hcju/isabelleps huggingface dataset:
rel_spmf_bindI1: fixes p :: "'a spmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'c spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf R (f x) q"   and "lossless_spmf p" shows "rel_spmf R (p \<bind> f) q"
part_equiv_sym: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(y, x) \<in> R"
union_inter: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "finite B" shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"
Y: shows "Y = u\<^sub>Y *\<^sub>R S + (1 - u\<^sub>Y) *\<^sub>R T"
suntil_abss: fixes y :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "\<forall>s\<in>sset y. s \<in> S"   and "(holds \<phi> suntil holds \<psi>) (s ## y)" shows "(holds \<phi>' suntil holds \<psi>') (s' ## smap abss y)"
rank_new_mat: shows "vec_space.rank (dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))) (take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) = dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
well_com_if: shows "well_com (Measure_P vars1 N testN (replicate N SKIP))"
col_ket_vec_index: fixes i :: "nat"   and v :: "complex Matrix.mat" assumes "i < dim_row v" shows "|Matrix.col v 0\<rangle> $$ (i, 0) = v $$ (i, 0)"
0: shows "monom (UP R) d 1 (deg R (monom (UP R) d 1)) = d"
fmdom_fmupd: fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "fmdom (m(a $$:= b)) = finsert a (fmdom m)"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = inf a b" shows "a \<le> b"
closure_is_closed_csubspace: fixes S :: "'d set" assumes "csubspace S" shows "closed_csubspace (closure S)"
get_elements_by_class_name_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_elements_by_class_name ptr' name \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
assms_6: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
real_vector_eq_affinity: fixes m :: "real"   and y :: "'a"   and x :: "'a"   and c :: "'a" assumes "m \<noteq> 0" shows "(y = m *\<^sub>R x + c) = (y /\<^sub>R m - c /\<^sub>R m = x)"
intro: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'd" assumes "open X"   and "\<And>x. x \<in> X \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"   and "continuous_on X f'" shows "c1_on_open f f' X"
erfc_real_eq_iff: fixes a :: "real"   and b :: "real" shows "(erfc a = erfc b) = (a = b)"
distinct_disc_8: fixes event :: "('a, 'b, 'c) event" assumes "isRecvMarker event" shows "\<not> isTrans event"
match_def: fixes \<Delta> :: "'state \<Rightarrow> 'a \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'state"   and vl1 :: "'value list"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'a" shows "match \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<exists>a1 ou1 s1' vl1'. let trn = Trans s a ou s'; trn1 = Trans s1 a1 ou1 s1' in validTrans trn1 \<and> consume trn1 vl1 vl1' \<and> \<gamma> trn = \<gamma> trn1 \<and> (\<gamma> trn \<longrightarrow> g trn = g trn1) \<and> \<Delta> s' vl' s1' vl1'"
simps_6: fixes t :: "'a local.term" shows "Arr \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Arr t"
nstut_or: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "NSTUTINV F"   and "NSTUTINV G" shows "NSTUTINV (F \<or> G)"
language_def: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and final :: "'a \<Rightarrow> bool" shows "language step final \<equiv> semantics step final"
rrsmult: shows "(\<currency>) = Ring1.rightreg_scalar_mult FH"
subfieldE_6: fixes K :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subfield K R" shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
stream_all_def: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "stream_all P s = (\<forall>p. P (s !! p))"
