hcju/isabelleps huggingface dataset:
rGeqF: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r1 < r2" shows "[Geq \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> False"
use_fps: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> fps L f" shows "f x .= x"
from_hma\<^sub>v_def: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v = Matrix.vec CARD('e) (\<lambda>i. v $h Bij_Nat.from_nat i)"
Rats_def: shows "field_char_0.Rats (/) (1::'a) (+) (0::'a) (-) = range (field_char_0.of_rat (/) (1::'a) (+) (0::'a) (-))"
mult_le_one: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq a one"   and "less_eq zero b"   and "less_eq b one" shows "less_eq (times a b) one"
2: shows "coreflexive ((f \<sqinter> - q) * (f \<sqinter> q))"
abs_prod: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<bar>prod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
Or: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and \<psi> :: "('msaux, 'muaux) mformula"   and \<psi>' :: "Formula.formula"   and buf :: "event_data option list set list \<times> event_data option list set list" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j P V n R \<psi> \<psi>'"   and "fv \<phi>' = fv \<psi>'"   and "wf_mbuf2' \<sigma> P V j n R \<phi>' \<psi>' buf" shows "wf_mformula \<sigma> j P V n R (MOr \<phi> \<psi> buf) (formula.Or \<phi>' \<psi>')"
lesso_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f <o g \<equiv> \<lambda>x. max (f x - g x) (0::'d)"
assms: fixes a :: "'a"   and p :: "'b"   and F :: "assn"   and a' :: "'a"   and F' :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F'" shows "a = a'"
fmdrop_irrelevant: fixes x11 :: "'a"   and A :: "('a, 'b) fmap" assumes "x11 \<notin># dom_m A" shows "fmdrop x11 A = A"
fin_2: shows "finite {x \<in> C. u x < 0}"
path_conc_conv: fixes E :: "('c \<times> 'c) set"   and u :: "'c"   and la :: "'c list"   and lb :: "'c list"   and w :: "'c" shows "path E u (la @ lb) w = (\<exists>v. path E u la v \<and> path E v lb w)"
idem: fixes i :: "nat"   and j :: "nat" assumes "i < length fas"   and "j < max_Var_floatarith (fas ! i)" shows "vs ! s j = vs ! j"
affine_hull_insert_subset_span: fixes a :: "'a"   and S :: "'a set" shows "affine hull insert a S \<subseteq> {a + v |v. v \<in> span {x - a |x. x \<in> S}}"
cycle_rotate_len_arcs: fixes xs :: "nat list"   and i :: "nat"   and j :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and a :: "nat" assumes "1 < length xs"   and "(i, j) \<in> arcs' xs" shows "\<exists>ys zs. len M a a xs = len M i i (j # ys @ a # zs) \<and> set (arcs a a xs) = set (arcs i i (j # ys @ a # zs)) \<and> xs = zs @ i # j # ys"
map_ds_def: fixes f :: "'a \<Rightarrow> 'b" shows "map_ds f = ds \<circ> f \<circ> SSup"
closedin_limpt: fixes T :: "'a set"   and S :: "'a set" shows "closedin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S))"
aux: fixes k :: "'a"   and l :: "'a" shows "sum f2 {q. f2 q \<noteq> (0::'b) \<and> k = l + q} = (\<Sum>q. f2 q when k = l + q)"
dghm_ArrMap_is_arr: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>op_dg \<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
NewInitThrowReds: fixes s :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "\<nexists>sfs. shp s C = \<lfloor>(sfs, Done)\<rfloor>"   and "is_class P C"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,s,False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>new C,s,False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
butlast_prefix: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" shows "prefix (butlast xs) (butlast ys)"
neq_LeafD: fixes t :: "('b \<times> 'c) tree" assumes "t \<noteq> \<langle>\<rangle>" shows "\<exists>l x c r. t = \<langle>l, (x, c), r\<rangle>"
inner_prod_with_row_bra_vec_col_ket_vec: fixes u :: "complex vec"   and v :: "complex vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = row \<langle>|u\<rangle>| 0 \<bullet> col |v\<rangle> 0"
test_import_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot" shows "sup x (inf (il_semiring.neg inf sup bot top x) y) = sup x y"
assms_2: assumes "0 < degree p"   and "\<And>q r. \<lbrakk>degree q < degree p; degree r < degree p\<rbrakk> \<Longrightarrow> p \<noteq> q * r" shows "thesis"
