hcju/isabelleps huggingface dataset:
a_evalF_def: shows "\<aF> \<equiv> fst (fix_syn (\<lambda>(a_evalF, a_evalC). (\<Lambda> fstate. case undiscr fstate of (PC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> b); ve' = ve \<union>. \<Union>.map2 (\<lambda>v. smap_singleton (v, b)) vs as in a_evalC\<cdot>(Discr (c, \<beta>', ve', b)) else \<bottom> | (PP (Plus c), [], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x, xa], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x, xa, cnts], ve, b) \<Rightarrow> let b' = \<anb> b c; \<beta> = [c \<mapsto> b] in (\<Union>cnt\<in>cnts. a_evalF\<cdot>(Discr (cnt, [{}], ve, b'))) \<union> {((c, \<beta>), cont) |cont. cont \<in> cnts} | (PP (Plus c), x # xa # cnts # ac # lista, bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x, cntts], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x, cntts, cntfs], ve, b) \<Rightarrow> (let b' = \<anb> b ct; \<beta> = [ct \<mapsto> b] in (\<Union>cnt\<in>cntts. a_evalF\<cdot>(Discr (cnt, [], ve, b'))) \<union> {((ct, \<beta>), cnt) |cnt. cnt \<in> cntts}) \<union> (let b' = \<anb> b cf; \<beta> = [cf \<mapsto> b] in (\<Union>cnt\<in>cntfs. a_evalF\<cdot>(Discr (cnt, [], ve, b'))) \<union> {((cf, \<beta>), cnt) |cnt. cnt \<in> cntfs}) | (PP (prim.If ct cf), x # cntts # cntfs # ac # lista, bb) \<Rightarrow> \<bottom> | (AStop, [], bb) \<Rightarrow> \<bottom> | (AStop, [x], xa, xb) \<Rightarrow> {} | (AStop, x # a # lista, bb) \<Rightarrow> \<bottom>, \<Lambda> cstate. case undiscr cstate of (App lab f vs, \<beta>, ve, b) \<Rightarrow> let fs = \<aA> f \<beta> ve; as = map (\<lambda>v. \<aA> v \<beta> ve) vs; b' = \<anb> b lab in (\<Union>f'\<in>fs. a_evalF\<cdot>(Discr (f', as, ve, b'))) \<union> {((lab, \<beta>), f') |f'. f' \<in> fs} | (call.Let lab ls c', \<beta>, ve, b) \<Rightarrow> let b' = \<anb> b lab; \<beta>' = \<beta>(lab \<mapsto> b'); ve' = ve \<union>. \<Union>.map (\<lambda>(v, l). { (v, b') := \<aA> (L l) \<beta>' ve}.) ls in a_evalC\<cdot>(Discr (c', \<beta>', ve', b')))))"
ts_c_def_3: shows "i < length (s_inst \<S>)"
dom_trm_lemma: fixes t :: "'a trm"   and C :: "'a literal set" assumes "dom_trm t C" shows "\<exists>u. u \<in> subterms_of_cl C \<and> (u = t \<or> (t, u) \<in> trm_ord)"
2: shows "{y \<in> \<pi> ` A. (y, \<pi> x) \<in> s} = \<pi> ` A \<inter> {y. (y, \<pi> x) \<in> s}"
widens_refl: fixes P :: "'a prog"   and Ts :: "ty list" shows "P \<turnstile> Ts [\<le>] Ts"
steps_Single_True: fixes x :: "'a" shows "local.steps [x] = True"
asIMOD_igSubstIGAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstIGAbs (asIMOD SEM)"
witness'_is_deep_model: fixes Y :: "nat"   and rs :: "nat list" shows "remove_weights (witness' Y rs) = deep_model' Y rs"
left_diff_distrib': 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 b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times (minus b c) a = minus (times b a) (times c a)"
cases: fixes a :: "'a"   and s :: "'a + 'b" assumes "a \<in> Basic_BNFs.setl s" obtains x :: "'a" where "a = x" and "s = Inl x"
Ca_def: shows "Ca = (1 / (max D__ 1 * real DIM('a))) ^ DIM('a)"
non_empty_newQuery: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and Q1 :: "(nat set \<times> 'b option list set) set"   and J :: "nat set"   and Q0 :: "(nat set \<times> 'b option list set) set"   and Q2 :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and t :: "'b option list"   and n :: "nat" assumes "getIJ getIJ"   and "Q1 = New_max.filterQuery J Q0"   and "Q2 = New_max.newQuery J Q1 (I, t)"   and "\<forall>X\<in>Q0. wf_atable n X" shows "non_empty_query Q2"
flist_arb_lists: fixes A :: "'b set" shows "flist_arb ` Fow A \<subseteq> lists A"
is_type_def: 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 T :: "ty" shows "is_type P T \<equiv> case T of Class C \<Rightarrow> is_class P C | _ \<Rightarrow> True"
