hcju/isabelleps huggingface dataset:
linear_split_gr: fixes xs :: "('c \<times> 'd) list"   and p :: "'d"   and ys :: "('c \<times> 'd) list"   and ls :: "('c \<times> 'd) list"   and rs :: "('c \<times> 'd) list" assumes "linear_split_help xs p ys = (ls, rs)"   and "sorted_less (separators (ys @ xs))"   and "\<forall>(sub, sep)\<in>set ys. sep < p" shows "case rs of [] \<Rightarrow> True | (x, sep) # xa \<Rightarrow> p \<le> sep"
trapped_def: fixes x :: "'a"   and K :: "'a set" shows "rev.trapped x K = (rev.trapped_forward x K \<and> rev.trapped_backward x K)"
erHomTr2: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and a :: "'a"   and b :: "'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<noteq> \<zero>"   and "b \<noteq> \<zero>"   and "a \<plusminus> b \<noteq> \<zero>"   and "deg_n R S X a < deg_n R S X b" shows "erH R S X A B Y h (a \<plusminus> b) = erH R S X A B Y h a \<plusminus>\<^bsub>A\<^esub> erH R S X A B Y h b"
intros_2: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list" assumes "P \<turnstile> C \<prec>\<^sub>R D"   and "Subobjs\<^sub>R P D Cs" shows "Subobjs\<^sub>R P C (C # Cs)"
Abrupt_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate"   and s' :: "'b" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Abrupt s'" shows "t = Abrupt s'"
inf_Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "AlternativeSmallStep.inf \<Gamma> (Catch c1 c2 # cs) css (Normal s) = AlternativeSmallStep.inf \<Gamma> [c1] ((cs, c2 # cs) # css) (Normal s)"
emeasure_bounded_finite: fixes A :: "'c set" assumes "bounded A" shows "emeasure lborel A < \<infinity>"
less_mod_0_imp_div_less: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a < b"   and "b mod m = 0" shows "a div m < b div m"
is12_def_2: shows "is1 \<lhd> input_sizes m1"
valid_strategy_in_V: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "strategy p \<sigma>"   and "v \<in> VV p"   and "\<not> deadend v" shows "\<sigma> v \<in> V"
distinct_11: fixes x3 :: "real mpoly"   and x4 :: "real mpoly" shows "Leq x3 \<noteq> Neq x4"
sum_upto_ln_conv_ln_fact: fixes x :: "real" shows "sum_upto (\<lambda>x. ln (real x)) x = ln (fact (nat \<lfloor>x\<rfloor>))"
results_converter_def: fixes \<I> :: "('b, 'a) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('b, 'a, 'c, 'd) converter" shows "results_converter \<I> \<I>' conv = {b. resultsp_converter \<I> \<I>' b conv}"
DX_1: shows "(3, Dec 2 7 0) \<in> X"
formal_out_THE: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs" shows "(THE outs. \<exists>ins. (p, ins, outs) \<in> set procs) = outs"
has_codomain_iff_arr: fixes f :: "'a1" shows "(C1.codomains f \<noteq> {}) = C1.arr f"
C: assumes "\<not> P \<sigma>"   and "\<sigma> \<Turnstile> B\<^sub>2" shows "Q"
r_one: fixes x :: "'a set" assumes "x \<in> carrier (R Quot J)" shows "x \<otimes>\<^bsub>R Quot J\<^esub> \<one>\<^bsub>R Quot J\<^esub> = x"
compact_space_def: fixes X :: "'c topology" shows "compact_space X \<equiv> compactin X (topspace X)"
conseq_extract_state_indep_prop: fixes P :: "'d set"   and R :: "bool"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. R"   and "R \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" shows "fast.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> fast.check_eqvRE enum_class.enum n (rexp_of'' n (norm \<phi>)) (rexp_of'' n (norm \<psi>)))"
no_unacquired_write_to_read_only: fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<O> :: "nat set"   and a :: "nat" assumes "no_write_to_read_only_memory \<S> sb"   and "sharing_consistent \<S> \<O> sb"   and "a \<in> read_only \<S>"   and "a \<notin> \<O> \<union> all_acquired sb" shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b sb"
that: fixes r :: "recf" assumes "recfn 2 r"   and "\<forall>x y. eval r [x, y] = f x y" shows "thesis"
restrict_mono_aux: fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and x :: "nat set"   and C :: "nat set" assumes "dom A \<subseteq> dom B" shows "dom (A \<ominus>\<^bsub>x\<^esub> C) \<subseteq> dom (B \<ominus>\<^bsub>x\<^esub> C)"