Abs_hf_0: shows "Abs_hf 0 = 0"
language_def: shows "MN.language \<equiv> {xs. MN.nextl (dfa.init Canon.DFA) xs \<in> dfa.final Canon.DFA}"
arcs_decomp_tail: fixes j :: "nat"   and l :: "nat"   and ys :: "nat list"   and i :: "nat" shows "arcs j l (ys @ [i]) = arcs j i ys @ [(i, l)]"
insertion_sum: fixes f :: "nat \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "insertion f (sum g A) = (\<Sum>x\<in>A. insertion f (g x))"
real_transfer_operator_cmult: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f" shows "AE x in M. real_transfer_operator (\<lambda>x. c * f x) x = c * real_transfer_operator f x"
Sup_set_def: fixes A :: "'d set set" shows "\<Union> A = {x. Sup ((\<in>) x ` A)}"
Limit_\<alpha>: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "Limit \<alpha>"
face_of_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and c :: "'b set"   and S :: "'b set" assumes "linear f"   and "inj f" shows "(f ` c face_of f ` S) = (c face_of S)"
atoms_nTimes: fixes r :: "nat rexp"   and s :: "nat rexp" shows "atoms (nTimes r s) \<subseteq> atoms r \<union> atoms s"
le_less: fixes x :: "'a list"   and y :: "'a list" shows "border x y = (strict_border x y \<or> x = y)"
s_topTr6: fixes H1 :: "'a set"   and H2 :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H1 "   and "G \<guillemotright> H2 "   and "G \<guillemotright> K "   and "H1 \<subseteq> K" shows "(H1 \<diamondop> H2) \<inter> K = H1 \<diamondop> H2 \<inter> K"
ker_subset_ker_restrict0: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "order.greater_eq (ker (restrict0 f A)) (ker f)"
master2_2_automation: assumes "g \<in> \<Theta>(MASTER_BOUND' p (- 1))" shows "f \<in> \<Theta>(MASTER_BOUND p 0 1)"
intro: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2" shows "delay_bisimulation_diverge_final trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 final1 final2"
Not_Rep_bit_iff_even: fixes b :: "bit" shows "(\<not> Rep_bit b) = even b"
comp_ide_arr: fixes b :: "Arr"   and f :: "Arr" assumes "ide b"   and "seq b f" shows "local.comp b f = f"
rl: shows "inner_prod v (A *\<^sub>v v) < 0"
assms_3: shows "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
lcs_def: fixes G :: "('c, 'd) Group_scheme"   and a :: "'c"   and H :: "'c set" shows "a \<diamondsuit>\<^bsub>G\<^esub> H = {b. \<exists>h\<in>H. a \<cdot>\<^bsub>G\<^esub> h = b}"
assms_6: fixes c :: "'a" assumes "c \<in> carrier R" shows "f c = g (h c)"
divide_inverse_commute: fixes a :: "'a"   and b :: "'a" shows "a / b = inverse b * a"
Rep_inject: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(sint x = sint y) = (x = y)"
assms: shows "fmdom A |\<inter>| fmdom B = {||}"
trans_RuR': fixes R :: "('a list \<times> 'a list) set"   and R' :: "('a list \<times> 'a list) set" assumes "trans R"   and "sym R"   and "trans R'"   and "sym R'"   and "\<forall>(V, V')\<in>R. length V = length V'"   and "\<forall>(V, V')\<in>R'. length V = length V'"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "trans (R \<union> R')"
simps_7: fixes i :: "int"   and a :: "MIR.num" shows "numbound0 (Mul i a) = numbound0 a"
graeffe_poly_def: fixes c :: "'b"   and as :: "'b list"   and m :: "nat" shows "graeffe_poly c as m = Polynomial.smult (c ^ 2 ^ m) (\<Prod>a\<leftarrow>as. [:- (a ^ 2 ^ m), 1::'b:])"
\<Sigma>_prot1: shows "\<Sigma>1.\<Sigma>_protocol"
assms: fixes j :: "nat" shows "ordertype (F j) (lenlex less_than) = \<omega> \<up> ord_of_nat j"
simps: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "joinR l x r = (if bheight l \<le> bheight r then R l x r else case l of \<langle>l', (x', Red), r'\<rangle> \<Rightarrow> R l' x' (joinR r' x r) | \<langle>l', (x', Black), r'\<rangle> \<Rightarrow> baliR l' x' (joinR r' x r))"
isoE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" obtains g :: "'b" where "category.inverse_arrows C f g"
that: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> vinsert \<langle>vcard xs, x\<rangle> xs" shows "thesis"
desargues_config_not_col_4: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col A A' C'"
first_less: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar (t # bq)" shows "\<forall>t'\<in>set bq. SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t'"
