hcju/isabelleps huggingface dataset:
pref_comp_not_pref: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie> v"   and "\<not> v \<le>p u" shows "u <p v"
isomorphic_group_cyclicity: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H"   and "Group.group G"   and "Group.group H" shows "cyclic_group G = cyclic_group H"
contour_integral_rmul: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "f contour_integrable_on g" shows "contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
from_hma\<^sub>v_diff: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v - Mod_Type_Connect.from_hma\<^sub>v w = Mod_Type_Connect.from_hma\<^sub>v (v - w)"
alive_trivial_simps_5: fixes s :: "Store" shows "alive nullV s"
preserves_reflects_arr: fixes f :: "'d \<times> 'c" shows "S.arr (\<Phi>.map f) = DopxC.arr f"
new_xcpt_var_def2: fixes vn :: "vname"   and x :: "abopt"   and s :: "st" shows "new_xcpt_var vn (x, s) = Norm (lupd(VName vn\<mapsto>Addr (the_Loc (the_Xcpt (the x)))) s)"
Legendre_1mod4: fixes k :: "nat" assumes "prime (4 * k + 1)" shows "Legendre (- 1) (int (4 * k + 1)) = 1"
w_simps_5: shows "local.dom \<mu>.chine = \<mu>.chine"
l9: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
Least_le: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "P k" shows "(LEAST x. P x) \<le> k"
polar_Ex: fixes x :: "real"   and y :: "real" shows "\<exists>r a. x = r * cos a \<and> y = r * sin a"
equal_trie_def: fixes t :: "('a, 'b) trie"   and t' :: "('a, 'b) trie" shows "equal_class.equal t t' = (impl_of t = impl_of t')"
override_on_eqvt: fixes \<pi> :: "perm"   and m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "\<pi> \<bullet> m1 ++\<^bsub>S\<^esub> m2 = (\<pi> \<bullet> m1) ++\<^bsub>\<pi> \<bullet> S\<^esub> (\<pi> \<bullet> m2)"
remove_code_1: fixes x :: "'a"   and xs :: "'a list" shows "Set.remove x (set xs) = set (removeAll x xs)"
pow_nat: fixes i :: "int"   and G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "0 \<le> i" shows "x [^]\<^bsub>G\<^esub> nat i = x [^]\<^bsub>G\<^esub> i"
WT_elim_cases_11: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and c :: "expr"   and T :: "ty" assumes "P,E \<turnstile> while (e) c :: T" obtains Ta :: "ty" where "T = Void" and "P,E \<turnstile> e :: Boolean" and "P,E \<turnstile> c :: Ta"
ACI_ACI_norm: fixes r :: "'a rexp" shows "ACI \<guillemotleft>r\<guillemotright> r"
TS_inv_sym: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and c :: "('a list \<times> 'a list) pmf" assumes "a \<noteq> b"   and "{a, b} = {x, y}"   and "z \<in> {x, y}" shows "TS_inv c z [a, b] = TS_inv c z [x, y]"
simps_1: fixes carrier :: "'b set"   and more :: "'c" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
maintain_invar2_ctd_1: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and d :: "nat" assumes "Prim2.prim_invar2_ctd w g r Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" shows "Prim2.prim_invar2_ctd w g r (Prim2.Q' w g r Q \<pi> u) (Prim2.\<pi>' w g r Q \<pi> u)"
Let_2: fixes \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "eq\<rho> (\<A>\<^bsub>fst (a, Aheap \<Gamma> e\<cdot>a)\<^esub> e) \<rho>1 \<rho>2" shows "eq (fst (a, Aheap \<Gamma> e\<cdot>a)) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>1\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>2\<^esub>)"
tendsto_iff: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" shows "(f \<longlongrightarrow> l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
that: fixes h :: "'a" assumes "h \<in> R\<setminus>I"   and "h \<cdot> (s' \<cdot> r - s \<cdot> r') = \<zero>" shows "thesis"
h2: shows "\<forall>x. (x * B ^ n * B ^ (n mod 2) < 0) = (x < 0)"
stream_pred_cases: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream" assumes "pred_stream P xs" obtains y :: "'a" and ys :: "'a stream" where "xs = y ## ys" and "P y" and "pred_stream P ys"
