hcju/isabelleps huggingface dataset:
read_tmps_append: fixes xs :: "'a memref list"   and ys :: "'a memref list" shows "read_tmps (xs @ ys) = read_tmps xs \<union> read_tmps ys"
SUP_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
listset_closed_map: fixes ys :: "'a list"   and xs :: "'a set list"   and f :: "'a \<Rightarrow> 'a" assumes "ys \<in> listset xs"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> x\<rbrakk> \<Longrightarrow> f y \<in> x" shows "map f ys \<in> listset xs"
minus_carrier_vec: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 - v\<^sub>2 \<in> carrier_vec n"
float_neg_numeral: fixes x :: "num" shows "real_of_float (- numeral x) = - numeral x"
that: fixes h :: "real \<Rightarrow> 'a" assumes "path h"   and "pathstart h = pathstart g"   and "path_image h \<subseteq> path_image g \<inter> S"   and "pathfinish h \<in> frontier S" shows "thesis"
insert_3: assumes "f \<noteq> {}"   and "\<forall>s\<in>f. \<exists>p. p division_of s" shows "\<exists>p. p division_of \<Inter> f"
not_simple_if_snd_fun: fixes S :: "('c, 'd) strand_step list" assumes "\<exists>S' S'' f X. S = S' @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S''" shows "\<not> simple S"
M_0: fixes m :: "int" shows "poly_mod.M m 0 = 0"
is_strong_nat_string_def: fixes s :: "char list" shows "is_strong_nat_string s \<equiv> is_nat_string s \<and> s \<noteq> [] \<and> (hd s = CHR ''0'' \<longrightarrow> length s = 1)"
sound_FEx: fixes s :: "form list"   and f :: "form"   and u :: "nat" assumes "Svalid (s @ [finst f u, FEx f])" shows "Svalid (FEx f # s)"
logical_result_def: fixes instr_name :: "sparc_operation"   and rs1_val :: "word32"   and operand2 :: "word32" shows "logical_result instr_name rs1_val operand2 \<equiv> if instr_name = logic_type ANDs \<or> instr_name = logic_type ANDcc then rs1_val AND operand2 else if instr_name = logic_type ANDN \<or> instr_name = logic_type ANDNcc then rs1_val AND NOT operand2 else if instr_name = logic_type ORs \<or> instr_name = logic_type ORcc then rs1_val OR operand2 else if instr_name \<in> {logic_type ORN, logic_type ORNcc} then rs1_val OR NOT operand2 else if instr_name \<in> {logic_type XORs, logic_type XORcc} then rs1_val XOR operand2 else rs1_val XOR NOT operand2"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
subst_fvs: fixes M :: "'a trm"   and z :: "'a"   and N :: "'a trm" shows "fvs M[z ::= N] \<subseteq> fvs M - {z} \<union> fvs N"
R1: fixes u :: "'a set" assumes "u \<in> nhds x"   and "y \<notin> u" shows "R"
alignUp_ge: fixes n :: "nat"   and a :: "'b word" assumes "n < LENGTH('b)"   and "alignUp a n \<noteq> 0" shows "a \<le> alignUp a n"
order_trans: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
cycle_free_upto_diag_equiv: fixes k :: "nat"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "k \<le> n" shows "cycle_free_up_to m k n = (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs)"
normalizability_I: fixes a :: "'d"   and b :: "'d"   and A :: "('d \<times> 'd) set" assumes "(a, b) \<in> A\<^sup>*"   and "b \<in> NF A" shows "(a, b) \<in> A\<^sup>!"