distinct_2: fixes tyvrs' :: "tyvrs"   and ty' :: "ty"   and vrs :: "vrs"   and ty :: "ty" shows "TVarB tyvrs' ty' \<noteq> VarB vrs ty"
coeff_C_gt0: fixes n :: "nat"   and x :: "'b" assumes "0 < n" shows "coeff (C x) n = (0::'b)"
piecewise_continuous_on_insert_leftI: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "piecewise_continuous_on a b I f" shows "piecewise_continuous_on a b (insert a I) f"
simps: fixes a1 :: "'b set"   and a2 :: "'b \<Rightarrow> nat"   and a3 :: "'b \<Rightarrow> intv"   and a4 :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and a5 :: "('b \<times> 'b) set" shows "valid_region a1 a2 a3 a4 a5 = (\<exists>X\<^sub>0 X I r k J. a1 = X \<and> a2 = k \<and> a3 = I \<and> a4 = J \<and> a5 = r \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d} \<and> refl_on X\<^sub>0 r \<and> trans r \<and> total_on X\<^sub>0 r \<and> (\<forall>x\<in>X. valid_intv (k x) (I x)) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> valid_intv' (int (k y)) (int (k x)) (J x y)))"
D: shows "xs = us @ y # vs \<and> (\<forall>u\<in>set us. key u \<noteq> k) \<and> key y = k \<and> ys = filter (\<lambda>x. key x = k) vs"
or_complement_overlap: fixes x :: "i" shows "\<forall>y. O y x \<or> O y (\<midarrow> x)"
h1a: shows "((\<Sum>i = 0..<deg + 1. f i / D ^ i) < 0) = ((\<Sum>i = 0..<deg + 1. f i * D ^ (deg - i)) * D ^ (deg mod 2) < 0)"
meet_right: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "le (dlat.meet le x y) y"
has_limits_of_shape_if_target_does: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "functor_category A B"   and "category J"   and "category.has_limits_of_shape B J" shows "category.has_limits_of_shape (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>)) J"
refl: fixes a :: "'g multiset" shows "a \<subseteq># a"
m_lcancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> \<zero>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
field_cont_on_left_edge_image: shows "continuous_on {p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (y, b)} F"
that: fixes p :: "('a \<times> 'a set) set" assumes "p tagged_division_of cbox a b"   and "g fine p" shows "thesis"
lemma4: fixes r :: "dB"   and r' :: "dB"   and r'' :: "dB" assumes "r \<rightarrow>\<^sub>s r'"   and "r' \<rightarrow>\<^sub>\<beta> r''" shows "r \<rightarrow>\<^sub>s r''"
remdups_adj_append_two: fixes xs :: "'g list"   and x :: "'g"   and y :: "'g" shows "remdups_adj (xs @ [x, y]) = remdups_adj (xs @ [x]) @ (if x = y then [] else [y])"
essential_class_def: fixes C :: "'s set" shows "essential_class C = (C \<in> UNIV // communicating \<and> acc `` C \<subseteq> C)"
lens_prism_def: fixes X :: "'c \<Longrightarrow> 'd" shows "lens_prism X = \<lparr>prism_match = \<lambda>s. if s \<in> \<S>\<^bsub>X\<^esub> then Some (get\<^bsub>X\<^esub> s) else None, prism_build = create\<^bsub>X\<^esub>\<rparr>"
set_empty: fixes xs :: "'a list" shows "(list.set xs = {}) = (xs = [])"
simps_2: fixes k :: "nat"   and u :: "nat" shows "\<J> k (Suc u) = \<pi> k (\<T> k (Suc u)) (\<J> k u)"
assms: shows "Y \<in> set (all_partitions_alg X)"
mult_vec_distrib_left: fixes r :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "r ** (p + q) = r ** p + r ** q"
finsum_vec_insert: fixes F :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'c vec"   and n :: "nat" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec n"   and "f a \<in> carrier_vec n" shows "finsum_vec TYPE('c) n f (insert a F) = f a + finsum_vec TYPE('c) n f F"
split_time_mod: fixes t :: "nat"   and n :: "nat"   and s :: "nat \<Rightarrow> 'a list" assumes "t mod n \<noteq> 0" shows "split_time s n t = []"
assms: shows "(f, g) \<in> [P]\<^sub>a RS \<rightarrow> T"
lift_envs_def: fixes R :: "'b list set" shows "lift_envs R = (\<lambda>(a, b). a # b) ` (UNIV \<times> R)"
subset_refl: fixes A :: "'c set" shows "A \<subseteq> A"
