hcju/isabelleps huggingface dataset:
subspace_add: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "csubspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
Point_eq_0: fixes xA :: "real"   and yA :: "real" shows "(Point xA yA = 0) = (xA = 0 \<and> yA = 0)"
current: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> Y" shows "current \<Gamma> f"
zneq_aneq: fixes n :: "int"   and m :: "int" shows "(n \<noteq> m) = (ant n \<noteq> ant m)"
convergent: fixes f :: "'d \<Rightarrow> 'e"   and X :: "nat \<Rightarrow> 'd" assumes "bounded_linear f"   and "convergent X" shows "convergent (\<lambda>n. f (X n))"
ball_set_intros_1: fixes P :: "'a \<Rightarrow> bool" shows "\<forall>x\<in>set []. P x"
case_prodD': fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and c :: "'d" assumes "(case (a, b) of (c, d) \<Rightarrow> R c d) c" shows "R a b c"
global_reg_mod_privilege: fixes s' :: "'b sparc_state"   and w1 :: "word32"   and n :: "nat"   and w2 :: "word5"   and s :: "'b sparc_state" assumes "s' = global_reg_mod w1 n w2 s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
ctermsDI: fixes pp :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "pp \<in> cterms \<Gamma>"   and "p \<in> dterms \<Gamma> pp" shows "p \<in> cterms \<Gamma>"
simps_5: fixes x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com"   and y61 :: "'d set"   and y62 :: "('d, 'e, 'f) com" shows "(While x61 x62 = While y61 y62) = (x61 = y61 \<and> x62 = y62)"
Sup_spec_def: fixes X :: "'b set" shows "Sup_spec X = SPEC (\<lambda>r. \<forall>x\<in>X. x \<le> r)"
splitFace_new_f21_norm: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "normFace f21 \<notin> set (normFaces (faces g))"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>A gt ys xs. \<lbrakk>\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x; ys \<in> lists A; xs \<in> lists A\<rbrakk> \<Longrightarrow> ext gt ys xs \<or> ext gt xs ys \<or> ys = xs" shows "ext_total_axioms ext"
stl_sset: fixes x :: "'a"   and a :: "'a stream" assumes "x \<in> sset (stl a)" shows "x \<in> sset a"
divide_le_eq_numeral1_1: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / numeral w \<le> a) = (b \<le> a \<cdot> numeral w)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> pol \<Rightarrow> 'c"   and f3 :: "pol \<Rightarrow> nat \<Rightarrow> pol \<Rightarrow> 'c"   and pol :: "pol" shows "P (case pol of Pc x \<Rightarrow> f1 x | Pinj x xa \<Rightarrow> f2 x xa | PX x xa xb \<Rightarrow> f3 x xa xb) = ((\<forall>x1. pol = Pc x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. pol = Pinj x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32 x33. pol = PX x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)))"
weakCongSimI: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
vD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "vD rt \<equiv> {dip. flag rt dip = Some val}"
continuous_on_eq_continuous_within: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "continuous_on s f = (\<forall>x\<in>s. continuous (at x within s) f)"
cone_\<chi>: shows "D.cone b (D.mkCone F)"
IDirProds_eq_finprod_Pi: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G" shows "IDirProds G S I = (\<lambda>x. finprod G x I) ` Pi I S"
