hcju/isabelleps huggingface dataset:
count_image_mset_ge_count: fixes A :: "'a multiset"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "count A b \<le> count (image_mset f A) (f b)"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e option \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> ('g \<Rightarrow> ('e, 'f, 'g) Resumption.resumption) \<Rightarrow> 'd"   and resumption :: "('e, 'f, 'g) Resumption.resumption" shows "P (case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. resumption = Resumption.resumption.Done x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. resumption = Resumption.resumption.Pause x21 x22 \<longrightarrow> P (f2 x21 x22)))"
i1: assumes "a \<sharp> t1" shows "a \<sharp> t2"
assms: shows "(ip, ip') \<in> rt_graph \<sigma> dip"
a_inv_closed: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<in> carrier G"
tendsto_zero_mult_left_iff: fixes c :: "'e"   and a :: "nat \<Rightarrow> 'e" assumes "c \<noteq> (0::'e)" shows "(\<lambda>n. c * a n) \<longlonglongrightarrow> (0::'e) = a \<longlonglongrightarrow> (0::'e)"
set_integrable_sum: fixes B :: "'a set"   and M :: "'b measure"   and A :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite B"   and "\<And>x. x \<in> B \<Longrightarrow> set_integrable M A (f x)" shows "set_integrable M A (\<lambda>y. \<Sum>x\<in>B. f x y)"
hyps_7: fixes b :: "lam" assumes " s1 \<longrightarrow>\<^sub>1 b" shows " b \<longrightarrow>\<^sub>1 s2"
lin_dense: fixes p :: "fm"   and l :: "real"   and u :: "real"   and x :: "real"   and bs :: "real list"   and y :: "real" assumes "isrlfm p"   and "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (uset p)"   and "l < x"   and "x < u"   and "Ifm (x # bs) p"   and "l < y"   and "y < u" shows "Ifm (y # bs) p"
prems_5: shows "\<Gamma> \<turnstile>\<^sub>c e : REAL"
idempotent_imp_retraction: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "continuous_on S f"   and "f ` S \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> f (f x) = f x" shows "retraction S (f ` S) f"
assms_1: shows "approx_floatarith p ra VS = Some X"
CX_KB_is_pubKB: fixes A :: "agent"   and Ba :: "agent"   and B :: "agent"   and KB :: "nat"   and evs :: "event list" assumes "Says A Ba (certificate B KB) \<in> set evs"   and "evs \<in> tls" shows "KB = pubK B"
poly_map_poly: fixes a :: "'c"   and p :: "'c poly poly"   and x :: "'c poly" shows "poly (map_poly (\<lambda>p. poly p a) p) (poly x a) = poly (poly p x) a"
gen_cond_def: fixes s :: "'s" shows "gen_cond s \<equiv> (V0 \<subseteq> gen_discovered s \<longrightarrow> \<not> gds_is_empty_stack gds s) \<and> \<not> gds_is_break gds s"
simps_1: fixes ys :: "'a list" shows "local.merge [] ys = ys"
less_eq_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs \<le> ys) = list_emb (\<le>) xs ys"
valid2_hd_Rcons: fixes tr :: "('state, 'act, 'out) trans list"   and tran :: "('state, 'act, 'out) trans" assumes "valid2 tr" shows "hd (tr ## tran) = hd tr"
le_take_is_prefix: fixes k :: "nat"   and n :: "nat"   and ws :: "'b list" assumes "k \<le> n" shows "take k ws \<le>p take n ws"
simps_3: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Neg \<phi>) = (\<not> MFOTL.sat \<sigma> v i \<phi>)"
widen1: fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.wn narrow widen less_eq less" shows "less_eq x (widen x y)"
isomorphism_def: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e" shows "ciso\<^bsub>C\<^esub> f \<equiv> \<exists>g. cinv\<^bsub>C\<^esub> f g"
disc_eq_case_2: fixes nat :: "nat" shows "(nat \<noteq> 0) = (case nat of 0 \<Rightarrow> False | Suc uu_ \<Rightarrow> True)"
If: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and c1 :: "stmt"   and C1 :: "assigned"   and c2 :: "stmt"   and C2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1"   and "Env\<turnstile> B \<union> assigns_if False e \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2"   and "nrm A = nrm C1 \<inter> nrm C2"   and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2" shows "Env\<turnstile> B \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A"
