hcju/isabelleps huggingface dataset:
ereal_liminf_limsup_add: fixes u :: "nat \<Rightarrow> ereal"   and v :: "nat \<Rightarrow> ereal" shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
fds_abs_summable_zeta_iff: fixes s :: "'a" shows "fds_abs_converges fds_zeta s = (1 < s \<bullet> (1::'a))"
alphaFrameRes: fixes y :: "name"   and F :: "'d frame"   and x :: "name" assumes "y \<sharp> F" shows "\<lparr>\<nu>x\<rparr>F = \<lparr>\<nu>y\<rparr>[(x, y)] \<bullet> F"
G_fact: fixes n :: "nat" shows "G (real n + 1) = fact n"
that: fixes v :: "'c" assumes "f v = v'"   and "(u, v) \<in> (\<Union> (relations M ` as))\<^sup>+" shows "thesis"
default_def: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) mapping" shows "Mapping.default k v m = (if k \<in> keys m then m else update k v m)"
rep_eq: fixes x :: "'g \<Rightarrow>\<^sub>L 'f"   and xa :: "'e \<Rightarrow>\<^sub>L 'g" shows "blinfun_apply (x o\<^sub>L xa) = (blinfun_apply x \<circ>\<circ> blinfun_apply) xa"
gcd_exp: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (power.power one times a n) (power.power one times b n) = power.power one times (gcd a b) n"
ab_group_add_axioms_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" shows "class.ab_group_add_axioms plus zero minus uminus \<equiv> (\<forall>a. plus (uminus a) a = zero) \<and> (\<forall>a b. minus a b = plus a (uminus b))"
comp_Rel_components_3: fixes S :: "V"   and T :: "V" shows "(S \<circ>\<^sub>R\<^sub>e\<^sub>l T)\<lparr>ArrCod\<rparr> = S\<lparr>ArrCod\<rparr>"
valid_expansion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<tturnstile> \<phi>" shows "K \<oplus> \<phi> = K"
padic_mult_comm0: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "prime p" shows "padic_mult p x y = padic_mult p y x"
ground_instance: fixes t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_on (vars_of t) \<sigma>" shows "ground_term (t \<lhd> \<sigma>)"
f: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" assumes "comp_fun_commute_on f A" shows "f \<in> A \<rightarrow> A \<rightarrow> A"
shift2: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and I :: "nat set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and ii :: "nat" assumes "PL_Indis cval indis"   and "I \<subseteq> {..<PL.brn (cl ! n)}"   and "ii \<in> PL_Indis.shift cl n I" shows "PL.brnL cl n \<le> ii \<and> ii < PL.brnL cl n + PL.brn (cl ! n)"
simps_1: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalQuadratic var a b c d (Eq p) = allZero p var"
inject_1: fixes x1 :: "'b \<times> 'b fs_set \<times> 'a"   and y1 :: "'b \<times> 'b fs_set \<times> 'a" shows "(AC x1 = AC y1) = (x1 = y1)"
degree_aform_inverse_aform: fixes p :: "nat"   and X :: "real \<times> real pdevs"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "inverse_aform p X = Some Y"   and "degree_aform X \<le> n" shows "degree_aform_err Y \<le> n"
simps_7: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (Eq a) = (Inum bs a = 0)"
vec_norm_def: fixes v :: "complex vec" shows "vec_norm v \<equiv> csqrt (inner_prod v v)"
qGoodInp_iff_goodInp_asInp: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" shows "goodInp (asInp qinp) = qGoodInp qinp"
mem_set_imp_le_max_list: fixes xs :: "'a list"   and a :: "'a" assumes "\<And>a b. \<lbrakk>a \<in> set xs; b \<in> set xs\<rbrakk> \<Longrightarrow> a \<le> b \<or> b \<le> a"   and "a \<in> set xs" shows "a \<le> Missing_Lemmas.max_list xs"
