hcju/isabelleps huggingface dataset:
focus_in_Polys: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "focus X p \<in> P[X]"
tllist_all2_TNil1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'c"   and ts :: "('b, 'd) tllist" shows "tllist_all2 P Q (TNil b) ts = (\<exists>b'. ts = TNil b' \<and> Q b b')"
assms: shows "(f expands_to MS (MSLCons (MS ys h, e) xs) g) (b # basis)"
cost_mergeSibling: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t"   and "sibling t a \<noteq> a" shows "cost (mergeSibling t a) + freq t a + freq t (sibling t a) = cost t"
simps_2: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (receive\<langle>t\<rangle> # S) = (fv t \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t V S)"
eqI: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd"   and \<sigma> :: "'b \<Rightarrow> 'd"   and \<sigma>' :: "'b \<Rightarrow> 'd" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation A B F G \<sigma>'"   and "\<And>a. partial_magma.ide A a \<Longrightarrow> \<sigma> a = \<sigma>' a" shows "\<sigma> = \<sigma>'"
simps_1: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and a :: "'e"   and x :: "'d"   and xs :: "'d list" shows "applied_rule C a (x # xs) = (if a \<in> dom (C x) then \<lfloor>x\<rfloor> else applied_rule C a xs)"
keys_eq_supp: fixes s :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys s = supp_fun (lookup s)"
pmc_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<not> K \<turnstile> \<phi>" shows "K \<div>\<^bsub>\<gamma>\<^esub> \<phi> = K"
Y_carrier: shows "Y \<subseteq> carrier_vec (n + 1)"
1: shows "\<forall>i j. (i, j) \<in> r \<and> i \<noteq> j \<longrightarrow> As i \<subset> As j"
collapse: fixes prod :: "'a \<times> 'b" shows "(fst prod, snd prod) = prod"
assoc_simps_1: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.arr \<a>\<^sub>C[f, g, h]"
assms_1: shows "reduced_row_echelon_form_upt_k A k"
cod_mkArr: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and img :: "'a \<Rightarrow> 'a"   and \<AA> :: "('b \<times> 'b) set"   and A :: "'a set"   and B :: "'a set"   and F :: "'a \<Rightarrow> 'a" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "partial_magma.cod S (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkIde S img \<AA> B"
v0_lset_P: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "v0 \<in> lset P"
prv_rawpsubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "prv \<phi>" shows "prv (rawpsubst \<phi> txs)"
fields_rec: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "ws_prog G" shows "DeclConcepts.fields G C = map (\<lambda>(fn, ft). ((fn, C), ft)) (cfields c) @ (if C = Object then [] else DeclConcepts.fields G (super c))"
rcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<oplus> x \<in> H #>\<^bsub>add_monoid G\<^esub> x"
set_of_map_alt_def: fixes m :: "'c \<Rightarrow> 'd option" shows "set_of_map m = (\<lambda>k. (k, the (m k))) ` dom m"
vars_subst: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and T :: "'c trm" shows "vars (subst \<pi> T) = (\<Union>x\<in>vars T. vars (\<pi> x))"
f'_def: fixes k :: "nat" shows "f' k = (if k = j then 0\<^sub>m d d else adjoint (f k) * f k)"
Abs_inverse: fixes Rep :: "'e \<Rightarrow> 'f"   and Abs :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and y :: "'f" assumes "type_definition Rep Abs A"   and "y \<in> A" shows "Rep (Abs y) = y"
Arr_rep: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "Arr (\<F>C.rep f)"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>B c\<guillemotright>" shows "B.seq g f"
span_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "More_Modules.ideal A \<subseteq> More_Modules.ideal B"
less_sub: fixes n :: "nat"   and m :: "nat"   and k :: "nat"   and m' :: "nat" assumes "n < m - k"   and "m \<le> m'" shows "n < m' - k"
arr_member_of_Ext: shows "tprg \<turnstile> (Base, fdecl (arr, \<lparr>access = Public, static = True, type = PrimT Boolean.[]\<rparr>)) member_of Ext"