expand: fixes tllist :: "('g, 'h) tllist"   and tllist' :: "('g, 'h) tllist" assumes "is_TNil tllist = is_TNil tllist'"   and "\<lbrakk>is_TNil tllist; is_TNil tllist'\<rbrakk> \<Longrightarrow> terminal tllist = terminal tllist'"   and "\<lbrakk>\<not> is_TNil tllist; \<not> is_TNil tllist'\<rbrakk> \<Longrightarrow> thd tllist = thd tllist' \<and> ttl tllist = ttl tllist'" shows "tllist = tllist'"
mult_poly_full_mult_poly_p': fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset"   and q :: "(char list list \<times> int) list"   and q' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> sorted_poly_rel"   and "(q, q') \<in> sorted_poly_rel" shows "mult_poly_full p q \<le> \<Down> sorted_poly_rel (mult_poly_p' p' q')"
I1: shows "I1 = (\<lambda>l s. 0 \<le> s ''a'' \<and> 0 \<le> s ''b'' \<and> s ''z'' \<le> s ''a'' \<and> 0 \<le> s ''z'' \<and> s ''y'' = s ''z'' * s ''b'')"
bounded: fixes x :: "nat" shows "bounded (range (\<lambda>n. f n x))"
lift_def: fixes f :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and v :: "'d + 'f" shows "lift f v \<equiv> case v of Inl e \<Rightarrow> throwError e | Inr v' \<Rightarrow> f v'"
fmfilter_alt_defs_2: fixes A :: "'c set" shows "fmdrop_set A = (\<lhd>/) A"
cons_cluster_wb: fixes p :: "'node"   and C :: "'node set" assumes "p \<in> C"   and "is_cons_cluster C" shows "p \<in> W"
total_sur: fixes x :: "'a" shows "is_total x = is_sur (x\<^sup>T)"
assms: fixes A :: "'b set" shows "times_abc a (Inf_b A) = Inf_c (times_abc a ` A)"
enum_all_char_def: fixes P :: "char \<Rightarrow> bool" shows "enum_class.enum_all P = list_all P enum_class.enum"
pos_ex: shows "{0..} \<subseteq> existence_ivl0 x"
chop_App_Hd: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" assumes "is_Hd s" shows "chop (App s t) = t"
vfsequence_vcons_vdistinct: fixes x :: "V" assumes "vdistinct (xs #\<^sub>\<circ> x)" shows "vdistinct xs"
OneAncestor_def: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" shows "OneAncestor F G = (\<forall>A\<in>F - {Root F G}. \<exists>!s. s \<in> \<Union> (States ` (F - {A})) \<and> A \<in> the (G s))"
pcr_cr_eq: shows "pcr_complex_homo = cr_complex_homo"
list_of_llist_of: fixes xs :: "'c list" shows "list_of (llist_of xs) = xs"
assms_2: shows "v *\<^sub>v\<^sub>m A = v' *\<^sub>v\<^sub>m A"
col_col_per_per: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and X :: "'a"   and C :: "'a"   and U :: "'a"   and V :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> X"   and "C \<noteq> X"   and "Tarski_neutral_dimensionless.Col Bet U A X"   and "Tarski_neutral_dimensionless.Col Bet V C X"   and "Tarski_neutral_dimensionless.Per Bet Cong A X C" shows "Tarski_neutral_dimensionless.Per Bet Cong U X V"
AE: shows "AE x in M. (\<lambda>i. X i x) \<longlonglongrightarrow> Y x"
stationary_distribution_pmf_of_st_vec: fixes x :: "'i st_vec" shows "stationary_distribution (pmf_of_st_vec x) = (A *st x = x)"
kruskal_tree_axioms: shows "kruskal_tree F mk root args trees"
r: shows "approx_slp' p ord I i ras (a # VS) = Some X"
comp_cong: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d"   and f' :: "'e \<Rightarrow> 'b"   and g' :: "'f \<Rightarrow> 'e"   and x' :: "'f" assumes "f (g x) = f' (g' x')" shows "(f \<circ> g) x = (f' \<circ> g') x'"
last_message_not_NoMsg_conv: fixes xs :: "'b message_af list" shows "(last_message xs \<noteq> NoMsg) = (\<exists>i<length xs. xs ! i \<noteq> NoMsg)"