map_of_defined_if_constructed_from_list_of_constant_assignments: fixes l :: "('c \<times> 'd) list"   and a :: "'d"   and xs :: "'c list" assumes "l = map (\<lambda>x. (x, a)) xs" shows "\<forall>x\<in>set xs. map_of l x = Some a"
floor_add: fixes x :: "'b"   and y :: "'b" shows "\<lfloor>x + y\<rfloor> = (if frac x + frac y < (1::'b) then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> + 1)"
assms: shows "|x\<acute>=f & G on U S @ t\<^sub>0] C = (\<lambda>s. True)"
StateParallel_form: fixes P :: "'d hrel"   and a :: "'e \<Longrightarrow> 'd"   and b :: "'f \<Longrightarrow> 'd"   and Q :: "'d hrel" shows "P |a|b|\<^sub>\<sigma> Q = (\<^bold>\<exists> (st\<^sub>0, st\<^sub>1) \<bullet> P\<lbrakk>\<guillemotleft>st\<^sub>0\<guillemotright>/$\<^bold>v\<acute>\<rbrakk> \<and> Q\<lbrakk>\<guillemotleft>st\<^sub>1\<guillemotright>/$\<^bold>v\<acute>\<rbrakk> \<and> $\<^bold>v\<acute> =\<^sub>u bop (\<triangleleft>\<^bsub>&b\<^esub>) (bop (\<triangleleft>\<^bsub>&a\<^esub>) $\<^bold>v \<guillemotleft>st\<^sub>0\<guillemotright>) \<guillemotleft>st\<^sub>1\<guillemotright>)"
eq_general: fixes B :: "'b set"   and A :: "'c set"   and h :: "'c \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> 'd"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "sum \<phi> A = sum \<gamma> B"
lookup_pair_sort_oalist_aux: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "lookup_pair ko (sort_oalist_aux ko xs) = lookup_raw xs"
assms_2: shows "(v, w, v') \<in> E \<or> (v', w, v) \<in> E"
ccpo_Sup_least: fixes A :: "'b set"   and z :: "'b" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z" shows "\<Or> A \<sqsubseteq> z"
N_of_state_Liminf: shows "N_of_state (Liminf_state Sts) = Liminf_llist (lmap N_of_state Sts)"
TI: fixes u :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set"   and a :: "'c set"   and b :: "'c set"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "u \<in> timpl_closure t TI"   and "(a, b) \<in> TI"   and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s" shows "s \<in> timpl_closure t TI"
simps_2: fixes n :: "nat" shows "rqs_cost_exp (Suc n) = real n + (\<Sum>i\<le>n. rqs_cost_exp i + rqs_cost_exp (n - i)) / real (Suc n)"
finite_ordLess_var: fixes S :: "'b set" shows "(|S| <o |UNIV| \<or> finite S) = (|S| <o |UNIV|)"
divide_pos_pos: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) < x"   and "(0::'a) < y" shows "(0::'a) < x / y"
tendsto_cmult_ereal_not_0: fixes x :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and c :: "ereal" assumes "x \<noteq> 0"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
dgrad_set_exhaust_expl: fixes F :: "'a set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" shows "F \<subseteq> dgrad_set d (Max (d ` F))"
boolean_unop_type_det: fixes op :: "boolean_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "boolean_unop_type op \<tau> \<sigma>\<^sub>1"   and "boolean_unop_type op \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
exec: shows "exec_move ci P t e3 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
inner_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "x \<bullet> y = (\<Sum>i\<in>UNIV. x $ i \<bullet> y $ i)"
mod_mult_self1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times c b)) b = modulo a b"
transaction_send_def: fixes prot_transaction :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_send prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x6)"
