hcju/isabelleps huggingface dataset:
dim_sumlist: fixes xs :: "'a vec list" assumes "\<forall>x\<in>set xs. dim_vec x = n" shows "dim_vec (M.sumlist xs) = n"
real_sqrt_gt_zero: fixes x :: "real" assumes "0 < x" shows "0 < sqrt x"
valid_of_sound: fixes \<phi> :: "('sf, 'sc, 'sz) formula" assumes "sound ([], [], [\<phi>])" shows "valid \<phi>"
has_derivative_bounded_linear: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "(f has_derivative f') F" shows "bounded_linear f'"
intro: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list" assumes "LLL.LLL_invariant_weak n m fs_init fs" shows "fs_int' n m fs_init fs"
add_diff_cancel_left': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
strict_trans2: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "lt_of_nat_term_order cmp_term b a"   and "le_of_nat_term_order cmp_term c b" shows "lt_of_nat_term_order cmp_term c a"
in_segment_2: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> open_segment a b) = (a \<noteq> b \<and> (\<exists>u>0. u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b))"
closure_ic_union_ci: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b"   and "b < c" shows "closure ({a..<b} \<union> {b<..c}) = {a..c}"
Cons_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n'' :: "'b"   and as :: "'a list"   and n' :: "'b"   and a :: "'a"   and n :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n'' as n'"   and "valid_edge a"   and "sourcenode a = n"   and "targetnode a = n''" shows "CFG.path sourcenode targetnode valid_edge n (a # as) n'"
no\<phi>_def: fixes K :: "'state \<Rightarrow> bool" shows "no\<phi> K \<equiv> \<forall>s a ou s'. reachNT s \<and> K s \<and> step s a = (ou, s') \<longrightarrow> \<not> \<phi> (Trans s a ou s')"
vc: shows "vc C (\<lambda>a. enat k * Q a)"
rel_conversep: fixes R1 :: "'j \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'k \<Rightarrow> 'd \<Rightarrow> bool" shows "rell_G R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rell_G R1 R2)\<inverse>\<inverse>"
associated_iff: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "monoid_cancel G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<sim>\<^bsub>G\<^esub> b = (\<exists>c\<in>Units G. a = b \<otimes>\<^bsub>G\<^esub> c)"
mod_ring_rel_integer_def: fixes pp :: "integer"   and x :: "integer"   and y :: "'a mod_ring" assumes "p = int_of_integer pp" shows "mod_ring_rel_integer x y = (\<exists>z. urel_integer x z \<and> mod_ring_rel z y)"
snd_scaleR: fixes r :: "real"   and A :: "'b \<times> 'a" shows "snd (r *\<^sub>R A) = r *\<^sub>R snd A"
eq: shows "{(a # as) ! i |i. i < length (a # as) \<and> (x # xs) ! i \<noteq> 0} = (if 0 < x then {a} else {}) \<union> {as ! i |i. i < length as \<and> xs ! i \<noteq> 0}"
collapse_1: fixes event :: "('a, 'b, 'c) event" assumes "isTrans event" shows "Trans (occurs_on event) (un_Trans2 event) (un_Trans3 event) = event"
s_space: shows "s \<subseteq> {..<n} \<rightarrow> {..p}"
dghm_proj_def: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i = [\<lambda>a\<in>\<^sub>\<circ>dg_prod I \<AA>\<lparr>Obj\<rparr>. a\<lparr>i\<rparr>, \<lambda>f\<in>\<^sub>\<circ>dg_prod I \<AA>\<lparr>Arr\<rparr>. f\<lparr>i\<rparr>, dg_prod I \<AA>, \<AA> i]\<^sub>\<circ>"
that: fixes f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<Gamma> s = Fun f T" shows "thesis"
inf_par_distrib2: fixes par :: "'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice"   and skip :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice"   and c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice" assumes "par_distrib (par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (skip::'a::refinement_lattice)" shows "(par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (d::'a::refinement_lattice) ((c\<^sub>0::'a::refinement_lattice) \<sqinter> (c\<^sub>1::'a::refinement_lattice)) = par d c\<^sub>0 \<sqinter> par d c\<^sub>1"
prems: shows "rsquarefree ([:- a, 1::'a:] * p * q)"
injective_cntxt: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr" shows "(\<E> [e1] = \<E> [e2]) = (e1 = e2)"
