hcju/isabelleps huggingface dataset:
f_Exec_Comp_Stream_Acc_LocalState_def: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c \<equiv> map localState (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) \<div>\<^sub>l k"
valid_pub_def: shows "valid_pub = carrier \<G> \<times> carrier \<G>"
simps_2: fixes \<mu> :: "'c" shows "Dom \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>C.dom \<mu>\<^bold>\<rangle>"
ground_vars_term_empty: fixes t :: "('c, 'd) Term.term" shows "ground t = (vars_term t = {})"
to_nat_1: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep one = 1"
Bet_perm: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Bet A B C \<and> Bet C B A"
dist_increases_online: fixes d :: "'b"   and a :: "'b"   and b :: "'b" assumes "d \<noteq> (0::'b)" shows "dist a b < dist a (b + d) \<or> dist a b < dist a (b - d)"
spanset_contains_basis: fixes vs :: "'v list" assumes "set vs \<subseteq> V" shows "\<exists>us. set us \<subseteq> set vs \<and> set us \<subseteq> RSpan vs \<and> RSpan vs = RSpan us \<and> R_lin_independent us"
intros_6: fixes s1 :: "'b list"   and r :: "'b rexp"   and v :: "'b val"   and s2 :: "'b list"   and vs :: "'b val list" assumes "s1 \<in> r \<rightarrow> v"   and "s2 \<in> Star r \<rightarrow> Stars vs"   and "flat v \<noteq> []"   and "\<nexists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r)" shows "(s1 @ s2) \<in> Star r \<rightarrow> Stars (v # vs)"
unilateral_substitutes_onD: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set"   and B :: "'x set"   and a :: "'x"   and b :: "'x" assumes "unilateral_substitutes_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "b \<in> A"   and "Xd b \<notin> Xd ` B"   and "b \<notin> f (insert b B)" shows "b \<notin> f (insert a (insert b B))"
inj_GmodH_M: shows "\<exists>g\<in>rcosets H \<rightarrow> M. inj_on g (rcosets H)"
simps_1: fixes return :: "unit \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> (unit \<Rightarrow> 'a) \<Rightarrow> 'a"   and mx :: "'a" shows "hanoi_base.repeat return bind 0 mx = return ()"
assms_1: shows "tm_floatarith prec ord I xs f = Some t"
IH_1: fixes ds :: "bool list" shows "treesize (cutoff red ds T1) \<le> treesize T1"
simps_1: fixes \<phi>\<^sub>1 :: "nat formula"   and \<phi>\<^sub>2 :: "nat formula" shows "cnf_to_dimacs (\<phi>\<^sub>1 \<^bold>\<and> \<phi>\<^sub>2) = cnf_to_dimacs \<phi>\<^sub>1 @ cnf_to_dimacs \<phi>\<^sub>2"
exu: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "exu x \<phi> \<in> fmla"
subst_idemp_iff: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "(\<sigma> \<circ>\<^sub>s \<sigma> = \<sigma>) = (subst_domain \<sigma> \<inter> range_vars \<sigma> = {})"
dims_Aw'_pow_2: shows "dim_col Aw' = last rs ^ N_half"
fmlookup_fmmap_keys: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and m :: "('d, 'e) fmap"   and x :: "'d" shows "fmmap_keys f m $$ x = map_option (f x) (m $$ x)"
that: fixes as' :: "'edge list"   and a' :: "'edge" assumes "as = as' @ [a']"   and "n -as'\<rightarrow>* (_Low_)"   and "kind a' = (\<lambda>s. True)\<^sub>\<surd>" shows "thesis"
elims_3: fixes x :: "Core.term" assumes "\<not> eta_reducible x" obtains uu_ :: "typ" and t :: "Core.term" where "x = Abs uu_ (t__ $ Bv 0)" and "\<not> (\<not> is_dependent t__ \<or> eta_reducible t__)"   | uv_ :: "typ" and v :: "String.literal" and va :: "typ" where "x = Abs uv___ (Ct v__ va__)" and "\<not> eta_reducible (Ct v__ va__)"   | uv_ :: "typ" and v :: "variable" and va :: "typ" where "x = Abs uva___ (Fv vb__ vaa__)" and "\<not> eta_reducible (Fv vb__ vaa__)"   | uv_ :: "typ" and v :: "nat" where "x = Abs uvb___ (Bv vc__)" and "\<not> eta_reducible (Bv vc__)"   | uv_ :: "typ" and v :: "typ" and va :: "Core.term" where "x = Abs uvc___ (Abs vd__ vab__)" and "\<not> eta_reducible (Abs vd__ vab__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = Abs uvd___ (ve__ $ Ct vba__ vca__)" and "\<not> eta_reducible (ve__ $ Ct vba__ vca__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = Abs uve___ (vf__ $ Fv vbb__ vcb__)" and "\<not> eta_reducible (vf__ $ Fv vbb__ vcb__)"   | uv_ :: "typ" and v :: "Core.term" and vc :: "nat" where "x = Abs uvf___ (vg__ $ Bv (Suc vcc__))" and "\<not> eta_reducible (vg__ $ Bv (Suc vcc__))"   | uv_ :: "typ" and v :: "Core.term" and vb :: "typ" and vc :: "Core.term" where "x = Abs uvg___ (vh__ $ Abs vbc__ vcd__)" and "\<not> eta_reducible (vh__ $ Abs vbc__ vcd__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = Abs uv_ (vi__ $ (vb $ vc))" and "\<not> eta_reducible (vi__ $ (vb $ vc))"   | t :: "Core.term" and u :: "Core.term" where "x = t $ u" and "\<not> (eta_reducible t \<or> eta_reducible u)"   | v :: "String.literal" and va :: "typ" where "x = Ct vj__ vac__"   | v :: "variable" and va :: "typ" where "x = Fv vk__ va"   | v :: "nat" where "x = Bv v"
fv_close_Var: fixes i :: "nat"   and xT :: "char list \<times> Unary_PCF.type"   and t :: "expr" shows "fv (close_Var i xT t) = fv t |-| {|xT|}"
dup_simps_2: fixes f :: "'a" assumes "arr f" shows "local.dom (ECC.dup f) = local.dom f"
removeDuplicateLiteralsEquivalentClause: fixes clause :: "Literal list" shows "equivalentFormulae [remdups clause] [clause]"
iptables_goto_bigstepD_1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = []" shows "s = t"
br_sv: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "single_valued (br \<alpha> I)"
preprm_unit_equal_id: fixes a :: "'b" shows "preprm_unit a a =p preprm_id"
semilattice_order_axioms: shows "semilattice_order gcd (dvd) (\<lambda>m n. m dvd n \<and> m \<noteq> n)"
assms_2: shows "u \<le>f v \<^sup>@ k"
rep_eq: shows "pfun_of {}\<^sub>f = {}\<^sub>p"
disjoint_family_on_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "A \<subseteq> B"   and "disjoint_family_on f B" shows "disjoint_family_on f A"
mat_append_row_dims: fixes A :: "'a mat"   and c :: "'a vec" shows "mat_append_row A c \<in> carrier_mat (dim_row A + 1) (dim_col A)"
open_ne_zero: fixes c :: "'b" shows "open {x. x \<noteq> c}"
list_ex_length: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_ex P xs = (\<exists>n<length xs. P (xs ! n))"
is_extensional: fixes f :: "(bool, bool list) concrete_category.arr" assumes "\<not> J.arr f" shows "\<chi> f = C.null"
unfold: shows "reverse = foldl\<cdot>(flip\<cdot>Cons)\<cdot>[]"
nth_default_beyond: fixes xs :: "'f list"   and n :: "nat"   and dflt :: "'f" assumes "length xs \<le> n" shows "nth_default dflt xs n = dflt"
at_map_sup_pres: fixes x :: "'a"   and y :: "'a" shows "at_map (x \<squnion> y) = at_map x \<squnion> at_map y"
pth_9_2: fixes c :: "real"   and x :: "'a"   and d :: "real"   and z :: "'a" shows "c *\<^sub>R x + (d *\<^sub>R x + z) \<equiv> (c + d) *\<^sub>R x + z"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "l_known_ptr\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some shadow_root_ptr \<Rightarrow> is_shadow_root_ptr shadow_root_ptr))"
hyps: assumes "s \<noteq> []" shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"
assms: shows "vdistinct (xs #\<^sub>\<circ> x)"
assms_3: shows "ring_hom_ring (K [X]) A g"
scaleC: fixes c :: "complex"   and x :: "'a ell2" shows "C1 (c *\<^sub>C x) = c *\<^sub>C C1 x"
simps_2: fixes p1 :: "'a"   and p2 :: "'a"   and v :: "'b" shows "isReceiverOf p1 <p2, v> = (p1 = p2)"
finite_Collect_bounded_ex: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite {y. P y}" shows "finite {x. \<exists>y. P y \<and> Q x y} = (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
B_carrier: shows "B \<in> carrier_mat nr (n + 1)"
fall_elim: fixes F :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" assumes "fall F \<pi>l" obtains "\<And>\<pi>. \<lbrakk>wfp UNIV \<pi>; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0\<rbrakk> \<Longrightarrow> F \<pi> \<pi>l"
monom_list_mult_list_vars: fixes m1 :: "('a \<times> nat) list"   and m2 :: "('a \<times> nat) list" shows "fst ` set (monom_mult_list m1 m2) = fst ` set m1 \<union> fst ` set m2"