kernel_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" shows "kernel G H h = {x \<in> carrier G. h x = \<one>\<^bsub>H\<^esub>}"
nth_take_lemma: fixes k :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "k \<le> \<^bold>|xs\<^bold>|"   and "k \<le> \<^bold>|ys\<^bold>|"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
pure_assn_eq_false_iff: fixes P :: "bool" shows "(\<up> P = false) = (\<not> P)"
undual: fixes x :: "'a dual" shows "undual x \<in> UNIV"
var_inj_term: fixes x :: "Term.var"   and y :: "Term.var" shows "\<langle>x\<rangle> = \<langle>y\<rangle> \<equiv> x = y"
sums_2: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "(\<lambda>n. g n * (a * z) ^ n) sums G z"
oInv_bounds: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "normal F"   and "F 0 \<le> a" shows "F (oInv F a) \<le> a \<and> a < F (oSuc (oInv F a))"
E6: shows "vpeq_local u (set_object_value x y s1) (set_object_value x y s2)"
simps: fixes xs :: "(int \<times> int) list" shows "closest_pair_rec xs = (let n = length xs in if n \<le> 3 then (mergesort snd xs, closest_pair_bf xs) else let (xs\<^sub>L, xs\<^sub>R) = split_at (n div 2) xs; (ys\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) = closest_pair_rec xs\<^sub>L; (ys\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) = closest_pair_rec xs\<^sub>R; ys = Common.merge snd ys\<^sub>L ys\<^sub>R in (ys, combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) (fst (hd xs\<^sub>R)) ys))"
map_of_zip_is_Some: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c" assumes "length xs = length ys" shows "(x \<in> set xs) = (\<exists>y. map_of (zip xs ys) x = Some y)"
dirichlet_prod'_inversion2: fixes x :: "real" assumes "\<forall>x\<ge>1. f x = dirichlet_prod' ainv g x"   and "1 \<le> x"   and "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))" shows "g x = dirichlet_prod' a f x"
prems: shows "\<forall>\<psi>\<in>\<^bold>G \<phi>. w' \<Turnstile> \<psi> = ((x # xs) \<frown> w') \<Turnstile> \<psi>"
prems_2: shows "D \<in> \<C> - f \<turnstile> \<C>"
HaddP_Mem_contra: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm"   and A :: "fm" assumes "H \<turnstile> HaddP x y z"   and "H \<turnstile> z IN x"   and "H \<turnstile> OrdP x" shows "H \<turnstile> A"
fequalityCE: fixes A :: "'a fset"   and B :: "'a fset"   and c :: "'a" assumes "A = B" obtains "c |\<in>| A" and "c |\<in>| B"   | "c |\<notin>| A" and "c |\<notin>| B"
tree_order: shows "h \<turnstile> to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document) \<rightarrow>\<^sub>r tree_order"
subseq_append': fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" shows "(zs @ xs \<preceq> zs @ ys) = (xs \<preceq> ys)"
6: fixes x :: "'a" shows "x \<squnion> bot = x"
line_integral_degenerate_chain: fixes chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and basis :: "(real \<times> real) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "\<forall>(k, \<gamma>)\<in>chain. point_path \<gamma>"   and "finite basis" shows "one_chain_line_integral F basis chain = 0"
bounded_def: shows "bounded TYPE('b) = {n. finite UNIV \<longrightarrow> n < card UNIV}"
described_perp_foot_cross_ratio_formula: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2"   and p :: "proj2"   and q :: "proj2"   and l :: "proj2_line"   and m :: "proj2_line"   and d :: "proj2" assumes "a \<noteq> b"   and "c \<in> hyp2"   and "are_endpoints_in_S p q a b"   and "proj2_incident p l"   and "proj2_incident q l"   and "M_perp l m"   and "proj2_incident d l"   and "proj2_incident d m"   and "proj2_incident c m" shows "cross_ratio p q d a = (cosh_dist b c * sqrt (cross_ratio p q a b) - cosh_dist a c) / (cosh_dist a c * cross_ratio p q a b - cosh_dist b c * sqrt (cross_ratio p q a b))"