cases: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" assumes "a1 is-a2\<rightarrow>\<^sub>d* a3" obtains n :: "'node SDG_node" where "a1 = na__" and "a2 = []" and "a3 = na__" and "valid_SDG_node na__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a1 = nb__" and "a2 = nsa__ @ [n''__]" and "a3 = n'a__" and "nb__ is-nsa__\<rightarrow>\<^sub>d* n''__" and "n''__ s\<longrightarrow>\<^bsub>cd\<^esub> n'a__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a1 = nc__" and "a2 = nsb__ @ [n''a__]" and "a3 = n'b__" and "nc__ is-nsb__\<rightarrow>\<^sub>d* n''a__" and "n''a__ s-V\<rightarrow>\<^sub>d\<^sub>d n'b__" and "n''a__ \<noteq> n'b__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a1 = n" and "a2 = ns @ [n'']" and "a3 = n'" and "n is-ns\<rightarrow>\<^sub>d* n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"
listset_Nil_mem_conv: fixes AVS :: "'c set list" shows "([] \<in> listset AVS) = (AVS = [])"
safe_regex_safe_formula: fixes m :: "modality"   and g :: "safety"   and r :: "formula Regex.regex"   and \<phi> :: "formula" assumes "safe_regex m g r"   and "\<phi> \<in> regex.atms r" shows "safe_formula \<phi> \<or> (\<exists>\<psi>. \<phi> = Neg \<psi> \<and> safe_formula \<psi>)"
Q_def: shows "Q = P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:]"
matrix_to_iarray_echelon_form_of_upt_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "k < ncols A" shows "matrix_to_iarray (echelon_form_of_upt_k A k bezout) = echelon_form_of_upt_k_iarrays (matrix_to_iarray A) k bezout"
iMODb_Suc: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m, c ]"   and "x < r + m * c" shows "(Suc x \<in> [ r, mod m, c ]) = (m = Suc 0)"
elims_2: fixes x :: "('e, 'f, 'g) FSM" assumes "finite_FSM x" obtains M :: "('e, 'f, 'g) FSM" where "x = M" and "finite (nodes M) \<and> finite (inputs M) \<and> finite (outputs M)"
rec_accum1_def: fixes f :: "recf" shows "rec_accum1 f = PR (CN f [CN Z [recf.Id 1 0], recf.Id 1 0]) (CN rec_mult [recf.Id 3 1, CN f [CN S [recf.Id 3 0], recf.Id 3 2]])"
dg_prod_2_def: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<AA> \<times>\<^sub>D\<^sub>G \<BB> \<equiv> \<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>"
ordered_insert_pairs_sorted: fixes ys :: "('c \<times> 'd) list"   and x :: "'c"   and v :: "'d" assumes "Lists_Ex.strict_sorted (map fst ys)" shows "Lists_Ex.strict_sorted (map fst (ordered_insert_pairs x v ys))"
hoare_tvalid_def: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>t {P} c {Q}) = (\<forall>l s. P l s \<longrightarrow> (\<exists>t. (c, s) \<Rightarrow> t \<and> Q l t))"
Id_on_int: fixes A :: "'e set"   and f :: "('e \<times> 'f) set" shows "Id_on A O f = A \<times> UNIV \<inter> f"
intro: fixes G :: "'a set" assumes "finite G" shows "FinGroup_axioms G"
sup_state_trans: fixes P :: "'a prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list"   and c :: "ty list \<times> ty err list" assumes "P \<turnstile> a \<le>\<^sub>i b"   and "P \<turnstile> b \<le>\<^sub>i c" shows "P \<turnstile> a \<le>\<^sub>i c"
triv_init_disc_filtr_prob_space_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" shows "triv_init_disc_filtr_prob_space M F \<equiv> prob_space M \<and> triv_init_disc_filtr_prob_space_axioms M F"
conn: fixes y :: "'b" assumes "y \<in> T" shows "connected (S \<inter> f -` {y})"
inv_f_f: fixes x :: "'a" shows "inv f (f x) = x"
U: shows "U \<in> carrier_mat (dim_col A) (dim_col A)"
take_append: shows "take j \<beta> @ [\<alpha> ! i] = take (Suc j) \<beta>"
sup_eq_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y = bot) = (x = bot \<and> y = bot)"
update_convs_3: fixes N' :: "nat \<Rightarrow> nat"   and I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "N_update N' \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = \<lparr>vars.I = I, M = M, N = N' N, S = S, \<dots> = more\<rparr>"
span_union_is_sum: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and A :: "'g set"   and B :: "'g set" assumes "module R M"   and "A \<subseteq> carrier M"   and "B \<subseteq> carrier M" shows "module.span R M (A \<union> B) = module.submodule_sum M (module.span R M A) (module.span R M B)"