continuous_on: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. prod (f x) (g x))"
PPf: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "PPf t1 t2 \<in> fmla"
Baroco: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A Q B"   and "A Z C" shows "B Z C"
fpxs_shift_uminus: fixes r :: "rat"   and f :: "'b fpxs" shows "fpxs_shift r (- f) = - fpxs_shift r f"
contains_def: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and nA :: "nat \<times> formula"   and n :: "nat" shows "contains f nA n = (nA \<in> set (snd (f n)))"
ag_plus_zero: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "(x = -\<^sub>a\<^bsub>A\<^esub> y) = (x \<plusminus>\<^bsub>A\<^esub> y = \<zero>\<^bsub>A\<^esub>)"
sums_Uniq: fixes f :: "nat \<Rightarrow> 'b" shows "\<exists>\<^sub>\<le>\<^sub>1 a. f sums a"
setdist_eq_0_sing_2: fixes S :: "'a set"   and x :: "'a" shows "(setdist S {x} = 0) = (S = {} \<or> x \<in> closure S)"
ordLess_or_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "r <o r' \<or> r' \<le>o r"
islimpt_greaterThanLessThan1: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a islimpt {a<..<b}"
prems: shows "a # xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last (a # xs) = hd ys \<and> last (a # xs) \<noteq> (0::'a)"
trad_foundation: fixes z :: "V" assumes "z \<noteq> []\<^sub>\<circ>" shows "\<exists>w. w \<in>\<^sub>\<circ> z \<and> w \<inter>\<^sub>\<circ> z = []\<^sub>\<circ>"
check_eqv_sound: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "fast.check_eqvRE \<Sigma> n r s"   and "M2L_Equivalence_Checking.wf \<Sigma> n r"   and "M2L_Equivalence_Checking.wf \<Sigma> n s" shows "lang \<Sigma> n r = lang \<Sigma> n s"
set_zero_plus2: fixes A :: "'c set"   and B :: "'c set" assumes "(0::'c) \<in> A" shows "B \<subseteq> A + B"
el: fixes w :: "'a list" assumes "w \<in> G" shows "w \<in> \<langle>H\<rangle>\<^sub>F"
decrease_def: fixes f :: "('c \<Rightarrow> 'd list) \<times> (('d list \<Rightarrow> 'd list) \<Rightarrow> 'c \<Rightarrow> 'c)" shows "decrease f \<equiv> snd f tl"
convex_polygon_skip: fixes x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and z :: "(real \<times> real) \<times> real \<times> real"   and w :: "(real \<times> real) \<times> real \<times> real"   and ws :: "((real \<times> real) \<times> real \<times> real) list" assumes "convex_polygon (x # y # z # w # ws)"   and "ccw'.sortedP (fst x) (map snd (butlast (x # y # z # w # ws)))" shows "convex_polygon ((fst x, snd y) # z # w # ws)"
netFlow_def: shows "netFlow \<equiv> sum f (outgoing' (cf.reachableNodes s)) - sum f (incoming' (cf.reachableNodes s))"
ex1E: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'a" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
curry_case_prod: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "curry (\<lambda>(x, y). f x y) = f"
compose_id_inv_into: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" assumes "f ` A = B" shows "compose B f (restrict (inv_into A f) B) = identity B"
\<alpha>'_simp: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.arr f"   and "D.arr g"   and "D.arr h" shows "D.\<alpha>' (f, g, h) = ((f \<otimes>\<^sub>D g) \<otimes>\<^sub>D h) \<cdot>\<^sub>D D.assoc' (D.dom f) (D.dom g) (D.dom h)"
lesubstep_type_simple: fixes xs :: "(nat \<times> 'a) list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "(nat \<times> 'a) list" assumes "xs [\<sqsubseteq>\<^bsub>Product.le (=) r\<^esub>] ys" shows "set xs {\<sqsubseteq>\<^bsub>r\<^esub>} set ys"
assms: shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> 1\<^sub>\<nat>"
eqvt_fresh_star_cong1: fixes f :: "'a \<Rightarrow> 'b"   and a :: "atom set"   and x :: "'a" assumes "\<And>p x. p \<bullet> f x = f (p \<bullet> x)"   and "a \<sharp>* x" shows "a \<sharp>* f x"
MOST_neq_1: fixes a :: "'b" shows "\<forall>\<^sub>\<infinity>x. x \<noteq> a"