convex_hull_list_def: fixes Vs :: "'a vec list" shows "n.convex_hull_list Vs = {x. \<exists>c. n.convex_lincomb_list c Vs x}"
inv_inv: fixes f :: "'b" assumes "D.iso f" shows "D.inv (D.inv f) = f"
suppSubst: fixes M :: "'a"   and xvec :: "name list"   and Tvec :: "'b list" shows "supp (M[xvec::=Tvec]) \<subseteq> supp M \<union> supp xvec \<union> supp Tvec"
B_def: shows "B = (\<lambda>j. mat (length ws) (length ws) (\<lambda>(i, j'). ws ! i $ (if j' < j then j' else Suc j')))"
dim_rows_le_dim_columns: fixes A :: "((real, 'a) vec, 'b) vec" shows "dim (rows A) \<le> dim (columns A)"
seq_comp_prop_2_aux: fixes P :: "'c option process"   and ws :: "'c option list"   and X :: "'c option set"   and Q :: "'c option process"   and xs :: "'c option list"   and x :: "'c option" assumes "weakly_sequential P"   and "(ws, X) \<in> seq_comp_failures P Q"   and "ws = xs @ [x]" shows "(xs, {}) \<in> seq_comp_failures P Q"
maxsimp_in_opp_half_apartment: fixes C :: "'a set" assumes "SimplicialComplex.maxsimp folding.Y C" shows "C \<in> folding.\<C> - f \<turnstile> folding.\<C>"
scaleR_conv_octo_of_real: fixes r :: "real"   and x :: "octo" shows "r *\<^sub>R x = octo_of_real r * x"
right_total_rel_set: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "right_total A" shows "right_total (rel_set A)"
assms_2: shows "\<A> \<subset> \<B> - {{}}"
mangoldt_1_def: fixes d :: "nat" shows "mangoldt_1 d = (if prime d then ln (real d) else 0)"
Un_Int_eq_3: fixes S :: "'b set"   and T :: "'b set" shows "S \<inter> (S \<union> T) = S"
ipt_iprange_to_set_uncurry_IpAddrNetmask: fixes a :: "'b word \<times> nat" shows "ipt_iprange_to_set (uncurry IpAddrNetmask a) = uncurry ipset_from_cidr a"
ln_upper_9_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_9 x"
Maclaurin_sin_expansion2: fixes x :: "real"   and n :: "nat" shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + sin (t + 1 / 2 * real n * pi) / fact n * x ^ n"
N0': shows "n0 = 0 \<or> execMsg (n0 - 1) = msg"
p_one_def: shows "p_one \<equiv> {p. \<exists>q. p = [q]} \<union> {[]}"
le_neq_trans: fixes a :: "nat list"   and b :: "nat list" assumes "a \<le>\<^sub>v b"   and "a \<noteq> b" shows "a <\<^sub>v b"
col_conjugate: fixes i :: "nat"   and m :: "'b mat" assumes "i < dim_col m" shows "col (conjugate m) i = conjugate (col m i)"
hom_0_iff: fixes x :: "'b"   and d :: "nat" shows "(monom x d = 0) = (x = (0::'b))"
usubst_upd_var_id: fixes x :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x" shows "[x \<mapsto>\<^sub>s x] = id"
preserves_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.map (CC.inv f) = CC.inv (CC.map f)"
atMost_shift: fixes g :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "prod g {..n} = g 0 * (\<Prod>i<n. g (Suc i))"
negligible_atLeastAtMostI: fixes b :: "real"   and a :: "real" assumes "b \<le> a" shows "negligible {a..b}"
cases: fixes x :: "exp" obtains xa :: "nat" where "x = EVar xa__"   | n :: "nat" where "x = ENat n"   | xa :: "nat" and e :: "exp" where "x = ELam xa e"   | e1 :: "exp" and e2 :: "exp" where "x = EApp e1__ e2__"   | f :: "nat \<Rightarrow> nat \<Rightarrow> nat" and e1 :: "exp" and e2 :: "exp" where "x = EPrim f e1a__ e2a__"   | e1 :: "exp" and e2 :: "exp" and e3 :: "exp" where "x = EIf e1 e2 e3"
perm_2: shows "\<forall>i<length ys. ys ! i = zs ! g i"
hom_power: fixes x :: "'b poly"   and n :: "nat"   and a :: "'b" shows "poly (x ^ n) a = poly x a ^ n"
