hcju/isabelleps huggingface dataset:
adj_path_butlast: fixes v :: "'v"   and ps :: "'v list" assumes "adj_path v ps" shows "adj_path v (butlast ps)"
negligible_diff: fixes s :: "'a set"   and t :: "'a set" assumes "negligible s" shows "negligible (s - t)"
subQ: shows "submatrix Q UNIV J \<in> carrier_mat n k"
zero_vec_def: fixes n :: "nat" shows "0\<^sub>v n \<equiv> vec n (\<lambda>i. 0::'c)"
e: shows "\<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>) \<inter> (K .\<bottom>. \<phi>) \<noteq> {}"
eq: shows "trace (outer_prod v v) = (\<Sum>i = 0..<n. v $ i * conjugate (v $ i))"
simps_2: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "qsort (x # xs) l = qsort (filter (\<lambda>y. \<not> smaller x y l) xs) l @ [x] @ qsort (filter (\<lambda>y. smaller x y l) xs) l"
mult_dvd_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a dvd b"   and "c dvd d" shows "a * c dvd b * d"
Icc_eq_Icc: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and h :: "'c"   and l' :: "'c"   and h' :: "'c" assumes "class.order less_eq less" shows "(ord.atLeastAtMost less_eq l h = ord.atLeastAtMost less_eq l' h') = (l = l' \<and> h = h' \<or> \<not> less_eq l h \<and> \<not> less_eq l' h')"
prod_fun_closed: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "\<And>x. f x \<in> A"   and "\<And>x. g x \<in> A" shows "prod_fun f g x \<in> A"
freshAbs_substAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "freshAbs zs z (A $[Y / y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> fresh zs z Y))"
S_transition_cases: fixes P :: "'state"   and \<alpha>\<^sub>S :: "(('act, 'pred) S_action, unit) S_action"   and P' :: "'state" assumes "S_transform.S_transition P \<langle>\<alpha>\<^sub>S,P'\<rangle>" obtains \<alpha> :: "('act, 'pred) S_action" where "\<alpha>\<^sub>S = S_action.Act \<alpha>" and "P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle>"   | \<phi> :: "unit" where "\<alpha>\<^sub>S = S_action.Pred \<phi>" and "P' = P" and "P \<turnstile>\<^sub>S \<phi>"
progress_set_lemma: fixes C :: "'b set set"   and F :: "'b program"   and T :: "'b set"   and B :: "'b set"   and r :: "'b set" assumes "C \<in> progress_set F T B"   and "r \<in> wens_set F B"   and "F \<in> stable T" shows "T \<inter> r \<in> C"
simps_1: fixes ob :: "'b set"   and ar :: "'c set"   and dom :: "'c \<Rightarrow> 'b"   and cod :: "'c \<Rightarrow> 'b"   and id :: "'b \<Rightarrow> 'c"   and comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "Ob\<^bsub>\<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr>\<^esub> = ob"
supt_subset_RPO_S: shows "{\<rhd>} \<subseteq> RPO_S"
assms_2: assumes "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A" shows "P"
swap_id_common: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<noteq> c"   and "b \<noteq> c" shows "Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
GSubspaces_1: shows "RModule FG (\<cdot>) W \<and> W \<subseteq> V"
exec_tmp: fixes x :: "nat"   and k :: "nat" shows "rec_exec (rec_all (Cn 2 rec_minus [recf.id 2 0, Cn 2 (constn (Suc 0)) [recf.id 2 0]]) (Cn 3 rec_noteq [Cn 3 rec_mult [recf.id 3 (Suc 0), recf.id 3 2], recf.id 3 0])) [x, k] = (if \<forall>w\<le>rec_exec (Cn 2 rec_minus [recf.id 2 0, Cn 2 (constn (Suc 0)) [recf.id 2 0]]) [x, k]. 0 < rec_exec (Cn 3 rec_noteq [Cn 3 rec_mult [recf.id 3 (Suc 0), recf.id 3 2], recf.id 3 0]) ([x, k] @ [w]) then 1 else 0)"
redT_updWs_None_implies_None: fixes t' :: "'d"   and ws :: "'d \<Rightarrow> 'e wait_set_status option"   and was :: "('d, 'e) wait_set_action list"   and ws' :: "'d \<Rightarrow> 'e wait_set_status option"   and t :: "'d" assumes "redT_updWs t' ws was ws'"   and "t \<noteq> t'"   and "ws t = None" shows "ws' t = None"
lc_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "i.lcf p = lookup p (i.lpp p)"