sets_bind': fixes A :: "'b measure"   and M :: "'b measure"   and B :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure" assumes "A \<in> space (prob_algebra M)"   and "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "sets (A \<bind> B) = sets N"
card_cosets_equal: fixes R :: "'a set"   and H :: "'a set" assumes "R \<in> rcosets H"   and "H \<subseteq> carrier G" shows "\<exists>f. bij_betw f H R"
Inv_swap: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat" assumes "dist_perm xs ys" shows "Inv xs (swap n ys) = (if Suc n < length xs then if ys ! n < ys ! Suc n in xs then Inv xs ys \<union> {(ys ! n, ys ! Suc n)} else Inv xs ys - {(ys ! Suc n, ys ! n)} else Inv xs ys)"
conj_left_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<^bold>\<squnion> x \<^bold>\<squnion> y = x \<^bold>\<squnion> y"
eOp_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'd option"   and einp :: "'index \<Rightarrow> 'c withERR option" assumes "\<not> igWlsBinp MOD delta binp" shows "eOp MOD delta einp (OKI binp) = ERR"
ope: fixes B :: "'a set" assumes "B \<in> \<C>" shows "open B"
emeasure_eq_measure: fixes A :: "'b set" shows "emeasure p A = ennreal (prob A)"
assoc: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and c :: "'c \<times> 'd" shows "times_prod' (times_prod' a b) c = times_prod' a (times_prod' b c)"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_def: fixes A :: "'b set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> x. (\<sqinter>xa\<in>A \<rightarrow>  x) |-| SKIP"
dg_in_Obj_in_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V" assumes "digraph \<alpha> \<CC>"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
quantale_hom_radj: fixes f :: "'c \<Rightarrow> 'd" assumes "f \<in> quantale_homset" shows "f \<stileturn> radj f"
dghm_singleton_ObjMap_app: fixes x :: "V"   and \<CC> :: "V"   and j :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "dghm_singleton j \<CC>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = ZFC_in_HOL.set {\<langle>j, x\<rangle>}"
prj_simps_10: shows "C.arr doms.Prj\<^sub>1"
entropy_le_space: fixes MX :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> Px x"   and "finite_measure MX"   and "integrable MX (\<lambda>x. - Px x * log b (Px x))" shows "entropy b MX X \<le> log b (Sigma_Algebra.measure MX (space MX))"
IC_2: shows "the (rcm r) = card (set (rhsq r) - (Q - W))"
IH: assumes "ref = None \<or> ref = Some r \<and> r \<in> set xs" shows "insert_spec (xs @ xsa) (oid, ref) = insert_spec xs (oid, ref) @ xsa"
perm_1: fixes pi :: "(name \<times> name) list"   and x1 :: "nat" shows "pi \<bullet> Atom x1 = Atom (pi \<bullet> x1)"
euler_criterion: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p" shows "[Legendre a (int p) = a ^ ((p - 1) div 2)] (mod int p)"
topology\<^sub>N_def: fixes N :: "'c quasinorm" shows "topology\<^sub>N N = topology (\<lambda>U. \<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
comm_groupE_2: fixes G :: "('e, 'f) monoid_scheme" assumes "comm_group G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
subtermeq_Var_const_2: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and f :: "'a" assumes "s \<sqsubseteq> t"   and "t = Fun f []" shows "s = Fun f []"
map\<^sub>1_simps_3: fixes f :: "'a" assumes "C.ide f" shows "trg\<^sub>D (\<tau>\<^sub>1 f) = G.map\<^sub>0 (trg\<^sub>C f)"
hyps: fixes thesis :: "bool" assumes "\<And>f g. \<lbrakk>(f, g) \<in> set ps \<or> (g, f) \<in> set ps; h = monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)\<rbrakk> \<Longrightarrow> thesis"   and "h \<in> set (pdata_pairs_to_list ps)" shows "thesis"
simply_connected_eq_contractible_loop_some: fixes S :: "'b set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a))))"
runit'_naturality: fixes f :: "'a" assumes "arr f" shows "(f \<otimes> \<I>) \<cdot> \<r>\<^sup>-\<^sup>1[local.dom f] = \<r>\<^sup>-\<^sup>1[cod f] \<cdot> f"
induct: fixes P :: "Unary_PCF.type \<Rightarrow> bool"   and type :: "Unary_PCF.type" assumes "P \<B>"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<rightarrow> x2)" shows "P type"
