hcju/isabelleps huggingface dataset:
lift_character_def: shows "lift_character = (\<lambda>(\<chi>, z) x. if x \<in> adjoin G H a then \<chi> (fst (unadjoin x)) * z ^ snd (unadjoin x) else 0)"
iim_def: fixes G :: "('e, 'f) Group_scheme"   and F :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and K :: "'g set" shows "iim G F f K = {x \<in> carrier G. f x \<in> K}"
SubstAssoc: fixes a :: "('a, 'b) Expression"   and b :: "('a, 'b) Expression"   and c :: "('a, 'b) Expression" shows "sub a in sub b in c = sub sub a in b in c"
widens_trans: fixes P :: "'b prog"   and Ss :: "ty list"   and Ts :: "ty list"   and Us :: "ty list" assumes "P \<turnstile> Ss [\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Us" shows "P \<turnstile> Ss [\<le>] Us"
big_refl: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<in> O[F](f)"
wt_elim_cases_3: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and i :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In2 (e.[i])\<Colon>T" obtains T :: "ty" where "Tb__ = Inl T" and "E,dt\<Turnstile>e\<Colon>-T.[]" and "E,dt\<Turnstile>i\<Colon>-PrimT Integer"
LAT4: fixes p :: "L"   and q :: "L" shows "LUB p q = LUB q p"
Gen_implies_pvalid: fixes x :: "item"   and P :: "(symbol \<times> character list) list set" assumes "x \<in> Gen P" shows "\<exists>p\<in>P. pvalid p x"
natural_isomorphism_\<alpha>: shows "natural_isomorphism VVV.comp (\<cdot>) HoHV HoVH (\<lambda>\<mu>\<nu>\<tau>. \<alpha> (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>)))"
lsimp: shows "length [Suc 0..<Suc (length L)] = length L"
finfun_All_All: fixes P :: "'c \<Rightarrow>f bool" shows "finfun_All P = All (($) P)"
Sup_distr: fixes X :: "('c \<times> 'd) set"   and y :: "'c \<times> 'd" shows "times_prod' (Sup_prod X) y = Sup_prod ((\<lambda>x. times_prod' x y) ` X)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)" shows "P a0"
ccProd_empty: fixes S :: "Vars.var set" shows "{} G\<times> S = \<bottom>"
accept: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m action.Accept], Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
absorb_iff2: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
conventional: fixes g :: "'a"   and n :: "'b"   and ns :: "'b list"   and m :: "'b"   and v :: "'e"   and x :: "'b"   and v' :: "'e" assumes "old.path2 g n ns m"   and "n \<notin> set (tl ns)"   and "v \<in> allDefs g n"   and "v \<in> allUses g m"   and "x \<in> set (tl ns)"   and "v' \<in> allDefs g x" shows "var g v' \<noteq> var g v"
hom_nat_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "nat" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) = hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub> n"
DeltaExists: fixes A :: "('a, 'b) form"   and n :: "'a"   and G :: "('a, 'b) form list" assumes "\<stileturn> A[App n []/0] # G"   and "news n (A # G)" shows "\<stileturn> Exists A # G"
WhileTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> t"
complement_disjointness: fixes x :: "i" assumes "\<exists>v. \<not> O v x" shows "\<not> O x (\<midarrow> x)"
l4_16R2: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.FSC Bet Cong A B C D A' B' C' D'"   and "Bet B C A" shows "Cong C D C' D'"
l_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
scale_minus_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b" assumes "module scale" shows "scale a (- x) = - scale a x"
simps_8: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
App: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and U :: "ty"   and u :: "trm" assumes "\<Gamma> \<turnstile> t : T \<rightarrow> U"   and "\<Gamma> \<turnstile> u : T" shows "\<Gamma> \<turnstile> t \<cdot> u : U"