IH: fixes r'' :: "'a"   and A'' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r'', A'')"   and "l + k < n" shows "r * det A'' = r'' * det A"
cwiseext_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "cwiseext gt ys xs" shows "cwiseext gt (map f ys) (map f xs)"
thetaParWL1_def: shows "thetaParWL1 \<equiv> {(c1 | c2, d) |c1 c2 d. c1 \<approx>w d \<and> discr c2}"
mono_neutral_right': fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'c)" shows "sum' g T = sum' g S"
bind_spmf_cong: fixes p :: "'b spmf"   and q :: "'b spmf"   and f :: "'b \<Rightarrow> 'd spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
wt_elim_cases_2: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and accC :: "qtname"   and statDeclC :: "qtname"   and s :: "bool"   and e :: "expr"   and fn :: "vname"   and T :: "tys" assumes "E,dt\<Turnstile>In2 ({accC,statDeclC,s}e..fn)\<Colon>T" obtains C :: "qtname" and f :: "Decl.field" where "T = Inl (type f)" and "E,dt\<Turnstile>e\<Colon>-Class C" and "accfield (prg E) (cls E) C fn = Some (statDeclC, f)" and "accC = cls E" and "s = is_static f"
wp_under_wlp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "wp_under_wlp (Apply f)"
extract_IPSets_def: fixes rs :: "'a simple_rule list" shows "extract_IPSets rs \<equiv> map ipcidr_tuple_to_wordinterval (mergesort_remdups (extract_src_dst_ips rs []))"
that: fixes l' :: "'a list" assumes "(tsl s, l') \<in> \<langle>Rk\<rangle>list_rel"   and "RETURN l' \<le> it_to_sorted_list ordR s'" shows "thesis"
mem_interior_closure_convex_shrink: fixes S :: "'a set"   and c :: "'a"   and x :: "'a"   and e :: "real" assumes "convex S"   and "c \<in> interior S"   and "x \<in> closure S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> interior S"
pell_mul_commutes: fixes D :: "nat"   and z1 :: "'a \<times> 'a"   and z2 :: "'a \<times> 'a" assumes "pell D" shows "pell.pell_mul D z1 z2 = pell.pell_mul D z2 z1"
diff_fun_compose: fixes k :: "enat"   and M1 :: "('c, 'd) chart set"   and M2 :: "('f, 'g) chart set"   and f :: "'c \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'h" assumes "diff k M1 M2 f"   and "diff_fun k M2 g" shows "diff_fun k M1 (g \<circ> f)"
Nil_imp_Nil: fixes aa :: "'a \<times> 'a"   and asa :: "('a \<times> 'a) list"   and va :: "'a"   and ba :: "'a \<times> 'a"   and bsa :: "('a \<times> 'a) list"   and wa :: "'a" assumes "awalk u (aa # asa) va"   and "awalk u (ba # bsa) wa"   and "set (inner_verts (ba # bsa)) \<inter> V = {}"   and "va \<in> V"   and "aa = ba"   and "asa = []" shows "bsa = []"
F_def: shows "F = filtermap (sum f) (finite_subsets_at_top A)"
less_eq_Max_lit: fixes C :: "'a multiset"   and D :: "'a multiset" assumes "C \<noteq> {#}"   and "C \<le> D" shows "Max_mset C \<le> Max_mset D"
foldSetDp_foldSetD_eq: fixes D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" shows "foldSetDp (\<lambda>x. x \<in> D) f e = (\<lambda>x xa. (x, xa) \<in> foldSetD D f e)"
subcls1_def2: shows "subcls1 = (SIGMA C:{C. is_class C}. {D. C \<noteq> Object \<and> super (the (class C)) = D})"
simps_5: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "safe_formula (Neg (Or (Neg \<phi>) \<psi>)) = (safe_formula \<phi> \<and> (safe_formula \<psi> \<and> fv \<psi> \<subseteq> fv \<phi> \<or> (case \<psi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)))"
d: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>s t"   and "s = Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t"
fp_cop_F_range_inv': fixes x :: "'a"   and ds :: "'b set" assumes "x \<in> TruePref_tax.fp_cop_F ds" shows "x \<in> Field (Pd'_tax (Xd x)) \<and> Xd x \<in> ds"
relH_D2_1: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and a :: "nat"   and t :: "typerep" assumes "relH as h h'"   and "a \<in> as" shows "refs h t a = refs h' t a"