intros: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
HFail_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2c s"   and "HFail s s' p"   and "Inv2a s" shows "Inv2c s'"
ex_nonseparating_wall_imp_wall_crossings_not_distinct: fixes X :: "'b set set"   and C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set"   and H :: "'b set set set" assumes "ThinChamberComplex X"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])"   and "H \<in> set (ThinChamberComplex.wall_crossings X (C # Cs @ [D]))"   and "H \<noteq> {}"   and "\<not> separated_by H C D" shows "\<not> distinct (ThinChamberComplex.wall_crossings X (C # Cs @ [D]))"
A_ncong_p: fixes x :: "int" assumes "x \<in> A" shows "[x \<noteq> 0] (mod int p)"
cblinfun_bij2: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"   and g :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "f o\<^sub>C\<^sub>L g = id_cblinfun" shows "bij (cblinfun_apply g)"
in_keys_plusI1: fixes t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<in> keys p"   and "t \<notin> keys q" shows "t \<in> keys (p + q)"
frontier_Un_core: fixes s :: "'a set" shows "frontier s \<union> core s = closure s"
complex_div_cnj: fixes a :: "complex"   and b :: "complex" shows "a / b = a * cnj b / cor ((cmod b)\<^sup>2)"
of_int_ceiling_le_add_one: fixes r :: "'a" shows "of_int \<lceil>r\<rceil> \<le> r + (1::'a)"
fst_divide_out: fixes p :: "'a"   and x :: "'a" shows "fst (divide_out p x) = x div p ^ multiplicity p x"
disj_introC: fixes Q :: "'c \<Rightarrow> bool"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "\<not> Q x \<Longrightarrow> P x" shows "(P \<or> Q) x"
ucast_up_mono: fixes x :: "'b word"   and y :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)"   and "x < y" shows "UCAST('b \<rightarrow> 'c) x < UCAST('b \<rightarrow> 'c) y"
cost_swapLeaves: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat" assumes "consistent t"   and "a \<noteq> b" shows "if a \<in> alphabet t then if b \<in> alphabet t then cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a * depth t a + freq t b * depth t b = cost t + w\<^sub>a * depth t b + w\<^sub>b * depth t a else cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a * depth t a = cost t + w\<^sub>b * depth t a else if b \<in> alphabet t then cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t b * depth t b = cost t + w\<^sub>a * depth t b else cost (swapLeaves t w\<^sub>a a w\<^sub>b b) = cost t"
ennreal_sum_eq_top: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I = \<top>) = (\<exists>i\<in>I. f i = \<top>)"
sepwall_chain3_fg_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and B :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "B \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "separated_by {f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} B C"   and "separated_by {f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} C D" shows "D \<in> f \<turnstile> ChamberComplex.chamber_system X"
isLim_def: fixes i :: "'a" shows "isLim i \<equiv> \<not> isSucc i"
mono_neutral_cong: fixes T :: "'c set"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "finite S"   and "\<And>i. i \<in> T - S \<Longrightarrow> h i = \<^bold>1"   and "\<And>i. i \<in> S - T \<Longrightarrow> g i = \<^bold>1"   and "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x" shows "F.F g S = F.F h T"
dom_comp: fixes g :: "'a1"   and f :: "'a1" assumes "C1.seq g f" shows "C1.dom (g \<cdot>\<^sub>1 f) = C1.dom f"
1: fixes m :: "nat"   and k :: "nat" shows "s m k = index_to_residue s k m"
DEFL_eq_llist: shows "DEFL('a\<cdot>llist) = DEFL(one \<oplus> 'a\<^sub>\<bottom> \<otimes> ('a\<cdot>llist)\<^sub>\<bottom>)"
sup_ty_opt_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "sup_ty_opt P \<equiv> Err.le (subtype P)"
