hcju/isabelleps huggingface dataset:
known_ptr_def: shows "ShadowRootClass.known_ptr \<equiv> l_known_ptr\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptr"
normalize_alt_simps_2: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and t :: "'a OCL_Types.type option"   and init :: "'a expr"   and body :: "'a expr"   and b :: "'a expr" shows "(\<Gamma> \<turnstile> expr.Let v t init body \<Rrightarrow> b) = ((\<exists>init\<^sub>2 \<tau>. t = Some \<tau> \<and> (\<exists>body\<^sub>2. b = expr.Let v (Some \<tau>) init\<^sub>2 body\<^sub>2 \<and> \<Gamma> \<turnstile> init \<Rrightarrow> init\<^sub>2 \<and> \<Gamma>(v \<mapsto>\<^sub>f \<tau>) \<turnstile> body \<Rrightarrow> body\<^sub>2)) \<or> t = None \<and> (\<exists>init\<^sub>2 \<tau> body\<^sub>2. b = expr.Let v (Some \<tau>) init\<^sub>2 body\<^sub>2 \<and> \<Gamma> \<turnstile> init \<Rrightarrow> init\<^sub>2 \<and> \<Gamma> \<turnstile>\<^sub>E init\<^sub>2 : \<tau> \<and> \<Gamma>(v \<mapsto>\<^sub>f \<tau>) \<turnstile> body \<Rrightarrow> body\<^sub>2))"
assms_2: assumes "P x y" shows "\<forall>x y. P x y \<longrightarrow> Q P (s1 x) (s2 y)"
maxdeg_def: fixes A :: "(('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd) set" shows "maxdeg A = Max (poly_deg ` A)"
path_components_of_subset: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> path_components_of X" shows "C \<subseteq> topspace X"
mset_le_subtract: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" assumes "A \<subseteq># B" shows "A - C \<subseteq># B - C"
code: fixes gpv :: "('a, 'c, 'e) gpv" shows "right_gpv gpv = GPV (map_spmf (map_generat id Inr right_rpv) (the_gpv gpv))"
arith_simps_29: shows "neg_numeral_class.dbl (0::'a) = (0::'a)"
l_one: fixes x :: "'a mod_ring poly" assumes "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
int_distrib_3: fixes z1 :: "int"   and z2 :: "int"   and w :: "int" shows "(z1 - z2) * w = z1 * w - z2 * w"
HSem_merge: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "atom ` domA \<Gamma> \<sharp>* \<Delta>" shows "\<N>\<lbrace>\<Gamma>\<rbrace>\<N>\<lbrace>\<Delta>\<rbrace>\<rho> = \<N>\<lbrace>\<Gamma> @ \<Delta>\<rbrace>\<rho>"
set_mult_one_imp_inc: fixes A :: "'a set"   and B :: "'a set" assumes "\<one> \<in> A"   and "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "B \<subseteq> B <#> A"
Figure_7_trans: fixes k :: "nat"   and js :: "nat list"   and r :: "nat"   and i :: "nat"   and xs :: "nat list"   and ys :: "nat list" assumes "\<And>i xs ys. \<lbrakk>i < k; js = xs @ ys; xs \<noteq> []; i = last xs\<rbrakk> \<Longrightarrow> i + 1 \<in> set ys"   and "0 < r"   and "i + r \<le> k"   and "js = xs @ ys"   and "xs \<noteq> []"   and "i = last xs" shows "i + r \<in> set ys"
unit_disc_fix_f_circline_def: fixes f :: "complex_homo \<Rightarrow> complex_homo"   and H :: "circline" shows "unit_disc_fix_f_circline f H = (if \<exists>M. unit_disc_fix M \<and> f = moebius_pt M then moebius_circline (THE M. unit_disc_fix M \<and> f = moebius_pt M) H else if \<exists>M. unit_disc_fix M \<and> f = moebius_pt M \<circ> conjugate then (moebius_circline (THE M. unit_disc_fix M \<and> f = moebius_pt M \<circ> conjugate) \<circ> conjugate_circline) H else H)"
proper_designI: assumes "int (size (del_point_blocks p)) \<noteq> 0" shows "proper_design (del_point p) (del_point_blocks p)"
simps_1: fixes bs :: "(nat \<times> (nat \<times> 'a) list) list" shows "add_spmat [] bs = bs"
cyclic_on_inI: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and s :: "'e" assumes "cyclic_on f S"   and "s \<in> S" shows "f s \<in> S"