Im_power_real: fixes x :: "complex"   and n :: "nat" assumes "Im x = 0" shows "Im (x ^ n) = 0"
dom_cod: fixes f :: "'t" assumes "S'.arr f" shows "S'.dom (S'.cod f) = S'.cod f"
assms_1: shows "continuous_map T1 T2 (f \<circ> g)"
exec_ss2: fixes s :: "'b sparc_state" assumes "snd (execute_instruction () s) = False" shows "snd (execute_instruction () s) = False"
is_bezout_ext_euclid_ext2: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and gcd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and lcm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and Gcd :: "'d set \<Rightarrow> 'd"   and Lcm :: "'d set \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "bezout_ring.is_bezout_ext times plus uminus (euclidean_ring_gcd.euclid_ext2 divide minus zero times one modulo unit_factor normalize gcd uminus)"
elims: fixes x :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'num, 'bool) instr fundef"   and y :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'num, 'bool) instr fundef" assumes "generalize_fundef x = y" obtains xs :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'num, 'bool) instr list" and ar :: "nat" where "x = Fundef xs ar" and "y = Fundef (map generalize_instr xs) ar"
sum_weights_append_singleton: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and xs :: "'b list"   and x :: "'b" assumes "graph weights" shows "foldr (+) (xs @ [x]) (0::'b) = foldr (+) xs (0::'b) + x"
count_image_mset_le_imp_lt: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and a :: "'c"   and N :: "'c multiset" assumes "count (image_mset f M) (f a) \<le> count (image_mset f N) (f a)"   and "count N a < count M a" shows "\<exists>b. f b = f a \<and> count M b < count N b"
assms: shows "x \<in>\<^sub>i range_tm e t"
Sup_upper: fixes x :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set" assumes "x \<in> A" shows "less_eq_prod x (Sup_prod A)"
elims_3: fixes x :: "('form, 'rule) graph_node" assumes "\<not> terminal_node x" obtains n :: "('form, 'rule) graph_node" where "x = n" and "outPorts n \<noteq> {||}"
star_transpose_swap: fixes rtc :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>x. 1' + x ; rtc x \<le> rtc x"   and "\<And>x y z. x + y ; z \<le> z \<Longrightarrow> rtc y ; x \<le> z" shows "rtc (x\<^sup>T) = rtc x\<^sup>T"
cod_cod: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.cod (CopxC.cod f) = CopxC.cod f"
cp_def: fixes P :: "('d state \<times> 'd state \<Rightarrow> 'e) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'f" shows "cp P \<equiv> \<exists>f. \<forall>X \<tau>. P X \<tau> = f (X \<tau>) \<tau>"
that: fixes b :: "'action list"   and b\<^sub>1 :: "'action list"   and b\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "R.path (b @ [a]) q"   and "Ind {a} (set b)"   and "set b \<subseteq> invisible"   and "b =\<^sub>F b\<^sub>1 @ b\<^sub>2"   and "b\<^sub>1 @- u =\<^sub>I v"   and "Ind (set b\<^sub>2) (sset u)" shows "thesis"
compatible_find_match: fixes cs :: "(Term_Class.term \<times> 'a) list"   and pat :: "Term_Class.term"   and t :: "'a"   and env :: "(name, 'a) fmap"   and rhs :: "'a" assumes "pattern_compatibles (fset_of_list cs)"   and "list_all (linear \<circ> fst) cs"   and "is_fmap (fset_of_list cs)"   and "match pat t = Some env"   and "(pat, rhs) \<in> set cs" shows "find_match cs t = Some (env, pat, rhs)"
2: shows "real (size1 r) \<le> (1 - \<alpha>) * real (size1 \<langle>l, a, r\<rangle>)"
Arg2pi_gt_0: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0" shows "0 < Arg2pi z"
to_nat_on_from_nat_into: fixes n :: "nat"   and A :: "'b set" assumes "n \<in> to_nat_on A ` A" shows "to_nat_on A (from_nat_into A n) = n"
All_def: fixes P :: "'b \<Rightarrow> bool" shows "All P \<equiv> P = (\<lambda>x. True)"
image_Pow_mono: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A \<subseteq> B" shows "(`) f ` Pow A \<subseteq> Pow B"