zero_le: fixes x :: "'e multiset" shows "{#} \<subseteq># x"
unfold: shows "op2 = (\<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))))"
\<delta>: shows "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
cardinal_le_lepoll: fixes A :: "V"   and \<alpha> :: "V" assumes "vcard A \<subseteq>\<^sub>\<circ> \<alpha>" shows "A \<lesssim>\<^sub>\<circ> \<alpha>"
\<mu>_to_M: shows "weak_conv_m \<mu> M"
inj_on_iff_card_le: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (card A \<le> card B)"
snd_eq_Range: fixes R :: "('e \<times> 'd) set" shows "snd ` R = Range R"
gc_W_empty_invL_def: shows "gc.gc_W_empty_invL = (\<lambda>s. (atS gc gc.get_roots_UN_get_work_locs s \<longrightarrow> (\<forall>x. mut_m.gc_W_empty_mut_inv x s\<down>)) \<and> (at gc gc.mark_loop_get_roots_load_W s \<longrightarrow> sys_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (at gc gc.mark_loop_get_work_load_W s \<longrightarrow> sys_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (at gc gc.mark_loop s \<longrightarrow> gc_W s\<down> = {} \<longrightarrow> no_grey_refs s\<down>) \<and> (atS gc gc.no_grey_refs_locs s \<longrightarrow> no_grey_refs s\<down>) \<and> (atS gc gc.gc_W_empty_locs s \<longrightarrow> gc_W s\<down> = {}))"
f4: shows "4 \<le> rk {A, B, C, a, b, c, P}"
sum_pow2_insert0: fixes X :: "nat set" assumes "finite X"   and "0 \<notin> X" shows "sum ((^) 2) (insert 0 X) = Suc (sum ((^) 2) X)"
supt_trans: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term"   and u :: "('d, 'e) Term.term" assumes "s \<rhd> t"   and "t \<rhd> u" shows "s \<rhd> u"
chain3: shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. f x x) ` Y)"
obj_at_def: fixes P :: "('f, 'g, 'h) object \<Rightarrow> bool"   and r :: "'h" shows "obj_at P r \<equiv> \<lambda>s. case sys_heap s r of None \<Rightarrow> False | Some obj \<Rightarrow> P obj"
dom_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C.VVV.dom f = (if C.VVV.arr f then C.VxVxV.dom f else C.VxVxV.null)"
assms_3: shows "f1 x1 y1 z1 = (0::'a)"
sets_measure_of_eq: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "sigma_algebra \<Omega> M" shows "sets (measure_of \<Omega> M \<mu>) = M"
transformation_by_components_axioms: shows "transformation_by_components (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G t"
set_mset_subset_singletonD: fixes A :: "'a multiset"   and x :: "'a" assumes "set_mset A \<subseteq> {x}" shows "A = replicate_mset (size A) x"
weakenU: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "U' \<sigma> \<sigma>'" shows "U \<sigma> \<sigma>'"
absorb2: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" shows "a \<union># b = b"
prems_4: shows "\<forall>zs z zs'. w # ws = zs @ z # zs' \<and> obs_intra z S \<noteq> {} \<longrightarrow> (\<exists>z''\<in>set (zs' @ [x]). \<exists>nx. call_of_return_node z'' nx \<and> nx \<notin> S)"
assms_2: shows "b * \<^bold>|y\<^bold>| \<le> a * \<^bold>|x\<^bold>|"
reachable_def: fixes E :: "('b \<times> 'b) set"   and src :: "'b" shows "reachable E src \<equiv> {v. (src, v) \<in> E\<^sup>*}"
simps_2: fixes P :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool" shows "list_all2_shortcircuit P [] [] = True"
Suc_2: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>T. x \<in> T \<and> open T \<and> higher_differentiable_on T f (Suc n)"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> VoV.arr f" shows "VoV.map f = VoV.null"
