hcju/isabelleps huggingface dataset:
game1''_def: shows "game1'' = key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (oracle_encrypt1'' key b \<oplus>\<^sub>O oracle_decrypt2 key) \<A> () \<bind> (\<lambda>(b', D). return_spmf (b = b'))))"
In1_def: fixes M :: "('c, 'd) Old_Datatype.node set" shows "In1 M \<equiv> Scons (Numb 1) M"
snd_vimage_eq_Times: fixes S :: "'d set" shows "snd -` S = UNIV \<times> S"
order_refl: fixes x :: "'a \<times> 'b" shows "less_eq_prod x x"
subgraph_refl: fixes G :: "('e, 'f) pre_digraph" assumes "wf_digraph G" shows "subgraph G G"
vbrelation_vintersection_vdomain: fixes s :: "V" assumes "vdisjnt (\<D>\<^sub>\<circ> r) (\<D>\<^sub>\<circ> s)" shows "vdisjnt r s"
map_upd_eq_restrict: fixes m :: "'a \<Rightarrow> 'b option"   and x :: "'a" shows "m(x := None) = m |` (- {x})"
CompDefinedE: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "f \<approx>>\<^bsub>C\<^esub> g" obtains "f \<in> mor\<^bsub>C\<^esub>" and "g \<in> mor\<^bsub>C\<^esub>" and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g"
ass_function_0': fixes ass :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "ass_function ass" shows "(ass x div x = (0::'a)) = (x = (0::'a))"
cos_minus1_imp_pi: fixes x :: "real" assumes "cos x = - 1"   and "0 \<le> x"   and "x < 3 * pi" shows "x = pi"
noteq_def: fixes r :: "'b array"   and s :: "'c array" shows "r =!!= s = (TYPEREP('b) \<noteq> TYPEREP('c) \<or> addr_of_array r \<noteq> addr_of_array s)"
arr_hcomp: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "hseq \<mu> \<nu>"
non_empty_ith_1: fixes i :: "nat" shows "ith (eq_upto (BAD P) lb i) i \<subseteq> A"
assms: fixes s :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "\<not> cond s" shows "P s"
le_infI: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "\<And>v\<^sub>1. v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 \<Longrightarrow> traces.le_fininf ind v\<^sub>1 w\<^sub>2" shows "traces.le_inf ind w\<^sub>1 w\<^sub>2"
expand_fps_eq: fixes p :: "'b fps"   and q :: "'b fps" shows "(p = q) = (\<forall>n. p $ n = q $ n)"
image_mult_atLeastAtMost: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero d" shows "times d ` ord.atLeastAtMost less_eq a b = ord.atLeastAtMost less_eq (times d a) (times d b)"
check_bounds_id: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<B>\<^sub>i (check s) = \<B>\<^sub>i s"
from_to_Gromov_completion: fixes x :: "'b" shows "from_Gromov_completion (to_Gromov_completion x) = x"
srev_scons: fixes x :: "'a"   and xs :: "[:'a:]" shows "srev\<cdot>(x :# xs) = srev\<cdot>xs :@ [:x:]"
preserves_arr: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.arr (FG.map f)"
induct: fixes P :: "('q \<Rightarrow> 'r fundef option) \<Rightarrow> ('s, 't, 'q, 'e, 'f, 'g, 'u, 'v) instr \<Rightarrow> Unboxed.type option list \<Rightarrow> bool"   and a0 :: "'q \<Rightarrow> 'r fundef option"   and a1 :: "('s, 't, 'q, 'e, 'f, 'g, 'u, 'v) instr"   and a2 :: "Unboxed.type option list" assumes "\<And>uu_ uv_ \<Sigma>. P uu_ (IPush uv_) \<Sigma>"   and "\<And>uw_ ux_ \<Sigma>. P uw_ (IPushUbx1 ux_) \<Sigma>"   and "\<And>uy_ uz_ \<Sigma>. P uy_ (IPushUbx2 uz_) \<Sigma>"   and "\<And>va_ vb_ \<Sigma>. P va_ IPop (vb_ # \<Sigma>)"   and "\<And>vc_ vd_ \<Sigma>. P vc_ (ILoad vd_) (None # \<Sigma>)"   and "\<And>ve_ \<tau> vf_ \<Sigma>. P ve_ (ILoadUbx \<tau> vf_) (None # \<Sigma>)"   and "\<And>vg_ vh_ \<Sigma>. P vg_ (IStore vh_) (None # None # \<Sigma>)"   and "\<And>vi_ \<tau>\<^sub>1 vj_ \<tau>\<^sub>2 \<Sigma>. P vi_ (IStoreUbx \<tau>\<^sub>1 vj_) (None # Some \<tau>\<^sub>2 # \<Sigma>)"   and "\<And>vk_ op \<Sigma>. P vk_ (IOp op) \<Sigma>"   and "\<And>vl_ opinl \<Sigma>. P vl_ (IOpInl opinl) \<Sigma>"   and "\<And>vm_ opubx \<Sigma>. P vm_ (IOpUbx opubx) \<Sigma>"   and "\<And>F f \<Sigma>. P F (ICall f) \<Sigma>"   and "\<And>vn_. P vn_ IPop []"   and "\<And>vn_ v. P vn_ (ILoad v) []"   and "\<And>vn_ v vc vb. P vn_ (ILoad v) (Some vc # vb)"   and "\<And>vn_ v va. P vn_ (ILoadUbx v va) []"   and "\<And>vn_ v va vd vc. P vn_ (ILoadUbx v va) (Some vd # vc)"   and "\<And>vn_ v. P vn_ (IStore v) []"   and "\<And>vn_ v vc vb. P vn_ (IStore v) (Some vc # vb)"   and "\<And>vn_ v va. P vn_ (IStore v) [va]"   and "\<And>vn_ v va ve vd. P vn_ (IStore v) (va # Some ve # vd)"   and "\<And>vn_ v va. P vn_ (IStoreUbx v va) []"   and "\<And>vn_ v va vd vc. P vn_ (IStoreUbx v va) (Some vd # vc)"   and "\<And>vn_ v va vb. P vn_ (IStoreUbx v va) [vb]"   and "\<And>vn_ v va vb ve. P vn_ (IStoreUbx v va) (vb # None # ve)"   and "\<And>vn_ v vp_. P vn_ (ICJump v) vp_" shows "P a0 a1 a2"
iLastWeak_False: fixes t0 :: "nat"   and I :: "nat set" shows "(\<not> (\<ominus>\<^sub>W t t0 I. False)) = (I \<down>< t0 \<noteq> {} \<and> t0 \<in> I)"
run_altc_sample_writer: fixes altc_sample :: "'b \<Rightarrow> ('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'c \<Rightarrow> ('w, 'a, 'm) writerT" shows "run_writer (local.altc_sample_writer altc_sample p f) = altc_sample p (\<lambda>p. run_writer (f p))"
main: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c" shows "Q A (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update\<B>\<I> (UBI_upd dir) i x c s)"
compact_lemma_cart: fixes f :: "nat \<Rightarrow> ('a, 'b) vec"   and d :: "'b set" assumes "bounded (range f)" shows "\<exists>l r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e)"
singleton_iff: fixes b :: "'d"   and a :: "'d" shows "(b \<in> {a}) = (b = a)"
NSCauchy_def: fixes X :: "nat \<Rightarrow> 'a" shows "NSCauchy X = (\<forall>M\<in>HNatInfinite. \<forall>N\<in>HNatInfinite. (*f* X) M \<approx> (*f* X) N)"
Uvariation_sym: fixes \<omega> :: "variable \<Rightarrow> real"   and \<nu> :: "variable \<Rightarrow> real"   and U :: "variable set" shows "Uvariation \<omega> \<nu> U = Uvariation \<nu> \<omega> U"
simps_128: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x7 :: "32 prefix_match" shows "(case IPv4Src x7 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f7 x7"
inject_1: fixes x1 :: "nonce_t"   and y1 :: "nonce_t" shows "(epublK x1 = epublK y1) = (x1 = y1)"
mult_assoc: fixes c :: "'a"   and b :: "'a"   and a :: "'a" shows "c \<cdot> (b \<cdot> a) = c \<cdot> b \<cdot> a"
less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "lt x y = (le x y \<and> \<not> le y x)"
Ex1_unfold: fixes P :: "'b \<Rightarrow> bool" shows "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
assms_19: shows "isUIP (opposite (getCl state)) (getC state) (getM state)"
proots_card_ball_plane_eq: fixes p :: "complex poly" assumes "p \<noteq> 0" shows "card (proots_within p (ball 0 1)) = card (proots_within (fcompose p [:\<i>, - 1:] [:\<i>, 1:]) {x. 0 < Im x})"
Cidm: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X"
ffold_finsert_idem: fixes z :: "'m"   and x :: "'m"   and A :: "'m fset" shows "ffold funionM z (finsert x A) = funionM x (ffold funionM z A)"
hyps_1: fixes ym :: "'c \<rightharpoonup>\<^sub>f 'b" assumes "fmrel R xm ym"   and "fmdom xm = fmdom ym" shows "P xm ym"
DDH_advantage_bound: fixes \<G> :: "'d cyclic_group"   and \<A> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool spmf" assumes "ddh_ext \<G>" shows "DH_Ext.ddh.DDH_advantage \<G> \<A> \<le> Diffie_Hellman.ddh.advantage \<G> \<A> + Diffie_Hellman.ddh.advantage \<G> (DH_Ext.ddh.DDH_\<A>' \<G> \<A>)"