arg_mult: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 * z2 \<noteq> 0" shows "arg (z1 * z2) = \<downharpoonright>arg z1 + arg z2\<downharpoonleft>"
hx: fixes x :: "complex" assumes "1 + x = 0"   and "0 < Re x" shows "False"
check_nsc_def: fixes check_s :: "'b \<Rightarrow> 'c list + 'b" shows "check_nsc check_s = sum_wrap (\<lambda>asitv. check_s)"
vrange_vid_on: fixes A :: "V" shows "\<R>\<^sub>\<circ> (vid_on A) = A"
disc_9: fixes x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap" shows "is_Vrecabs (Vrecabs x31 x32 x33)"
Awhile_8: shows "\<forall>l s1 s2. s1 = s2 on LVE \<longrightarrow> postList upds l s1 = postList upds l s2"
nat_ivl_Suc': fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> Suc n" shows "prod g {m..Suc n} = g (Suc n) * prod g {m..n}"
assms_3: shows "S\<lparr>ArrVal\<rparr> = T\<lparr>ArrVal\<rparr>"
lens_put_comm: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'c"   and u :: "'a" assumes "X \<bowtie> Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"
simps_2: fixes x :: "'b" shows "ptrm_fvs (PVar x) = {x}"
close: fixes P :: "complex mat" assumes "is_quantum_predicate P" shows "is_quantum_predicate (WS P)"
thetaFDW0_sym: shows "sym thetaFDW0"
det_state_cover_initial: fixes M :: "('d, 'e, 'f) FSM"   and V :: "'d list set" assumes "is_det_state_cover M V" shows "[] \<in> V"
domsMT_notND_DD: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "dom (C (DenyAllFromTo a b)) \<inter> dom (C (DenyAllFromTo c d)) \<noteq> {}" shows "\<not> netsDistinct a c"
distinct_65: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "[name1\<frown>name2]pi \<noteq> pi1' \<parallel> pi2'"
map_last: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
TINV: shows "\<forall>t\<in>set ts. invar_tree t"
some_nan_def: shows "some_nan = (SOME a. is_nan a)"
indoc_appendI1: fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "indoc a u" shows "indoc a (u @ v)"
map_poly_gcd: fixes p :: "rat poly"   and q :: "rat poly" shows "map_poly of_rat (gcd p q) = gcd (map_poly of_rat p) (map_poly of_rat q)"
preR: shows "pre_adjust \<langle>l, (a, lv), r'\<rangle>"
star_plus_one: fixes x :: "'a" shows "1' + x\<^sup>\<star> = x\<^sup>\<star>"
homeomorphic_maps_relative_homology_group_isomorphisms: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'c set"   and p :: "int" assumes "homeomorphic_maps X Y f g"   and "f ` S \<subseteq> T"   and "g ` T \<subseteq> S" shows "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T) (hom_induced p X S Y T f) (hom_induced p Y T X S g)"
inf1: fixes A :: "'c"   and k :: "'c"   and a :: "'c"   and b :: "'c" assumes "A \<equiv> inf k a" shows "inf A b \<equiv> inf k (inf a b)"
alt: shows "image_mset block_complement \<B> rep x = int (size (filter_mset ((\<notin>) x) \<B>))"
SN_on_Image_rtrancl: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r A" shows "SN_on r (r\<^sup>* `` A)"
semInt_prWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prWls (semInt SEM) SEM"
\<tau>red1t_into_\<tau>Red1t: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e'' :: "'addr expr1"   and xs'' :: "'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list" assumes "\<tau>red1gt uf P t h (e, xs) (e'', xs'')" shows "\<tau>Red1gt uf P t h ((e, xs), exs) ((e'', xs''), exs)"
simps_1: fixes carrier :: "'a set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "(\<sqsubseteq>\<^bsub>\<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr>\<^esub>) = le"
map_pmf_eq_return_pmf_iff: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c pmf"   and x :: "'b" shows "(map_pmf f p = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = x)"
abs_INV_init_transfer: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme"   and I :: "'a set" assumes "PO_refines R Ta Tc"   and "init Ta \<subseteq> I" shows "init Tc \<subseteq> R `` I"