classical_operator_ket: fixes \<pi> :: "'c \<Rightarrow> 'd option"   and x :: "'c" assumes "classical_operator_exists \<pi>" shows "classical_operator \<pi> *\<^sub>V ket x = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
last_fst_esp: fixes a :: "'b"   and s :: "'c"   and xs :: "('b option \<times> 'c) list" assumes "fst (((Some a, s) # xs) ! length xs) = None" shows "fst (xs ! (length xs - Suc 0)) = None"
eq': shows "image_mset (map ((`) (permutation_of_list xs))) (anonymous_profile R) = anonymous_profile R"
assms_2: shows "monotone (\<le>) (\<le>) g"
LSM_inc_T: fixes M :: "('f, 'a, 'g) Module_scheme"   and T :: "'f set" assumes "R module M"   and "T \<subseteq> carrier M" shows "T \<subseteq> LSM\<^bsub>R\<^esub> M T"
to_mregex_nonempty_progress: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)"   and "\<phi>s \<noteq> []" shows "progress_regex \<sigma> P r j = (MIN \<phi>\<in>set \<phi>s. Monitor.progress \<sigma> P \<phi> j)"
vertical_chop_add1: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<longrightarrow> |lan v| = |lan u| + |lan w|"
const_poly_nonzero_coeff: fixes p :: "'a poly" assumes "degree p = 0"   and "p \<noteq> 0" shows "coeff p 0 \<noteq> (0::'a)"
subset_equiv_class: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and b :: "'b"   and a :: "'b" assumes "equiv A r"   and "r `` {b} \<subseteq> r `` {a}"   and "b \<in> A" shows "(a, b) \<in> r"
preserves_reflects_arr: fixes f :: "'j" shows "A_B.arr (\<chi>' f) = J.arr f"
nsqn\<^sub>r_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "nsqn\<^sub>r r \<equiv> if \<pi>\<^sub>4 r = val \<or> \<pi>\<^sub>2 r = 0 then \<pi>\<^sub>2 r else \<pi>\<^sub>2 r - 1"
cring_simprules_2: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
assms_1: fixes i :: "nat" assumes "i < n" shows "f i = g i"
numfloor_def: fixes t :: "MIR.num" shows "numfloor t = (let (tv, ti) = split_int t in case tv of C i \<Rightarrow> numadd (tv, ti) | _ \<Rightarrow> numadd (CF 1 tv (C 0), ti))"
ht_at_phl_while: fixes p :: "'a"   and s :: "'a"   and x :: "'a"   and w :: "'a" assumes "\<lparr>t p \<cdot> t s\<rparr>x\<lparr>x \<cdot> t p\<rparr>" shows "\<lparr>t p\<rparr>(t s \<cdot> x)\<^sup>\<infinity> \<cdot> t w\<lparr>(t s \<cdot> x)\<^sup>\<infinity> \<cdot> t w \<cdot> (t p \<cdot> t w)\<rparr>"
1: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) * (\<k> - 1) = \<Lambda> * (\<v> - 1)"
simps_2: fixes T :: "POPLmark.type" shows "type_ofB (TVarB T) = T"
eq_Maybe_simps_1: shows "eq\<cdot>Nothing\<cdot>Nothing = TT"
szip_unfold: fixes a :: "'d"   and s1 :: "'d stream"   and b :: "'e"   and s2 :: "'e stream" shows "szip (a ## s1) (b ## s2) = (a, b) ## szip s1 s2"
generate_incl: fixes H :: "int set" assumes "H \<subseteq> carrier (Z n)" shows "generate (Z n) H \<subseteq> carrier (Z n)"
vsv_eqI: fixes r :: "V"   and s :: "V" assumes "vsv r"   and "vsv s"   and "\<D>\<^sub>\<circ> r = \<D>\<^sub>\<circ> s"   and "\<And>a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<Longrightarrow> r\<lparr>a\<rparr> = s\<lparr>a\<rparr>" shows "r = s"
simps_1: fixes P :: "'c"   and x :: "'d"   and \<Pi> :: "('c, 'd) chain" shows "nthtriple (\<lbrace> P \<rbrace> \<cdot> x \<cdot> \<Pi>) 0 = (P, x, pre \<Pi>)"
prems_8: shows "fmrel_on_fset (consts (name $$ ts)) vrelated (fmap_of_list rs) \<Gamma>\<^sub>2"
r: fixes u :: "'a"   and v :: "'a" assumes "u \<in> F"   and "v \<in> F" shows "a u \<le> b v"
slength_0_conv_1: fixes xs :: "[:'a:]" shows "(slength\<cdot>xs = 0) = (xs = [::])"
assms_3: shows "\<forall>y<z. eval f (y # xs) \<notin> {None, Some 0}"
is_group: shows "Group.group (G\<lparr>carrier := P\<rparr>)"
simps: fixes t :: "name"   and u :: "name"   and y :: "tm" assumes "atom t \<sharp> (u, y)"   and "atom u \<sharp> y" shows "AtomicP y = SyntaxN.Ex t (SyntaxN.Ex u (TermP (Var t) AND TermP (Var u) AND (y EQ Q_Eq (Var t) (Var u) OR y EQ Q_Mem (Var t) (Var u))))"