simps_2: fixes x31 :: "'b local.term"   and x32 :: "'b local.term"   and y31 :: "'b local.term"   and y32 :: "'b local.term" shows "(x31 \<^bold>\<otimes> x32 = y31 \<^bold>\<otimes> y32) = (x31 = y31 \<and> x32 = y32)"
MaxR_opt_eq_fold': fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "MaxR r" shows "MaxR.MaxR_opt r A = Finite_Set.fold (MaxR.MaxR_f r) None A"
S_transform_formula_hereditarily_fs: fixes x :: "('a, 'c, 'b) formula" shows "hereditarily_fs (S_transform_formula x)"
vfiniteI: fixes A :: "V" assumes "\<exists>n\<in>elts \<omega>. n \<approx>\<^sub>\<circ> A" shows "vfinite A"
B: fixes y :: "'b process" assumes "cont f" shows "cont (\<lambda>x. y [+] f x)"
DBM_and_complete: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M1"   and "u \<turnstile>\<^bsub>v,n\<^esub> M2" shows "u \<turnstile>\<^bsub>v,n\<^esub> And M1 M2"
simps_5: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "ttl (unfold_tllist p g1 g21 g22 a) = unfold_tllist p g1 g21 g22 (g22 a)"
t_idem: fixes x :: "('a \<times> 'a) set" shows "rel_kat.t_op x ; rel_kat.t_op x = rel_kat.t_op x"
forest_maximally_connected_incl_max2: fixes G :: "('a, 'b) graph"   and H :: "('a, 'b) graph" assumes "valid_graph G"   and "forest H"   and "subgraph H G"   and "maximally_connected H G" shows "\<forall>(a, w, b)\<in>edges G - edges H. \<not> forest (add_edge a w b H)"
in_set_product_lists_length: fixes xs :: "'b list"   and xss :: "'b list list" assumes "xs \<in> set (product_lists xss)" shows "length xs = length xss"
assms_1: shows "Complete_Partial_Order.chain (\<le>) C"
cobounded2: fixes b :: "'f"   and a :: "'f" shows "b \<le> a \<squnion> b"
obtuse_per__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Obtuse A B C"   and "D \<noteq> E"   and "E \<noteq> F"   and "Per D E F" shows "D E F LtA A B C"
assms_2: shows "Domainp A = (\<lambda>x. x \<in> \<UU>A)"
Step: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and tl :: "'b"   and s' :: "'a"   and tls :: "'b llist" assumes "trsys s tl s'"   and "trsys.Runs trsys s' tls" shows "trsys.Runs trsys s (LCons tl tls)"
obj_fields_marked_locs_subseteq_hp_IdleMarkSweep_locs: shows "obj_fields_marked_locs \<subseteq> hp_IdleMarkSweep_locs"
indep_in_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>' \<subseteq> \<E>" shows "indep_system.indep_in (\<E>.indep_in \<E>) \<E>' X = \<E>.indep_in \<E>' X"
basis_reduction_swap_def: fixes i :: "nat"   and fs :: "'a list" shows "basis_reduction_swap i fs = (False, i - 1, fs[i := fs ! (i - 1), i - 1 := fs ! i])"
Invariants_is_subalg: shows "subalgebra M Invariants"
mix_pmf_comp_with_dif_equiv: fixes \<alpha> :: "real"   and \<beta> :: "real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> < \<alpha>" shows "mix_pmf (\<beta> / \<alpha>) (mix_pmf \<alpha> p q) q = mix_pmf \<beta> p q"
test_kat_1': fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(p \<cdot> x \<cdot> !q = (0::'a)) = (p \<cdot> x = p \<cdot> x \<cdot> q)"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<zero> [^] n = \<zero>"
oSuc_plus_ordinal_of_nat: fixes x :: "ordinal"   and n :: "nat" shows "oSuc x + ordinal_of_nat n = oSuc (x + ordinal_of_nat n)"
big_small_trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
length_update: fixes b :: "'b array"   and i :: "nat"   and v :: "'b"   and h :: "heap" shows "Array.length (Array.update b i v h) = Array.length h"
simps_1: fixes p :: "('a, 'b) form" shows "vars_for_consts p [] = p"
closure_idem: fixes T :: "'b set set"   and s :: "'b set" shows "closure\<^bsub>T\<^esub> (closure\<^bsub>T\<^esub> s) = closure\<^bsub>T\<^esub> s"
Nats_le_HNatInfinite: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> \<nat>"   and "y \<in> HNatInfinite" shows "x \<le> y"
inv_to_loop_freedom: fixes \<sigma> :: "nat \<Rightarrow> state" assumes "\<forall>i dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)" shows "\<forall>dip. irrefl ((rt_graph \<sigma> dip)\<^sup>+)"
numeral_uint16: fixes n :: "num" shows "numeral n = Uint16 (numeral n)"