assoc: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
assms_6: shows "\<forall>m. is_zero_row_upt_k m k A"
pre: shows "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf"
that: assumes "lookup m k = Some p"   and "\<forall>k' p'. lookup m k' = Some p' \<longrightarrow> p \<le> p'" shows "thesis"
neg: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "module_hom s1 s2 f" shows "f (- x) = - f x"
complex_eq_cancel_iff2: fixes x :: "real"   and y :: "real"   and xa :: "real" shows "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)"
triangle_in_hom_1: shows "\<guillemotleft>(\<xi> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<zeta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
inner_Pair_0_2: fixes x :: "'c \<times> 'd"   and a :: "'c" shows "inner x (a, 0::'d) = inner (fst x) a"
wave_plus_current: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g"   and "wave (quotient_web \<Gamma> f) g" shows "wave \<Gamma> (plus_current f g)"
abrupt_if_True_not_None: fixes x :: "abopt"   and y :: "abopt" assumes "x \<noteq> None" shows "abrupt_if True x y \<noteq> None"
as_tree_empty: fixes t :: "'b" shows "(as_tree t = E) = is_empty t"
widen_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "P \<turnstile> S \<le> U"   and "P \<turnstile> U \<le> T" shows "P \<turnstile> S \<le> T"
intro: fixes S :: "complex set" assumes "open S" shows "SC_Chain S"
simps_18: fixes f1 :: "i32 \<Rightarrow> 'a"   and f2 :: "i64 \<Rightarrow> 'a"   and f3 :: "f32 \<Rightarrow> 'a"   and f4 :: "f64 \<Rightarrow> 'a"   and x2 :: "i64" shows "(case ConstInt64 x2 of ConstInt32 x \<Rightarrow> f1 x | ConstInt64 x \<Rightarrow> f2 x | ConstFloat32 x \<Rightarrow> f3 x | ConstFloat64 x \<Rightarrow> f4 x) = f2 x2"
assms_6: shows "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
of_complex_inner_1': fixes x :: "complex" shows "(1::'a) \<bullet>\<^sub>C of_complex x = x"
next_action_def: fixes current :: "'b \<Rightarrow> 'c"   and control :: "'b \<Rightarrow> 'c \<Rightarrow> 'a list list \<Rightarrow> 'a option \<times> 'a list list \<times> 'b"   and s :: "'b"   and execs :: "'c \<Rightarrow> 'a list list" shows "Kernel.next_action current control s execs = fst (control s (current s) (execs (current s)))"
space_Sup_eq_UN: fixes M :: "'c measure set" shows "space (Sup M) = \<Union> (space ` M)"
oalist_inv_raw_sort_oalist: fixes ko :: "'a key_order"   and xs :: "('a \<times> 'b) list" shows "ko.oalist_inv_raw ko (sort_oalist_ko' ko xs)"
simps: fixes f :: "heap \<Rightarrow> ('c \<times> heap) option" shows "execute (Heap.Heap f) = f"
FDOM_pre_subset_prob_dom_pair: fixes a :: "('b, 'c) fmap \<times> ('b, 'd) fmap"   and PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" assumes "a \<in> PROB" shows "fmdom' (fst a) \<subseteq> prob_dom PROB"
cptn_onlyif_cptn_mod: fixes c :: "('b com option \<times> 'b) list" assumes "c \<in> cptn" shows "c \<in> cptn_mod"
hyps: assumes "\<not> is_final (steps0 (1, tp) A stp)" shows "fst (steps0 (1, tp) A stp) \<le> length A div 2"
simps_8: fixes x1 :: "'b"   and x2 :: "'b stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
less_irrefl: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k" assumes "class.preorder less_eq less" shows "\<not> less x x"
dghm_up_ObjMap_app_vdomain: fixes a :: "V"   and \<CC> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "\<D>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) = I"
ide_char: fixes f :: "'a \<times> 'a" shows "VxV.ide f = (ide (fst f) \<and> ide (snd f))"
\<E>\<tau>_in_Fa: fixes \<tau> :: "'c \<Rightarrow> 's" assumes "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) (Y a') \<tau>" shows "\<E> \<tau> \<in> Ya'.SET a"