glb_special_subset_coset: fixes w :: "'w"   and T :: "'w set"   and T' :: "'w set" assumes "w \<in> W"   and "T \<in> Pow S"   and "T' \<in> Pow S" shows "dual_order.glbound_in_of \<P> \<langle>T\<rangle> (w +o \<langle>T'\<rangle>) \<langle>T \<union> T' \<union> reduced_letter_set S w\<rangle>"
neg_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.neg \<gamma> \<equiv> \<lambda>t n. \<not> \<gamma> t n"
IH: shows "vars\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<union> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q assign) S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q check) S)) \<union> \<Union> (set (map fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q S)) \<union> \<Union> (set (map (set \<circ> bvars\<^sub>s\<^sub>t\<^sub>p) S))"
Q_hit'_notin_Q_minus_Q_hit: shows "Q_hit' \<notin> Q - {Q_hit}"
br_sv: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "single_valued (br \<alpha> I)"
test_iteration_sim: fixes p :: "'a"   and x :: "'a" assumes "test p"   and "p \<cdot> x \<le> x \<cdot> p" shows "p \<cdot> x\<^sup>\<infinity> \<le> x\<^sup>\<infinity> \<cdot> p"
whileAnnoFix: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P guards gs (whileAnnoFix b J V (\<lambda>Z. Seq (c Z) (guards gs Skip))) Q,A"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x div xa) = int_of_integer x div int_of_integer xa"
pbdom_def: fixes f :: "'a"   and g :: "'a" shows "f \<down>\<down> g \<equiv> local.dom \<p>\<^sub>0[f, g]"
prj_chine_assoc_1: shows "\<mu>\<nu>\<pi>.Prj\<^sub>1 \<cdot> \<mu>\<nu>\<pi>.chine_assoc = \<mu>\<nu>\<pi>.Prj\<^sub>1\<^sub>1"
SMP_consts: fixes M :: "('fun, 'var) Term.term set" assumes "\<forall>t\<in>M. \<exists>c. t = Fun c []"   and "\<forall>t\<in>M. Ana t = ([], [])" shows "SMP M = M"
assms: fixes u :: "'t"   and v :: "'t" assumes "component_of_term u < component_of_term v" shows "u \<prec>\<^sub>t v"
skipping_ok: fixes j :: "nat"   and s :: "'a list"   and i :: "nat" assumes "1 < j"   and "j \<le> length s"   and "s ! (i - 1) \<noteq> s ! (j - 1)"   and "\<ff> s j \<le> i + 1"   and "strict_border (take (i - 1) s) (take (j - 1) s)" shows "\<ff> s j \<le> \<ff> s (i - 1) + 1"
foldD_insert_aux: fixes x :: "'b"   and A :: "'b set"   and e :: "'a"   and v :: "'a" assumes "x \<notin> A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "((insert x A, v) \<in> foldSetD D (\<cdot>) e) = (\<exists>y. (A, y) \<in> foldSetD D (\<cdot>) e \<and> v = x \<cdot> y)"
Dom_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "Dom t = t"
d_diff: fixes t :: "real" shows "d t = other.s t - ego.s t"
ipurge_tr_rev_t_setp_ipurge_tr_rev_t_set_eq: fixes X :: "('c, 'd) ipurge_rec" shows "ipurge_tr_rev_t_setp X = (\<lambda>x. x \<in> ipurge_tr_rev_t_set X)"
word_ao_absorbs_5: fixes y :: "'a word"   and x :: "'a word" shows "(y OR x) AND x = x"
OrdP_linear: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and A :: "fm" assumes "H \<turnstile> OrdP x"   and "H \<turnstile> OrdP y"   and "insert (x IN y) H \<turnstile> A"   and "insert (x EQ y) H \<turnstile> A"   and "insert (y IN x) H \<turnstile> A" shows "H \<turnstile> A"
dependent_and_consistent_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "dependent_and_consistent_iarrays A b = (let GJ = Gauss_Jordan_iarrays_PA A; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))); P_mult_b = fst GJ *iv b; consistent_A = (if \<not> is_zero_iarray P_mult_b then greatest_not_zero P_mult_b + 1 else 0) \<le> rank_A; dim_solution_set = ncols_iarray A - rank_A in consistent_A \<and> 0 < dim_solution_set)"
SIod_Worder: fixes D :: "'c Order"   and T :: "'c set" assumes "Worder D"   and "T \<subseteq> carrier D" shows "Worder (SIod D T)"
UP_zero_closed: shows "\<zero>\<^bsub>UP Zp\<^esub> \<in> carrier (UP Zp)"
lex_ord_pair_EqI: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> f k (lookup_pair x xs k) (lookup_pair x ys k) = Some Eq" shows "lex_ord_pair x f xs ys = Some Eq"
