hcju/isabelleps huggingface dataset:
simps_31: fixes vd :: "char list"   and ve :: "aexp"   and vf :: "'b"   and v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b" shows "(vd ::= ve {vf} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
poly_deg_max_keys: fixes t :: "'c \<Rightarrow>\<^sub>0 'd"   and p :: "('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'e" assumes "t \<in> keys p" shows "deg_pm t \<le> poly_deg p"
heap_is_wellformed_def: shows "heap_is_wellformed = (\<lambda>h. heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h \<and> acyclic (parent_child_rel h \<union> local.a_host_shadow_root_rel h) \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_shadow_root_valid h)"
mono_Ndet3: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D (P |-| S)) \<subseteq> T (Q |-| S)"
incl: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> H" shows "a \<in> additive.generate H"
"1_2": shows "[a] \<preceq>\<^sub>F v\<^sub>2"
f15: shows "t' \<otimes> (snd x' \<otimes> r') = t' \<otimes> (s' \<otimes> fst x')"
inj2_def: fixes M1 :: "('i, 'h) module"   and M2 :: "('i, 'g) module" shows "inj2 M1 M2 = Pair \<zero>\<^bsub>M1\<^esub>"
\<R>_div_def: fixes \<omega> :: "('c \<Rightarrow> 'b) set stream" shows "\<R>_div \<omega> \<equiv> \<forall>x\<in>\<X>. (\<forall>i. (\<exists>j\<ge>i. zero x (\<omega> !! j)) \<and> (\<exists>j\<ge>i. \<not> zero x (\<omega> !! j))) \<or> (\<exists>i. \<forall>j\<ge>i. unbounded x (\<omega> !! j))"
Units_l_inv_ex: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> Units (DirProds G I)" shows "\<exists>y\<in>carrier (DirProds G I). y \<otimes>\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>"
vals_nonequiv_valuation: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and m :: "nat" assumes "vals_nonequiv K n vv"   and "m \<le> n" shows "valuation K (vv m)"
units_of_def: fixes G :: "('c, 'd) monoid_scheme" shows "units_of G = \<lparr>carrier = Units G, monoid.mult = (\<otimes>\<^bsub>G\<^esub>), one = \<one>\<^bsub>G\<^esub>\<rparr>"
linear: fixes x :: "'a"   and y :: "'a" shows "cmp.le x y \<or> cmp.le y x"
convex_on_diff_1: fixes I :: "real set"   and f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real"   and t :: "real" assumes "convex_on I f"   and "x \<in> I"   and "y \<in> I"   and "x < t"   and "t < y" shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
eval_def: fixes f :: "recf"   and xs :: "nat list" assumes "recfn (length xs) f" shows "eval f xs \<equiv> eval_wellf f xs"
tm_comp_second: fixes A :: "(action \<times> nat) list"   and l :: "cell list"   and r :: "cell list"   and B :: "(action \<times> nat) list"   and stp :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "tm_wf0 A"   and "steps0 (1, l, r) B stp = (s', l', r')" shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (if s' = 0 then 0 else s' + length A div 2, l', r')"
cmp_simps'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "src\<^sub>D (\<Phi>\<^sub>F (\<mu>, \<nu>)) = F.map\<^sub>0 (src\<^sub>C \<nu>)"
op_eucl_of_list_image_def: fixes X :: "real list set" shows "op_eucl_of_list_image X = eucl_of_list ` X"
put_borel: fixes T :: "nat"   and K :: "real" shows "put_option T K \<in> borel_measurable (G T)"
drop_derivation_shift: fixes n :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "drop n (derivation_shift D left right) = derivation_shift (drop n D) left right"
whisker_right: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.ide f"   and "B.seq \<nu> \<mu>" shows "(\<nu> \<cdot>\<^sub>B \<mu>) \<star>\<^sub>B f = (\<nu> \<star>\<^sub>B f) \<cdot>\<^sub>B \<mu> \<star>\<^sub>B f"
simps: fixes fs :: "'b list"   and rs :: "'b list" shows "is_empty (fs, rs) = (fs = [])"