ford_fulkerson_method_def: shows "ford_fulkerson_method \<equiv> let f\<^sub>0 = \<lambda>(u, v). 0::'capacity in while (\<lambda>(f, brk). \<not> brk) (\<lambda>(f, brk). (select p. NPreflow.isAugmentingPath c s t f p) \<bind> (\<lambda>p. case p of None \<Rightarrow> return (f, True) | Some p \<Rightarrow> return (NFlow.augment_with_path c f p, False))) (f\<^sub>0, False) \<bind> (\<lambda>(f, brk). return f)"
tail_edge_succ: fixes a :: "'b" shows "tail H (edge_succ HM a) = tail H a"
jkbpTn_jkbpCn_represents: fixes n :: "nat" shows "jkbpT\<^bsub>n\<^esub> jkbpC = jkbpC\<^bsub>n\<^esub>"
g_sg: shows "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (g x - sg i x)) \<partial>M) \<longlonglongrightarrow> 0"
awalkI_apath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "apath u p v" shows "awalk u p v"
card_of_Card_order: fixes A :: "'c set" shows "Card_order |A|"
lcos_m_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier H'"   and "g \<in> carrier H'"   and "h \<in> carrier H'" shows "g <#\<^bsub>H'\<^esub> (h <#\<^bsub>H'\<^esub> M) = g \<otimes>\<^bsub>H'\<^esub> h <#\<^bsub>H'\<^esub> M"
echelon_form_intro: fixes A :: "(('e, 'f) vec, 'd) vec" assumes "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"   and "\<forall>i j. i < j \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'e)) < (LEAST n. A $ j $ n \<noteq> (0::'e))" shows "echelon_form A"
delete_def: fixes a :: "'c"   and t :: "('c \<times> 'd) bro" shows "delete a t = tree (del a t)"
value_j: shows "value_flow \<Psi> j = \<alpha>"
semiring_simprules_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
assms_3: shows "{x0} \<times> K \<subseteq> W"
InvariantConflictFlagCharacterizationAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)" shows "let state' = assertLiteral literal decision state in InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
1: shows "wf\<^sub>M\<^sub>G\<^sub>U \<theta> s t"
simps_2: fixes f :: "'s"   and i :: "nat" shows "coef_hd (Sym f) i = PNum (coef_sym f i)"
ord_of_nat_inject: fixes i :: "nat"   and j :: "nat" shows "(i\<^sub>\<nat> = j\<^sub>\<nat>) = (i = j)"
finprod_insert: fixes G :: "('a, 'b) monoid_scheme"   and F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "Group.comm_monoid G"   and "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finprod G f (insert a F) = f a \<otimes>\<^bsub>G\<^esub> finprod G f F"
intros_1: fixes G :: "('c, 'd) monoid_scheme"   and gens :: "'c set" shows "\<one>\<^bsub>G\<^esub> \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
A: shows "\<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)"
limproc_Seq_F2: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "F (S `;` lim_proc (range Y)) = F (lim_proc (range (\<lambda>i. S `;` Y i)))"
blocks_subset: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>" shows "\<A> \<subseteq># \<B>"
Good_def: shows "Good = (\<lambda>c. {x \<in> G. upper_asymptotic_density (V c x) < d})"
elims_3: fixes x :: "'a tree list" assumes "\<not> sortedByWeight x" obtains t\<^sub>1 :: "'a tree" and t\<^sub>2 :: "'a tree" and ts :: "'a tree list" where "x = t\<^sub>1 # t\<^sub>2 # ts" and "\<not> (weight t\<^sub>1 \<le> weight t\<^sub>2 \<and> sortedByWeight (t\<^sub>2 # ts))"
rbt_comp_map_entry: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_map_entry c = ord.rbt_map_entry (lt_of_comp c)"
AE_space: fixes M :: "'b measure" shows "AE x in M. x \<in> space M"
Vinv2I: fixes x :: "v_state" assumes "\<And>r. no_defection x (votes x r) r" shows "x \<in> Vinv2"
induct: fixes P :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and a0 :: "'b option"   and a1 :: "'b option" assumes "\<And>x y. P (Some x) (Some y)"   and "\<And>y. P None y"   and "\<And>uu_. P (Some uu_) None" shows "P a0 a1"
of_circline_of_ocircline_pos_oriented: fixes H :: "ocircline" assumes "pos_oriented H" shows "of_circline (of_ocircline H) = H"
assms_3: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "extra U f n (\<psi> (the (s (f \<triangleright> n))))"
termMOD_igSwapCls: shows "igSwapCls termMOD"