Vfrom_def: fixes a :: "V" shows "Vfrom a \<equiv> transrec (\<lambda>f x. a \<squnion> (SUP y\<in>elts x. VPow (f y)))"
opp_mult_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" shows "times.opp_mult times x y \<equiv> times y x"
rank_eq_card_iarrays: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = card {vec_to_iarray (row i (Gauss_Jordan A)) |i. \<not> is_zero_iarray (vec_to_iarray (row i (Gauss_Jordan A)))}"
is_subseq_of_id: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_subseq_of s s"
Re_winding_number_half_lower: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "\<forall>p\<in>path_image \<gamma>. Im p \<le> Im z"   and "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "Re (winding_number \<gamma> z) = (Im (Ln (\<i> * pathfinish \<gamma> - \<i> * z)) - Im (Ln (\<i> * pathstart \<gamma> - \<i> * z))) / (2 * pi)"
axioms_3: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
simps_4: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and p :: "real mpoly" shows "linear_substitution var a b (Neq p) = (let d = MPoly_Type.degree p var in Neq (\<Sum>i = 0..<d + 1. isolate_variable_sparse p var i * a ^ i * b ^ (d - i)))"
add_node_set_edges: fixes v :: "'b"   and G :: "'b list_graph" shows "set (edgesL (FiniteListGraph.add_node v G)) = set (edgesL G)"
one_star_trans: fixes M1 :: "lam"   and M2 :: "lam"   and M3 :: "lam" assumes " M1 \<longrightarrow>\<^sub>1\<^sup>* M2"   and " M2 \<longrightarrow>\<^sub>1\<^sup>* M3" shows " M1 \<longrightarrow>\<^sub>1\<^sup>* M3"
pOp_assocTr41: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus> b \<plusminus> c \<plusminus> d = a \<plusminus> b \<plusminus> (c \<plusminus> d)"
deriv_compose_linear: fixes f :: "'a \<Rightarrow> 'a"   and c :: "'a"   and z :: "'a" assumes "f field_differentiable at (c * z)" shows "deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
f_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')" shows "map of_int_hom.vec_hom fs' ! i \<in> Rn"
gdpr_EF': shows "gdpr_Kripke \<turnstile> EF sgdpr"
Diff_insert0: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "A - insert x B = A - B"
distinct_9: fixes x2 :: "('b, 'c, 'd, 'e) Tree"   and x41 :: "'e"   and x42 :: "'d"   and x43 :: "('b, 'c, 'd, 'e) Tree" shows "tNot x2 \<noteq> tAct x41 x42 x43"
fun_upd_env_delete: fixes x :: "'d"   and \<Gamma> :: "'d \<Rightarrow> 'e" assumes "x \<in> edom \<Gamma>" shows "(env_delete x \<Gamma>)(x := \<Gamma> x) = \<Gamma>"
Liminf_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" shows "Liminf F f = (SUP P\<in>{P. eventually P F}. Inf (f ` {x. P x}))"
methd_wf_mdecl: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and y :: "class" assumes "methd G C sig = Some m"   and "wf_prog G"   and "class G C = Some y" shows "G\<turnstile>C\<preceq>\<^sub>C declclass m \<and> is_class G (declclass m) \<and> wf_mdecl G (declclass m) (sig, mthd m)"
simps_10: fixes b :: "nat"   and \<phi> :: "Formula.formula" shows "Formula.fvi b (formula.Exists \<phi>) = Formula.fvi (Suc b) \<phi>"
set_eq: shows "{0..<int CARD('a)} = int ` {0..<CARD('a)}"
simps_6: fixes ux :: "'b"   and pr :: "('a, 'b) Digit"   and m :: "('a, 'b) FingerTreeStruc"   and a :: "('a, 'b) Node" shows "FingerTreeStruc.viewRn (Deep ux pr m (One a)) = (case FingerTreeStruc.viewRn m of None \<Rightarrow> Some (a, FingerTreeStruc.digitToTree pr) | Some (b, m2) \<Rightarrow> Some (a, FingerTreeStruc.deep pr m2 (FingerTreeStruc.nodeToDigit b)))"
igSubstIGVar2STR_imp_igSubstIGVar2: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSubstIGVar2STR MOD" shows "igSubstIGVar2 MOD"
asymptotics1': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>0 \<le> x"   and "i < k" shows "1 + ln x powr (- e / 2) \<le> (1 + hb * inverse (bs ! i) * ln x powr - (1 + e)) powr p * (1 + ln (bs ! i * x + hb * x / ln x powr (1 + e)) powr (- e / 2))"
inject: fixes x :: "real"   and ya :: "real" shows "(Rmax x = Rmax ya) = (x = ya)"