If: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and b :: "val"   and s1 :: "state"   and c1 :: "stmt"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2\<rightarrow> s2"
A1n_def: shows "A1n = matpow (A + mat 1) N"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" shows "f\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f"
ImpNext_def: fixes p :: "PrIds" shows "ImpNext p = ([ENext p]_(e p) \<and> [MClkNext memCh crCh cst p]_(c p) \<and> [RPCNext crCh rmCh rst p]_(r p) \<and> [RNext rmCh mm ires p]_(m p))"
division_segment_not_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment a \<noteq> zero"
comp_simps_2: shows "composition \<equiv> hide (ioa 0 id1 \<parallel> ioa id1 id2) {act. \<exists>p c av. act = Switch id1 p c av}"
rel_pmf_inf: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'c pmf" assumes "rel_pmf R p q"   and "rel_pmf R q p"   and "reflp R"   and "transp R" shows "rel_pmf (R \<sqinter> R\<inverse>\<inverse>) p q"
pairwiseD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and m :: "'a"   and n :: "'a" assumes "Compiler_Utils.pairwise P M"   and "m \<in> M"   and "n \<in> M" shows "P m n"
assms: shows "\<forall>i\<ge>j. P (\<omega> !! i)"
simps_2: fixes c :: "tcolor"   and l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and uu :: "tcolor"   and mkp :: "'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "paint c \<langle>l, (a, uu, mkp), r\<rangle> = \<langle>l, (a, c, mkp), r\<rangle>"
resl_unit: fixes x :: "'a" shows "x \<leftarrow> (1::'a) = x"
distinct_7: fixes x1 :: "b_e"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "v list"   and x54 :: "e list" shows "$x1 \<noteq> Local x51 x52 x53 x54"
monoD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
dist_triangle_ineq: fixes x :: "fbint"   and z :: "fbint"   and y :: "fbint" shows "fb.dist x z \<le> fb.dist x y + fb.dist y z"
OclImplies_true1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(true implies X) = X"
space_empty: fixes M :: "'c measure" assumes "space M = {}" shows "M = count_space {}"
sum_rel_def: fixes Rl :: "('d \<times> 'f) set"   and Rr :: "('e \<times> 'g) set" shows "\<langle>Rl, Rr\<rangle>sum_rel \<equiv> {(Inl a, Inl a') |a a'. (a, a') \<in> Rl} \<union> {(Inr a, Inr a') |a a'. (a, a') \<in> Rr}"
last_message_conv: fixes m :: "'b message_af"   and xs :: "'b message_af list" assumes "m \<noteq> NoMsg" shows "(last_message xs = m) = (\<exists>i<length xs. xs ! i = m \<and> (\<forall>j<length xs. i < j \<longrightarrow> xs ! j = NoMsg))"
elims: fixes x :: "'b tree"   and xa :: "nat \<times> nat"   and y :: "'b tree list" assumes "reusables x xa = y" obtains t :: "'b tree" and w :: "nat \<times> nat" where "x = t" and "xa = w" and "y = (if r t < fst w then [] else if fst w = l t then [t] else let v = lchild t; u = rchild t in if l u \<le> fst w then reusables u w else u # reusables v w)"
eOp_nchotomy: fixes einp :: "'index \<Rightarrow> 'a withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and ebinp :: "'bindex \<Rightarrow> 'b withERR option" shows "(\<exists>inp binp. einp = OKI inp \<and> igWlsInp MOD delta inp \<and> ebinp = OKI binp \<and> igWlsBinp MOD delta binp) \<or> eOp MOD delta einp ebinp = ERR"
not_less_iff_gr_or_eq: fixes x :: "'d DBMEntry"   and y :: "'d DBMEntry" shows "(\<not> x \<prec> y) = (y \<prec> x \<or> x = y)"
consistent_swapSyms: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" assumes "consistent t" shows "consistent (swapSyms t a b)"
runit_char_2: fixes f :: "'a" assumes "B.ide f" shows "L (B.runit f) = (\<i>[trg f] \<star> f) \<cdot> \<a>\<^sub>B[f, trg f, trg f]"