P_mod_mod: fixes I :: "'a set"   and p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "ideal S I"   and "p \<in> carrier R"   and "pol_coeff S c"   and "p = polyn_expr R X (fst c) c" shows "(\<forall>j\<le>fst c. snd c j \<in> I) = P_mod R S X I p"
left_adjoints_compose: fixes f :: "'a"   and f' :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint f'"   and "src f = trg f'" shows "is_left_adjoint (f \<star> f')"
enn2real_le: fixes e :: "ennreal"   and r :: "real" assumes "enn2real e \<le> r"   and "e \<noteq> \<top>" shows "e \<le> ennreal r"
Domainp_conversep: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "Domainp R\<inverse>\<inverse> = Rangep R"
assms: shows "\<Gamma> \<turnstile> s \<leftrightarrow> t : T"
spr_jview_tObsC: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t = spr_jview a t'" shows "tObsC t = tObsC t'"
treap: shows "treap \<langle>l, (k1, p1), r\<rangle>"
r_totalizer_recfn: fixes d :: "nat" shows "recfn 2 (r_totalizer d)"
prems_5: shows "compress_normalize_primitive_monad (f # fs) m = None"
inject_2: fixes x3 :: "nat"   and y3 :: "nat" shows "(DNum x3 = DNum y3) = (x3 = y3)"
perm_supp_eq: fixes p :: "perm"   and x :: "'d" assumes "supp p \<sharp>* x" shows "p \<bullet> x = x"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = Set.insert x21 (set x22)"
elims_3: fixes x :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and xa :: "'c \<Rightarrow> 'b" assumes "\<not> wf_disc_sel x xa" obtains disc :: "'b \<Rightarrow> bool" and sel :: "'b \<Rightarrow> 'c" and C :: "'c \<Rightarrow> 'b" where "x = (disc, sel)" and "xa = C" and "\<not> ((\<forall>a. disc a \<longrightarrow> C (sel a) = a) \<and> (\<forall>a. sel (C a) = a))"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "C.dom (F f) = F (D.dom f)"
Clean_PPf_encPf: fixes \<phi> :: "'fmla"   and p :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "p \<in> num"   and "p \<notin> encPf ` proof" shows "bprv (neg (PPf p \<langle>\<phi>\<rangle>))"
comm_semiring_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.comm_semiring plus times \<equiv> class.ab_semigroup_add plus \<and> class.ab_semigroup_mult times \<and> class.comm_semiring_axioms plus times"
real_le_intI: fixes x :: "real"   and y :: "real" assumes "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"   and "x \<in> \<int>" shows "x \<le> y"
P_def: fixes k :: "nat" shows "P k \<equiv> snd ((p ## r) !! k)"
hyp: fixes x :: "lam"   and z :: "lam" assumes "\<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> SN y"   and "\<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> P y z"   and "\<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> SN u"   and "\<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> P x u" shows "P x z"
concatenates_Nil: fixes ys :: "brick list" shows "[] \<otimes> ys = ys"
monoid: shows "Group.monoid (G\<lparr>carrier := H\<rparr>)"
gfp_rolling: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'a" assumes "mono g"   and "mono f" shows "g (gfp (\<lambda>x. f (g x))) = gfp (\<lambda>x. g (f x))"
elims: fixes x :: "('b \<times> nat) tree"   and y :: "('b \<times> nat) tree" assumes "skew x = y" obtains t1 :: "('b \<times> nat) tree" and b :: "'b" and lvb :: "nat" and t2 :: "('b \<times> nat) tree" and a :: "'b" and lva :: "nat" and t3 :: "('b \<times> nat) tree" where "x = \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle>" and "y = (if lva = lvb then \<langle>t1, (b, lvb), \<langle>t2, (a, lva), t3\<rangle>\<rangle> else \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle>)"   | "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | va :: "'b \<times> nat" and vb :: "('b \<times> nat) tree" where "x = \<langle>\<langle>\<rangle>, va, vb\<rangle>" and "y = \<langle>\<langle>\<rangle>, va, vb\<rangle>"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> Pds'.Cd d X) = (Pds'.Cd d X = {x})"