partially_vec_diff: shows "\<forall>a\<in>Dx_pair. partially_vector_differentiable (\<lambda>a. F a \<bullet> j) i a"
trans_tm_graph_aux_def: shows "trans_tm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>trans_tm_aux e. x1 = (e, Zero) \<and> x2 = DBZero) \<or> (\<exists>trans_tm_aux e k. x1 = (e, Var k) \<and> x2 = lookup e 0 k) \<or> (\<exists>trans_tm_aux e t u. x1 = (e, Eats t u) \<and> x2 = DBEats (((\<lambda>uu_. uu_) \<circ> trans_tm_aux) (e, t)) (((\<lambda>uu_. uu_) \<circ> trans_tm_aux) (e, u)) \<and> p (e, t) (((\<lambda>uu_. uu_) \<circ> trans_tm_aux) (e, t)) \<and> p (e, u) (((\<lambda>uu_. uu_) \<circ> trans_tm_aux) (e, u))))"
size_18: fixes x7 :: "'b ltln" shows "size (X\<^sub>n x7) = size x7 + Suc 0"
ordinal_times_mono: fixes x :: "ordinal"   and x' :: "ordinal"   and y :: "ordinal"   and y' :: "ordinal" assumes "x \<le> x'"   and "y \<le> y'" shows "x * y \<le> x' * y'"
sub_composition_closed: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "submonoid N M composition unit"   and "a \<in> N"   and "b \<in> N" shows "composition a b \<in> N"
\<alpha>_l: fixes p :: "fm" assumes "iszlfm p" shows "\<forall>b\<in>set (\<alpha> p). numbound0 b"
p_eq: shows "p = [:- x, 1::'a:] ^ n * q"
restrict_return_spmf: fixes x :: "'a"   and A :: "'a set" shows "return_spmf x \<upharpoonleft> A = (if x \<in> A then return_spmf x else return_pmf None)"
horner_eval_eq_setsum: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" shows "horner_eval f x n = (\<Sum>i<n. f i * x ^ i)"
da_elim_cases_16: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (Throw e)\<guillemotright> A" obtains E :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "nrm A = UNIV" and "brk A = (\<lambda>l. UNIV)"
chine_assoc_props_2: shows "doms.Prj\<^sub>1 \<cdot> doms.chine_assoc = doms.Prj\<^sub>1\<^sub>1"
intros: fixes L1 :: "'n \<Rightarrow> 't \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'v \<Rightarrow> bool"   and Contra1 :: "'q \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2 :: "'r \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "(('n, 'o, 'p, 'q, 'r, 's) T, 'n, 'o, 'p, 'q, 'r, 's) G"   and y :: "(('t, 'u, 'v, 'w, 'x, 's) T, 't, 'u, 'v, 'w, 'x, 's) G" assumes "rel_G (rel_T L1 Co1 Co2 Contra1 Contra2) L1 Co1 Co2 Contra1 Contra2 x y" shows "rel_T L1 Co1 Co2 Contra1 Contra2 (C_T x) (C_T y)"
vars0_def: shows "st2.vars0 = {k} \<union> (vars1 - {k})"
Max_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq (insert x A) = ord.max less_eq x (linorder.Max less_eq A)"
le_less_linear: fixes x :: "'a"   and y :: "'a" shows "x \<preceq> y \<or> y \<prec> x"
ln_ln_nonneg': fixes x :: "nat" assumes "3 \<le> x" shows "0 \<le> ln (ln (real x))"
chamber_some_third_chamber: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "chamber (some_third_chamber C D z)"
max_value_of_not_const_less: assumes "\<not> const 0"   and "\<not> const 1" shows "(cmod (of_int (\<Sum>k<2 ^ n. (- int 1) ^ f k)))\<^sup>2 < real (2 ^ (2 * n))"