rep_eq: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c \<Rightarrow> 'd"   and xb :: "'c \<Rightarrow> 'c" shows "lgenerator (unfold_llist_prod x xa xb) = unfold_llist_raw x xa xb"
LPD_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex" assumes "ok m mr"   and "ms \<in> LPD mr" shows "ok m ms"
lasso_cycle_def: fixes L :: "('d, 'e) lasso_scheme" shows "lasso_cycle L = lasso_va L # lasso_cysfx L"
inverse_distrib_swap: fixes a :: "'c"   and b :: "'c" shows "- (a + b) = - b + - a"
Cn_disj_bis: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<or>. q) = (Cn (A \<union> {q}) \<subseteq> Cn (A \<union> {.\<not> p}))"
msubset_of_1: fixes F :: "('c \<Rightarrow> 'd) set"   and A :: "'c set"   and B :: "'d set" assumes "F \<in> (A \<rightarrow>\<^sub>E B) // domain_permutation A B" shows "size (msubset_of A F) = card A"
non_neg_mat_mult_non_neg_vec: fixes a :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('a, 'b) Finite_Cartesian_Product.vec" assumes "non_neg_mat a"   and "non_neg_vec v" shows "non_neg_vec (a *v v)"
fold_deriv_FNot: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and xs :: "'x list"   and \<phi> :: "('a, 'k) aformula" shows "fold (deriv d0) xs (FNot \<phi>) = FNot (fold (deriv d0) xs \<phi>)"
distinct_3: fixes x1 :: "name"   and x31 :: "name"   and x32 :: "Lambda_Z.term" shows "Var x1 \<noteq> Abs x31 x32"
Sup_measure'_def: fixes M :: "'b measure set" shows "Sup_measure' M = measure_of (\<Union> (space ` M)) (\<Union> (sets ` M)) (\<lambda>X. SUP P\<in>{P. finite P \<and> P \<subseteq> M}. emeasure (comm_monoid_set.F sup bot id P) X)"
stabilizer_m_closed: fixes x :: "'c"   and g1 :: "'a"   and g2 :: "'a" assumes "x \<in> E"   and "g1 \<in> stabilizer G \<phi> x"   and "g2 \<in> stabilizer G \<phi> x" shows "g1 \<otimes> g2 \<in> stabilizer G \<phi> x"
extf_wf: fixes f :: "'s" shows "ext_wf (extf f)"
VSEqualityBlocksLimited_def: shows "VSEqualityBlocksLimited = opt_group \<circ> QE_dnf opt_group (step_augment qe_eq_repeat IdentityHeuristic) \<circ> VSLuckiestBlocks \<circ> opt_group"
induced_hom_def: shows "F \<equiv> restrict0 (quotientfun (res_freeword_funlift f S) \<circ> the_inv_into (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+))) (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) ` FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S))"
inorder_balL: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "Tree2.inorder (balL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
eme1p: fixes n :: "int"   and d :: "int" assumes "even n"   and "even d"   and "0 \<le> d" shows "(1 + n) mod d = 1 + n mod d"
X: shows "range (\<lambda>i s. 0::'a) = {\<lambda>s. 0::'a}"
orthogonal_complement_subset: fixes A :: "'b vec set"   and B :: "'b vec set" assumes "A \<subseteq> B" shows "orthogonal_complement B \<subseteq> orthogonal_complement A"
psym_lens_def: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" shows "psym_lens S \<equiv> (mwb_lens \<V>\<^bsub>S\<^esub> \<and> mwb_lens \<C>\<^bsub>S\<^esub>) \<and> \<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub> \<and> pbij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)"
InvariantWatchCharacterizationAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantUniq (getM state @ [(literal, decision)])"   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 "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)" shows "let state' = assertLiteral literal decision state in InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