set_of_variables_is_finite_eq: fixes e :: "'b equation" shows "finite (vars_of_eq e)"
LowSame: shows "low_proc_no_input_change s\<^sub>L"
lookup_pair_sort_oalist_aux: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ko :: "'a" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "comparator.lookup_pair (key_compare (rep_key_order ko)) (oalist_raw.sort_oalist_aux rep_key_order ko xs) = oalist_raw.lookup_raw rep_key_order xs"
qring_ring: fixes I :: "'a set" assumes "ideal R I" shows "Ring (R /\<^sub>r I)"
bis_image2: fixes Aa :: "'d set"   and A :: "'e set"   and s1'a :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2'a :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1 :: "'g set"   and B2 :: "'h set"   and s1 :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2 :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and fa :: "'d \<Rightarrow> 'g"   and f :: "'e \<Rightarrow> 'h"   and B1' :: "'i set"   and B2' :: "'j set"   and s1' :: "'i \<Rightarrow> ('f, 'i, 'j) F1"   and s2' :: "'j \<Rightarrow> ('f, 'i, 'j) F2"   and ga :: "'d \<Rightarrow> 'i"   and g :: "'e \<Rightarrow> 'j" assumes "coalg Aa A s1'a s2'a"   and "mor Aa A s1'a s2'a B1 B2 s1 s2 fa f"   and "coalg Aa A s1'a s2'a"   and "mor Aa A s1'a s2'a B1' B2' s1' s2' ga g" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' (BNF_Greatest_Fixpoint.image2 Aa fa ga) (BNF_Greatest_Fixpoint.image2 A f g)"
monomial_uminus: fixes s :: "'a"   and c :: "'b" shows "- monomial c s = monomial (- c) s"
rcosetsI: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>add_monoid G\<^esub> x \<in> rcosets\<^bsub>add_monoid G\<^esub> H"
chain_boundary_def: fixes p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p c \<equiv> if p = 0 then 0 else frag_extend (\<lambda>f. \<Sum>k\<le>p. frag_cmul ((- 1) ^ k) (frag_of (singular_face p k f))) c"
simps_10: fixes f1 :: "'a"   and f2 :: "num \<Rightarrow> 'a"   and f3 :: "num \<Rightarrow> 'a"   and x2 :: "num" shows "(case num.Bit0 x2 of num.One \<Rightarrow> f1 | num.Bit0 x \<Rightarrow> f2 x | num.Bit1 x \<Rightarrow> f3 x) = f2 x2"
take_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list"   and y :: "'g" assumes "n \<le> m" shows "take n (xs[m := y]) = take n xs"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "min a b \<le> a"
isolate_zero: fixes var :: "nat"   and n :: "nat" shows "isolate_variable_sparse 0 var n = 0"
closed_complex_atMost: fixes a :: "complex" shows "closed {..a}"
subsumes_def: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" shows "(t <\<cdot> s) = (t \<le>\<cdot> s \<and> \<not> s \<le>\<cdot> t)"
mopup_bef_erase_b_Bk_via_a_Oc: fixes s :: "nat"   and l :: "cell list"   and xs :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" assumes "mopup_bef_erase_a (s, l, Oc # xs) lm n ires" shows "mopup_bef_erase_b (Suc s, l, Bk # xs) lm n ires"
AE_T_reachable: fixes s :: "'s" shows "almost_everywhere (T s) (alw (HLD (acc `` {s})))"
lossless_\<Sigma>_S_2: fixes x0 :: "'g"   and e0 :: "'e" shows "lossless_spmf (\<Sigma>0.S x0 e0)"
mat_hom_pow: fixes hom :: "'a \<Rightarrow> 'b"   and A :: "'a mat"   and n :: "nat"   and k :: "nat" assumes "semiring_hom hom"   and "A \<in> carrier_mat n n" shows "map_mat hom (A ^\<^sub>m k) = map_mat hom A ^\<^sub>m k"
elims_2: fixes x :: "'b \<times> 'c"   and xa :: "'b \<times> 'c" assumes "Psi.subset x xa" obtains l :: "'b" and u :: "'c" and l' :: "'b" and u' :: "'c" where "x = (l, u)" and "xa = (l', u')" and "l' \<le> l \<and> u \<le> u'"
isCont_def: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f a = f \<midarrow>a\<rightarrow> f a"
full_abstraction_wrt_preorders_impl_indRelSTEQ_to_SRel_and_TRel_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "trans TRel"   and "refl SRel"   and "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q" shows "\<forall>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<longrightarrow> (SP, SQ) \<in> SRel"
assms_2: fixes x :: "'a" assumes "\<Phi> x" shows "f x \<le> M"
generator_pow_mult_order: shows "\<^bold>g [^] (order G * order G) = \<one>"