finite_UN: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A" shows "finite (\<Union> (B ` A)) = (\<forall>x\<in>A. finite (B x))"
involved_ifp: shows "\<forall>s a. \<forall>d\<in>involved a. kprecondition s (the a) \<longrightarrow> ifp d (current s)"
UnI1: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A" shows "c \<in> A \<union> B"
dim_col_transfer_rule: fixes A :: "'e mat"   and A' :: "(('e, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_col A = CARD('f)"
zero_linear_poly_def: shows "0 \<equiv> Abs_linear_poly fun_zero"
simps_4: fixes X :: "tyvrs"   and T :: "ty"   and t' :: "trm"   and t :: "trm"   and y :: "vrs" assumes "X \<sharp> (T, t')" shows "(\<lambda>X<:T. t)[y \<mapsto> t'] = (\<lambda>X<:T. t[y \<mapsto> t'])"
ordLess_or_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "r <o r' \<or> r' \<le>o r"
arr_mkIde: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set" shows "S.arr (S.mkIde A) = (A \<subseteq> S.Univ)"
dghm_cn_comp_ObjMap_vdomain: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
runiq_paste1: fixes Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "runiq Q"   and "runiq (P outside Domain Q)" shows "runiq (P +* Q)"
distinct_1: fixes x1 :: "POPLmarkRecord.type"   and x2 :: "(char list \<times> pat) list" shows "PVar x1 \<noteq> PRcd x2"
tail_def: fixes S :: "'b list" shows "tail S \<equiv> if S = [] then [] else tl S"
split_trace: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and t' :: "('p, 's, 'm) event list"   and t'' :: "('p, 's, 'm) event list" assumes "trace c t c'"   and "t = t' @ t''" shows "\<exists>c''. trace c t' c'' \<and> trace c'' t'' c'"
hyps: shows "changes_poly_pos_inf (pders (pderiv p)) = 0"
upred_weak_complete_lattice: shows "weak_complete_lattice \<P>"
under_Under_trans: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and C :: "'c set" assumes "trans r"   and "a \<in> under r b"   and "b \<in> Under r C" shows "a \<in> Under r C"
IH: assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t S"   and "fv t \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}"   and "fv (t \<cdot> \<theta>) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}" shows "fv (t \<cdot> \<theta>) \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
lfp_exp_lemma3: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "mono_trans t"   and "t R \<tturnstile> R"   and "sound R" shows "lfp_exp t \<tturnstile> t (lfp_exp t)"
has_integral_bound_real: fixes B :: "real"   and S :: "real set"   and f :: "real \<Rightarrow> 'a"   and i :: "'a"   and a :: "real"   and b :: "real" assumes "0 \<le> B"   and "finite S"   and "(f has_integral i) {a..b}"   and "\<And>x. x \<in> {a..b} - S \<Longrightarrow> norm (f x) \<le> B" shows "norm i \<le> B * content {a..b}"
na_def: fixes x :: "ant" shows "na x = (if x < 0 then 0 else if x \<noteq> \<infinity> then nat (tna x) else undefined)"
elims_3: fixes x :: "('b, 'c) Combinators list" assumes "\<not> wellformed_policy1 x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "\<not> DenyAll \<notin> set xs"
meet_def: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = (THE inf. is_inf x y inf)"
prob_sum_is_one: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" assumes "state n v" shows "prob1 n v i + prob0 n v i = 1"
WtFT_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "WtFT cl \<equiv> \<Sum>n\<in>theFT cl. 1 / real (length cl)"
strictly_preserves_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "E.map (\<F>C.tensor f g) = E.map f \<otimes>\<^sub>D E.map g"
strict_order_iff: fixes b :: "'a"   and a :: "'a" shows "lt b a = (a = local.max a b \<and> a \<noteq> b)"
spectrum_imp_pr: fixes \<pp> :: "('a \<times> 'a) set set" assumes "\<pp> \<in> pi.spectrum" shows "pr_ideal pi.carrier_local_ring_at \<pp> pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
FL_valid_Pred: fixes P :: "'state"   and f :: "'effect"   and \<phi> :: "'pred" shows "P \<Turnstile> Pred f \<phi> = \<langle>f\<rangle>P \<turnstile> \<phi>"
