hcju/isabelleps huggingface dataset:
unit_factor_lcm: 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 b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "unit_factor (lcm a b) = (if a = zero \<or> b = zero then zero else one)"
FG_consitutents_n0: assumes "of_nat (card G) \<noteq> (0::'a)" shows "0 \<notin> set FG_constituents"
hsplit: fixes c :: "hf \<Rightarrow> hf \<Rightarrow> 'b"   and a :: "hf"   and b :: "hf" shows "hsplit c \<langle>a, b\<rangle> = c a b"
Ord\<pi>: fixes x :: "V" assumes "x \<in> elts \<alpha>" shows "Ord (\<pi> x)"
change_eval_leq: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real"   and y :: "real" assumes "aEvalUni (LeqUni (a, b, c)) x \<noteq> aEvalUni (LeqUni (a, b, c)) y \<and> x < y" shows "\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0"
append_split_vec_distrib_scalar_prod: fixes u :: "'a vec"   and w :: "'a vec"   and x :: "'a vec" assumes "dim_vec (u @\<^sub>v w) = dim_vec x" shows "(u @\<^sub>v w) \<bullet> x = u \<bullet> vec_first x (dim_vec u) + w \<bullet> vec_last x (dim_vec w)"
HEndPhase2_HInv5_q1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "p \<noteq> q"   and "maxBalInp s (bal (dblock s q)) (inp (dblock s q))" shows "maxBalInp s' (bal (dblock s' q)) (inp (dblock s' q))"
dim_vec_last: fixes v :: "'a vec"   and n :: "nat" shows "dim_vec (vec_last v n) = n"
Exh_Up: fixes z :: "'a\<^sub>\<bottom>" shows "z = \<bottom> \<or> (\<exists>x. z = up\<cdot>x)"
\<S>_not_empty: shows "\<S> \<noteq> {}"
a_export': fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> y) = ans_d x + ad y"
replete_concrete_set_category_axioms: shows "replete_concrete_set_category (\<cdot>\<^sub>S) UNIV \<iota>"
lemma_2_5_13_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b r\<rightarrow> c \<le> a r\<rightarrow> c"
sum_apply: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and X :: "'d set"   and i :: "'e" shows "sum f X i = (\<Sum>x\<in>X. f x i)"
WEAK_EX_def: fixes I :: "'c \<Rightarrow> bool"   and X :: "'c set"   and P :: "'c \<Rightarrow> bool nres" shows "WEAK_EX\<^bsup>I\<^esup> X P = WHILE\<^bsup>\<lambda>(Y, b). Y \<subseteq> X \<and> (b \<longrightarrow> (\<exists>x\<in>X. I x))\<^esup> (\<lambda>(X, b). \<not> b \<and> X \<noteq> {}) (\<lambda>(X, b). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(x, X'). P x \<bind> (\<lambda>b'. RETURN (X', b' \<or> b))))) (X, False) \<bind> (\<lambda>(uu_, b). RETURN b)"
aext_uop: fixes f :: "'e \<Rightarrow> 'b"   and u :: "('e, 'd) uexpr"   and a :: "'d \<Longrightarrow> 'c" shows "uop f u \<oplus>\<^sub>p a = uop f (u \<oplus>\<^sub>p a)"
negatively_invariant_eq_rev: fixes M :: "'a set" shows "negatively_invariant M = rev.positively_invariant M"
rep_eq: fixes x :: "rat"   and xa :: "nat" shows "Rep_linear_poly (lp_monom x xa) = (\<lambda>y. if xa = y then x else 0)"
in_set_sublists: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<in> set (sublists ys)) = sublist xs ys"
red_single_nonzero2: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single p q f t" shows "f \<noteq> 0"
dbproj_subset: fixes i :: "'b"   and D :: "('b \<times> 'a) list" shows "set (unlabel (dbproj i D)) \<subseteq> set (unlabel D)"
l_get_parent_axioms: shows "l_get_parent ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
im_diff: fixes g :: "'g"   and g' :: "'g" assumes "g \<in> G"   and "g' \<in> G" shows "T (g - g') = T g - T g'"
less_QF_step: fixes j :: "nat"   and k :: "nat"   and i' :: "nat"   and i :: "nat" assumes "0 < j"   and "j \<le> k"   and "i' < m" shows "QF k (j - Suc 0, i') \<lless> QF k (j, i)"
m_def: shows "m = (LEAST n. 0 < n \<and> int n \<in> group_closure S)"
antisym: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<preceq>\<^sub>p y"   and "y \<preceq>\<^sub>p x" shows "x = y"
left_neutral: fixes a :: "'a \<times> 'b" shows "times_prod' one_prod a = a"
if_SE_D2': fixes P :: "'a \<Rightarrow> bool"   and \<sigma> :: "'a"   and B\<^sub>1 :: "'a \<Rightarrow> ('b \<times> 'a) option"   and B\<^sub>2 :: "'a \<Rightarrow> ('b \<times> 'a) option"   and M :: "'a \<Rightarrow> (bool \<times> 'a) option" assumes "\<not> P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = (\<sigma> \<Turnstile> B\<^sub>2 ;- M)"
inv_msog_def: fixes M :: "((accent \<times> 'a) \<times> (accent \<times> 'a) list) multiset" shows "inv_msog M = {#(inv_letter x, inv_greek t). (x, t) \<in># M#}"