roots_mem_iff_is_root: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and x :: "'c" assumes "domain R"   and "p \<in> carrier (poly_ring R)" shows "(x \<in># ring.roots R p) = ring.is_root R p x"
hyps: shows "Q \<longmapsto> \<alpha> \<prec> Q'"
Lp_Lq_duality_bound_2: fixes p :: "ennreal"   and q :: "ennreal"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "1 / p + 1 / q = 1"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "g \<in> space\<^sub>N (\<LL> q M)" shows "\<bar>LINT x|M. f x * g x\<bar> \<le> Norm (\<LL> p M) f * Norm (\<LL> q M) g"
tllist_of_llist_cong: fixes xs :: "'f llist"   and xs' :: "'f llist"   and b :: "'g"   and b' :: "'g" assumes "xs = xs'"   and "lfinite xs' \<Longrightarrow> b = b'" shows "tllist_of_llist b xs = tllist_of_llist b' xs'"
intvs_lower_incr: fixes l :: "int"   and h :: "int" assumes "l < h" shows "{l..<h} = Set.insert l {l + 1..<h}"
infinite_super: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C" shows "identity_functor C"
comp_in_homI: fixes f :: "'s"   and a :: "'s"   and b :: "'s"   and g :: "'s"   and c :: "'s" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>S c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>S f : a \<rightarrow>\<^sub>S c\<guillemotright>"
pref_iffD2: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "|~ F = G" shows "|~ G \<longrightarrow> F"
supp_perm_eq: fixes x :: "'b"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
simps: fixes ds :: "'d set"   and a :: "'x set"   and aa :: "'x set" shows "COP.iterates_abovep ds a aa = (aa = a \<or> (\<exists>x. aa = cop_F ds x \<and> COP.iterates_abovep ds a x) \<or> (\<exists>M. aa = \<Union> M \<and> M \<in> Chains {(x, y). x \<subseteq> y} \<and> M \<noteq> {x. bot x} \<and> (\<forall>x. x \<in> M \<longrightarrow> COP.iterates_abovep ds a x)))"
snapshot_event: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "ps (distributed_system.s channel trans send recv init t (i + 1)) p \<noteq> None" shows "isSnapshot (t ! i) \<or> isRecvMarker (t ! i)"
ntsem_def: fixes \<sigma> :: "'tp" shows "ntsem \<sigma> \<equiv> if unprot \<sigma> \<or> protFw \<sigma> then id else iimg {b. eintT \<sigma> b} (eintF (Tag \<sigma>) \<circ> singl)"
step_Suc: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and i :: "nat"   and t :: "('d, 'e, 'f) event list"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "i < length t"   and "distributed_system.trace channel trans send recv c t c'" shows "distributed_system.next channel trans send recv (distributed_system.s channel trans send recv c t i) (t ! i) (distributed_system.s channel trans send recv c t (Suc i))"
nn_integral_eq_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal (integral\<^sup>L M f)"
apply_cltn2_line_def: fixes l :: "proj2_line"   and A :: "cltn2" shows "apply_cltn2_line l A \<equiv> P2L (apply_cltn2 (L2P l) (cltn2_transpose (cltn2_inverse A)))"
lsub_def: fixes interval_tree :: "interval_tree" shows "lsub interval_tree = (case interval_tree of Node x21 x22 x23 x24 \<Rightarrow> x21)"
eq_implies: fixes x :: "'a"   and y :: "'a" assumes "x = y"   and "x \<in> A" shows "\<not> x \<bowtie> y"
Sup_Continuous_closed: fixes H :: "'a upred \<Rightarrow> 'a upred"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'a upred" assumes "Continuous H"   and "\<And>i. i \<in> A \<Longrightarrow> P i is H"   and "\<not> A = {}" shows "\<Or>(P ` A) is H"
1: shows "rv1 = src re \<or> (rv1, src re) \<in> (subs prb)\<^sup>+"
ldropn_ltake: fixes n :: "nat"   and m :: "enat"   and xs :: "'a llist" shows "ldropn n (ltake m xs) = ltake (m - enat n) (ldropn n xs)"
conjugate_inf: shows "conjugate \<infinity>\<^sub>h = \<infinity>\<^sub>h"
