<lbrakk>n = CFG_node m; n' = CFG_node m'; m controls m'\<rbrakk> \<Longrightarrow> P n None None False n'"   and "\<And>a Q r p fs n a' n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; n = CFG_node (targetnode a); a' \<in> get_return_edges a; n' = CFG_node (sourcenode a')\<rbrakk> \<Longrightarrow> P n None None False n'"   and "\<And>n' m n. \<lbrakk>valid_SDG_node n'; m = parent_node n'; n = CFG_node m; n \<noteq> n'\<rbrakk> \<Longrightarrow> P n None None False n'"   and "\<And>n V n'. n influences V in n' \<Longrightarrow> P n (Some V) None False n'"   and "\<And>a Q r p fs n n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; n = CFG_node (sourcenode a); n' = CFG_node (targetnode a)\<rbrakk> \<Longrightarrow> P n None (Some (p, True)) False n'"   and "\<And>a Q p fs n n'. \<lbrakk>valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>fs; n = CFG_node (sourcenode a); n' = CFG_node (targetnode a)\<rbrakk> \<Longrightarrow> P n None (Some (p, False)) False n'"   and "\<And>a Q r p fs ins outs V x n n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; (p, ins, outs) \<in> set procs; V = ins ! x; x < length ins; n = Actual_in (sourcenode a, x); n' = Formal_in (targetnode a, x)\<rbrakk> \<Longrightarrow> P n (Some V) (Some (p, True)) False n'"   and "\<And>a Q p f ins outs V x n n'. \<lbrakk>valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; (p, ins, outs) \<in> set procs; V = outs ! x; x < length outs; n = Formal_out (sourcenode a, x); n' = Actual_out (targetnode a, x)\<rbrakk> \<Longrightarrow> P n (Some V) (Some (p, False)) False n'"   and "\<And>a Q r p fs a' n n'. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; a' \<in> get_return_edges a; n = CFG_node (sourcenode a); n' = CFG_node (targetnode a')\<rbrakk> \<Longrightarrow> P n None (Some (p, True)) True n'"   and "\<And>a Q r p fs a' x ns x' n n' ins outs. \<lbrakk>valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; a' \<in> get_return_edges a; matched (Formal_in (targetnode a, x)) ns (Formal_out (sourcenode a', x')); n = Actual_in (sourcenode a, x); n' = Actual_out (targetnode a', x'); (p, ins, outs) \<in> set procs; x < length ins; x' < length outs\<rbrakk> \<Longrightarrow> P n None (Some (p, True)) True n'" shows "P x1 x2 x3 x4 x5"
simps_2: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set" shows "CRotate (DenyAllFromTo x y) = C (DenyAllFromTo x y)"
Le_le_LtI: fixes a :: "'d"   and b :: "'d" assumes "a < b" shows "Le a \<le> Lt b"
binomial_mono: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m \<le> n" shows "m choose k \<le> n choose k"
select_convs_11: fixes igWls :: "'g \<Rightarrow> 'b \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'g \<Rightarrow> 'd \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and igAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'd"   and igOp :: "'h \<Rightarrow> ('e \<Rightarrow> 'b option) \<Rightarrow> ('f \<Rightarrow> 'd option) \<Rightarrow> 'b"   and igFresh :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and igSwapAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubst :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and igSubstAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'i" shows "igSubstAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igSubstAbs"
permutationss_def: shows "permutationss = product_lists \<circ> map permutations"
quat_inner_i_left: fixes x :: "quat" shows "\<i> \<bullet> x = Im1 x"
lessThan_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a" shows "ord.lessThan less u \<equiv> {x. less x u}"
composites_are_arrows_2: shows "hseq (f \<star> g) h"
distinct_disc_13: fixes event :: "('a, 'b, 'c) event" assumes "isSend event" shows "\<not> isRecvMarker event"
Min_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "Min A \<in> A"
rec_fac_def: shows "rec_fac = Cn 1 rec_dummyfac [recf.id 1 0, recf.id 1 0]"
subset_image_inj: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and T :: "'b set" shows "(S \<subseteq> f ` T) = (\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U)"
poly_mult_r_distr': fixes p1 :: "'a list"   and p2 :: "'a list"   and p3 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R"   and "set p3 \<subseteq> carrier R" shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)"