minimal_element_axioms: shows "minimal_element (Prec_F_g D) UNIV"
mult_sgn_abs: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (sgn x) (abs x) = x"
f_def: fixes i :: "nat"   and x :: "'a" shows "f i x = (if x \<in> W i then H (apply_chart (\<psi> i) x) else 0)"
TensorDiag_preserves_Diag_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "Diag (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
DONE_neq_ABORT: fixes a :: "'f" shows "DONE a \<noteq> ABORT"
in_fst_imageE: fixes x :: "'d"   and S :: "('d \<times> 'e) set" assumes "x \<in> fst ` S" obtains y :: "'e" where "(x, y) \<in> S"
finite_finfun_dom: fixes f :: "('a, 'b) finfun" shows "finite {x. finfun_apply (finfun_dom f) x}"
prems: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>T. x \<in> T \<and> open T \<and> higher_differentiable_on T f (Suc n)"
erf_has_field_derivative: fixes z :: "'a"   and A :: "'a set" shows "(erf has_field_derivative of_real (2 / sqrt pi) * exp (- z\<^sup>2)) (at z within A)"
simps: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" shows "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n = {s. \<exists>y. s = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n \<and> s < semi_mojmir_def.senior \<delta> q\<^sub>0 w x n \<and> \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w s n)}"
the_RES_inv: fixes m :: "'b nres" assumes "nofail m" shows "RES (the_RES m) = m"
assms_3: shows "\<forall>i\<in>T - S. h i = \<^bold>1"
distinct_6: fixes x3 :: "'c"   and x2 :: "'b" shows "In3 x3 \<noteq> In2 x2"
gg_def: shows "gg \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then g x else x"
assms: shows "infer \<Gamma> (App A B) = Some \<tau>"
pivot_tableau_equiv: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'b" assumes "Pivot eq_idx_for_lvar pivot"   and "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (pivot x\<^sub>i x\<^sub>j s)"
assms_4: shows "z \<noteq> inversion (of_complex a)"
free_tv_nth_A_impl_free_tv_A: fixes n :: "nat"   and A :: "'b list"   and x :: "nat" assumes "n < length A"   and "x \<in> free_tv (A ! n)" shows "x \<in> free_tv A"
CSP1_join: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is CSP1 healthy"   and "y is CSP1 healthy" shows "x \<or> y is CSP1 healthy"
symKeys_invKey_iff: fixes K :: "key" shows "(invKey K \<in> symKeys) = (K \<in> symKeys)"
worker_wrapper_fusion_new: fixes wrap :: "'c \<rightarrow> 'b"   and unwrap :: "'b \<rightarrow> 'c"   and body :: "'b \<rightarrow> 'b"   and body' :: "'c \<rightarrow> 'c" assumes "wrap oo unwrap = ID"   and "unwrap\<cdot>\<bottom> = \<bottom>"   and "unwrap oo body oo wrap = body' oo unwrap oo wrap" shows "fix\<cdot>body = wrap\<cdot>(fix\<cdot>body')"
e1: shows "\<forall>w\<in>ball p e1. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)"
evaldjf_ex: fixes vs :: "'b list"   and bs :: "'b list"   and f :: "'c \<Rightarrow> fm"   and ps :: "'c list" shows "Ifm vs bs (evaldjf f ps) = (\<exists>p\<in>set ps. Ifm vs bs (f p))"
simps: fixes fmask1 :: "tcp_flag set"   and c1 :: "tcp_flag set"   and fmask2 :: "tcp_flag set"   and c2 :: "tcp_flag set" shows "match_tcp_flags_conjunct (TCP_Flags fmask1 c1) (TCP_Flags fmask2 c2) = (if c1 \<subseteq> fmask1 \<and> c2 \<subseteq> fmask2 \<and> fmask1 \<inter> fmask2 \<inter> c1 = fmask1 \<inter> fmask2 \<inter> c2 then TCP_Flags (fmask1 \<union> fmask2) (c1 \<union> c2) else ipt_tcp_flags_NoMatch)"
iIN_inext_nth_if: fixes n :: "nat"   and d :: "nat"   and a :: "nat" shows "[n\<dots>,d] \<rightarrow> a = (if a \<le> d then n + a else n + d)"
mem_set_subseqs_append: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "xs \<in> set (subseqs ys)" shows "xs \<in> set (subseqs (zs @ ys))"
simps: fixes r :: "'b wordinterval" shows "wordinterval_invert r = wordinterval_setminus wordinterval_UNIV r"