app_insf_LstSeq: fixes s :: "hf"   and k :: "hf"   and z :: "hf"   and y :: "hf" assumes "LstSeq s k z" shows "app (insf s (succ k) y) (succ k) = y"
solves_ode_domainD: fixes y :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
in_set_drop_conv_nth: fixes x :: "'c"   and n :: "nat"   and l :: "'c list" shows "(x \<in> set (drop n l)) = (\<exists>i\<ge>n. i < length l \<and> x = l ! i)"
sorted_group: fixes xs :: "'a list" assumes "sorted (map f xs)" shows "sorted (map f (local.group xs))"
m2: shows "m2 = \<lparr>iiface = iif2, oiface = oif2, src = sip2, dst = dip2, proto = p2, sports = sps2, dports = dps2\<rparr>"
division_of_finite: fixes s :: "'a set set"   and i :: "'a set" assumes "s division_of i" shows "finite s"
awalk_verts_ne_eq: fixes p :: "'b list"   and u :: "'a"   and v :: "'a" assumes "p \<noteq> []" shows "awalk_verts u p = awalk_verts v p"
bind_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'c \<Rightarrow> 'b set" shows "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
verts_add_arc: fixes a :: "'b" assumes "tail G a \<in> verts G"   and "head G a \<in> verts G" shows "verts (add_arc a) = verts G"
deriv: fixes x :: "'d" shows "((\<lambda>x. 0::'e) has_derivative (\<lambda>x. 0::'e)) (at x)"
choice_iff: fixes Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
V_eqvt: fixes x :: "trm"   and T :: "ty"   and pi :: "(name \<times> name) list" assumes "x \<in> V T" shows "pi \<bullet> x \<in> V T"
lift_power: fixes a :: "'a"   and n :: "nat" shows "lift (a ^ n) = lift a ^ n"
simps_1: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and uu :: "variable"   and S :: "String.literal set"   and S' :: "String.literal set" shows "exehas_sort oss (Tv uu S) S' = exesort_leq (fst oss) S S'"
finite_UNIV_fun: shows "finite UNIV = (finite UNIV \<and> finite UNIV \<or> CARD('b) = 1)"
dvd_mult_unit_iff': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (times b c) = dvd.dvd times a c"
eval\<^sub>G_And_map: fixes S :: "'b ltl set"   and xs :: "'b ltl list" shows "eval\<^sub>G S (And xs) = And (map (eval\<^sub>G S) xs)"
row_rank_def_gen: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A \<equiv> vec.dim (rows A)"
continuous_map_pairedI: fixes Z :: "'d topology"   and X :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and Y :: "'f topology"   and g :: "'d \<Rightarrow> 'f" assumes "continuous_map Z X f"   and "continuous_map Z Y g" shows "continuous_map Z (prod_topology X Y) (\<lambda>x. (f x, g x))"
fmdrop_fmrestrict_fset: fixes b :: "'a"   and S :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdrop b (fmrestrict_fset S m) = fmrestrict_fset (S |-| {|b|}) m"
bounded_piecewise_continuous_image: fixes a :: "real"   and b :: "real"   and I :: "real set"   and f :: "real \<Rightarrow> 'a" assumes "piecewise_continuous_on a b I f" shows "bounded (f ` {a..b})"
red_minus_singleton_zero: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.red (F - {0}) = punit.red F"
execn_Normal_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> =n\<Rightarrow> t" obtains s' :: "('e, 'f) xstate" where "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> =n\<Rightarrow> t"
mono_SucI1: fixes X :: "nat \<Rightarrow> 'e" assumes "\<forall>n. X n \<le> X (Suc n)" shows "monoseq X"
subset_image_inj: fixes S :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and T :: "'e set" shows "(S \<subseteq> f ` T) = (\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U)"
lemma3_6: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list"   and \<upsilon> :: "'b list" assumes "trans r"   and "\<tau> \<noteq> []"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<sigma>'| + r|\<upsilon>|, r|\<sigma>| + r|\<tau> @ \<upsilon>|) \<in> mul r"