range_constant: fixes x :: "'d" shows "range (\<lambda>_. x) = {x}"
\<rho>_def: shows "\<rho> = last (renamings_apart [mset D, mset C])"
interior_of_interior_of: fixes X :: "'d topology"   and S :: "'d set" shows "X interior_of X interior_of S = X interior_of S"
intface\<M>: fixes C1 :: "'a set"   and C2 :: "'a set" assumes "C1 \<in> \<M>"   and "C2 \<in> \<M>" shows "C1 \<inter> C2 face_of C1"
rev: shows "Cs' @ [C''] = rev Cs @ [C'']"
wf_Cons: fixes \<Gamma> :: "binding list"   and B :: "binding" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
simps: fixes cs :: "('i \<times> constraint) list"   and J :: "'i set"   and ds :: "'a ns_constraint list"   and s :: "'s" shows "invariant_cs cs J (ds, s) = (ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s)"
simps_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and l :: "'c btree"   and a :: "'c"   and r :: "'c btree" assumes "split split" shows "bal_up\<^sub>i (Up\<^sub>i l a r) = (height l = height r \<and> bal l \<and> bal r)"
strict_dominate_def: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat" assumes "cfg_doms G" shows "cfg_doms.strict_dominate G n1 n2 \<equiv> \<forall>pa. path_entry (g_E G) pa n2 \<longrightarrow> n1 \<in> set pa \<and> n1 \<noteq> n2"
sheaf_spec_extensional: fixes a :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" assumes "a \<in> \<O> U" shows "a \<in> extensional U"
replace_Goto_with_Call_in_terminal_chain: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and s :: "state"   and t :: "state" assumes "\<Gamma> chain = Some rs"   and "terminal_chain rs" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Goto chain)], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Call chain)], s\<rangle> \<Rightarrow> t"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'd"   and xa :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "blinfun_cblinfun_eq x xa = ((*\<^sub>v) x = (*\<^sub>V) xa)"
cInf_eq_minimum: fixes z :: "'a multiset"   and X :: "'a multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "\<Sqinter> X = z"
lcp_first_letters: fixes u :: "'b list"   and v :: "'b list" assumes "u ! 0 \<noteq> v ! 0" shows "u \<and>\<^sub>p v = \<epsilon>"
w_def: shows "w = P *\<^sub>v unit_vec n i"
F_def: shows "F \<equiv> \<lambda>ldropn xs. case xs of LNil \<Rightarrow> \<lambda>_. LNil | LCons x xs \<Rightarrow> \<lambda>n. if n = 0 then LCons x xs else ldropn xs (n - 1)"
elims: fixes x :: "('b \<times> 'c) tree"   and y :: "'b list" assumes "Tree2.inorder x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "('b \<times> 'c) tree" and a :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "y = Tree2.inorder l @ a # Tree2.inorder r"
norm_cos_squared: fixes z :: "complex" shows "(cmod (cos z))\<^sup>2 = (cos (Re z))\<^sup>2 + (exp (Im z) - inverse (exp (Im z)))\<^sup>2 / 4"
finite_measure_distr: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b"   and M' :: "'b measure" assumes "M'.random_variable M' f" shows "finite_measure (distr M'__ M' f)"
assms_1: fixes x :: "nat" shows "g x \<in> carrier R"
fdisjnt_subset_left: fixes N' :: "'b fset"   and N :: "'b fset"   and M :: "'b fset" assumes "N' |\<subseteq>| N"   and "fdisjnt N M" shows "fdisjnt N' M"
coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> a" shows "c \<le> a \<squnion> b"