simps_10: fixes jmps :: "jump set"   and c1 :: "stmt"   and c2 :: "stmt" shows "jumpNestingOkS jmps (c1 Finally c2) = (jumpNestingOkS jmps c1 \<and> jumpNestingOkS jmps c2)"
cil_contains_empty: fixes wa :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and wb :: "'a list" shows "([] \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb) = (wa = [] \<and> wb = [])"
g_def: shows "g = smult (inverse (lead_coeff f)) f"
B: shows "card {i. \<exists>N. 220 = N * 4 \<and> prime i \<and> i dvd N \<and> \<not> i dvd 4} = 2"
simps_13: fixes i :: "int"   and a :: "MIR.num" shows "simpfm (NDvd i a) = (if i = 0 then simpfm (NEq a) else if \<bar>i\<bar> = 1 \<and> check_int a then F else let a' = simpnum a in case a' of C v \<Rightarrow> if \<not> i dvd v then T else F | _ \<Rightarrow> let (x, y) = simpdvd i a' in NDvd x y)"
ts_lose_ack_st2: fixes lose :: "'a \<Rightarrow> bool list"   and t :: "'a"   and ack :: "'a \<Rightarrow> 'b list"   and x :: "'b"   and i1 :: "'a \<Rightarrow> 'c list"   and vc :: "'a \<Rightarrow> 'd list"   and st_out :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "lose t = [True] \<or> lose t = [False]"   and "lose t = [True] \<longrightarrow> ack t = [x] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = x"   and "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = y" shows "ack t = [st_out t]"
unwinding_def: fixes z :: "complex" shows "unwinding z \<equiv> THE k. of_int k = (z - Ln (exp z)) / (complex_of_real (2 * pi) * \<i>)"
POR_sat: shows "appFLv (ValF\<cdot>(\<Lambda> x. ValF\<cdot>(\<Lambda> y. x por y))) [POR_arg1_rel, POR_arg2_rel] = POR_result_rel"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> kerbIV" shows "K \<notin> keysFor (parts (knows Spy evs))"
all_partitions_paper_equiv_alg: fixes xs :: "'c list" assumes "distinct xs" shows "set (map set (all_partitions_list xs)) = all_partitions (set xs)"
lprefix_code_2: fixes x :: "'a"   and xs :: "'a llist" shows "lprefix (x #\<^sub>l xs) []\<^sub>l = False"
SumIdemp': fixes P :: "pi"   and P' :: "pi" assumes "P \<equiv>\<^sub>e P'" shows "P \<oplus> P' \<equiv>\<^sub>e P"
simps: fixes f :: "'b list \<Rightarrow> (bool \<times> term list) option"   and i :: "natural" shows "exhaustive_class.exhaustive f i = (if 0 < i then Quickcheck_Exhaustive.orelse (f []) (exhaustive_class.exhaustive (\<lambda>uu_. exhaustive_class.exhaustive (\<lambda>uua_. f (uu_ # uua_)) (i - 1)) (i - 1)) else None)"
prod: fixes J :: "'i set" assumes "J \<subseteq> insert i I" shows "integrable (Pi\<^sub>M J M) (\<lambda>x. \<Prod>i\<in>J. f i (x i))"
uv: shows "Diag (u \<^bold>\<otimes> v)"
is_pr_Hom_trans: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a list set"   and B :: "'b list set"   and g :: "'b \<Rightarrow> 'c"   and C :: "'c list set" assumes "is_pr_Hom f A B"   and "is_pr_Hom g B C" shows "is_pr_Hom (g \<circ> f) A C"
zero_le_ceiling: fixes x :: "'a" shows "(0 \<le> \<lceil>x\<rceil>) = (- (1::'a) < x)"
hyps: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length ps = length qs" shows "tab(ps [\<mapsto>] qs @ zs) = tab(ps [\<mapsto>] qs)"
run_option_def: fixes optionT :: "('e, 'd) optionT" shows "run_option optionT = (case optionT of OptionT x \<Rightarrow> x)"
join_ac_2: fixes La :: "('c, 'd) gorder_scheme"   and L :: "('e, 'f) gorder_scheme"   and x :: "'e"   and y :: "'e" assumes "bounded_lattice La" shows "x \<squnion>\<^bsub>L\<^esub> y = y \<squnion>\<^bsub>L\<^esub> x"
inj_map: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "inj (map_bset f)"
someI_ex: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
mop_prio_pop_min_def: fixes prio :: "'a \<Rightarrow> 'b"   and b :: "'a multiset" shows "mop_prio_pop_min prio b = ASSERT (b \<noteq> {#}) \<bind> (\<lambda>_. SPEC (\<lambda>(v, b'). v \<in># b \<and> b' = b - {#v#} \<and> (\<forall>v'\<in>#b. prio v \<le> prio v')))"
wc_def: shows "[Contingent (F\<^sup>-) \<^bold>& (\<^bold>\<forall>x. \<^bold>\<diamond>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) in v]"
eval_tot: fixes \<theta> :: "trm"   and \<nu>' :: "fin_string \<Rightarrow> real" shows "\<exists>r. [\<theta>]\<nu>' \<down> r"
intros_3: fixes \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and v :: "'b" assumes "(\<pi>, m) \<in> IFC_def.scp prob"   and "IFC_def.is_ddi prob n \<pi> v m" shows "(\<pi>, n) \<in> IFC_def.scp prob"