a_l_coset_defs_1: fixes G :: "('c, 'd) ring_scheme" shows "(<+\<^bsub>G\<^esub>) = (<#\<^bsub>add_monoid G\<^esub>)"
cases: fixes a :: "('g, 'h, 'i, 'j) com" assumes "basic_com a" obtains l :: "'h" and action :: "'j \<Rightarrow> 'i" and val :: "'g \<Rightarrow> 'j \<Rightarrow> 'j set" where "a = \<lbrace>l__\<rbrace> Request action__ val"   | l :: "'h" and action :: "'i \<Rightarrow> 'j \<Rightarrow> ('j \<times> 'g) set" where "a = \<lbrace>la__\<rbrace> Response action"   | l :: "'h" and R :: "'j \<Rightarrow> 'j set" where "a = \<lbrace>lb__\<rbrace> LocalOp R"   | l :: "'h" and b :: "'j \<Rightarrow> bool" and c :: "('g, 'h, 'i, 'j) com" where "a = \<lbrace>lc__\<rbrace> IF b__ THEN c__ FI"   | l :: "'h" and b :: "'j \<Rightarrow> bool" and c1 :: "('g, 'h, 'i, 'j) com" and c2 :: "('g, 'h, 'i, 'j) com" where "a = \<lbrace>ld__\<rbrace> IF ba__ THEN c1 ELSE c2 FI"   | l :: "'h" and b :: "'j \<Rightarrow> bool" and c :: "('g, 'h, 'i, 'j) com" where "a = \<lbrace>l\<rbrace> WHILE b DO c OD"
dbm_abstr_completeness: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and cc :: "('c, 'd) cconstraint"   and n :: "nat" assumes "u \<turnstile>\<^bsub>v,n\<^esub> abstr cc (\<lambda>i j. \<infinity>) v"   and "\<forall>c. 0 < v c"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "u \<turnstile> cc"
similar_mat_wit_kernel_dim: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat n n"   and "similar_mat_wit A B P Q" shows "vectorspace.dim class_ring (module_vec TYPE('b) n\<lparr>carrier := mat_kernel A\<rparr>) = vectorspace.dim class_ring (module_vec TYPE('b) n\<lparr>carrier := mat_kernel B\<rparr>)"
ChamberComplexMorphism_def: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" shows "ChamberComplexMorphism X Y f \<equiv> ChamberComplex X \<and> ChamberComplex Y \<and> ChamberComplexMorphism_axioms X Y f"
stake_Suc: fixes n :: "nat"   and s :: "'c stream" shows "stake (Suc n) s = stake n s @ [s !! n]"
hyps: fixes used :: "'a set" shows "\<exists>extra. witness_list (ps @ qs) used = witness_list ps used @ witness_list qs (extra \<union> used)"
card_Pow: fixes A :: "'a set" assumes "finite A" shows "card (Pow A) = 2 ^ card A"
new_shadow_root_types_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
range_oVeblen: fixes x :: "ordinal" shows "range (oVeblen F x) = critical_set (range F) x"
rcis_mult: fixes r1 :: "real"   and a :: "real"   and r2 :: "real"   and b :: "real" shows "rcis r1 a * rcis r2 b = rcis (r1 * r2) (a + b)"
assms: shows "abs_convergent_prod (\<lambda>n. f (n + m))"
enum_all_Agent_def: fixes P :: "Agent \<Rightarrow> bool" shows "enum_class.enum_all P = (P Child0 \<and> P Child1 \<and> P Child2)"
simple_state_relI: fixes s' :: "('e, 'f) state_scheme"   and m' :: "'g"   and erel :: "('g \<times> 'f) set" assumes "dom (finished s') = dom (discovered s') - set (stack s')"   and "set (stack s') \<subseteq> dom (discovered s')"   and "(m', state.more s') \<in> erel" shows "(\<lparr>ss_stack = map (\<lambda>u. (u, pending s' `` {u})) (stack s'), on_stack = set (stack s'), visited = dom (discovered s'), \<dots> = m'\<rparr>, s') \<in> \<langle>erel\<rangle>simple_state_rel"
vars_subst_in_left_only: fixes x :: "nat"   and p :: "int mpoly" assumes "x \<notin> vars p" shows "x \<in> vars (p - Var x)"
ind_sheaf_is_presheaf: shows "presheaf_of_rings X (ind_topology.ind_is_open X is_open X) local.ind_sheaf ind_ring_morphisms b ind_add_str ind_mult_str ind_zero_str ind_one_str"
h: shows "\<forall>x. undo_factorize_polys data (sign_vec ftrs x) = sign_vec ps x"
poincare_on_ray_poincare_collinear: fixes p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" assumes "p \<in> unit_disc"   and "a \<in> unit_disc"   and "b \<in> unit_disc"   and "poincare_on_ray p a b" shows "poincare_collinear {p, a, b}"
g_2: fixes x :: "nat" shows "(x \<in> A) = (\<exists>y. g x y = 0)"