ac: shows "AE x in S \<Otimes>\<^sub>M T. Px (fst x) * Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
TR3_sub_carr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR3 R M N P \<subseteq> carrier P"
refl_onI: fixes r :: "('e \<times> 'e) set"   and A :: "'e set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
insert: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite A"   and "x \<notin> A" shows "abel_semigroup_set.F f z g (insert x A) = f (g x) (abel_semigroup_set.F f z g A)"
matchNilLeft: fixes a :: "name"   and b :: "name"   and Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "a \<noteq> b" shows "\<zero> \<leadsto>[Rel] [a\<frown>b]P"
in_diff: fixes x :: "'c"   and a :: "'c"   and A :: "'c set" assumes "x \<noteq> a"   and "x \<in> A" shows "x \<in> A - {a}"
a: shows "\<Gamma> \<turnstile> Ax x a \<turnstile> ((a, B) # \<Delta>)"
pvalid_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and p :: "(symbol \<times> character list) list"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.pvalid \<NN> \<TT> \<RR> \<SS> Doc p x = (\<exists>u \<gamma>. LocalLexing.wellformed_tokens \<TT> p \<and> LocalLexing.wellformed_item \<RR> Doc x \<and> u \<le> length p \<and> charslength p = item_end x \<and> charslength (take u p) = item_origin x \<and> CFG.is_derivation \<NN> \<TT> \<RR> \<SS> (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> CFG.derives \<NN> \<TT> \<RR> (item_\<alpha> x) (terminals (drop u p)))"
heads_def: fixes xs :: "bool list" shows "heads xs = length (filter id xs)"
simps_1: fixes e :: "'c \<times> 'd \<times> bool"   and n :: "'c option"   and xs :: "('c \<times> 'd \<times> bool) list" shows "\<langle>Insert e n\<rangle> xs = Ordered_List.insert xs e n"
snd_netgmap_subnet: fixes sr :: "'e \<Rightarrow> 'd \<times> 'c"   and s2 :: "'e net_state"   and s1 :: "'e net_state" shows "snd (case netgmap sr s1 of (\<sigma>\<^sub>1, ss) \<Rightarrow> case netgmap sr s2 of (\<sigma>\<^sub>2, tt) \<Rightarrow> (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt)) = SubnetS (snd (netgmap sr s1)) (snd (netgmap sr s2))"
map_resource_id: fixes res :: "('c, 'd) resource" shows "map_resource id id res = res"
sup_bot_right: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> mbot = x"
exsplit: fixes p :: "fm"   and x :: "real"   and y :: "real"   and bs :: "real list" assumes "qfree p" shows "Ifm (x # y # bs) (exsplit p) = Ifm ((x + y) # bs) p"
induct: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "well_founded R"   and "\<And>x. (\<And>y. R y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
disjCases3: fixes A :: "bool"   and B :: "bool"   and C :: "bool" assumes "A \<or> B \<or> C" obtains "A"   | "B"   | "C"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and ternaryvalue :: "ternaryvalue" shows "P (case ternaryvalue of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = (\<not> (ternaryvalue = TernaryTrue \<and> \<not> P f1 \<or> ternaryvalue = TernaryFalse \<and> \<not> P f2 \<or> ternaryvalue = TernaryUnknown \<and> \<not> P f3))"
toList_empty: fixes t :: "('c, 'd) FingerTreeStruc" shows "(toList t = []) = (t = Empty)"
restr_def: fixes r :: "('b \<times> 'b) set"   and m :: "'b \<Rightarrow> bool" shows "(r | m) = {(x, y). (x, y) \<in> r \<and> \<not> m x}"
emeasure_uniform_count_measure: fixes A :: "'b set"   and X :: "'b set" assumes "finite A"   and "X \<subseteq> A" shows "emeasure (uniform_count_measure A) X = ennreal (real (card X) / real (card A))"
l2_can_signal_trans: fixes s' :: "l2_state"   and A :: "agent"   and B :: "agent"   and s :: "l2_state" assumes "can_signal s' A B"   and "(s, s') \<in> TS.trans l2" shows "can_signal s A B"
stakeWhile_SCons: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c stream" shows "stakeWhile P (x ## xs) = (if P x then LCons x (stakeWhile P xs) else LNil)"
that: fixes U :: "'a set"   and V :: "'a set" assumes "openin X U"   and "openin X V"   and "S \<subseteq> U"   and "T \<subseteq> V"   and "disjnt U V" shows "thesis"