induct: fixes P :: "nat \<Rightarrow> rat vec list \<Rightarrow> (rat vec \<times> rat) list \<Rightarrow> rat list list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "rat vec list"   and a2 :: "(rat vec \<times> rat) list"   and a3 :: "rat list list" assumes "\<And>n n_gs mus. P n [] n_gs mus"   and "\<And>n f fs n_gs mus. (\<And>x xa y xb. \<lbrakk>x = mus_adjuster_rat f n_gs [] (0\<^sub>v n); (xa, y) = x; xb = y + f\<rbrakk> \<Longrightarrow> P n fs ((xb, \<parallel>xb\<parallel>\<^sup>2) # n_gs) (xa # mus)) \<Longrightarrow> P n (f # fs) n_gs mus" shows "P a0 a1 a2 a3"
borel_measurable_iff_halfspace_greater: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)"
subst_mgt_refl: fixes \<theta> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta>"
p_def: fixes k :: "nat" shows "p k = (\<Sum>i<3. f i ^ k)"
star_trans_eq: fixes x :: "'b" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
BothFalse: assumes "\<not> P \<and> \<not> Q" shows "R"
tree_recurse_fusion: fixes h :: "'a \<Rightarrow> 'b"   and l :: "'a \<Rightarrow> 'a"   and l' :: "'b \<Rightarrow> 'b"   and r :: "'a \<Rightarrow> 'a"   and r' :: "'b \<Rightarrow> 'b"   and x :: "'a" assumes "h \<circ> l = l' \<circ> h"   and "h \<circ> r = r' \<circ> h" shows "map_tree h (tree_recurse l r x) = tree_recurse l' r' (h x)"
subst_bvs_empty: fixes t :: "Core.term" shows "subst_bvs [] t = t"
expand_incoming_name_exist: fixes n_ns :: "'b node \<times> 'b node set" assumes "expand_init < name (fst n_ns) \<and> (\<forall>nm\<in>incoming (fst n_ns). nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd n_ns) \<and> ((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"   and "\<forall>nd\<in>snd n_ns. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd n_ns)" shows "expand n_ns \<le> SPEC (\<lambda>nm_nds. \<forall>nd\<in>snd nm_nds. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd nm_nds))"
ENR_imp_locally_path_connected: fixes S :: "'b set" assumes "ENR S" shows "locally path_connected S"
last_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
gram_schmidt_code: fixes n :: "nat"   and ws :: "'c vec list" shows "gram_schmidt n ws = gram_schmidt_sub2 n [] ws"
simps: fixes t :: "'b" shows "- \<tau>\<^sub>c\<^sub>s\<^sub>t t = \<tau>\<^sub>c\<^sub>s\<^sub>t (- t)"
Upper_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Upper L A \<subseteq> carrier L"
prefixE: fixes xs :: "'e list"   and ys :: "'e list" assumes "prefix xs ys" obtains zs :: "'e list" where "ys = xs @ zs"
Int_assoc: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
dependent_eq_real1: shows "module_on.dependent (*\<^sub>R) = dependent"
chine_simps_3: shows "C.cod trg_f.chine = trg_f.cod.apex"
simps_11: fixes c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" shows "redexes (Catch c\<^sub>1 c\<^sub>2) = {Catch c\<^sub>1 c\<^sub>2} \<union> redexes c\<^sub>1"
\<K>_pow: shows "\<K> = (\<Sum>i\<in>L. 1 / b powr real (cw_len i))"
subst_semBV: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf, 'sz) ODE" shows "semBV (local.adjoint I \<sigma> \<nu>') ODE = semBV I (Osubst ODE \<sigma>)"
lActive_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf" shows "\<langle>c \<and> t\<rangle> \<equiv> GREATEST i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
AXIOMS_preserved_2: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and M' :: "trm" assumes "x:M \<in> AXIOMSn B"   and "M \<longrightarrow>\<^sub>a* M'" shows "x:M' \<in> AXIOMSn B"
full_abstraction_impl_SRel_iff_TRel_is_sym: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)" shows "sym SRel = sym (_Collect (T1, T2) (\<exists>S1 S2. T1 = \<lbrakk>S1\<rbrakk> \<and> T2 = \<lbrakk>S2\<rbrakk> \<and> (T1, T2) \<in> TRel))"
conc_fun_fail_iff_2: fixes R :: "('b \<times> 'c) set"   and S :: "'c nres" shows "(FAIL = \<Down> R S) = (S = FAIL)"