simps_26: fixes x6 :: "'b ptrm"   and x2 :: "'b" shows "PFst x6 \<noteq> PVar x2"
mem\<^sub>A_of_def: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and mem\<^sub>C :: "'e \<Rightarrow> 'd" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of" shows "sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>C \<equiv> \<lambda>x\<^sub>A. mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
chamberD_simplex: fixes x :: "'a set" assumes "folding.chamber x" shows "x \<in> X"
simpfm_rl: fixes p :: "fm" assumes "isrlfm p" shows "isrlfm (simpfm p)"
len_cut_le: fixes n :: "nat"   and P :: "'v" shows "len (cut n P) \<le> n"
semilat_Def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semilat (A, r, f) = (order r \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
sides'_3: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist b1 c1 = dist b2 c2"
dimension_independent: fixes K :: "'a list set"   and Us :: "'a list list" assumes "UP.independent K Us" shows "UP.dimension (length Us) K (UP.Span K Us)"
open_product_open_eq: shows "arbitrary union_of (\<lambda>U. \<exists>S T. U = S \<times> T \<and> open S \<and> open T) = open"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<in> o[F](g)"
rename_rename: fixes p1 :: "'var \<Rightarrow> 'var"   and p2 :: "'var \<Rightarrow> 'var"   and f :: "'form" shows "renameLCs p1 (renameLCs p2 f) = renameLCs (p1 \<circ> p2) f"
mset_insort: fixes x :: "'c"   and xs :: "'c list" shows "mset (insort x xs) = {#x#} + mset xs"
simps_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and \<phi> :: "'c"   and t1 :: "'b"   and x1 :: "'a"   and txs :: "('b \<times> 'a) list" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst" shows "Generic_Syntax.rawpsubst subst \<phi> ((t1, x1) # txs) = Generic_Syntax.rawpsubst subst (subst \<phi> t1 x1) txs"
lfinite_prepend: fixes xs :: "'b list"   and ys :: "'b llist" shows "lfinite (prepend xs ys) = lfinite ys"
dletrec1: fixes funs :: "(char list \<times> char list \<times> exp0) list"   and ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and locs :: "locn \<times> locn" assumes "allDistinct (map (\<lambda>x. case x of (x, y, z) \<Rightarrow> x) funs)" shows "evaluate_dec ck mn env s (Dletrec locs funs) (s, Rval (make_sem_env (build_rec_env funs env nsEmpty) nsEmpty))"
finite_induct: fixes F :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
subterms_in_cterms: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a" assumes "wellformed \<Gamma>"   and "p \<in> subterms (\<Gamma> pn)"   and "not_call p"   and "not_choice p" shows "p \<in> cterms \<Gamma>"
Call: fixes r :: "((pname \<times> state) \<times> pname \<times> state) set"   and P :: "pname \<Rightarrow> 'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "pname \<Rightarrow> 'a \<Rightarrow> state \<Rightarrow> bool" assumes "wf r"   and "\<forall>q pre. (\<Union>p. {(\<lambda>z s. P p z s \<and> ((p, s), q, pre) \<in> r, CALL p, Q p)}) \<turnstile>\<^sub>t {\<lambda>z s. P q z s \<and> s = pre} body q {Q q}" shows "{} |\<turnstile>\<^sub>t \<Union>p. {(P p, CALL p, Q p)}"
assms_4: shows "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"
coprime_generic: fixes bs :: "('a poly \<times> nat) set"   and f :: "nat \<Rightarrow> nat" assumes "bs \<subseteq> as"   and "\<And>a i. (a, i) \<in> bs \<Longrightarrow> 0 < f i" shows "coprime (\<Prod>(a, i)\<in>bs. a) (\<Sum>(a, i)\<in>bs. (\<Prod>(b, j)\<in>bs - {(a, i)}. b) * smult (of_nat (f i)) (pderiv a))"
int_mult_mono: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i < j"   and "0 < k" shows "k * i < k * j"
24: shows "padic_mult p f g (Suc (nat (padic_val p f) + nat (padic_val p g))) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc (nat (padic_val p f + padic_val p g)))\<^esub>"
stable_translation_length_def: fixes f :: "'b \<Rightarrow> 'b" shows "stable_translation_length f = Inf {translation_length (f ^^ n) / real n |n. 0 < n}"