cinner_mult_right': fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and a :: "'b"   and b :: "'b"   and m :: "complex" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "a \<bullet>\<^sub>C (b * of_complex m) = a \<bullet>\<^sub>C b * m"
lookup_fresh': fixes z :: "name"   and \<theta> :: "(name \<times> trm) list" assumes "z \<sharp> \<theta>" shows "\<theta> maps z to Var z"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D E.GF.map E.GF.cmp"
filter_cong: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
power_mult: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a ^ (m * n) = (a ^ m) ^ n"
suc_greater: fixes B :: "(nat + 'a Bonk_Schramm_extension_unfolded set) set"   and b :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "B \<subseteq> Field wo_aux"   and "AboveS wo_aux B \<noteq> {}"   and "b \<in> B" shows "suc wo_aux B \<noteq> b \<and> (b, suc wo_aux B) \<in> wo_aux"
ContractsWithLAD_axioms: shows "ContractsWithLAD Xd Xh Pd Ch"
e_lam_elim: fixes v :: "val"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E (ELam x e) \<rho>" obtains f :: "(val \<times> val) fset" where "v = VFun f" and "\<forall>v1 v2. (v1, v2) \<in> fset f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)"
fold_ins_bst_of_list: fixes zs :: "'b list"   and p :: "'b \<Rightarrow> 'c" assumes "distinct zs"   and "inj_on p (set zs)" shows "map_tree fst (foldl (\<lambda>t (x, p). ins x p t) \<langle>\<rangle> (map (\<lambda>x. (x, p x)) zs)) = bst_of_list (sort_key p zs)"
MkIde_Cod': fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "MkIde (Cod a) = a"
collect_clkt_def: fixes S :: "('g \<times> ('e, 'f) cconstraint \<times> 'h \<times> 'e list \<times> 'g) set" shows "collect_clkt S = \<Union> {collect_clock_pairs (fst (snd t)) |t. t \<in> S}"
l1_inv5I: fixes x :: "l1_state" assumes "\<And>gny A B Ra. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd}; guessed_frame Ra xgny = Some gny; can_signal x A B\<rbrakk> \<Longrightarrow> 0 < signalsResp x (Running A B (Exp gny (NonceF (Ra $ nx))))" shows "x \<in> l1_inv5"
wf_mbuf2_def: fixes i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and buf :: "'b option list set list \<times> 'b option list set list" shows "wf_mbuf2 i ja jb P Q buf = (i \<le> ja \<and> i \<le> jb \<and> (case buf of (xs, ys) \<Rightarrow> list_all2 P [i..<ja] xs \<and> list_all2 Q [i..<jb] ys))"
case_prodE': fixes c :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and p :: "'f \<times> 'g"   and z :: "'h" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'f" and y :: "'g" where "p = (x, y)" and "c x y z"
Sup_upper2: fixes u :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set"   and v :: "'c \<times> 'd" assumes "u \<in> A"   and "less_eq_prod v u" shows "less_eq_prod v (Sup_prod A)"
find_non0_all0: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and l :: "nat"   and A :: "'a mat" assumes "find_non0 sel_fun l A = None" shows "triangular_column l A"
dpWeak: fixes \<Gamma> :: "'a form multiset"   and E :: "'a form"   and n :: "nat"   and R :: "('a sequent list \<times> 'a sequent) set"   and R' :: "('a sequent list \<times> 'a sequent) set"   and \<Gamma>' :: "'a form multiset" assumes "( \<Gamma> \<Rightarrow>* E, n) \<in> derivable R*"   and "R' \<subseteq> upRules"   and "R = Ax \<union> R'" shows "( \<Gamma> + \<Gamma>' \<Rightarrow>* E, n) \<in> derivable R*"
llength_lappend_startE: fixes n :: "enat"   and w :: "'b llist" assumes "eSuc n \<le> llength w" obtains a :: "'b" and v :: "'b llist" where "w = <a> $ v" and "n \<le> llength v"
Rep_eqD: fixes x :: "'b word"   and y :: "'b word" assumes "unat x = unat y" shows "x = y"