cases: fixes x :: "'a formula" obtains k :: "'a" where "x = Atom k"   | "x = \<bottom>"   | F :: "'a formula" where "x = \<^bold>\<not> F__"   | F :: "'a formula" and G :: "'a formula" where "x = Fa__ \<^bold>\<and> G__"   | F :: "'a formula" and G :: "'a formula" where "x = Fb__ \<^bold>\<or> Ga__"   | F :: "'a formula" and G :: "'a formula" where "x = F \<^bold>\<rightarrow> G"
case_3: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a"   and x31 :: "'b pred"   and x32 :: "'b seq" shows "(case Join x31 x32 of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
fromFull_inj: fixes \<sigma> :: "'tp"   and a :: "univ"   and b :: "univ" shows "(fromFull \<sigma> a = fromFull \<sigma> b) = (a = b)"
cp_StrongEq: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" shows "(X \<triangleq> Y) \<tau> = ((\<lambda>_. X \<tau>) \<triangleq> (\<lambda>_. Y \<tau>)) \<tau>"
tensor_preserves_can: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "can f"   and "can g" shows "can (f \<otimes> g)"
bottom_edge_explicit: shows "(\<lambda>x. twoC (x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
wordinterval_invert_set_eq: fixes r :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_invert r) = UNIV - wordinterval_to_set r"
surjective: fixes r :: "'a thread_t_scheme" shows "r = \<lparr>ev_counter = ev_counter r, \<dots> = thread_t.more r\<rparr>"
\<alpha>: fixes M :: "nat \<Rightarrow> 'a" assumes "decseq M"   and "\<And>i. P (M i)" shows "\<alpha> (Inf (range M)) = (INF i. \<alpha> (M i))"
foldli_concat: fixes xs :: "'e list list"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli (concat xs) c f \<sigma> = foldli xs c (\<lambda>x. foldli x c f) \<sigma>"
cod_cod: fixes f :: "'a2" assumes "C2.arr f" shows "C2.cod (C2.cod f) = C2.cod f"
assms_6: fixes c :: "'a"   and b :: "bool" shows "P c (Bool b)"
NOTRIGHT_eqvt_name: fixes pi :: "(name \<times> name) list"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> NOTRIGHT (NOT B) X = NOTRIGHT (NOT B) (pi \<bullet> X)"
assms_2: shows "usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta> \<noteq> undeft"
interval_subst: fixes S :: "'b set"   and x :: "'b"   and y :: "'b \<Rightarrow> 'b"   and j :: "'b" assumes "is_interval S"   and "x \<in> S"   and "y j \<in> S"   and "j \<in> Basis" shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
simps_2: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list"   and x :: "'e" shows "ipurge_tr I D u (xs @ [x]) = (if D x \<in> sinks I D u (xs @ [x]) then ipurge_tr I D u xs else ipurge_tr I D u xs @ [x])"
module_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "module scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'c) x = x)"
trg_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (src \<mu>) = src \<mu>"
assms_7: shows "\<forall>n. \<forall>asset\<in>support_set pf. finite (pf asset n ` space M)"
map_spmf_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'c spmf"   and g :: "'c \<Rightarrow> 'b spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
sorted_list_of_set_3: fixes A :: "'a word set" shows "distinct (signed.sorted_list_of_set A)"
G2_def: shows "G2 \<equiv> fps_expansion (\<lambda>s. (s - 1) * pre_zeta 1 s + 1) 1"
Gcd_0_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "(Gcd A = zero) = (A \<subseteq> {zero})"
step: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and x :: "'b" assumes "segmentp f a b x"   and "f x \<noteq> b" shows "segmentp f a b (f x)"
filter_map: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
continuous_dist: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. dist (f x) (g x))"
val_Zp_dist_res_eq2: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k" shows "eint (int k) \<le> val_Zp_dist a b"