inv\<^sub>1_def: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and n :: "nat"   and T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "LoadBalancing m" shows "LoadBalancing.inv\<^sub>1 t m n T A j = (LoadBalancing.lb t m T A j \<and> j \<le> n \<and> (\<forall>T' A'. LoadBalancing.lb t m T' A' j \<longrightarrow> f_Max\<^sub>0 T m \<le> 2 * f_Max\<^sub>0 T' m))"
sum_nonneg_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "(comm_monoid_add.sum plus zero f A = zero) = (\<forall>x\<in>A. f x = zero)"
lub_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" assumes "class.po below" shows "po.lub below S = (THE x. po.is_lub below S x)"
open_path_connected_component: fixes S :: "'b set"   and x :: "'b" assumes "open S" shows "path_component S x = connected_component S x"
summable_complex_iff: fixes f :: "nat \<Rightarrow> complex" shows "summable f = (summable (\<lambda>x. Re (f x)) \<and> summable (\<lambda>x. Im (f x)))"
h1: fixes x :: "real" shows "h (1, x) = g2 x"
toplevel_summands_ACI_norm_lderiv: fixes as :: "'a"   and r :: "'b rexp" shows "(\<Union>a\<in>toplevel_summands r. toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>r\<guillemotright>\<guillemotright>"
Ri_def: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" shows "standard_redundancy_criterion.Ri \<Gamma> N = {\<gamma> \<in> \<Gamma>. standard_redundancy_criterion.redundant_infer N \<gamma>}"
stails_sdrop': fixes xs :: "[:'r:]" shows "stails\<cdot>xs = smap\<cdot>(\<Lambda> i. sdrop\<cdot>i\<cdot>xs)\<cdot>(supto\<cdot>0\<cdot>(slength\<cdot>xs))"
mset_subset_eq_single: fixes a :: "'b"   and B :: "'b multiset" assumes "a \<in># B" shows "a, {#} \<subseteq># B"
cont_on_bot: shows "continuous_on (path_image (\<lambda>x. twoC (x, 0))) F"
DOM_dom: fixes f :: "'c local.term set" assumes "arr f" shows "DOM (local.dom f) = DOM f"
permutations_of_set_list_def: fixes xs :: "'a list" shows "permutations_of_set_list xs = permutations_of_set_aux_list [] xs"
assms_2: fixes x :: "'n"   and k :: "nat" assumes "x \<in> S" shows "f k x \<le> f (Suc k) x"
reduce_system_matrix_match: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "M_mat (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs))) (get_subsets (reduce_system p (qs, M_mat signs subsets, subsets, signs))) = get_matrix (reduce_system p (qs, M_mat signs subsets, subsets, signs))"
member_rec_1: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" shows "List.member (x # xs) y = (x = y \<or> List.member xs y)"
par_assoc: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "parallel (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (par (a::'b::refinement_lattice) (b::'b::refinement_lattice)) (c::'b::refinement_lattice) = par a (par b c)"
linorder_not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> x < y) = (y \<le> x)"
discriminant_zero: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "a \<noteq> 0"   and "discrim a b c = 0" shows "(a * x\<^sup>2 + b * x + c = 0) = (x = - b / (2 * a))"
length_tl: fixes xs :: "'c list" shows "length (tl xs) = degree xs"
in_FvarsT_rawpsubstT_imp: fixes y :: "'var"   and r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "y \<in> FvarsT (rawpsubstT r txs)"   and "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm" shows "y \<in> FvarsT r - snd ` set txs \<or> y \<in> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
all_zero: shows "\<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'b)"
Arccos_def: shows "Arccos \<equiv> \<lambda>z. - \<i> * Ln (z + \<i> * csqrt (1 - z\<^sup>2))"
Span_consistent: fixes K :: "'a set" assumes "subring K (L\<lparr>carrier := A\<rparr>)" shows "ring.Span (L\<lparr>carrier := A, carrier := K\<rparr>) = Hom.Span"
assms_2: shows "list_comb f xs = list_comb g ys"