isOK_bind: fixes m :: "'c + 'e"   and f :: "'e \<Rightarrow> 'c + 'd" shows "isOK (m \<bind> f) = (isOK m \<and> isOK (f (projr m)))"
FIRSTSPLIT_4: shows "(([entry fg p], {#}), w, [v], ch) \<in> trcl (trss fg)"
apply_ncrit_subset: fixes ic :: "bool"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "(ic, p, q) \<in> set (apply_ncrit crit data gs bs hs ps)" obtains q_in_bs :: "bool" where "(ic, q_in_bs, p, q) \<in> set ps"
abs_update_map_of': fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b"   and x :: "'a" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat (abs_update k v l)) x = (map_of (concat l)(k \<mapsto> v)) x"
simps_2: fixes f :: "'c"   and Tl :: "'c trm list" shows "nv2T (Fn f Tl) = {}"
trd_def: fixes cmp_term :: "('a, 'b) pp nat_term_order"   and fs :: "(('a, 'b) pp \<Rightarrow>\<^sub>0 'c) list"   and p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "trd_punit cmp_term fs p = trd_aux_punit cmp_term fs p 0"
dec: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> y" shows "f y \<le> f x"
assms_3: shows "b \<cdot> l \<le> l \<cdot> b"
the_def: fixes option :: "'f option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
hbase_hd: fixes ns :: "C list"   and b :: "nat" assumes "C ns \<in> hbase b"   and "ns \<noteq> []" shows "hd ns \<in> hbase b"
subset_class_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "subset_class \<Omega> M \<equiv> M \<subseteq> Pow \<Omega>"
compute_bound_poly_mono: fixes p :: "float interval poly"   and I :: "float interval list"   and J :: "float interval list"   and a :: "float interval list"   and prec :: "nat" assumes "num_params p \<le> length I"   and "I all_subset J"   and "a all_subset I" shows "set_of (compute_bound_poly prec p I a) \<subseteq> set_of (compute_bound_poly prec p J a)"
happens_before_axioms: shows "happens_before (\<preceq>) (\<prec>)"
terminates_elim_cases_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Spec r \<down> s" obtains s :: "'e" where "sb__ = Normal sc__"   | f :: "'f" where "sb__ = Fault f"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Abrupt s"
hom_from_subgroup_generated: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and A :: "'a set" assumes "h \<in> hom G H" shows "h \<in> hom (subgroup_generated G A) H"
xF_ortho: shows "\<langle>f x, sum f F\<rangle> = 0"
vfsequence_vdomain_in_omega: shows "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
map_comp: fixes g :: "'h \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'h"   and v :: "'i llist" shows "lmap g (lmap f v) = lmap (g \<circ> f) v"
psubst_vsubst_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "goodEnv rho" shows "X #[y1 // y]_ys #[rho] = X #[rho [y \<leftarrow> Var ys y1 #[rho]]_ys]"
outer_measure_of_def: fixes M :: "'b measure"   and A :: "'b set" shows "outer_measure_of M A = Inf (emeasure M ` {B \<in> sets M. A \<subseteq> B})"
cont_spmf: fixes x :: "'h" shows "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. spmf p x)"
without_ccTTree: fixes x :: "Vars.var"   and S :: "Vars.var set"   and G :: "CoCalls" shows "without x (ccTTree S G) = ccTTree (S - {x}) G"
Units_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<in> carrier G"
homomorphism: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "pure f \<diamondop> pure x = pure (f x)"
Cd_greatest: fixes d :: "'d"   and X :: "'x set" shows "Cd d X = greatest (Pd d) (X \<inter> Field (Pd d))"
hermitean_diagonizable: fixes H :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H" shows "\<exists>k1 k2 M. mat_det M \<noteq> 0 \<and> unitary M \<and> congruence M H = (k1, 0, 0, k2) \<and> is_real k1 \<and> is_real k2 \<and> sgn (Re k1 * Re k2) = sgn (Re (mat_det H))"
ODList_def: shows "ODList \<equiv> odlist_Abs \<circ> msort"