eval_cases_15: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains s\<^sub>1 :: "heap \<times> sfields \<times> sheap" where "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1__\<rangle>" and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1__\<rangle> \<Rightarrow> \<langle>e'a__,s'\<rangle>"   | s\<^sub>1 :: "heap \<times> sfields \<times> sheap" where "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e'a__,s'\<rangle>"   | e' :: "char list exp" where "e'a__ = throw e'" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e',s'\<rangle>"
numer_cf5_deriv_pos: fixes x :: "real" shows "0 \<le> numer_cf5_deriv x"
G_perm_apply: fixes g :: "'a"   and x :: "'a" assumes "g \<in> G"   and "x \<in> G" shows "\<pp> g \<rightarrow> x = binop g x"
unifiers_unifiert: fixes \<sigma> :: "char list \<Rightarrow> (char list, char list) Term.term"   and ts :: "fterm set" assumes "\<sigma> \<in> unifiers (fterm_to_iterm ` ts \<times> fterm_to_iterm ` ts)" shows "unifier\<^sub>t\<^sub>s (\<lambda>x. iterm_to_fterm (\<sigma> x)) ts"
Basis_list: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Basis_list :: "'e list"   and eucl_down :: "int \<Rightarrow> 'e \<Rightarrow> 'e"   and eucl_truncate_down :: "nat \<Rightarrow> 'e \<Rightarrow> 'e"   and eucl_truncate_up :: "nat \<Rightarrow> 'e \<Rightarrow> 'e" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "set Basis_list = Basis"
po_onI: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "irreflp_on P A"   and "transp_on P A" shows "po_on P A"
P': shows "P' = (\<lambda>X. if X \<subseteq> B then {} else X) ` Q' - {{}}"
quotient_eqI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and Y :: "'a set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r"   and "x \<in> X"   and "y \<in> Y"   and "(x, y) \<in> r" shows "X = Y"
emeasure_uniform_measure: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
pre_nts: shows "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<j'] nts"
FactRing_def: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
Inf_lower2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and u :: "'d"   and A :: "'d set"   and v :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq u v" shows "less_eq (Inf A) v"
left_tree_separates: fixes G :: "('c, 'd) Graph_scheme"   and s :: "'c"   and t :: "'c"   and u :: "'c"   and u' :: "'c" assumes "Tree G"   and "s \<rightarrow>\<^bsub>G\<^esub> t"   and "u \<in> Tree.left_tree G s t"   and "u' \<in> Tree.left_tree G t s" shows "s \<in> set (u \<leadsto>\<^bsub>G\<^esub> u')"
contf: shows "continuous_on (S \<inter> {z. 0 \<le> Im z}) f"
game2a_2b: shows "\<bar>spmf (map_spmf fst game2_a) True - spmf (map_spmf fst game2_b) True\<bar> \<le> spmf (map_spmf snd game2_a) True"
lin_poly_to_vec_coeff_access: fixes x :: "nat"   and y :: "linear_poly" assumes "x < dim_poly y" shows "lpoly_to_vec y $ x = Abstract_Linear_Poly.coeff y x"
While: fixes p :: "'a set"   and b :: "'a set"   and c :: "'a com"   and i :: "'a set" assumes "\<parallel>- (p \<inter> b) c p" shows "\<parallel>- p While b i c (p \<inter> - b)"
actrans_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "_actrans P v \<equiv> P \<or> Unchanged v"
q_ends_2: shows "pathfinish q = a - complex_of_real d"
Un_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
filterlim_tendsto_pos_mult_at_top_iff: fixes f :: "'c \<Rightarrow> real"   and c :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c" shows "(LIM x F. f x * g x :> at_top) = filterlim g at_top F"
vcg_intro_frame: fixes m :: "'a nres" shows "m \<equiv> VCG_FRAME m"
isos_compose: fixes f :: "('a, 'a set) Maps.arr"   and f' :: "('a, 'a set) Maps.arr" assumes "Maps.iso f"   and "Maps.iso f'"   and "Maps.seq f' f" shows "Maps.iso (f' \<odot> f)"
None: fixes xs :: "('d, 'c) bintree option list" shows "pqueue (None # xs) = pqueue xs"
AppL: fixes t :: "trm"   and t' :: "trm"   and u :: "trm" assumes "t \<longmapsto> t'" shows "t \<cdot> u \<longmapsto> t' \<cdot> u"
coprime_absorb_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x y" shows "algebraic_semidom.coprime times one x y = dvd.dvd times x one"
dASSUME_def: shows "dASSUME \<equiv> iASSUME dRETURN"
PNT1_imp_PNT4: assumes "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)" shows "\<theta> \<sim>[at_top] (\<lambda>x. x)"
finite_permutations_of_set: fixes A :: "'a set" shows "finite (permutations_of_set A)"