drefinesD3: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and a :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and s :: "'c"   and Q :: "'d \<Rightarrow> real" assumes "drefines \<phi> G a b"   and "G s"   and "unitary Q"   and "well_def a" shows "wp a Q (\<phi> s) \<le> wp b (Q \<circ> \<phi>) s"
concat_map_filter: fixes f :: "'b \<Rightarrow> 'a list"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "concat (map f (filter P xs)) = concat (map (\<lambda>x. if P x then f x else []) xs)"
proj_k_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and x :: "nat" assumes "grover_state_sig n f dims R K" shows "grover_state_sig.proj_k K x = outer_prod (grover_state_sig.ket_k K x) (grover_state_sig.ket_k K x)"
abs_div: fixes y :: "int"   and x :: "int" assumes "y dvd x" shows "\<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
gbinomial_absorb_comp: fixes a :: "'a"   and k :: "nat" shows "(a - of_nat k) * (a gchoose k) = a * (a - (1::'a) gchoose k)"
thetaSeqTT_BisT: shows "thetaSeqTT \<subseteq> BisT"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_rexp_of_rexp_of': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (rexp_of n \<phi>) = lang n (rexp_of' n \<phi>)"
content_def: fixes p :: "'a poly" shows "Polynomial.content p = gcd_list (Polynomial.coeffs p)"
mod_minus: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "0 < a - b"   and "0 < c - d" shows "(a - b + (c - d mod q)) mod q = (a - b + (c - d)) mod q"
iD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> iD rt" shows "\<exists>dsn dsk hops nhip pre. rt dip = Some (dsn, dsk, Aodv_Basic.inv, hops, nhip, pre)"
logical_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (logical_instr instr s1)) \<and> t2 = snd (fst (logical_instr instr s2))" shows "low_equal t1 t2"
sees_field_i_i_i_o_o_oI: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xc :: "ty"   and xd :: "fmod"   and xe :: "String.literal" assumes "x \<turnstile> xa sees xb:xc (xd) in xe" shows "pred.eval (sees_field_i_i_i_o_o_o x xa xb) (xc, xd, xe)"
preserves_reflects_arr: fixes f :: "'c \<times> 'c" shows "C.CC.arr (C.CC.map f) = C.CC.arr f"
vec_eq: fixes m :: "'a"   and n :: "'a" shows "(Finite_Cartesian_Product.vec m = Finite_Cartesian_Product.vec n) = (m = n)"
distinct_53: fixes x6 :: "'b poly"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Neg x6 \<noteq> CN x81 x82 x83"
step: fixes \<xi> :: "'a"   and p :: "'b"   and a :: "'c"   and \<xi>' :: "'a"   and p' :: "'b" assumes "(\<xi>, p) \<in> reachable A I"   and "P (\<xi>, p)"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "I a" shows "P (\<xi>', p')"
action_loc_aux_intros_4: fixes ad :: "'a"   and al :: "addr_loc"   and P :: "'b prog"   and v :: "'a val" shows "(ad, al) \<in> action_loc_aux P (NormalAction (ReadMem ad al v))"
filter_empty_conv: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
short_cycles_def: fixes G :: "nat set \<times> nat set set"   and k :: "nat" shows "short_cycles G k \<equiv> {p \<in> ucycles G. uwalk_length p \<le> k}"
map_iterator_I: fixes l0 :: "('d \<times> 'e) list"   and m :: "'d \<Rightarrow> 'e option"   and iti :: "('f \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" assumes "distinct (map fst l0)"   and "m = map_of l0"   and "iti = foldli l0" shows "map_iterator iti m"
ereal_minus_le_minus_plus: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<le> b + c" shows "- b \<le> - a + c"
to_gbg_ext_def: fixes G :: "('e, 'f) b_graph_rec_scheme"   and m :: "'g" assumes "b_graph G" shows "b_graph.to_gbg_ext G m \<equiv> \<lparr>g_V = g_V G, g_E = g_E G, g_V0 = g_V0 G, gbg_F = if bg_F G = UNIV then {} else {bg_F G}, \<dots> = m\<rparr>"
rtrancl_finite_relpow: fixes a :: "'c"   and b :: "'c"   and rel :: "('c \<times> 'c) list" shows "((a, b) \<in> (set rel)\<^sup>*) = (\<exists>n\<le>length rel. (a, b) \<in> set rel ^^ n)"