trivial_combine_imp_independent: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "\<And>Ks. \<lbrakk>set Ks \<subseteq> K; combine Ks Us = \<zero>\<rbrakk> \<Longrightarrow> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "local.independent K Us"
components_are_iso: fixes a :: "'a" assumes "A.ide a" shows "B.iso (G a)"
empty_sets: fixes M :: "'b measure" shows "{} \<in> fmeasurable M"
regular_mult_closed: fixes x :: "'b"   and y :: "'b" assumes "regular x"   and "regular y" shows "regular (x * y)"
td_ext_sint: shows "td_ext sint word_of_int (sints LENGTH('a)) (\<lambda>w. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) - 2 ^ (LENGTH('a) - 1))"
any_closest_point_dot: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b" assumes "convex S"   and "closed S"   and "x \<in> S"   and "y \<in> S"   and "\<forall>z\<in>S. dist a x \<le> dist a z" shows "(a - x) \<bullet> (y - x) \<le> 0"
tab\<^sub>0_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg r) g"
nth_append_cond_1: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "i < length xs" shows "(xs @ ys) ! i = xs ! i"
Cons_2: fixes x :: "nat" assumes "x \<in> set (x__ # xs) - set ys" shows "\<exists>y. y \<in> set ys \<and> x \<le> y"
f_def: shows "f = (\<lambda>x. 1 - (1 - q ^ x) ^ n)"
closedin_imp_subset: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "closedin (subtopology U S) T" shows "T \<subseteq> S"
f_eq_1: fixes x :: "'a" shows "f (poly q x) = x"
insert: fixes A :: "'e set"   and x :: "'e"   and g :: "'e \<Rightarrow> 'f" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
add_col_sub_row_inv_from_uppert: fixes A :: "'a mat"   and j :: "nat"   and i :: "nat"   and v :: "'a" assumes "local.inv_from local.uppert A j"   and "A \<in> carrier_mat n n"   and "i < n"   and "i < j"   and "j < n" shows "local.inv_from local.uppert (add_col_sub_row v i j A) j"
separated_by_this_wall_fg: fixes C :: "'a set"   and D :: "'a set" assumes "separated_by {f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>} C D"   and "C \<in> f \<turnstile> folding_g.\<C>" shows "D \<in> g \<turnstile> folding_g.\<C>"
ODList_toList: fixes xs :: "'c odlist" shows "ODList (toList xs) = xs"
get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('c, 'a, 'b, 'j option) RDocument_scheme \<Rightarrow> 'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
less_iff_sym_greater: fixes b :: "'a"   and a :: "'a" shows "(cmp b a = Less) = (cmp a b = Greater)"
less_trans: fixes x :: "'t"   and y :: "'t"   and z :: "'t" assumes "x \<prec>\<^sub>t y"   and "y \<prec>\<^sub>t z" shows "x \<prec>\<^sub>t z"
ip0: fixes I :: "'d set"   and R :: "('d, 'e) Ring_scheme" shows "I \<^bsup>\<diamondsuit>R 0\<^esup> = carrier R"
step_5_correct: fixes p :: "prog" assumes "spec\<^sub>5 p" shows "spec\<^sub>4 p"
sappend_def: fixes Al :: "'c list set"   and Bl :: "'c list set" shows "Al @@ Bl = {al @ bl |al bl. al \<in> Al \<and> bl \<in> Bl}"
le_degree: fixes p :: "'a poly"   and n :: "nat" assumes "poly.coeff p n \<noteq> (0::'a)" shows "n \<le> degree p"
unit_simps_4: shows "src \<eta>\<^sub>1 = src e\<^sub>1"
iv_defined: shows "t0 \<in> T \<and> x0 \<in> X"
P0: shows "ns \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
pr_tuple_2: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "span f g"   and "cod f = a"   and "cod g = b" shows "\<pp>\<^sub>0[a, b] \<cdot> \<langle>f, g\<rangle> = g"
gp_real: fixes x :: "'a" shows "enn2real (gp x) = max (g x) 0"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category A"   and "category B"   and "isomorphic_categories_axioms A B" shows "isomorphic_categories A B"
convex_translation_eq: fixes a :: "'c"   and s :: "'c set" shows "convex ((+) a ` s) = convex s"
conj_simps_3: fixes Q :: "fm" shows "Ferrack.conj T Q = Q"