suppa2__conga456: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D' E' F'" shows "Tarski_neutral_dimensionless.CongA Bet Cong D E F D' E' F'"
smc_Arr_vempty_if_Obj_vempty: assumes "\<BB>\<lparr>Obj\<rparr> = []\<^sub>\<circ>" shows "\<BB>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
fls_X_nonzero: shows "fls_X \<noteq> 0"
permutes_induct: fixes S :: "'b set"   and P :: "('b \<Rightarrow> 'b) \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b" assumes "finite S"   and "P id"   and "\<And>a b p. \<lbrakk>a \<in> S; b \<in> S; P p; P p; permutation p\<rbrakk> \<Longrightarrow> P (Fun.swap a b id \<circ> p)"   and "p permutes S" shows "P p"
morphic_propE_2: fixes R :: "('e, 'f) ring_scheme"   and P :: "'e \<Rightarrow> bool"   and r :: "'e" assumes "morphic_prop R P"   and "r \<in> carrier R" shows "P r"
span_add: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x + y \<in> ideal S"
inv_stereographic_stereographic: fixes x :: "riemann_sphere" shows "inv_stereographic (stereographic x) = x"
Op_asInp_asTerm_qOp: fixes qinp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and qbinp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs option"   and delta :: "'c" assumes "qGoodInp qinp"   and "qGoodBinp qbinp" shows "Op delta (asInp qinp) (asBinp qbinp) = asTerm (qOp delta qinp qbinp)"
select_convs_2: fixes nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "edgesL \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = edgesL"
direct_path_return: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob" obtains \<pi> :: "nat \<Rightarrow> 'a" and n :: "nat" where "IFC_def.is_path prob \<pi>" and "\<pi> 0 = x" and "\<pi> n = IFC_def.return prob" and "\<forall>i>0. \<pi> i \<noteq> x"
diff_single_trivial: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<notin># M" shows "M - (x, {#}) = M"
lm058: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum snd (summedBidVectorRel bids N G || a)"
unique_circline_01inf': shows "0\<^sub>h \<in> circline_set x_axis \<and> 1\<^sub>h \<in> circline_set x_axis \<and> \<infinity>\<^sub>h \<in> circline_set x_axis \<and> (\<forall>H. 0\<^sub>h \<in> circline_set H \<and> 1\<^sub>h \<in> circline_set H \<and> \<infinity>\<^sub>h \<in> circline_set H \<longrightarrow> H = x_axis)"
base_case_sgas: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "coprime p q" shows "set (characterize_consistent_signs_at_roots_copr p [q]) \<subseteq> {[1], [- 1]}"
coeff_0_degree_minus_1: fixes rrr :: "'b poly"   and dr :: "nat" assumes "coeff rrr dr = (0::'b)"   and "degree rrr \<le> dr" shows "degree rrr \<le> dr - 1"
set_append: fixes xs :: "'g list"   and ys :: "'g list" shows "set (xs @ ys) = set xs \<union> set ys"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>l x r. \<lbrakk>P l; \<And>xa. P r\<rbrakk> \<Longrightarrow> P \<langle>l, x, r\<rangle>" shows "P a0"
pi_float_interval: fixes prec :: "nat" shows "pi \<in>\<^sub>r pi_float_interval prec"
prefix_dropWhile: fixes xs :: "'c list"   and ys :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "prefix xs ys" shows "prefix (dropWhile P xs) (dropWhile P ys)"
n_val_equiv_val: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v_equiv K v (n_val K v)"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "trg\<^sub>C f = C.null"
seq_pr_tuple: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.dom C f = partial_magma.dom C g" shows "partial_magma.arr C (C (pr0 (partial_magma.cod C f) (partial_magma.cod C g)) (elementary_category_with_binary_products.tuple C pr0 pr1 f g))"
comp_def: fixes P :: "'b \<times> 'd \<Rightarrow> bool"   and Q :: "'d \<times> 'c \<Rightarrow> bool" shows "P ;; Q \<equiv> \<lambda>r. r \<in> {p. P p} O {q. Q q}"