ins_invariant: fixes x :: "bool list"   and S :: "nat option bdd \<times> bool list list" assumes "nfa_is_node A x"   and "subset_invariant A S"   and "\<not> subset_memb x S" shows "subset_invariant A (subset_ins x S)"
pdevs_apply_pdevs_of_list_append: fixes xs :: "'b list"   and zs :: "'b list"   and i :: "nat" shows "pdevs_apply (pdevs_of_list (xs @ zs)) i = (if i < length xs then pdevs_apply (pdevs_of_list xs) i else pdevs_apply (pdevs_of_list zs) (i - length xs))"
wfactors_unique: fixes fs :: "'a list"   and a :: "'a"   and fs' :: "'a list" assumes "wfactors G fs a"   and "wfactors G fs' a"   and "a \<in> carrier G"   and "set fs \<subseteq> carrier G"   and "set fs' \<subseteq> carrier G" shows "essentially_equal G fs fs'"
eq_fold: fixes A :: "'a set" shows "sorted_list_of_set A = Finite_Set.fold insort [] A"
safe_max_Some_dest_le: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "finite X"   and "safe_max X = Some x"   and "y \<in> X" shows "y \<le> x"
MemoryInvariantAll: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" shows "\<turnstile> (\<forall>l. #l \<in> #MemLoc \<longrightarrow> MSpec ch mm rs l) \<longrightarrow> (\<forall>l. \<box>MemInv mm l)"
b_t: shows "T\<^sub>p_on_rand' BIT (type0 init x y) v = 75 / 10\<^sup>2 * real (length v) - 5 / 10"
tt_plus_lessE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p + q \<noteq> 0"   and "tt (p + q) \<prec>\<^sub>t tt p" shows "tt q \<prec>\<^sub>t tt p"
seqr_left_zero: fixes P :: "('d, 'c) urel" shows "false ;; P = false"
alice_pos_def: fixes \<phi> :: "complex Matrix.mat"   and q :: "complex Matrix.mat" shows "alice_pos \<phi> q \<equiv> q = Tensor.mat_of_cols_list 8 [[\<phi> $$ (0, 0), \<phi> $$ (1, 0), 0, 0, 0, 0, 0, 0]] \<or> q = Tensor.mat_of_cols_list 8 [[0, 0, \<phi> $$ (1, 0), \<phi> $$ (0, 0), 0, 0, 0, 0]] \<or> q = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, \<phi> $$ (0, 0), - \<phi> $$ (1, 0), 0, 0]] \<or> q = Tensor.mat_of_cols_list 8 [[0, 0, 0, 0, 0, 0, - \<phi> $$ (1, 0), \<phi> $$ (0, 0)]]"
lm081: fixes X :: "'c set"   and Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "X \<inter> Domain Q = {}" shows "(P +< Q) `` X = Outside' (Domain Q) P `` X"
assms_2: shows "mustT (if tst then c1 else c2) s"
Pi_pmf_bij_betw: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and B :: "'b set"   and dflt :: "'c"   and f :: "'c pmf" assumes "finite A"   and "bij_betw h A B"   and "\<And>x. x \<notin> A \<Longrightarrow> h x \<notin> B" shows "Pi_pmf A dflt (\<lambda>_. f) = map_pmf (\<lambda>g. g \<circ> h) (Pi_pmf B dflt (\<lambda>_. f))"
nsum_cmp_assoc: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "\<forall>j\<le>n. f j \<in> carrier A"   and "g \<in> {j. j \<le> n} \<rightarrow> {j. j \<le> n}"   and "h \<in> {j. j \<le> n} \<rightarrow> {j. j \<le> n}" shows "\<Sigma>\<^sub>e A cmp (cmp f h) g n = \<Sigma>\<^sub>e A cmp f (cmp h g) n"
seqSubst2: fixes x :: "name"   and \<sigma> :: "(name list \<times> 'a list) list"   and T :: "'c" assumes "x \<sharp> \<sigma>"   and "x \<sharp> T" shows "x \<sharp> substCond.seqSubst T \<sigma>"
distinct_263: shows "BinXor \<noteq> ShiftRightZeros"
Min_eqI: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> x \<le> y"   and "x \<in> A" shows "Min A = x"
exhaust: fixes y :: "'b ext0" obtains "y = ext0.Bot"   | x2 :: "'b" where "y = ext0.Val x2"   | "y = ext0.Top"
IH: fixes F :: "'a formula" shows "(set GS \<turnstile> F) = ({} \<turnstile> \<^bold>\<And>GS \<^bold>\<rightarrow> F)"