keys_dom_lookup: fixes m :: "('a, 'b) mapping" shows "Mapping.keys m = dom (Mapping.lookup m)"
sdrop_shift: fixes i :: "nat"   and w :: "'b list"   and s :: "'b stream" shows "sdrop i (w @- s) = drop i w @- sdrop (i - length w) s"
dlist_cases: fixes dl :: "'b dlist" shows "dl = DList.empty \<or> (\<exists>h t. dl = DList.insert h t \<and> \<not> member t h)"
prems_1: fixes s :: "'a set" assumes "s \<in> insert x F" shows "closed s"
subst_atm_mset_lists_comp_substs: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and AAs :: "'b multiset list"   and \<tau>s :: "'c list"   and \<sigma>s :: "'c list" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_atm_mset_lists subst_atm AAs (substitution_ops.comp_substs comp_subst \<tau>s \<sigma>s) = substitution_ops.subst_atm_mset_lists subst_atm (substitution_ops.subst_atm_mset_lists subst_atm AAs \<tau>s) \<sigma>s"
typeof_lit_eq_NT: fixes v :: "'a val" shows "(typeof v = \<lfloor>NT\<rfloor>) = (v = Null)"
exhaust: fixes y :: "Preliminaries.var" obtains x :: "nat" where "y = Variable x"
select_convs_4: fixes E_ES :: "'c set"   and I_ES :: "'c set"   and O_ES :: "'c set"   and Tr_ES :: "'c list set"   and more :: "'d" shows "Tr_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = Tr_ES"
translation_subtract_Compl: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and t :: "'b set" assumes "class.ab_group_add plus zero minus uminus" shows "(\<lambda>x. minus x a) ` (- t) = - (\<lambda>x. minus x a) ` t"
foldl_spmf_Cons': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a spmf"   and a :: "'a"   and x :: "'b"   and xs :: "'b list" shows "foldl_spmf f (return_spmf a) (x # xs) = f a x \<bind> (\<lambda>a'. foldl_spmf f (return_spmf a') xs)"
comm_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "Groups.comm_monoid f z" shows "f a z = a"
Union_SetCompr_eq: fixes f :: "'b \<Rightarrow> 'a set"   and P :: "'b \<Rightarrow> bool" shows "\<Union> {f x |x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
none: assumes "ioprog a \<sigma> = None"   and "\<sigma> \<Turnstile> P []" shows "Q"
lim_1_over_n: shows "(\<lambda>n. (1::'e) / of_nat n) \<longlonglongrightarrow> (0::'e)"
subset_subtract_listed_sorted: fixes xs :: "'c list"   and ys :: "'c list" shows "set (subtract_list_sorted xs ys) \<subseteq> set xs"
bdd_all_bdd_lookup_iff: fixes n :: "nat"   and bdd :: "'a bdd"   and P :: "'a \<Rightarrow> bool" assumes "bddh n bdd" shows "bdd_all P bdd = (\<forall>ws. length ws = n \<longrightarrow> P (bdd_lookup bdd ws))"
l_new_character_data_get_shadow_root_axioms_def: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_shadow_root_axioms get_shadow_root_locs \<equiv> \<forall>h new_character_data_ptr h' r ptr'. h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h'"
dummies_unchanged_invariant: fixes V :: "Vars.var set" shows "invariant (\<Rightarrow>) (\<lambda>(\<Gamma>, e, S). dummies S = V)"
BinOp: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and v :: "val" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>"
iaction_mono: fixes \<Delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'a"   and vl :: "'b"   and s1 :: "'state"   and vl1 :: "'value list"   and \<Delta>' :: "'a \<Rightarrow> 'b \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "iaction \<Delta> s vl s1 vl1"   and "\<And>s vl s1 vl1. \<Delta> s vl s1 vl1 \<Longrightarrow> \<Delta>' s vl s1 vl1" shows "iaction \<Delta>' s vl s1 vl1"
assms_2: fixes y :: "('b \<times> ('a \<times> 'b) list) \<times> 'b \<times> ('a \<times> 'b) list" assumes "(y, (s, [\<alpha>_step] @ \<rho>_step), s, [\<beta>_step] @ \<upsilon>_step) \<in> pex r"   and "peak ars y" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"
open_Collect_neg: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.topological_space open"   and "topological_space.closed open {x. P x}" shows "open {x. \<not> P x}"