valid_Rcons: fixes tr :: "('state, 'act, 'out) trans list"   and trn :: "('state, 'act, 'out) trans" assumes "valid tr"   and "tgtOf (last tr) = srcOf trn"   and "validTrans trn" shows "valid (tr ## trn)"
subst_apply_strand_def: fixes S :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
PO_refines_implies_Range_trans: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "{Range R} TS.trans Tc {> Range R}"
src_def: fixes i :: "nat"   and p :: "'a" shows "src i p \<equiv> \<exists>q. channel i = Some (p, q)"
e_hd_neq_0: fixes e :: "nat" assumes "e \<noteq> 0" shows "e_hd e = hd (Partial_Recursive.list_decode e)"
dfree_Fun: fixes args :: "'a \<Rightarrow> ('b, 'a) trm"   and i :: "'b" assumes "\<And>i. dfree (args i)" shows "dfree ($f i args)"
convex_contains_segment: fixes S :: "'b set" shows "convex S = (\<forall>a\<in>S. \<forall>b\<in>S. {a--b} \<subseteq> S)"
inject_4: fixes x41 :: "('a, 'b, 'c) formula"   and x42 :: "('a, 'b, 'c) formula"   and y41 :: "('a, 'b, 'c) formula"   and y42 :: "('a, 'b, 'c) formula" shows "((x41 && x42) = (y41 && y42)) = (x41 = y41 \<and> x42 = y42)"
ep: shows "edge_preserving (Id_on (vertices G\<^sub>1)) (edges G\<^sub>1) (edges G\<^sub>2)"
sorted_spvec_addmult_spvec_helper3: fixes y :: "'a"   and arr :: "(nat \<times> 'a) list"   and brr :: "(nat \<times> 'a) list"   and aa :: "nat"   and b :: "'a"   and ba :: "'a" assumes "sorted_spvec (addmult_spvec y arr brr)"   and "sorted_spvec ((aa, b) # arr)"   and "sorted_spvec ((aa, ba) # brr)" shows "sorted_spvec ((aa, b + y * ba) # addmult_spvec y arr brr)"
flat_lub_in_chain: fixes b :: "'a"   and A :: "'a set"   and a :: "'a" assumes "Complete_Partial_Order.chain (flat_ord b) A"   and "flat_lub b A = a" shows "a = b \<or> a \<in> A"
magQ_qinverse: fixes x :: "'a['b, 'c]" shows "\<lbrakk>x\<^sup>-\<^sup>\<one>\<rbrakk>\<^sub>Q = inverse \<lbrakk>x\<rbrakk>\<^sub>Q"
hyps_3: fixes b :: "name list"   and ba :: "fm" shows "(trans_fm b fm = trans_fm b ba) = (fm = ba)"
fps_of_poly_simps_15: fixes c :: "'b"   and p :: "'b poly" shows "fps_of_poly (pCons c p) = fps_const c + fps_of_poly p * fps_X"
rbt_insertw_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "rbt_insert_with f = rbt_insert_with_key (\<lambda>_. f)"
IH_2: fixes m :: "'node"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "m \<noteq> n"   and "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "Mapping.lookup u' m = Mapping.lookup u m"
minarc_below: fixes x :: "'a" shows "minarc x \<le> - - x"
dim_eq_card_independent: fixes B :: "('b, 'c) vec set" assumes "vec.independent B" shows "vec.dim B = card B"
trivial_ker_imp_inj: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h"   and "kernel G H h = {\<one>\<^bsub>G\<^esub>}" shows "inj_on h (carrier G)"
qFreshAbs_imp_ex_qAFreshAbs2: fixes V :: "'f set"   and A :: "('g, 'h, 'e, 'd, 'i) qAbs"   and xs :: "'e"   and x :: "'d"   and ys :: "'e"   and y :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "finite V"   and "FixVars.qGoodAbs A"   and "qFreshAbs xs x A"   and "qFreshAbs ys y A" shows "\<exists>A'. FixVars.qGoodAbs A' \<and> A $= A' \<and> qAFreshAbs xs x A' \<and> qAFreshAbs ys y A'"
assms_4: shows "b \<in> hyp2 \<union> S"
alw_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "alw \<phi> \<equiv> neg (ev (neg \<phi>))"
subst_freshfact2_ty: fixes X :: "W.tvar"   and S :: "ty"   and T :: "ty" assumes "X \<sharp> S" shows "X \<sharp> T[X::=S]"
simps_1: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and tset :: "('e, 'c, 'd) Tree set['e]" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid_Tree satisfies transition P (tConj tset) = Ball (set_bset tset) (nominal_ts.valid_Tree satisfies transition P)"