radical_sqrt_normal_form: fixes e :: "expr" assumes "radicals e \<noteq> {}" shows "\<exists>r\<in>radicals e. \<exists>a b. \<lbrace>e\<rbrace> = \<lbrace>Addition a (Multiplication b (Sqrt r))\<rbrace> \<and> 0 \<le> \<lbrace>r\<rbrace> \<and> radicals a \<union> radicals b \<union> radicals r \<subseteq> radicals e \<and> r \<notin> radicals a \<union> radicals b \<union> radicals r"
l\<^sub>2': shows "l\<^sub>2' = [this \<mapsto> Ref (a, Cs'), pns [\<mapsto>] vs']"
tell_env_def: fixes tell :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and w :: "'f"   and m :: "('d, 'e) envT" shows "tell_env tell w m = EnvT (\<lambda>r. tell w (run_env m r))"
simps_1: fixes f :: "'b"   and x :: "'b"   and xs :: "'b list" shows "pderiv_coeffs_code f (x # xs) = f * x ## pderiv_coeffs_code (f + (1::'b)) xs"
Cauchy_theorem_triangle_cofinite: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "continuous_on (convex hull {a, b, c}) f"   and "finite S"   and "\<And>x. x \<in> interior (convex hull {a, b, c}) - S \<Longrightarrow> f field_differentiable at x" shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
L_sim1: fixes r :: "'b \<Rightarrow> 'u"   and i :: "'b"   and l :: "'L" assumes "range r \<subseteq> V"   and "L (r i) l" shows "G'.L (f (r i)) l"
ls_rehash_map_of: fixes l :: "('a \<times> 'b) list list" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat (ls_rehash l)) = map_of (concat l)"
linear_map_Poly_list_of_vec': shows "linear_map class_ring local.V (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}) (Poly \<circ> list_of_vec)"
enum_strict_mono: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "i \<le> n"   and "j \<le> n" shows "(kuhn_simplex.enum base upd i < kuhn_simplex.enum base upd j) = (i < j)"
absolutely_integrable_on_const: fixes S :: "'a set"   and c :: "'b" assumes "S \<in> lmeasurable" shows "(\<lambda>x. c) absolutely_integrable_on S"
sq: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ P" shows "\<turnstile> \<box>[P]_v"
leq_head_def: fixes t :: "('c, 'd) Term.term"   and u :: "('c, 'd) Term.term" shows "leq_head t u = (case (root t, root u) of (None, xa) \<Rightarrow> True | (Some a, None) \<Rightarrow> False | (Some a, Some g) \<Rightarrow> a \<le> g)"
vector_eq_ldot: fixes y :: "'e"   and z :: "'e" shows "(\<forall>x. x \<bullet> y = x \<bullet> z) = (y = z)"
partial_latin_square_full: fixes s :: "(latin_type \<Rightarrow> nat) set"   and n :: "nat" shows "(partial_latin_square s n \<and> card s = n * n) = latin_square s n"
inf_ant_def: shows "\<infinity> = Abs_Ainteg (0, 1)"
C_cases: fixes r :: "C" obtains n :: "nat" where "r = C\<^bsup>n\<^esup>"   | "r = C\<^sup>\<infinity>"
in_hhomE: fixes V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and \<mu> :: "'d"   and a :: "'d"   and b :: "'d" assumes "horizontal_homs V src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b" obtains "partial_magma.arr V \<mu>" and "horizontal_homs.obj V src a" and "horizontal_homs.obj V src b" and "src \<mu> = a" and "trg \<mu> = b"
p'_proj_def: shows "p'_proj = reduce_fps_poly p'"
execute_while_unfolds_1: fixes h :: "heap"   and f :: "'b Heap" assumes "success p h"   and "cond p h" shows "execute (while p f) h = execute (f \<bind> (\<lambda>_. while p f)) h"
right_impl_ded: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b r\<rightarrow> c = b r\<rightarrow> a r\<rightarrow> c"
Node_3: shows "inorder \<langle>l, k, r\<rangle> = [i..j]"
pos_add_nonneg_ne_zero: fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 \<le> b" shows "a + b \<noteq> 0"