cancels_to_1_def: fixes l1 :: "(bool \<times> 'b) list"   and l2 :: "(bool \<times> 'b) list" shows "cancels_to_1 l1 l2 = (\<exists>i. cancels_to_1_at i l1 l2)"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus_ii: fixes \<psi> :: "('c, 'd) sas_plus_operator list"   and G :: "'c \<Rightarrow> 'd option"   and I :: "'c \<Rightarrow> 'd option" assumes "\<forall>op\<in>set \<psi>. are_operator_effects_consistent op op"   and "G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I \<psi>" shows "G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I (List_Supplement.embed \<psi>)"
trm_induct: fixes P :: "'b trm \<Rightarrow> bool"   and M :: "'b trm" assumes "P Unit"   and "\<And>x. P (Var x)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (App A B)"   and "\<And>x T A. P A \<Longrightarrow> P (Fn x T A)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (SimplyTyped.Pair A B)"   and "\<And>A. P A \<Longrightarrow> P (Fst A)"   and "\<And>A. P A \<Longrightarrow> P (Snd A)" shows "P M"
reachable_points_iff: fixes p :: "'a"   and x :: "'a" assumes "point p" shows "x\<^sup>T\<^sup>\<star> ; p \<cdot> x = (x\<^sup>T\<^sup>\<star> ; p \<cdot> 1') ; x"
prime_elements_with_ord_3: fixes d :: "nat"   and a :: "nat" assumes "\<psi> d \<noteq> 0"   and "prime n"   and "a \<in> totatives n"   and "ord n a = d"   and "a \<noteq> 1" shows "bij_betw (\<lambda>k. a ^ k mod n) (totatives d) {x \<in> {..<n}. ord n x = d}"
simps_3: fixes uv :: "'b extT" shows "extT.Top + uv = extT.Top"
SE1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(\<box>F \<and> G)"
1: shows "trg f \<star> f \<star> e \<cong> trg f \<star> f' \<star> e"
insert_disjoint_1: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
app_not_snd: fixes A :: "'b trm"   and B :: "'b trm"   and P :: "'b trm" shows "App A B \<noteq> Snd P"
Seq_WbisT_Wbis: fixes c1 :: "('test, 'atom) com"   and d1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and d2 :: "('test, 'atom) com" assumes "c1 \<approx>wT d1"   and "c2 \<approx>w d2" shows "c1 ;; c2 \<approx>w d1 ;; d2"
rep_vertices_in_blocks: fixes G :: "('a, 'b) pre_digraph"   and \<r> :: "int"   and x :: "'a" assumes "regular_graph G \<r>"   and "x \<in> verts G" shows "int (size (filter_mset ((\<in>) x) (graph.arcs_blocks G))) = \<r>"
subset_with_given_card_exists: fixes n :: "nat"   and A :: "'c set" assumes "n \<le> card A" shows "\<exists>B\<subseteq>A. card B = n"
monom_card_pow_mod_sum_berlekamp: fixes i :: "nat"   and u :: "'a mod_ring poly" assumes "i < degree u" shows "monom 1 (CARD('a) * i) mod u = (\<Sum>j<degree u. monom (local.berlekamp_mat u $$ (i, j)) j)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> bankerb_gets" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
steps_ConsD: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "Graph_Defs.steps E (x # xs)"   and "xs \<noteq> []" shows "Graph_Defs.steps E xs"
m\<^sub>a: shows "adjoint_pair m m\<^sup>* \<and> w \<cong> n \<star> m\<^sup>*"
catalan_Suc_aux: fixes n :: "nat" shows "(n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n"
inner_neg_move_base: fixes a :: "'b"   and b :: "'b"   and n :: "'b"   and x :: "'b"   and y :: "'b" assumes "(a - b) \<bullet> n = 0"   and "(x - y) \<bullet> n < 0"   and "y \<in> {a--b}" shows "(x - a) \<bullet> n < 0"
assms_6: fixes r :: "'a" assumes "I r s"   and "\<not> C r s" shows "Q (Some r)"