nyinitcls_le_SucD: fixes G :: "prog"   and x :: "abopt"   and s :: "st"   and n :: "nat"   and C :: "qtname"   and y :: "class" assumes "card (nyinitcls G (x, s)) \<le> Suc n"   and "\<not> inited C (globs s)"   and "class G C = Some y" shows "card (nyinitcls G (x, init_class_obj G C s)) \<le> n"
commute: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "sup_prod a b = sup_prod b a"
adj_idem1: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "f \<circ> g \<circ> (f \<circ> g) = f \<circ> g"
nfa_Reverse_nfa: shows "nfa (Reverse_nfa MS)"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "inf a b \<le> a"
is_ord_r_bal: fixes x :: "'a"   and l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0" assumes "is_ord (MKT\<^sub>0 x l r)"   and "height r = height l + 2" shows "is_ord (r_bal\<^sub>0 x l r)"
pr0_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr0 a b : local.prod a b \<rightarrow> b\<guillemotright>"
to_hma\<^sub>m_def: fixes a :: "'a mat" shows "to_hma\<^sub>m a = (\<chi>i j. a $$ (Bij_Nat.to_nat i, Bij_Nat.to_nat j))"
essentialE: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "essential G B S x" obtains p :: "'a list" and y :: "'a" where "path G x p y" and "y \<in> B" and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S"
collapse_7: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" assumes "is_NegChecks stateful_strand_step" shows "\<forall>bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p stateful_strand_step\<langle>\<or>\<noteq>: the_eqs stateful_strand_step \<or>\<notin>: the_ins stateful_strand_step\<rangle> = stateful_strand_step"
trapped_backward_iff_rev_trapped_forward: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b"   and K :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.trapped_backward f X x K = auto_ll_on_open.trapped_forward (- f) X x K"
LT2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> F \<longrightarrow> \<diamond>G"
c\<alpha>_is_\<gamma>c_i\<delta>: shows "[c] @ \<alpha> = \<gamma> @ [c_i] @ \<delta>"
IH: assumes "crdt_ops (xs @ [(oid, oper)] @ ys) deps"   and "\<And>op2 r. \<lbrakk>op2 \<in> snd ` set ys; r \<in> deps op2\<rbrakk> \<Longrightarrow> r \<noteq> oid" shows "crdt_ops (xs @ ys @ [(oid, oper)]) deps"
Disj_def: fixes xset :: "('d, 'e, 'f) formula set['d]" shows "Disj xset = Formula.Not (Conj (map_bset Formula.Not xset))"
ereal_less_minus_iff: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" shows "(x < z - y) = ((y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and> (y = - \<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y < z))"
preserves_ide: fixes a :: "'a" assumes "hom.ide a" shows "hom'.ide (e \<star> a \<star> src f)"
neq_le_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
ge: fixes i :: "'n" shows "HMA_Connect.spectral_radius A * v $h i \<le> (local.B *v v) $h i"
rel_setD1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set"   and x :: "'c" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
prj_simps_7: shows "C.cod Prj\<^sub>1\<^sub>1 = \<mu>.apex"
simps_3: fixes v :: "real" shows "mreal_max MInfty (mreal v) = mreal v"
wlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "wlsFSw MOD \<equiv> gWlsAllDisj MOD \<and> gWlsAbsIsInBar MOD \<and> gConsPresGWls MOD \<and> gSwapAllPresGWlsAll MOD \<and> gFreshCls MOD \<and> gSwapCls MOD \<and> gAbsCongS MOD"
isomorphicI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "B.VV.iso_in_hom f a b" shows "B.VV.isomorphic a b"
strip_guards_block: fixes F :: "'f set"   and init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "strip_guards F (block init bdy return c) = block init (strip_guards F bdy) return (\<lambda>s t. strip_guards F (c s t))"
bind_eq_SomeE: fixes x :: "'b option"   and f :: "'b \<Rightarrow> 'a option"   and y :: "'a" assumes "Option.bind x f = Some y" obtains x' :: "'b" where "x = Some x'" and "f x' = Some y"
pc_in_grid: shows "pc \<in> grid (child pd dir d) {d}"
not_all_zero: shows "\<not> (\<forall>m. is_zero_row_upt_k m k A)"
sel_3: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.rank (SkewBinomialHeapStruc.Node x1 x2 x3 x4) = x3"
set_spmf_return_pmf_None: shows "set_spmf (return_pmf None) = {}"
trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"