iso_upper_hemisphere_reduced_homology_group: fixes p :: "int"   and n :: "nat" shows "hom_boundary (1 + p) (subtopology (nsphere (Suc n)) {x. 0 \<le> x (Suc n)}) {x. x (Suc n) = 0} \<in> Group.iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. 0 \<le> x (Suc n)}) {x. x (Suc n) = 0}) (reduced_homology_group p (nsphere n))"
sup_left_zero: fixes x :: "'a" shows "top \<squnion> - x = top"
ex1_pole: fixes M :: "moebius" shows "\<exists>!z. is_pole M z"
c11: shows "lst \<delta>1''' = lst_conv (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))"
dvd_Lcm_fin: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times a (semiring_gcd.Lcm_fin one zero lcm A)"
dotprod_pointwise_le: fixes as :: "nat list"   and xs :: "nat list"   and i :: "nat" assumes "length as = length xs"   and "i < length as" shows "as ! i * xs ! i \<le> as \<bullet> xs"
exhaust: fixes y :: "'a tree" obtains "y = ET"   | x21 :: "'a" and x22 :: "'a tree" and x23 :: "'a tree" and x24 :: "nat" where "y = MKT x21 x22 x23 x24"
subdistl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<parallel> x + z \<parallel> (x + y) = z \<parallel> (x + y)"
bits_div_by_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide a zero = zero"
33: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "(x \<squnion> y) \<sqinter> z \<squnion> u = x \<sqinter> z \<squnion> (y \<sqinter> z \<squnion> u)"
mono: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (f a b) (f c d)"
Max_ge: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
a: shows "inj_on fa (Rep_zet a) \<and> fa ` Rep_zet a = explode za"
nop_onto_vdomain: shows "\<D>\<^sub>\<circ> vint_plus = \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
iFROM_iff: fixes x :: "nat"   and n :: "nat" shows "(x \<in> [n\<dots>]) = (n \<le> x)"
worklist_tree_aux_state_inv: fixes succs :: "'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d"   and ws' :: "'c list"   and s' :: "'d"   and I :: "'d \<Rightarrow> bool" assumes "worklist_tree_aux succs f (ws, s) = Some (ws', s')"   and "I s"   and "\<And>x s. I s \<Longrightarrow> I (f x s)" shows "I s'"
summable_ln_plus: fixes f :: "nat \<Rightarrow> real" assumes "summable f"   and "\<forall>n. 0 < f n" shows "summable (\<lambda>n. ln (1 + f n))"
finQBZ: shows "finite ((Q + B) \<inter> \<int>\<^sub>v)"
paths_single: fixes x :: "'b" shows "paths (single x) = {[], [x]}"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "domain_semigroup.ds_ord d (\<cdot>) b a"   and "domain_semigroup.ds_ord d (\<cdot>) c b" shows "domain_semigroup.ds_ord d (\<cdot>) c a"
det_Gauss_Jordan_in_ij: fixes A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "det (Gauss_Jordan_in_ij A i j) = det (mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)) i ((1::'b) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n) $ i $ j))"
simps_6: fixes f1 :: "'a"   and f2 :: "'a" shows "rec_T1 f1 f2 T1.B = f2"
he: shows "nsphere n homotopy_equivalent_space subtopology (nsphere n) {a}"
cases: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a \<times> nat" assumes "class.monoid_mult one times" obtains xa :: "'a" and n :: "nat" where "x = (xa, n)"
X: fixes i :: "'i" assumes "i \<in> I" shows "X i \<in> space M \<rightarrow> space (M' i)"
aff_dim_le_card: fixes V :: "'b set" assumes "finite V" shows "aff_dim V \<le> int (card V) - 1"
price_structure_maturity: fixes pyf :: "'a \<Rightarrow> real"   and T :: "nat"   and \<pi> :: "real"   and pr :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "price_structure pyf T \<pi> pr" shows "AEeq M (pr T) pyf"
add_decreasing: fixes x :: "'b"   and y :: "'b" shows "x + y \<le> x"
chop_nonempty: fixes P :: "'b list \<Rightarrow> bool"   and splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list"   and xs :: "'b list" assumes "is_maxsplitter P splitf" shows "\<forall>yss zs. chop splitf xs = (yss, zs) \<longrightarrow> (\<forall>ys\<in>set yss. ys \<noteq> [])"