map_id: fixes t :: "'b rexp" shows "map_rexp id t = t"
simps_2: fixes a :: "'b"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "(a ## w) (Suc i) = w i"
LIMSEQ_SUP: fixes X :: "nat \<Rightarrow> 'a" assumes "incseq X" shows "X \<longlonglongrightarrow> \<Squnion> range X"
inv_rule_from_INV_rule: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set"   and P :: "'e set" assumes "init T \<subseteq> I"   and "I \<subseteq> P"   and "{I} TS.trans T {> I}" shows "reach T \<subseteq> P"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c option"   and b :: "'d option" shows "rel_option R\<inverse>\<inverse> a b = rel_option R b a"
z: shows "z \<equiv> getFr ([xx, yy] @ map snd txs) ([t1, t2] @ map fst txs) []"
intros_6: fixes c :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and x :: "'d" shows "(c x, DynCom c) \<in> com_rel"
power_inductl_var: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<cdot> y \<le> y" shows "x\<^bsup>n\<^esup> \<cdot> y \<le> y"
decomposition_theorem_integer_hull_of_polyhedron: fixes A :: "'a mat"   and nr :: "nat"   and b :: "'a vec"   and P :: "'a vec set"   and Bnd :: "'a" assumes "A \<in> carrier_mat nr n"   and "b \<in> carrier_vec nr"   and "A \<in> \<int>\<^sub>m"   and "b \<in> \<int>\<^sub>v"   and "P = polyhedron A b"   and "Bnd = Max (abs ` (elements_mat A \<union> set\<^sub>v b))" shows "\<exists>H C. H \<union> C \<subseteq> carrier_vec n \<inter> \<int>\<^sub>v \<and> H \<subseteq> Bounded_vec (fact (n + 1) * max (1::'a) Bnd ^ n) \<and> C \<subseteq> Bounded_vec (fact n * max (1::'a) Bnd ^ n) \<and> finite (H \<union> C) \<and> integer_hull P = convex_hull H + cone C"
inf_bot_left: fixes x :: "'c multiset" shows "{#} \<inter># x = {#}"
less_imp_triv: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y" shows "(less y x \<longrightarrow> P) = True"
cyc_post_def: shows "cyc_post = (\<lambda>blues reds onstack u0 cyc. case cyc of NO_CYC \<Rightarrow> add_inv blues reds onstack | REACH u p \<Rightarrow> path E u0 p u \<and> u \<in> onstack - {u0} \<and> insert u (set p) \<inter> F \<noteq> {} | CIRC pr pl \<Rightarrow> \<exists>v. pl \<noteq> [] \<and> path E v pl v \<and> path E u0 pr v \<and> set pl \<inter> F \<noteq> {})"
subcls1_irrefl_lemma2: fixes G :: "prog" assumes "ws_prog G" shows "\<forall>x. (x, x) \<notin> (subcls1 G)\<^sup>+"
monom_1_b: fixes m' :: "nat"   and b :: "nat" assumes "monom 1 1 \<in> carrier R"   and "monom 1 1 [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom 1 1"   and "b < degree f" shows "monom 1 b [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom 1 b"
seq': fixes rs :: "'c rule list"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t"   and "\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow>\<^sub>\<alpha> t'" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t'"
sub_branch_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and branch :: "(('b, 'e) fm list \<times> 'e) list" shows "sub_branch f (sub_branch g branch) = sub_branch (f \<circ> g) branch"
top: shows "UNIV \<in> sigma_sets UNIV (range lessThan)"
succ_inv_1: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "\<triangle> (\<T> s')"
path_connectedin_subset_topspace: fixes X :: "'b topology"   and S :: "'b set" assumes "path_connectedin X S" shows "S \<subseteq> topspace X"
Deriv_inter: fixes a :: "'a"   and A :: "'a list set"   and B :: "'a list set" shows "Deriv a (A \<inter> B) = Deriv a A \<inter> Deriv a B"
pbeta_Fvar: fixes x :: "char list" shows "Fvar x => Fvar x"
less_next: fixes x :: "ordinal" shows "x < (LEAST z. z \<in> A \<and> x < z)"
sup_bot_left: fixes x :: "'a multiset" shows "{#} \<union># x = x"
a': shows "( \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>, n) \<in> derivable  R*"
hcomp_simps_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "hseq \<nu> \<mu>" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