lunit_hcomp_4: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<l>\<^sup>-\<^sup>1[f \<star> g] = \<a>[trg f, f, g] \<cdot> (\<l>\<^sup>-\<^sup>1[f] \<star> g)"
fprems_1: shows "\<forall>a. fmul (0::'b) a = (0::'c)"
hyps_3: shows "\<Gamma> \<turnstile> fs [:] fTs"
set_mono_prefix: fixes xs :: "'e list"   and ys :: "'e list" assumes "prefix xs ys" shows "set xs \<subseteq> set ys"
monoid_mult_ow: shows "class.monoid_mult = monoid_mult_ow UNIV"
control_vars_are_A_vars: shows "\<C>\<^sub>C = var\<^sub>C_of ` \<C>\<^sub>A"
uint_add_le: fixes x :: "'b word"   and y :: "'b word" shows "uint (x + y) \<le> uint x + uint y"
Ipoly_fst_tmf_polys: fixes xs :: "real list"   and z :: "float interval list" shows "Ipoly xs (map_poly real_of_float (fst (tmf_polys z))) = (\<Sum>i<length z. xs ! 0 ^ i * real_of_float (mid (z ! i)))"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c list_R1"   and b :: "'d list_R1" shows "rel_list_R1 R a b = rel_sum (=) (rel_prod R (rel_list_R1 R)) (unR a) (unR b)"
update_preserves_le: fixes mds\<^sub>1 :: "Mode \<Rightarrow> 'Var set"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and annos :: "'Var ModeUpd list" assumes "mds\<^sub>1 \<sqsubseteq> mds\<^sub>2" shows "mds\<^sub>1 \<oplus> annos \<sqsubseteq> mds\<^sub>2 \<oplus> annos"
true_def: shows "true \<equiv> \<lambda>\<tau>. \<lfloor>\<lfloor>True\<rfloor>\<rfloor>"
inv_COMB_def: fixes s :: "('a list \<times> (bool list \<times> 'a list + 'a list)) pmf"   and x :: "'a"   and i :: "'a list" shows "inv_COMB s x i \<equiv> \<exists>Da Db. finite (set_pmf Da) \<and> finite (set_pmf Db) \<and> map_pmf con s = Sum_pmf (8 / 10) Da Db \<and> BIT_inv Da x i \<and> TS_inv Db x i"
group_homomorphism_hom_relboundary: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "hom_relboundary p X S T \<in> hom (relative_homology_group p X S) (relative_homology_group (p - 1) (subtopology X S) T)"
3: assumes "\<nexists>t. b = send\<langle>t\<rangle> \<or> b = receive\<langle>t\<rangle>" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) # unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
replace_distinct: fixes fs :: "'a list"   and newfs :: "'a list"   and oldf :: "'a" assumes "distinct fs"   and "distinct newfs"   and "oldf \<in> set fs \<longrightarrow> set newfs \<inter> set fs \<subseteq> {oldf}" shows "distinct (replace oldf newfs fs)"
gen_def: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and cs :: "'d list"   and w :: "'e" shows "gen P cs w \<equiv> \<forall>c\<in>set cs. P c w"
disjoint: shows "\<forall>f\<in>A. \<forall>g\<in>A. f \<noteq> g \<longrightarrow> (\<forall>x y. f x \<noteq> g y)"
X: fixes i :: "'l" shows "G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
borel_measurable_abs: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
intros_2: fixes d :: "nat"   and u :: "'b \<Rightarrow> real"   and x :: "'b" assumes "real d < u x"   and "u x < real (d + 1)" shows "intv_elem x u (Intv d)"
BijGroup_def: fixes S :: "'c set" shows "BijGroup S = \<lparr>carrier = Bij S, monoid.mult = \<lambda>g\<in>Bij S. restrict (compose S g) (Bij S), one = \<lambda>x\<in>S. x\<rparr>"
null_space_orthogonal_complement_row_space: fixes A :: "((real, 'a) vec, 'b) vec" shows "null_space A = orthogonal_complement (row_space A)"
sums: shows "(\<lambda>k. 1 / fact (n + k)) sums (exp 1 - (\<Sum>k<n. 1 / fact k))"
scount_finite: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "ev (alw (\<lambda>xs. \<not> P xs)) \<omega>" shows "scount P \<omega> < \<infinity>"