contour_integrable_inversediff: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path g"   and "z \<notin> path_image g" shows "(\<lambda>w. 1 / (w - z)) contour_integrable_on g"
simps_8: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<rbrace> = \<rr>'.map \<lbrace>t\<rbrace>"
pratt_sound: fixes c :: "pratt list"   and t :: "pratt"   and p :: "nat"   and a :: "nat"   and x :: "nat" assumes "valid_cert c"   and "t \<in> set c" shows "(t = Prime p \<longrightarrow> prime p) \<and> (t = Triple p a x \<longrightarrow> (\<forall>q\<in>#prime_factorization x. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)) \<and> 0 < x)"
f_step: shows "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"
list_product_def: fixes A :: "'b list set"   and B :: "'b list set" shows "A \<otimes>\<^sub>l B = {a @ b |a b. a \<in> A \<and> b \<in> B}"
maximal_mem: fixes m :: "'a" assumes "maximal m" shows "m \<in> carrier D"
is_RB_in_def: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" shows "is_RB_in d rword G u = ((\<exists>g. is_canon_rewriter rword G u g \<and> \<not> is_sig_red (\<prec>\<^sub>t) (=) G (monom_mult (1::'b) (pp_of_term u - lp g) g)) \<or> is_syz_sig d u)"
rep_eq: fixes x :: "('a, 'b) RBT.rbt" shows "entries x = RBT_Impl.entries (impl_of x)"
denseI: fixes A :: "'a llist set" assumes "carrier \<subseteq> closure A" shows "A dense"
elims_2: fixes x :: "('a, 'b, 'c) SLin_state \<times> ('a, 'e, 'f, 'g) SLin_state_scheme" assumes "P8 x" obtains s1 :: "('a, 'b, 'c) SLin_state" and s2 :: "('a, 'e, 'f, 'g) SLin_state_scheme" where "x = (s1, s2)" and "\<forall>iv\<in>initVals s2. \<exists>rs\<in>pendingSeqs s1. iv = dstate s1 \<star> rs"
col_sylvester_sub: fixes j :: "nat"   and m :: "nat"   and n :: "nat"   and p :: "'b poly"   and q :: "'b poly" assumes "j < m + n" shows "col (sylvester_mat_sub m n p q) j = Missing_Lemmas.vec_of_poly_rev_shifted p n m j @\<^sub>v Missing_Lemmas.vec_of_poly_rev_shifted q m n j"
that: fixes r :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "r \<in> set gs \<union> set bs \<union> set hs"   and "fst r \<noteq> 0"   and "r \<noteq> p"   and "r \<noteq> q"   and "lt (fst r) adds\<^sub>t term_of_pair (lcs (lp (fst p)) (lp (fst q)), component_of_term (lt (fst p)))"   and "(p, r) \<in>\<^sub>p snd ` set ps"   and "r \<in> set gs \<union> set bs \<and> q_in_bs \<or> (q, r) \<in>\<^sub>p snd ` set ps" shows "thesis"
Plus_le_iff: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "(Plus r s \<le> t) = (r \<le> t \<and> s \<le> t)"
obtuse_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Obtuse A B C" shows "Obtuse C B A"
underS_incl_iff: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(underS r a \<subseteq> underS r b) = ((a, b) \<in> r)"
raw_frees_all_frees: fixes t :: "'a" shows "abs_pred (\<lambda>t. frees t |\<subseteq>| all_frees t) t"
less_numeral_extra_4: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
seq: shows "\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (a # S) ioprog) P"
rel_interior_openin: fixes s :: "'a set" assumes "openin (top_of_set (affine hull s)) s" shows "rel_interior s = s"
to_bl_nth: fixes n :: "nat"   and w :: "'b word" assumes "n < size w" shows "to_bl w ! n = w !! (size w - Suc n)"
simps_1: fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" shows "insert xs e None = Some (insert_body xs e)"
maximal_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "maximal t \<equiv> \<forall>c\<ge>0. t (\<lambda>_. c) = (\<lambda>_. c)"
matrix_add_commute: fixes A :: "('c, 'd, 'e) matrix"   and B :: "('c, 'd, 'e) matrix" shows "A + B = B + A"
spmf_map_pmf_Some: fixes p :: "'c pmf"   and i :: "'c" shows "spmf (map_pmf Some p) i = pmf p i"
finite_accessible_valuations: fixes s :: "('i, 'a) state" shows "finite (\<V> ` {s'. (s, s') \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*})"
convergesI: fixes x :: "'a"   and F :: "'a set set" assumes "nhds x \<subseteq> F" shows "F \<longlongrightarrow> x"