omit_all_is: fixes p :: "('c, 'd) fm"   and ps :: "('c, 'd) fm list"   and xs :: "nat set"   and q :: "('c, 'd) fm" assumes "all_is p ps xs"   and "q \<in>. ps"   and "q \<notin> set (omit xs ps)" shows "q = p"
uniform_unfold_point_set_mset: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (size {#p \<in># mset_set \<V>. p \<in> bl#}) = \<k>"
Union_empty_conv: fixes A :: "'b set set" shows "(\<Union> A = {}) = (\<forall>x\<in>A. x = {})"
lsorted_LCons: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" assumes "class.preorder less_eq less" shows "ord.lsorted less_eq (LCons x xs) = (ord.lsorted less_eq xs \<and> (\<forall>y\<in>lset xs. less_eq x y))"
snth_Stream: fixes x :: "'c"   and s :: "'c stream"   and i :: "nat" shows "(x ## s) !! Suc i = s !! i"
x_dom_3: shows "x \<in> subst_domain \<theta>"
isomorphicI': fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<cong>\<^sub>C b\<guillemotright>" shows "a \<cong>\<^sub>C b"
set_spmf_def: fixes p :: "'d spmf" shows "set_spmf p = set_pmf p \<bind> set_option"
form_cases: fixes l :: "nat" obtains "l = 0"   | ka :: "nat" and kb :: "nat" where "l = ka + kb - 1" and "0 < kb" and "kb \<le> ka" and "ka \<le> Suc kb"
all_in_listAppend: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list"   and ys :: "('a, 'b) Combinators list" assumes "all_in_list xs l"   and "all_in_list ys l" shows "all_in_list (xs @ ys) l"
simps_2: fixes i :: "nat"   and R :: "nat set" shows "wf_net_tree \<langle>i; R\<rangle> = True"
dom_gf: shows "\<D>\<^sub>\<circ> gf = 2\<^sub>\<nat>"
homeomorphism_on_UNIV_sequentially: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x u. u \<longlonglongrightarrow> x = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f x" shows "homeomorphism_on UNIV f"
2: shows "\<guillemotleft>cmp\<^sub>D\<^sub>N \<mu>\<nu> : DN (local.dom (fst \<mu>\<nu>)) \<star>\<^sub>B DN (local.dom (snd \<mu>\<nu>)) \<Rightarrow>\<^sub>B DN (cod (fst \<mu>\<nu>) \<star> cod (snd \<mu>\<nu>))\<guillemotright>"
error_reads: fixes e :: "'y"   and h :: "'x"   and h' :: "'x" shows "reads {} (error e) h h'"
continuous_eval_fps: fixes z :: "'a"   and F :: "'a fps"   and A :: "'a set" assumes "ereal (norm z) < fps_conv_radius F" shows "continuous (at z within A) (eval_fps F)"
simps: fixes s :: "'d"   and \<omega> :: "(real \<times> 'd) stream"   and j :: "real" shows "trace_at s \<omega> j = (case \<omega> of (t', s') ## \<omega> \<Rightarrow> if t' \<le> j then trace_at s' \<omega> j else s)"
assms_2: shows "AE x in M. \<bar>f x\<bar> \<le> C"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (t1__, aa__, R t2__ ba__ (R t3__ c__ t4))"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" where "x = (t1a__, ab__, R (R t2a__ bb__ t3a__) ca__ \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (t1b__, ac__, R (R t2 b t3) c (B v__ vc__ vb__))"   | t1 :: "('b \<times> color) tree" and a :: "'b" where "x = (t1c__, ad__, \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (t1d__, ae__, B va__ vca__ vba__)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and va :: "'b \<times> color" where "x = (t1e__, af__, \<langle>\<langle>\<rangle>, vaa__, \<langle>\<rangle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and vb :: "('b \<times> color) tree" and ve :: "'b" and vd :: "('b \<times> color) tree" and va :: "'b \<times> color" where "x = (t1f__, ag__, \<langle>B vbb__ ve__ vd, vab__, \<langle>\<rangle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vb :: "'b" where "x = (t1g__, ah__, B vf__ vbc__ \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and va :: "'b \<times> color" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1h__, ai__, \<langle>\<langle>\<rangle>, vac__, B vcb__ vfa__ vea__\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and vb :: "('b \<times> color) tree" and vh :: "'b" and vg :: "('b \<times> color) tree" and va :: "'b \<times> color" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1i__, aj__, \<langle>B vbd__ vh vg, va, B vcc__ vfb__ veb__\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vb :: "'b" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1, a, B v vb (B vc vf ve))"