guard_simps_1: fixes b :: "bool"   and x :: "unit" shows "(guard b = Some x) = b"
cov_invertible_nonneg_le: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and T :: "(real, 'a) vec set"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> real"   and b :: "real" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"   and "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h (g x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g (h y) = y"   and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g' (h y) = id" shows "(f integrable_on T \<and> integral T f \<le> b) = ((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) \<le> b)"
L_trans_2: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "l' \<in> L"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = \<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)"
sum_nonneg_eq_0_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "(comm_monoid_add.sum (+) {#} f A = {#}) = (\<forall>x\<in>A. f x = {#})"
wb_lens_weak: fixes x :: "'b \<Longrightarrow> 'c" assumes "wb_lens x" shows "weak_lens x"
abs_mult_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (abs a) c"   and "less (abs b) d" shows "less (times (abs a) (abs b)) (times c d)"
pdt_inv_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" assumes "IFC prob" shows "IFC.pdt_inv prob = (IFC_def.pdt prob)\<inverse>"
append_butlast_last_id: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "butlast xs \<cdot> [last xs] = xs"
tminus_zero_imp_le: fixes x :: "real"   and y :: "real" assumes "x \<ominus> y \<le> 0" shows "x \<le> y"
assms: fixes h' :: "'a" assumes "h ## h'"   and "P h'" shows "Q (h + h')"
domIff: fixes a :: "'d"   and m :: "'d \<Rightarrow> 'e option" shows "(a \<in> dom m) = (m a \<noteq> None)"
lists_def2: fixes A :: "'c set" shows "lists A = {l. set l \<subseteq> A}"
prems_7: shows "set uus \<inter> FvarsT r = {}"
prems_1: fixes t :: "('a, 'b) BinomialTree" assumes "t \<in> set (v # va)" shows "prio (getMinTree (v # va)) \<le> prio t"
Di_def: fixes i :: "nat"   and y :: "nat list" shows "Di i y = {dij i j - 1 |j. j < length y \<and> eij i j \<le> y ! j}"
n2_type3: fixes t1 :: "'b bro"   and h :: "nat"   and t2 :: "'b bro"   and a :: "'b" assumes "t1 \<in> T h"   and "t2 \<in> T h" shows "n2 t1 a t2 \<in> T (Suc h)"
hop: shows "the (dhops (rt2 nhip) dip) \<le> hops"
l4: shows "- (ordinate A - ordinate B) * abscissa X + (abscissa A - abscissa B) * ordinate X = - abscissa A * (ordinate A - ordinate B) + ordinate A * (abscissa A - abscissa B)"
nat_diff_reindex: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>i<n. g (n - Suc i)) = sum g {..<n}"
abs_abs: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (abs a) = abs a"
X: fixes \<tau> :: "'b state \<times> 'b state"   and B :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "\<tau> |\<noteq> invalid and B"
res_carrier_eq: fixes m :: "int" assumes "residues m" shows "carrier (residue_ring m) = {0..m - 1}"
consistent: shows "sign \<in> set (characterize_consistent_signs_at_roots p qs)"
comp_assoc: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and h :: "'c \<Rightarrow> 'e" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
lens_equiv_trans: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and Z :: "'d \<Longrightarrow> 'b" assumes "X \<approx>\<^sub>L Y"   and "Y \<approx>\<^sub>L Z" shows "X \<approx>\<^sub>L Z"
left_neq: fixes a :: "real"   and x :: "real"   and b :: "real"   and m :: "real" assumes "a \<le> x"   and "x < leftcut a b m"   and "a \<le> m"   and "m \<le> b"   and "{a..b} \<subseteq> {0..1}" shows "f x \<noteq> f m"