cases: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "ccpo.compact lub ord x" obtains "ccpo.admissible lub ord (\<lambda>y. \<not> ord x y)" and "ccpo.admissible lub ord ((\<noteq>) x)"
swap\<^sub>m_def: shows "swap\<^sub>m = (0-\<Sigma>), (1-\<Sigma>) := &1-\<Sigma>, &0-\<Sigma>"
gen_length_def: fixes n :: "nat"   and xs :: "'c list" shows "List.gen_length n xs = n + |xs|"
openI: fixes S :: "'c set" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"
in_close_world_conv: fixes \<phi> :: "object atom formula"   and M :: "object atom formula set" shows "(\<phi> \<in> close_world M) = (\<phi> \<in> M \<or> (\<exists>p as. \<phi> = \<^bold>\<not> (Atom (predAtm p as)) \<and> Atom (predAtm p as) \<notin> M) \<or> (\<exists>a. \<phi> = Atom (Eq a a)) \<or> (\<exists>a b. \<phi> = \<^bold>\<not> (Atom (Eq a b)) \<and> a \<noteq> b))"
mod_dist_4: fixes P :: "assn"   and Q :: "assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P \<and>\<^sub>A Q) = (h \<Turnstile> P \<and> h \<Turnstile> Q)"
simps_34: fixes vd :: "'b"   and v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b" shows "(SKIP {vd} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
a_consistent_thunk_0: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and S :: "stack_elem list"   and e :: "exp" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, Var x, S)"   and "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>0" shows "ArityAnalysisLetSafe.a_consistent Aexp (ae, 0, as) (delete x \<Gamma>, e, Upd x # S)"
sel_1: fixes x1 :: "'f"   and x2 :: "'g" shows "fst (x1, x2) = x1"
M: shows "\<forall>i m. M i \<le> m \<longrightarrow> (\<forall>j\<le>i. fps_X (M i) $ j = fps_X m $ j)"
retraction_comp: fixes S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and U :: "'b set"   and g :: "'b \<Rightarrow> 'b" assumes "retraction S T f"   and "retraction T U g" shows "retraction S U (g \<circ> f)"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a = inf a b \<and> a \<noteq> b)"
quasi_inverts_components_2: fixes a :: "'d" assumes "D.obj a" shows "\<eta>\<^sub>0 a \<star>\<^sub>D \<eta>'.map\<^sub>0 a \<cong>\<^sub>D GF.map\<^sub>0 a"
PAR_least: fixes y :: "'a"   and x :: "'a"   and t :: "real" assumes "y \<in> I x" shows "emeasure (PAR x) {p \<in> space (PAR x). t \<le> p y \<and> select_first x p y} = emeasure (exponential (escape_rate x)) {t..} * ennreal (pmf (J x) y)"
local_nonsatiation_def: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" shows "local_nonsatiation B P = (\<forall>x\<in>B. \<forall>e>0. \<exists>y\<in>B. norm (y - x) \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P)"
cbox_image_unit_interval: fixes a :: "'b"   and b :: "'b" assumes "cbox a b \<noteq> {}" shows "cbox a b = (+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox (0::'b) One"
hyps: shows "invertible (fst (Gauss_Jordan_upt_k_PA A k))"
HMA_eigen_value: shows "(HMA_M ===> (=) ===> (=)) eigenvalue eigen_value"
leg1_in_hom_1: shows "\<guillemotleft>tab\<^sub>1 f \<star> TfTgh.p\<^sub>1 : src ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<rightarrow> trg (f \<star> g \<star> h)\<guillemotright>"
param_prod_2: fixes Ra :: "('a \<times> 'd) set"   and Rb :: "('b \<times> 'e) set"   and Rr :: "('c \<times> 'f) set" shows "(case_prod, case_prod) \<in> (Ra \<rightarrow> Rb \<rightarrow> Rr) \<rightarrow> Ra \<times>\<^sub>r Rb \<rightarrow> Rr"
le_infI2: fixes b :: "'h multiset"   and x :: "'h multiset"   and a :: "'h multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
neg_smult: fixes r :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "m \<in> V" shows "- r \<cdot> m = - (r \<cdot> m)"
joinablep_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "joinablep P x y = (\<exists>z. P x z \<and> P y z)"