hyps_3: shows "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>\<^sub>2"
ring_simprules_18: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = y"
n1: shows "n1 = length (signs1 ! (j div length signs2))"
lb_def: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" shows "lb T A j = ((\<forall>x\<in>{1..m}. \<forall>y\<in>{1..m}. x \<noteq> y \<longrightarrow> A x \<inter> A y = {}) \<and> \<Union> (A ` {1..m}) = {1..j} \<and> (\<forall>x\<in>{1..m}. sum t (A x) = T x))"
lift_preserves_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "inj (lift f)"
cpo: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a" assumes "class.cpo below"   and "po.chain below S" shows "\<exists>x. po.is_lub below (range S) x"
FI_finalize: fixes m :: "(assn \<times> assn) list"   and p :: "assn"   and up :: "assn"   and q :: "assn"   and uq :: "assn"   and f :: "assn" assumes "FI_RESULT m (p * up) (q * uq) f" shows "FI m p q up uq f"
a_hd_b_neq_all: shows "a ! hd b \<noteq> rev [0..<n]"
same_append_eq: fixes xs :: "'f list"   and ys :: "'f list"   and zs :: "'f list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
le_imp_max_x_ge: fixes v :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "v \<le>\<^sub>v y"   and "length y \<le> n" shows "max_x y i \<le> max_x v i"
cases: fixes r :: "env" obtains prg :: "prog" and cls :: "qtname" and lcl :: "lenv" where "r = \<lparr>prg = prg, cls = cls, lcl = lcl\<rparr>"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e blindable\<^sub>h" shows "map_blindable\<^sub>h g (map_blindable\<^sub>h f v) = map_blindable\<^sub>h (g \<circ> f) v"
simps_2: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" shows "subst \<rho> (App s t) = App (subst \<rho> s) (subst \<rho> t)"
indRelLTPO_impl_TRel_weakly_reflects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelLTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
refl: fixes x :: "'a" assumes "x \<in> carrier" shows "le x x"
IH2: fixes m :: "nat"   and xa :: "taylor_model"   and z :: "nat" assumes "m < n"   and "num_params (tm_poly xa) \<le> z" shows "num_params (tm_poly (tm_pow prec ord I a xa m)) \<le> z"
unit_simps_3: shows "cod \<eta>\<^sub>1 = d\<^sub>1 \<star> e\<^sub>1"
min_correct_2: fixes s :: "'s"   and x :: "'u"   and P :: "'u \<Rightarrow> bool" assumes "invar s"   and "x \<in> \<alpha> s"   and "P x" shows "the (min s P) \<le> x"
Cons_1: fixes k :: "nat"   and thesis :: "bool" assumes "\<And>i j. \<lbrakk>i < j; j < length bs; v = ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i))\<rbrakk> \<Longrightarrow> thesis"   and "v \<in> set (Koszul_syz_sigs_aux bs k)" shows "thesis"
assms_2: shows "(\<^bold>\<diamond> Nom i) at a in (ps, a) # branch"
list_max_in_list: fixes xs :: "'t list" assumes "xs \<noteq> []" shows "list_max xs \<in> set xs"
integral_subset_le: fixes s :: "'c set"   and t :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "s \<subseteq> t"   and "f integrable_on s"   and "f integrable_on t"   and "\<forall>x\<in>t. 0 \<le> f x" shows "integral s f \<le> integral t f"
red_single_ord: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single p q f t" shows "punit.ord_strict_p q p"
dims1_def: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list" assumes "partial_state2 vars1 vars2" shows "partial_state2.dims1 dims vars1 = nths dims vars1"
split_int_eq_real': fixes a :: "int"   and b :: "real" shows "(real_of_int a + b = 0) = (a - \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b = 0)"
lc_monomial: fixes t :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "lcf (monomial c t) = c"
divmod_def: fixes p :: "'a mpoly"   and q :: "'a mpoly" shows "MPoly_Type.divmod p q = (p div q, p mod q)"
app_simps_3: fixes t :: "'a"   and u :: "'a" shows "is_app (app t u)"
hIm_hcis: fixes a :: "real star" shows "hIm (hcis a) = (*f* sin) a"