subcocycle_bounded_by_birkhoff1: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and x :: "'a" assumes "subcocycle u"   and "0 < n" shows "u n x \<le> birkhoff_sum (u 1) n x"
assms_1: shows "w\<^sub>1 =\<^sub>F w\<^sub>2"
num_ivl_def: shows "num_ivl \<equiv> map_fun id abs_ivl (\<lambda>i. (Fin i, Fin i))"
nullstellensatz_univariate: fixes p :: "complex poly"   and q :: "complex poly" shows "(\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0) = (p dvd q ^ degree p \<or> p = 0 \<and> q = 0)"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>) UP"
sdrop_snth: fixes n :: "nat"   and s :: "'a stream"   and m :: "nat" shows "sdrop n s !! m = s !! (n + m)"
card_roots_unity_eq: fixes n :: "nat" assumes "0 < n" shows "card {z. z ^ n = 1} = n"
r_cancel_one: fixes x :: "'a \<Rightarrow> 'b"   and a :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)"   and "a \<in> carrier (DirProds G I)" shows "(a \<otimes>\<^bsub>DirProds G I\<^esub> x = x) = (a = \<one>\<^bsub>DirProds G I\<^esub>)"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> (b \<squnion> c) = a \<cdot> b \<squnion> a \<cdot> c"
induct: fixes P :: "('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'c rule list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'c rule list" assumes "\<And>uu_ uv_. P uu_ uv_ []"   and "\<And>\<gamma> p m uw_ rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule m (Goto uw_) # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Accept # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Drop # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Log # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Reject # rs)"   and "\<And>\<gamma> p v vb rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v (Call vb) # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Return # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Empty # rs)"   and "\<And>\<gamma> p v rs. P \<gamma> p rs \<Longrightarrow> P \<gamma> p (Rule v Unknown # rs)" shows "P a0 a1 a2"
wf_digraph_axioms: shows "wf_digraph (Digraph_Component.union G H)"
LeftDerivation_append: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and c :: "symbol list" shows "LeftDerivation a (D @ E) c = (\<exists>b. LeftDerivation a D b \<and> LeftDerivation b E c)"
axioms_2: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e" assumes "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
image_inv_f_f: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj f" shows "inv f ` f ` A = A"
2: shows "bprv (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>\<rangle>))"
exI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "P x" shows "\<exists>x. P x"
M_rel_iI: fixes f1 :: "'i"   and f2 :: "'a mod_ring"   and f3 :: "int" assumes "R f1 f2"   and "M_Rel f3 f2" shows "M_rel_i f1 f3"
d_indistinguishable_sym: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and e :: "'a"   and e' :: "'a" assumes "Strongly_Secure_Programs.d_indistinguishable E DA d e e'" shows "Strongly_Secure_Programs.d_indistinguishable E DA d e' e"
less_eq_llist_def: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(xs \<le> ys) = lprefix xs ys"
bind_has_subprob_density': fixes M :: "'a measure"   and N :: "'a measure"   and \<delta>M :: "'a \<Rightarrow> ennreal"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b measure"   and \<delta>f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "has_subprob_density M N \<delta>M"   and "space R \<noteq> {}"   and "sigma_finite_measure R"   and "\<And>x. x \<in> space M \<Longrightarrow> has_subprob_density (f x) R (\<delta>f x)"   and "(\<lambda>(x, y). \<delta>f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M R)"   and "f \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "has_subprob_density (M \<bind> f) R (\<lambda>y. \<integral>\<^sup>+ x. \<delta>M x * \<delta>f x y \<partial>N)"