msubsteq2_def: fixes c :: "poly"   and t :: "tm"   and a :: "poly"   and b :: "tm" shows "msubsteq2 c t a b = Eq (tm.Add (tm.Mul a t) (tm.Mul c b))"
monad_state_prob_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and sample :: "'e pmf \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_state_prob return bind get put sample" shows "monad_state_prob (return_writer return) (bind_writer return bind) (get_writer get) (put_writer put) (sample_writer sample)"
steps_z_norm_complete': fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and Z :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>* \<langle>l', Z\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "\<exists>D'. A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle> \<and> Z \<subseteq> [D']\<^bsub>v,n\<^esub>"
subring_is_domain: fixes H :: "'a set" assumes "subring H R" shows "domain (R\<lparr>carrier := H\<rparr>)"
lprefix_llength_le: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys" shows "llength xs \<le> llength ys"
tag_name_eq_h2: fixes ptr' :: "(_) element_ptr"   and tag :: "char list" shows "h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag"
all_pairs_def: fixes xs :: "'b list" shows "all_pairs xs \<equiv> concat (map (\<lambda>x. map (Pair x) xs) xs)"
map_poly_smult: fixes f :: "'b \<Rightarrow> 'a"   and c :: "'b"   and p :: "'b poly" assumes "f (0::'b) = (0::'a)"   and "\<And>c x. f (c * x) = f c * f x" shows "map_poly f (smult c p) = smult (f c) (map_poly f p)"
simps_3: fixes uv :: "real" shows "evalUni FalseFUni uv = False"
a: shows "list_all (\<lambda>op'. list_all (are_operator_effects_consistent op') (op # ops)) (op # ops)"
Max: shows "Max_wrt_among le (\<Union> (set (A # As))) = A"
trivial_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and inEdges' :: "'b \<Rightarrow> 'c \<Rightarrow> ('c \<times> 'd) list"   and defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'e set"   and "uses" :: "'b \<Rightarrow> 'c \<Rightarrow> 'e set"   and phis :: "'b \<Rightarrow> 'c \<times> 'e \<Rightarrow> 'e list option"   and g :: "'b"   and v :: "'e" shows "CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g v \<equiv> \<exists>v'\<in>CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g. CFG_SSA_wf_base.isTrivialPhi \<alpha>n defs phis g v v'"
tail_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "tail G (edge_rev M a) = head G a"
exhaust_sel: fixes option :: "'a option" obtains "option = \<bottom>"   | "option = \<lfloor>the option\<rfloor>"
IH_1: assumes "poly p x\<^sub>0 \<noteq> 0"   and "poly q x\<^sub>0 = 0"   and "poly (hd (r # ps)) x\<^sub>0 \<noteq> 0" shows "sign_changes' (r # ps) x\<^sub>0 = sign_changes (r # ps) x\<^sub>0"
C: fixes x :: "real"   and y :: "real" assumes "C \<le> x"   and "x \<le> y" shows "f x \<le> f y"
sglob_ind_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "sglob_ind s i j = inst.globs (inst s ! i) ! j"
Spec: shows "\<lparr>c_sinvar = \<lambda>G. sinvar_spec G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec G nP, c_isIFS = nm_receiver_violation m\<rparr> = Spec"
deltafun_apply_neq: fixes x :: "'a"   and a :: "'a"   and b :: "'b" assumes "x \<noteq> a" shows "(b \<delta> a) x = (0::'b)"
assms_1: shows "(sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s"
sorted_list_of_set_mono_on: fixes A :: "'b set" assumes "finite A" shows "mono_on ((!) (sorted_list_of_set A)) {..<card A}"
with_optimal_paths_solves: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
lemma3: fixes r :: "lam"   and r' :: "lam"   and s :: "lam"   and s' :: "lam"   and x :: "name" assumes "r \<rightarrow>\<^sub>s r'"   and "s \<rightarrow>\<^sub>s s'" shows "r[x::=s] \<rightarrow>\<^sub>s r'[x::=s']"
finite_polyhedron_faces: fixes S :: "'b set" assumes "polyhedron S" shows "finite {F. F face_of S}"
SUP_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (f i) (g j)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (g j) (f i)" shows "Sup (f ` A) = Sup (g ` B)"
finter_left_commute: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)"