k: shows "(2 ^ 7 * (3 * 2 ^ (7 - 1) - 1) * (3 * 2 ^ 7 - 1)) Amic (2 ^ 7 * (9 * 2 ^ (2 * 7 - 1) - 1))"
HInit_HInv6: fixes s :: "state" assumes "HInit s" shows "HInv6 s"
I_minWeightBasisD_5: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "T \<subseteq> carrier"
Always_reachable: fixes F :: "'d program" shows "F \<in> Always (reachable F)"
unit_def: fixes a :: "'b" shows "GoH.unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : GoH.map\<^sub>0 a \<Rightarrow>\<^sub>D GoH.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[GoH.map\<^sub>0 a] = (GoH.map \<i>\<^sub>B[a] \<cdot>\<^sub>D GoH.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
tensor_non_empty: fixes A :: "'a list list"   and B :: "'a list list" assumes "A \<noteq> []"   and "B \<noteq> []" shows "A \<otimes> B \<noteq> []"
nn_integral_measure_pmf_finite: fixes M :: "'a pmf"   and f :: "'a \<Rightarrow> ennreal" assumes "finite (set_pmf M)"   and "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>set_pmf M. f x * ennreal (pmf M x))"
cCons_0_Nil_eq: shows "(0::'b) ## [] = []"
sup_top_right: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and top :: "'b::type"   and x :: "'b::type" assumes "class.bounded_lattice_top (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (top::'b::type)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (top::'b::type) = top"
scalar_def: fixes A :: "V"   and f :: "V" shows "scalar A f \<equiv> nop A []\<^sub>\<circ> f"
euler_witness_coset: fixes x :: "nat"   and y :: "nat" assumes "euler_witness (int x) p"   and "y \<in> H #>\<^bsub>G\<^esub> x" shows "euler_witness (int y) p"
no_abs_simps_3: fixes t :: "'c"   and u :: "'c" shows "no_abs (app t u) = (no_abs t \<and> no_abs u)"
p_dist_sup: fixes x :: "'c"   and y :: "'c" shows "- (x \<sqinter> y) = - x \<squnion> - y"
P1: shows "Bet B C' C \<and> Cong B A B C'"
sign_changes_neg_number_rootat: fixes ps :: "rat poly list" shows "sign_changes_neg_inf (map real_of_rat_poly ps) = sign_changes_neg_number_rootat ps"
implies_itself_same: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> x = - y \<squnion> y"
mem_Sigma_iff: fixes a :: "'c"   and b :: "'d"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
sd_chain_Suc: fixes m :: "nat"   and q :: "'alt pmf"   and r :: "'alt pmf" assumes "sd_chain m = Some q"   and "sd_chain (Suc m) = Some r" shows "q \<prec>[Pareto (SD \<circ> R)] r"
map_fun_def: fixes f :: "'k \<Rightarrow> 'm"   and g :: "'n \<Rightarrow> 'l"   and h :: "'m \<Rightarrow> 'n" shows "(f ---> g) h = g \<circ> h \<circ> f"
dist_fls_ge0: fixes a :: "'a fls"   and b :: "'a fls" shows "0 \<le> dist a b"
list_of_R1: fixes x :: "'b list_R1" shows "list_to_list_R1 (list_R1_to_list x) = x"
strongly_fair_def: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "strongly_fair enabled taken = \<box>(\<lambda>s. (\<box>\<diamond>enabled) s \<longrightarrow> (\<diamond>taken) s)"
minor_transpose: fixes A :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "minor (Square_Matrix.transpose A) i j = Square_Matrix.transpose (minor A j i)"
uniformity_Bonk_Schramm_extension_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
unowned_shared_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list" shows "unowned_shared \<S> ts \<equiv> - (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set ts. \<O>) \<subseteq> dom \<S>"
integral_snd: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "LINT y|M2. LINT x|M1. f x y = LINT (x, y)|(M1 \<Otimes>\<^sub>M M2). f x y"
divide_divide_times_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e"   and w :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide x y) (divide z w) = divide (times x w) (times y z)"
succs: fixes x :: "'a"   and y :: "'a" assumes "isNode x"   and "isNode y"   and "nodeAbs x = nodeAbs y" shows "nodeAbs ` set (succs x) = nodeAbs ` set (succs y)"