dagger_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<star>"
simps: fixes x :: "'a extC"   and y :: "'a extC" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "shd (siterate f x) = x"
surjective: fixes r :: "('b, 'c, 'd) lens_scheme" shows "r = \<lparr>lens_get = get\<^bsub>r\<^esub>, lens_put = put\<^bsub>r\<^esub>, \<dots> = more r\<rparr>"
equality: fixes r :: "('a, 'b, 'c) Quantity_scheme"   and r' :: "('a, 'b, 'c) Quantity_scheme" assumes "mag r = mag r'"   and "Quantity.dim r = Quantity.dim r'"   and "Quantity.more r = Quantity.more r'" shows "r = r'"
matchC_C_mono: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and theta' :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "matchC_C theta c d"   and "theta \<subseteq> theta'" shows "matchC_C theta' c d"
intros_2: fixes m :: "(nat \<times> nat) convnet"   and r1 :: "nat"   and r0 :: "nat" assumes "output_size m = r1"   and "valid_net m" shows "valid_net (Conv (r0, r1) m)"
acquired_reads_union: fixes a :: "nat"   and pending_writes :: "bool"   and sb :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "a \<in> acquired_reads pending_writes sb (A \<union> B)"   and "a \<notin> A" shows "a \<in> acquired_reads pending_writes sb B"
pseudo_proj_True_finite_image: fixes n :: "nat" shows "finite (range (pseudo_proj_True n))"
not_UNIF1_singleton_Var_right_1: fixes x :: "'c"   and y :: "'c" assumes "\<not> UNIF1 Var {#(Var x, Var y)#} {#}" shows "x \<noteq> y"
var_order_rel_var_order: fixes x :: "char list"   and y :: "char list" shows "((x, y) \<in> var_order_rel) = var_order x y"
subst_ident: fixes x :: "'d"   and t :: "('e, 'd) Term.term"   and u :: "('e, 'd) Term.term" assumes "x \<notin> vars_term t" shows "t \<cdot> subst x u = t"
subseq_to_Suc: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'b" assumes "i \<le> j" shows "w [i \<rightarrow> Suc j] = w [i \<rightarrow> j] @ [w j]"
\<I>_trivialD: fixes \<I> :: "('a, 'b) \<I>" assumes "\<I>_trivial \<I>" shows "outs_\<I> \<I> = UNIV"
sorted_wrt_mono_rel: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
H_2: fixes q :: "'a" assumes "q \<in> H" shows "infdist q G \<le> C"
top_greatest: fixes a :: "'a upred" shows "true \<sqsubseteq> a"
hoare_triple_def: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<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')"
raise_if_None: fixes c :: "bool"   and x :: "xname"   and y :: "abopt" shows "(raise_if c x y = None) = (\<not> c \<and> y = None)"
left_right_inverse_power: fixes x :: "'b list"   and y :: "'b list"   and n :: "nat" assumes "x \<cdot> y = \<epsilon>" shows "x \<^sup>@ n \<cdot> y \<^sup>@ n = \<epsilon>"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f))"
dims2_def: shows "stm2.dims2 = nths dims {Suc m}"
is_associative_weak_composition: shows "associative_weak_composition (\<cdot>) (\<star>) \<a>"
Sup_eq_top_iff: 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" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup A = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x i))"
in_prefix: fixes xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "prefix xs ys"   and "x \<in> set xs" shows "x \<in> set ys"
birkhoff_sum_small_asymp_pos: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "integrable M f"   and "0 < e" shows "AE x in M. infinite {n. birkhoff_sum f n x \<in> {real n * expectation f..real n * expectation f + e}}"
sorted_list_of_fset_simps_1: fixes S :: "'a fset" shows "set (sorted_list_of_fset S) = fset S"
lunit'_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[a] : a \<rightarrow> \<one> \<otimes> a\<guillemotright>"
phi_complemented: fixes x :: "'a" shows "complement (phi x) (phi (- x))"