pinduct: fixes a0 :: "(node \<times> node) list"   and a1 :: "node list"   and a2 :: "node list"   and P :: "(node \<times> node) list \<Rightarrow> node list \<Rightarrow> node list \<Rightarrow> bool" assumes "dfs2_dom (a0, a1, a2)"   and "\<And>g ys. dfs2_dom (g, [], ys) \<Longrightarrow> P g [] ys"   and "\<And>g x xs ys. \<lbrakk>dfs2_dom (g, x # xs, ys); List.member ys x \<Longrightarrow> P g xs ys; \<not> List.member ys x \<Longrightarrow> P g (nexts g x) (x # ys); \<not> List.member ys x \<Longrightarrow> P g xs (dfs2 g (nexts g x) (x # ys))\<rbrakk> \<Longrightarrow> P g (x # xs) ys" shows "P a0 a1 a2"
mult_p32_def: fixes p :: "uint32"   and x :: "uint32"   and y :: "uint32" shows "mult_p32 p x y = x * y mod p"
forrest2_E: fixes u :: "'b"   and v :: "'b"   and E :: "'b uprod set" assumes "(u, v) \<in> uconnected E"   and "Upair u v \<notin> E"   and "u \<noteq> v" shows "\<not> forest (insert (Upair u v) E)"
assms_1: shows "transition stateA stateB F0 decisionVars"
left_plus_below_circ: fixes x :: "'a" shows "x * x\<^sup>\<circ> \<le> x\<^sup>\<circ>"
mult_minus_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "times a (uminus b) = uminus (times a b)"
bound: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y" shows "norm (EM_remainder' n f x y) \<le> D' * (g y - g x)"
arr_cod_iff_arr: fixes f :: "'c \<times> 'c" shows "C.VV.arr (C.VV.cod f) = C.VV.arr f"
approx_def: shows "approx \<equiv> \<lambda>N. (\<Sum>n = 1..<N. s / of_nat n - Ln (1 + s / of_nat n)) - (euler_mascheroni * s + Ln s) - complex_of_real (ln_Gamma (real N) - ln (2 * pi / real N) / 2 - real N * ln (real N) + real N) - s * (harm (N - 1) - Ln (of_nat (N - 1)) - euler_mascheroni) + s * (Ln (of_nat N + s) - Ln (of_nat (N - 1))) - 1 / 2 * (Ln (of_nat N + s) - Ln (of_nat N)) + of_nat N * (Ln (of_nat N + s) - Ln (of_nat N)) - (s - 1 / 2) * Ln s - complex_of_real (ln (2 * pi) / 2)"
bot_meet_irreducible: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot"
lt_monom_mult_zero: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "punit.lt (local.punit.monom_mult c (0::'a) p) = punit.lt p"
IH: assumes "a < b" shows "(\<forall>x>a. x < b \<longrightarrow> x \<notin> I \<longrightarrow> (\<lambda>u. integral {a..u} f) differentiable at x) \<and> (\<forall>x\<ge>a. x < b \<longrightarrow> (\<lambda>t. integral {a..t} f) differentiable at_right x) \<and> (\<forall>x>a. x \<le> b \<longrightarrow> (\<lambda>t. integral {a..t} f) differentiable at_left x)"
invariant_run_index: fixes P :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and p :: "'c"   and A :: "('d, 'c) nba"   and Q :: "nat \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'c \<Rightarrow> bool" assumes "P n p"   and "\<And>n p. P n p \<Longrightarrow> \<exists>a. (fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) \<and> P (Suc n) (snd a) \<and> Q n p a" obtains r :: "('d \<times> 'c) stream" where "run A r p" and "\<And>i. P (n + i) (target (stake i r) p)" and "\<And>i. Q (n + i) (target (stake i r) p) (r !! i)"
oth_class_taut_3_c: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<phi> \<^bold>& (\<psi> \<^bold>& \<chi>) \<^bold>\<equiv> \<phi> \<^bold>& \<psi> \<^bold>& \<chi> in v]"
N_def: shows "N = (\<lambda>t. snd (index t))"
v12: shows "v1 \<squnion> v2 = Some v12"
sparsegrid_def: fixes dm :: "nat"   and lm :: "nat" shows "sparsegrid dm lm = lgrid (start dm) {0..<dm} lm"
hyps_2: fixes t' :: "trm" assumes "t \<longmapsto> t'" shows "\<Gamma> \<turnstile> t' : S"
s'_e_s'': shows "s' e\<longrightarrow>\<^bsub>SES\<^esub> s''"