trms_proj\<^sub>l\<^sub>s\<^sub>t_append: fixes l :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) strand_step) list"   and B :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t (proj l (A @ B)) = trms\<^sub>l\<^sub>s\<^sub>t (proj l A) \<union> trms\<^sub>l\<^sub>s\<^sub>t (proj l B)"
ext: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "B.VxVxV.comp g f \<noteq> B.VxVxV.null" shows "B.VxVxV.seq g f"
cbp: shows "cball z (3 / 4 * pe) \<subseteq> - path_image p"
msetext_dersh_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "msetext_dersh gt ys xs" shows "msetext_dersh gt (ys @ [x]) (xs @ [x])"
replacefacesAt2_notin: fixes i :: "nat"   and "is" :: "nat list"   and olfF :: "face"   and newFs :: "face list"   and Fss :: "face list list" assumes "i \<notin> set is" shows "replacefacesAt2 is olfF newFs Fss ! i = Fss ! i"
tm_wf_tcopy_3: shows "tm_wf0 tcopy_end"
nu_fusion_1: fixes l :: "'a \<Rightarrow> 'a"   and u :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "galois l u"   and "isotone h"   and "has_greatest_postfixpoint g"   and "has_greatest_fixpoint h"   and "h (u (l (\<nu> h))) \<le> u (g (l (\<nu> h)))" shows "\<nu> h \<le> u (p\<nu> g)"
init_fin_silent_movesD: fixes t :: "'t"   and s :: "status"   and x :: "'x2"   and m :: "'m2"   and s' :: "status"   and x' :: "'x2"   and m' :: "'m2" assumes "(\<tau>trsys.silent_move (r2.init_fin t) r2.init_fin_\<tau>move)\<^sup>*\<^sup>* ((s, x), m) ((s', x'), m')" shows "r2.silent_moves t (x, m) (x', m') \<and> s = s'"
not_symKeys_priK: fixes A :: "agent" shows "priK A \<notin> symKeys"
preserves_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (UP \<mu>) = UP.map\<^sub>0 (src\<^sub>B \<mu>)"
l_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "(x = x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> a) = (a = \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub>)"
monomials_def: shows "monomials \<equiv> map_fun mapping_of id keys"
a: shows "\<forall>n. ideal (I n) \<and> I n \<subset> I (Suc n)"
standard_open_is_zariski_open: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "x \<in> R" shows "comm_ring.is_zariski_open R addition multiplication zero unit (comm_ring.standard_open R addition multiplication zero unit x)"
preciseD: fixes a :: "'a"   and p :: "'b"   and a' :: "'a"   and h :: "heap \<times> nat set" assumes "precise (pure R)"   and "h \<Turnstile> pure R a p * emp \<and>\<^sub>A pure R a' p * emp" shows "a = a'"
sameDom_and_liftAll2_iff: fixes inp :: "'a \<Rightarrow> 'b option"   and inp' :: "'a \<Rightarrow> 'c option"   and phi :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(sameDom inp inp' \<and> liftAll2 phi inp inp') = (\<forall>i. inp i = None \<and> inp' i = None \<or> (\<exists>v v'. inp i = Some v \<and> inp' i = Some v' \<and> phi v v'))"
lookup_map_key_PP: fixes p :: "('b, 'c) pp \<Rightarrow>\<^sub>0 'a"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (Poly_Mapping.map_key PP p) t = lookup p (PP t)"
Seed_holds_edges_sym: fixes p :: "nat" shows "edges_sym Seed\<^bsub>p\<^esub>"
symvars_def: fixes c :: "('c, 'd) conf" shows "Conf.symvars c = Store.symvars (store c) \<union> Bexp.vars (conjunct (pred c))"
cs_implies_cbelow_on: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "relation_order r ord A"   and "relation.dw_closed r A"   and "relation.cs r a b"   and "a \<in> A"   and "b \<in> A"   and "ord a c"   and "ord b c" shows "cbelow_on A ord c (\<lambda>a b. r a b \<or> r b a) a b"
cases: fixes a1 :: "'k set"   and a2 :: "'k set"   and R :: "('k \<times> 'k) set" assumes "(a1, a2) \<in> max_ext R" obtains X :: "'k set" and Y :: "'k set" where "a1 = X" and "a2 = Y" and "finite X" and "finite Y" and "Y \<noteq> {}" and "\<And>x. x \<in> X \<Longrightarrow> \<exists>xa\<in>Y. (x, xa) \<in> R"
assms_2: shows "support_set pf2 \<subseteq> stocks Mkt"
len_bit0: fixes uu :: "'b bit0 itself" shows "len_of uu = 2 * LENGTH('b)"
in_small_setI: fixes A :: "V set"   and x :: "V" assumes "small A"   and "x \<in> A" shows "x \<in>\<^sub>\<circ> set A"
frees_def: fixes t :: "dB" shows "frees t = {i. free t i}"