simps: fixes A :: "('b \<times> nat) tree"   and a :: "'b"   and B :: "('b \<times> nat) tree"   and b :: "'b"   and uu :: "nat"   and C :: "('b \<times> nat) tree" shows "rotateL A a \<langle>B, (b, uu), C\<rangle> = (if single B C then rot1L A a B b C else rot2 A a B b C)"
p_star_elim: fixes x :: "'a list"   and X :: "'a list set" shows "(x \<in> p_star X) = (\<exists>n. x \<in> path_dioid.power X n)"
slice_def: fixes "from" :: "nat"   and to :: "nat"   and list :: "'c list" shows "Misc.slice from to list = take (to - from) (drop from list)"
is_natural_2: fixes f :: "'b" assumes "arr f" shows "\<ll> (cod f) \<cdot> L f = \<ll> f"
starset_def: fixes A :: "'c set" shows "*s* A = Iset (star_of A)"
indis_refl: fixes s :: "'state" shows "s \<approx> s"
ok: shows "\<turnstile> (\<Delta> @ TVarB X Q # \<Gamma>) ok"
predicate2D_vimage2p: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "R \<le> vimage2p f g S"   and "R x y" shows "S (f x) (g y)"
totatives_code: fixes n :: "nat" shows "totatives n = Set.filter (\<lambda>k. coprime k n) {0<..n}"
rev_iterateoi_def: fixes list_rev_it :: "'d \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list"   and S :: "'d" shows "poly_map_rev_iterateoi_defs.rev_iterateoi list_rev_it S \<equiv> it_to_it (list_rev_it S)"
representsE_full: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'a)" and "degree p \<noteq> 0"
set_6: fixes x2 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>t\<^sub>p (receive\<langle>x2\<rangle>\<^sub>s\<^sub>t) = fv x2"
supp_abs_residual_pair: fixes \<alpha> :: "'b"   and P :: "'c" shows "supp \<langle>\<alpha>,P\<rangle> = supp (\<alpha>, P) - bn \<alpha>"
ide_char': fixes a :: "'b" shows "D.ide a = (D.arr a \<and> (D.dom a = a \<or> D.cod a = a))"
simps_3: fixes n :: "nat"   and x :: "nat"   and xs :: "nat list" shows "tuple_encode (Suc n) (x # xs) = prod_encode (x, tuple_encode n xs)"
x_2: shows "(f b - f a) \<bullet> f b - (f b - f a) \<bullet> f a = (f b - f a) \<bullet> f' x (b - a)"
F1set3_image_Lev1: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xc :: "'d"   and xd :: "bd_type_F"   and xa :: "'d" assumes "xe \<in> Lev1 s1 s2 n xc"   and "xe @ [Inr xd] \<in> Lev1 s1 s2 (Suc n) xc"   and "rv1 s1 s2 xe xc = Inl xa" shows "xd \<in> tobd_F13 s1 xa ` F1set3 (s1 xa)"
in_enum_image: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and A :: "nat set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "A \<subseteq> {..n}"   and "i \<le> n" shows "(kuhn_simplex.enum base upd i \<in> kuhn_simplex.enum base upd ` A) = (i \<in> A)"
eval_\<pi>: shows "\<pi> = (\<lambda>x. \<Sum>p\<leftarrow>map fst (filter snd (List.enumerate 2 (sieve 1 (replicate (nat \<lfloor>x\<rfloor> - 1) True)))). 1)"
fmrestrict_set_null: fixes m :: "('c, 'd) fmap" shows "fmrestrict_set {} m = fmempty"
orderE: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" assumes "less_eq_prod b a" obtains "a = sup_prod a b"
rep_eq: fixes x :: "nat"   and xa :: "'a pdevs" shows "pdevs_apply (trunc_err_pdevs x xa) = trunc_err_pdevs_raw x (pdevs_apply xa)"
simps_2: fixes rs :: "'rule stream"   and s :: "'state" shows "cont (mkTree rs s) = mkTree (stl (trim rs s)) |`| pickEff (shd (trim rs s)) s"
endomorph: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'a" assumes "GroupEnd G T" shows "T ` G \<subseteq> G"
bij_1: shows "bij_betw f\<^sub>1 (uverts G\<^sub>1) (uverts G\<^sub>2)"
Nats_less_HNatInfinite: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> \<nat>"   and "y \<in> HNatInfinite" shows "x < y"