abort_def: fixes uu :: "String.literal"   and f :: "unit \<Rightarrow> 'b" shows "Code.abort uu f = f ()"
3961: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> - - x = - - x"
square_integrable_imp_absolutely_integrable: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S"   and "S \<in> lmeasurable" shows "f absolutely_integrable_on S"
rel_ex_def: fixes u :: "int"   and u' :: "int"   and x :: "bool"   and y :: "bool" shows "rel_ex u u' x y = ((x \<and> u' = u + 1 \<or> \<not> x \<and> u' = u - 1 \<or> u' = 0) \<and> y = (u' = 0))"
less_imp_le: fixes y :: "('a, 'b) Term.term"   and x :: "('a, 'b) Term.term" assumes "y \<rhd> x" shows "y \<unrhd> x"
map_M_pure_I: fixes xs :: "'m list"   and f :: "'m \<Rightarrow> ('n, 'o, 'p) prog"   and h :: "'n" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_M f xs) h"
is_gcd_is_bezout_ext: fixes 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 bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "class.bezout_ring times one plus zero minus uminus"   and "bezout_ring.is_bezout_ext times plus uminus bezout" shows "GCD_ring.is_gcd times (\<lambda>a b. case bezout a b of (x, xa, u, v, gcd') \<Rightarrow> gcd')"
floor_less_numeral: fixes x :: "'a"   and v :: "num" shows "(\<lfloor>x\<rfloor> < numeral v) = (x < numeral v)"
argmin_map: fixes xs :: "'a list"   and c :: "'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "argmin c (map f xs) = f (argmin (c \<circ> f) xs)"
cases: fixes x :: "'d list" obtains "x = []"   | s :: "'d" and ss :: "'d list" where "x = s # ss"
c_offending_flows_subseteq_edges: fixes m :: "'b SecurityInvariant_configured"   and G :: "'b graph" assumes "configured_SecurityInvariant m" shows "\<Union> (c_offending_flows m G) \<subseteq> edges G"
finite_V_Fe: fixes A :: "nat set" assumes "finite V"   and "A \<in> F" shows "finite A"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1. module_hom_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B\<^sub>2))"
singletonD: fixes b :: "'a"   and a :: "'a" assumes "b \<in> {a}" shows "b = a"
weakSimTauChain: fixes Q :: "pi"   and Q' :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "(P, Q) \<in> Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q" shows "\<exists>P'. P \<Longrightarrow>\<^sub>\<tau> P' \<and> (P', Q') \<in> Rel"
conway_c3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and star :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.commutative_kleene_algebra plus times one zero less_eq less star" shows "star (plus x y) = times (star x) (star y)"
cases: fixes x :: "nat" obtains "x = 0"   | "x = 1"   | n :: "nat" where "2 \<le> n" and "x = n"
path_image_rec_join: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real"   and l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "(k, \<gamma>) \<in> set l"   and "valid_chain_list l" shows "path_image \<gamma> \<subseteq> path_image (rec_join l)"
T_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "p \<leadsto> x \<leadsto> q \<equiv> ad p +  |x] d q"
generate_empty: shows "generate G {} = {\<one>\<^bsub>G\<^esub>}"
closed_uminus: fixes x :: "'b" assumes "x \<in> U\<^sub>M" shows "- x \<in> U\<^sub>M"
terminates_callUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and init :: "'d \<Rightarrow> 'd"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and result :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile>call init p return result \<down> Normal s"
prod_topology_homeomorphic_space_left: fixes Y :: "'d topology"   and b :: "'d"   and X :: "'e topology" assumes "topspace Y = {b}" shows "prod_topology X Y homeomorphic_space X"