lockstep_simulation: fixes s1 :: "'state1"   and s2 :: "'state2" assumes "match s1 s2"   and "step1 s1 s1'" shows "\<exists>s2'. step2 s2 s2' \<and> match s1' s2'"
convert_semantics_lem_R: fixes fml :: "real poly fml"   and ls :: "real list"   and ps :: "real poly list"   and x :: "real" assumes "\<And>p. p \<in> set (poly_list fml) \<Longrightarrow> ls ! index_of ps p = poly p x" shows "real_sem fml x = lookup_sem (map_fml (index_of ps) fml) ls"
similar'_up_le: fixes n :: "nat"   and m :: "nat"   and d :: "Value"   and e :: "CValue" assumes "n \<le> m"   and "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>m\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e"
inverse_functors_\<Phi>_\<Psi>: shows "inverse_functors (\<cdot>) (\<cdot>\<acute>) \<Psi> \<Phi>"
wf_monitor: fixes i :: "nat"   and v :: "'a option list"   and \<pi> :: "(char list \<times> 'a list) prefix" assumes "(i, v) \<in> M \<pi>" shows "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"
in_arcs_G: shows "{uv, rev_I uv} \<subseteq> arcs I"
div_add1_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (plus a b) c = plus (plus (divide a c) (divide b c)) (divide (plus (modulo a c) (modulo b c)) c)"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "sup_class.sup x y = sup_class.sup y x"
sym_order_trans: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "semiattractive A less_eq"   and "sympartp less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
sublist_code_1: fixes ys :: "'a list" shows "sublist [] ys = True"
region_continuous: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and t1 :: "real"   and t2 :: "real"   and u :: "'a \<Rightarrow> real" assumes "Regions.valid_region X k I r"   and "0 \<le> t1"   and "t1 \<le> t2"   and "u \<in> Regions.region X I r"   and "u \<oplus> t2 \<in> Regions.region X I r" shows "u \<oplus> t1 \<in> Regions.region X I r"
While_le: fixes I :: "'a"   and b :: "bexp"   and p :: "'a"   and C :: "'a acom"   and P :: "'a"   and W :: "'a acom" shows "({I} WHILE b DO {p} C {P} \<le> W) = (\<exists>I' p' C' P'. W = {I'} WHILE b DO {p'} C' {P'} \<and> C \<le> C' \<and> p \<le> p' \<and> I \<le> I' \<and> P \<le> P')"
continuous_map_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S' :: "'d set"   and is_open' :: "'d set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map S is_open S' is_open' f \<equiv> (topological_space S is_open \<and> topological_space S' is_open') \<and> Set_Theory.map f S S' \<and> continuous_map_axioms S is_open is_open' f"
Pxz_nn: fixes x :: "'c \<times> 'e" assumes "x \<in> space (S \<Otimes>\<^sub>M P)" shows "0 \<le> Pxz x"
map_to_set_inverse: fixes m :: "'e \<Rightarrow> 'f option" shows "set_to_map (map_to_set m) = m"
span_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel B" shows "KB.span S = KB.NC.span S"
z_def: shows "z \<equiv> the_elem (Set.filter (\<lambda>(x, y). x\<^sup>2 + y\<^sup>2 = p) (SIGMA x:{0..p}. {x..p}))"
prems_1: shows "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
assms_3: fixes xs :: "('a, 'b) tllist"   and y :: "'a" assumes "\<not> is_TNil xs"   and "y \<in> tset (ttl xs)"   and "P y (ttl xs)" shows "P y xs"
ev_blockD: fixes n :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "??.Jordan_Normal_Form_Existence.ev_block n A"   and "i < n"   and "j < n" shows "A $$ (i, i) = A $$ (j, j)"
XY: fixes U :: "'a set" assumes "openin X U" shows "openin Y U"