iteratei_abs_rule_insert_P: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and I :: "'f set \<Rightarrow> 'g \<Rightarrow> bool"   and \<sigma>0 :: "'g"   and c :: "'g \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'g \<Rightarrow> bool" assumes "set_iterator_abs_genord \<alpha> invar iti S0 R"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; invar x; \<alpha> x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {\<alpha> x}. R (\<alpha> x) y; \<forall>y\<in>S. R y (\<alpha> x)\<rbrakk> \<Longrightarrow> I (insert (\<alpha> x) S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
valid_abstraction_def: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> ('e set \<times> 'd) pmf) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and X :: "'e set"   and k :: "'e \<Rightarrow> 'f" shows "PTA.valid_abstraction A X k \<equiv> (\<forall>(x, m)\<in>PTA.clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>) \<and> PTA.collect_clkvt A \<subseteq> X \<and> finite X"
lunit_in_vhom: fixes f :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.trg f = b" shows "\<guillemotleft>S.lunit f : b \<star>\<^sub>S f \<Rightarrow>\<^sub>S f\<guillemotright>"
surj_diff_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "surj (\<lambda>x. minus x a)"
product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "product_topology X I = topology (arbitrary union_of (finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (X i) U) relative_to (\<Pi>\<^sub>E i\<in>I. topspace (X i))))"
nodes_restrict_nodes: fixes g :: "'b ugraph"   and V :: "'b set" shows "nodes (restrict_nodes g V) = nodes g \<inter> V"
bind_spmf_def: fixes x :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
sg_props_1: shows "arcs H \<subseteq> arcs G"
state_is_associated_string': fixes j :: "nat"   and i :: "'a"   and M :: "('a \<times> nat, 'b) message set" assumes "j \<le> length (events i)"   and "consistent M"   and "make_set j (\<lambda>k m. \<exists>s. event_at (i, k) (Receive s m)) \<subseteq> M" shows "is_certified_associated_string (make_set j (\<lambda>k m. \<exists>s. event_at (i, k) (Receive s m))) (state (i, j))"
is_alignedE': fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n" obtains q :: "nat" where "w = push_bit n (word_of_nat q)" and "q < 2 ^ (LENGTH('b) - n)"
bl_shiftl: fixes w :: "'b word"   and n :: "nat" shows "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False"
cdi: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
intros_6: fixes u :: "'e \<Rightarrow> 'd"   and r :: "'e"   and c :: "'e"   and d :: "'d" assumes "u r - u c < d" shows "dbm_entry_val u (Some r) (Some c) (Lt d)"
open_diff: fixes t :: "'a set"   and s :: "'a set" assumes "finite t"   and "open s" shows "open (s - t)"
has_bottom_def: fixes P :: "'a set" shows "has_bottom P \<equiv> \<exists>z\<in>P. \<forall>x\<in>P. x \<succeq> z"
take_rev: fixes n :: "nat"   and xs :: "'b list" shows "take n (rev xs) = rev (drop (\<^bold>|xs\<^bold>| - n) xs)"
take_take: fixes m :: "nat"   and n :: "nat"   and x :: "CValue" shows "\<psi>\<^sup>E\<^bsub>m\<^esub>\<cdot>(\<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>x) = \<psi>\<^sup>E\<^bsub>min m n\<^esub>\<cdot>x"
borel_measurable_continuous_on_indicator: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "A \<in> sets borel"   and "continuous_on A f" shows "(\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable borel"
fair_def: fixes rs :: "'rule stream" shows "fair rs \<equiv> sset rs \<subseteq> R \<and> (\<forall>r\<in>R. alw (ev (holds ((=) r))) rs)"
dgrad_p_setI: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "\<And>v. v \<in> keys p \<Longrightarrow> d (term_powerprod.pp_of_term pair_of_term v) \<le> m" shows "p \<in> gd_term.dgrad_p_set pair_of_term d m"
ilist_slice_nth_length: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and m :: "nat" shows "length (ilist_slice f k m) = k"