carr1_3: shows "map of_int_hom.vec_hom fs ! j \<in> Rn"
Ord_mult: fixes y :: "V"   and x :: "V" assumes "Ord y"   and "Ord x" shows "Ord (x * y)"
BK3_msg_in_parts_spies: fixes S :: "agent"   and A :: "agent"   and KA :: "nat"   and Timestamp :: "msg"   and B :: "msg"   and K :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says S A (Crypt KA \<lbrace>Timestamp, B, K, X\<rbrace>) \<in> set evs" shows "X \<in> parts (knows Spy evs)"
ntr_mon_s: fixes c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and c' :: "'n list multiset" assumes "(c, e, c') \<in> ntr fg" shows "mon_c fg c' = mon_c fg c \<union> fst (\<alpha>n fg e)"
plus_stream_shd: fixes x :: "'b stream"   and y :: "'b stream" shows "shd (x + y) = shd x + shd y"
rep_eq: fixes x :: "unit2"   and xa :: "unit2" shows "(x < xa) = (Rep_unit2 x < Rep_unit2 xa)"
abs_summable_on_comparison_test: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'd" assumes "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> norm (g x)" shows "f abs_summable_on A"
reorder_cyclic_list: fixes l :: "'a list"   and s :: "'a"   and sl :: "nat"   and r :: "('a \<times> 'a) set" assumes "hd l = s"   and "last l = s"   and "2 < length l"   and "sl + 1 < length l"   and "rel_chain l r" obtains l' :: "'a list" where "hd l' = l ! (sl + 1)" and "last l' = l ! sl" and "rel_chain l' r" and "length l' = length l - 1" and "\<forall>i. i + 1 < length l' \<longrightarrow> (\<exists>j. j + 1 < length l \<and> l' ! i = l ! j \<and> l' ! (i + 1) = l ! (j + 1))"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha> set['b]" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = abs_Tree\<^sub>\<alpha> (tConj (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
trans: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
inv_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>B.inv f : b \<Rightarrow>\<^sub>B a\<guillemotright>"
optimal_forest_def: fixes F :: "('a, 'b) graph"   and G :: "('a, 'b) graph" shows "optimal_forest F G \<equiv> \<forall>F'. spanning_forest F' G \<longrightarrow> edge_weight F \<le> edge_weight F'"
simple_cipher_def: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf" shows "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen \<equiv> (finite prf_domain \<and> prf_domain \<noteq> {} \<and> (\<forall>x. x \<in> prf_domain \<longrightarrow> length x = prf_dlen)) \<and> (\<forall>key_prf m. key_prf \<in> set_spmf prf_key_gen \<longrightarrow> m \<in> prf_domain \<longrightarrow> length (prf_fun key_prf m) = prf_clen) \<and> lossless_spmf prf_key_gen \<and> lossless_spmf upf_key_gen"
step_bounded_pres: shows "\<forall>(p, x)\<in>set ps. p < length xs \<and> x \<in> A"
gallery_remdup_adj: fixes xs :: "'a set list"   and x :: "'a set"   and ys :: "'a set list" assumes "gallery (xs @ [x, x] @ ys)" shows "gallery (xs @ [x] @ ys)"
m: shows "m = monom_mult (punit.lc (rep_list p) / punit.lc (rep_list g1)) (punit.lt (rep_list p) - punit.lt (rep_list g1)) g1"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'd" assumes "functor A A F" shows "endofunctor A F"
Float_num_2: shows "real_of_float (Float 1 1) = 2"
simps_1: fixes ops :: "'b arith_ops_record"   and r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "mod_poly_one_main_i ops r d (Suc n) = (let a = hd r; rr = tl (if a = arith_ops_record.zero ops then r else minus_poly_rev_list_i ops r (map (arith_ops_record.times ops a) d)) in mod_poly_one_main_i ops rr d n)"
pinf_7: fixes t :: "'c" shows "\<exists>z. \<forall>x>z. (t < x) = True"
WT_adv: fixes \<eta> :: "nat" shows "\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<A> \<eta> \<surd>"