jordan_block_poly_bound: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and a :: "'b"   and k :: "nat" assumes "i < n"   and "j < n"   and "norm a = 1" shows "norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)) \<le> max 1 (real k ^ (n - 1))"
restrict_to_halfspaces_clw: fixes X :: "'b set"   and sctns :: "'b sctn set" shows "sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. restrict_to_halfspaces sctns X \<bind> (\<lambda>R. RETURN (filter_empty_ivls (mk_coll R)))) (\<lambda>X Y. RETURN (Y \<union> X))) \<le> restrict_to_halfspaces_clw sctns X"
simps_2: fixes y :: "'a"   and P :: "'a pred"   and x :: "'a" shows "member (Insert y P) x = (x = y \<or> eval P x)"
em_inter: fixes a :: "'a"   and b :: "'a" assumes "a \<in> J"   and "b \<in> J"   and "a \<noteq> b" shows "(H #> a) \<inter> (H #> b) = {}"
map_prod_inj_on: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'f \<Rightarrow> 'g"   and B :: "'f set" assumes "inj_on f A"   and "inj_on g B" shows "inj_on (map_prod f g) (A \<times> B)"
power_commute_rewrite: shows "\<^bold>g [^] ((order \<G> - s + u1') mod order \<G>) = \<^bold>g [^] (u1' + (order \<G> - s))"
iter: shows "iter f step start (unstables r step start) = (ss, w)"
All_mapping_def: fixes m :: "('c, 'd) mapping"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "All_mapping m P = (\<forall>x. case lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
quat_of_real_times_left_commute: fixes r :: "real"   and p :: "quat"   and q :: "quat" shows "quat_of_real r * (p * q) = p * (quat_of_real r * q)"
suc: shows "path \<sigma> (Suc j) \<noteq> path \<sigma>' (Suc j')"
nonneg_cexpr_sem_marg_dens2: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and y :: "nat"   and \<rho> :: "nat \<Rightarrow> val"   and v :: "val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "x \<in> set vs"   and "y \<in> set vs"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)"   and "v \<in> type_universe (PRODUCT (\<Gamma> x) (\<Gamma> y))" shows "0 \<le> extract_real (cexpr_sem (case_nat v \<rho>) (marg_dens2_cexpr \<Gamma> vs x y \<delta>))"
x2: shows "e\<^sub>2 \<Down> f\<^sub>2"
cfE_of_ss_invE: fixes cf :: "nat \<times> nat \<Rightarrow> 'a" shows "cfE_of cf \<subseteq> E \<union> E\<inverse>"
drf_weak: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "correctly_synchronized P \<E>"   and "weakly_legal_execution P \<E> (E, ws)" shows "sequentially_consistent P (E, ws)"
coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<le> b" shows "c \<le> a \<squnion> b"
mono_Min_commute: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "finite A"   and "A \<noteq> {}" shows "f (linorder.Min less_eq A) = Min (f ` A)"
complex_In_mult_cnj_zero: fixes z :: "complex" shows "Im (z * cnj z) = 0"
Propagate_Black_aux_def: shows "Propagate_Black_aux \<equiv> \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>E INV \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind \<le> length \<acute>E\<rbrace>  DO \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E\<rbrace>  IF \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black THEN \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv \<acute>ind \<and> \<acute>ind < length \<acute>E \<and> \<acute>M ! fst (\<acute>E ! \<acute>ind) = Black\<rbrace> \<acute>M := \<acute>M[snd (\<acute>E ! \<acute>ind) := Black];; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> \<acute>PBInv (\<acute>ind + 1) \<and> \<acute>ind < length \<acute>E\<rbrace> \<acute>ind := \<acute>ind + 1 FI OD"
sint_lt: fixes x :: "'b word" shows "sint x < 2 ^ (LENGTH('b) - 1)"
convergent_Cauchy_dist: fixes u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b" assumes "Cauchy u"   and "Cauchy v" shows "convergent (\<lambda>n. dist (u n) (v n))"
get_root_node_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root" shows "(root \<in> set c) = (ptr' \<in> set c)"
1: shows "\<forall>ys. length ys < length p \<longrightarrow> P e w ys"