bin_nth_minus: fixes n :: "nat"   and w :: "int"   and b :: "bool" assumes "0 < n" shows "bit (w BIT b) n = bit w (n - 1)"
sup_bot_right: fixes x :: "'b upred" shows "(x \<or> false) = x"
emeasure_lborel_f_Suc: fixes A :: "(nat \<Rightarrow> real) set"   and n :: "nat" assumes "A \<in> sets (lborel_f (Suc n))"   and "\<And>y. {x \<in> space (lborel_f n). x(n := y) \<in> A} \<in> sets (lborel_f n)" shows "emeasure (lborel_f (Suc n)) A = \<integral>\<^sup>+ y. emeasure (lborel_f n) {x \<in> space (lborel_f n). x(n := y) \<in> A} \<partial>lborel"
ca_def: fixes \<phi> :: "cnf \<Rightarrow> bool" shows "ca \<phi> \<equiv> \<lambda>t n. \<phi> (t n)"
finite2: fixes A :: "'c set"   and f :: "'a \<Rightarrow> 'c" shows "(\<Sum>\<^sup>+ y. g (x, y) * indicator A (f y)) \<noteq> \<top>"
vreal_impl_def: shows "vreal_impl = (SOME y. range vreal_of_real_impl \<approx> elts y \<and> y \<inter>\<^sub>\<circ> \<omega> = []\<^sub>\<circ> \<and> y \<in>\<^sub>\<circ> Vset (\<omega> + \<omega>))"
Dom_TensorDiag: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u"
lookup_rule: fixes p :: "'b ref"   and x :: "'b" shows "<p \<mapsto>\<^sub>r x> !p <\<lambda>r. p \<mapsto>\<^sub>r x * \<up> (r = x)>"
val_simps_1: fixes x :: "'b" shows "Time_Monad.val (return x) = x"
maxBound_is_bound: fixes x :: "'a" shows "fromEnum x \<le> fromEnum maxBound"
simps_26: fixes x41 :: "exp"   and x42 :: "exp"   and x31 :: "nat"   and x32 :: "exp" shows "EApp x41 x42 \<noteq> ELam x31 x32"
f_make_mono_less: fixes f :: "nat \<Rightarrow> ordinal"   and n :: "nat" assumes "\<forall>n. f n < oLimit f" shows "f (make_mono f n) < f (make_mono f (Suc n))"
tendsto_lowerbound: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and F :: "'c filter"   and a :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> \<bottom>" shows "less_eq a x"
mult_one_associative: fixes x :: "'a"   and y :: "'a" shows "x * (1::'a) * y = x * y"
sat_consistency: fixes e :: "nat \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" shows "consistency {S. infinite (- \<Union> (params ` S)) \<and> (\<exists>f. \<forall>p\<in>S. eval e f g p)}"
is_vals_conv: fixes es :: "('e, 'f, 'g) exp list" shows "is_vals es = (\<exists>vs. es = map Val vs)"
assms_1: shows "threshold \<psi> w \<G>' = Some k"
measurable_shd: fixes M :: "'c measure" shows "shd \<in> stream_space M \<rightarrow>\<^sub>M M"
of_nat_neq_0: fixes n :: "nat" shows "of_nat (Suc n) \<noteq> (0::'a)"
while_eval\<^sub>w: fixes b :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>While b c, mds, mem\<rangle> \<leadsto> \<langle>Stmt.If b (c ;; While b c) Stop, mds, mem\<rangle>"
IH: fixes x :: "bool list" assumes "x \<in> gener_bool_list n" shows "length x = n"
fresh_subst_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y"   and "fresh xs x Y" shows "Y #[X / x]_xs = Y"
sorted_append: fixes xs :: "'a list"   and ys :: "'a list" shows "sorted (xs @ ys) = (sorted xs \<and> sorted ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le> y))"
size_last_index_conv: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "(n = last_index xs x) = (x \<notin> set xs)"
strict_prefixE': fixes xs :: "'b list"   and ys :: "'b list" assumes "xs < ys" obtains z :: "'b" and zs :: "'b list" where "ys = xs @ z # zs"
orthogonal_proj_set: fixes y :: "'a"   and C :: "'a set"   and a :: "'a" assumes "y \<in> C"   and "finite C"   and "pairwise orthogonal C" shows "orthogonal (a - proj_onto a C) y"
thenOrdering_LT_iff: fixes x :: "Ordering"   and y :: "Ordering" shows "(thenOrdering\<cdot>x\<cdot>y = LT) = (x = LT \<or> x = EQ \<and> y = LT)"
singular_chain_boundary: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "singular_chain (p - Suc 0) X (chain_boundary p c)"