crename_AndR_aux: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and d :: "coname"   and N :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = AndR <c>.M <d>.N e" shows "a = e \<and> a = b \<or> a \<noteq> e"
vsv_vunion: fixes r :: "V"   and s :: "V" assumes "vsv r"   and "vsv s"   and "vdisjnt (\<D>\<^sub>\<circ> r) (\<D>\<^sub>\<circ> s)" shows "vsv (r \<union>\<^sub>\<circ> s)"
scale_left_commute: fixes a :: "'d"   and b :: "'d"   and x :: "'d" shows "a * (b * x) = b * (a * x)"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'b" assumes "B.ide x" shows "F'G'.\<epsilon> x = F'G'.\<eta>\<epsilon>.\<psi> x (G' x)"
lift_clear_vars: fixes i :: "nat"   and j :: "nat"   and p :: "real mpoly" shows "vars (liftPoly i j p) \<inter> {i..<i + j} = {}"
prime_factors_power: fixes n :: "nat"   and x :: "'a" assumes "0 < n" shows "prime_factors (x ^ n) = prime_factors x"
simps_2: fixes \<rho> :: "(nat \<times> exp) list"   and x :: "nat" shows "psubst \<rho> (EVar x) = (case lookup \<rho> x of None \<Rightarrow> EVar x | Some v \<Rightarrow> v)"
eq_fin_execute: fixes w\<^sub>1 :: "'action list"   and p :: "'state"   and w\<^sub>2 :: "'action list" assumes "path w\<^sub>1 p"   and "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "target w\<^sub>1 p = target w\<^sub>2 p"
simps_1: fixes x :: "'b \<times> sort_dir"   and y :: "'b sort_key" shows "sort_key_embedding (NonFinal x y) = embed_dir x # sort_key_embedding y"
irreducible_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.irreducible times one zero p = (p \<noteq> zero \<and> \<not> dvd.dvd times p one \<and> (\<forall>a b. p = times a b \<longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one))"
lenlex_conv: fixes r :: "('g \<times> 'g) set" shows "lenlex r = {(xs, ys). length xs < length ys \<or> length xs = length ys \<and> (xs, ys) \<in> lex r}"
ld_ld_1_less: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "1 + log 2 x + log 2 y < 2 * log 2 (x + y)"
M_def: fixes \<pi> :: "(char list \<times> 'a list) prefix" shows "M \<pi> = {(i, v). i < pprogress \<phi> \<pi> \<and> wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> MFOTL.sat \<sigma> (map the v) i \<phi>)}"
const_OclAnd: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and X' :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "const X"   and "const X'" shows "const (X and X')"
eNorm_Norm: fixes x :: "'a"   and N :: "'a quasinorm" assumes "x \<in> space\<^sub>N N" shows "eNorm N x = ennreal (Norm N x)"
res_rc1: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<rightarrow> y \<cdot> x"
while_lemma: fixes s :: "state"   and w :: "com"   and n :: "nat"   and t :: "state"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and P :: "state \<Rightarrow> bool" assumes "s -w-n\<rightarrow> t"   and "w = WHILE b DO c \<and> P s \<and> (\<forall>s s'. P s \<and> b s \<and> s -c-n\<rightarrow> s' \<longrightarrow> P s')" shows "P t \<and> \<not> b t"
the_eq_trivial: fixes a :: "'b" shows "(THE x. x = a) = a"
fbd_Sup_pres: shows "Sup_pres bd\<^sub>\<F>"
prems_2: shows "\<exists>a\<in>insert a A. f a = \<zero>"
ArrayIndexOutOfBounds_def: shows "ArrayIndexOutOfBounds = STR ''java/lang/ArrayIndexOutOfBoundsException''"
left_commute: fixes b :: "'k"   and a :: "'k"   and c :: "'k" shows "max b (max a c) = max a (max b c)"
lin_indpt: shows "\<not> LinearCombinations.module.lin_dep K (vs (local.module.span S)) U"
closure_of_singleton: fixes X :: "'b topology"   and a :: "'b" assumes "t1_space X" shows "X closure_of {a} = (if a \<in> topspace X then {a} else {})"
f: fixes x :: "'a" assumes "x \<in> S" shows "f x = x"
greatest_ge_nonzero_row': fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "\<not> is_zero_row_upt_k i k A"
type_ok_def: fixes G :: "prog"   and t :: "Term.term"   and s :: "state" shows "type_ok G t s = (\<exists>L T C A. (normal s \<longrightarrow> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T \<and> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>t\<guillemotright> A) \<and> s\<Colon>\<preceq>(G, L))"
P'_Q_dim: shows "P' + Q \<in> carrier_mat d d"
induct: fixes P :: "typ \<Rightarrow> bool"   and "typ" :: "typ" assumes "\<And>x1 x2. (\<And>x2a. x2a \<in> set x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (Ty x1 x2)"   and "\<And>x1 x2. P (Tv x1 x2)" shows "P typ"
pdevs_apply_fst_trunc_bound_pdevs: fixes p :: "nat"   and x :: "'a pdevs" shows "pdevs_apply (fst (trunc_bound_pdevs p x)) = pdevs_apply (trunc_pdevs p x)"
row_echelon_form_berlekamp_resulting_mat: fixes u :: "'b mod_ring poly" shows "row_echelon_form (berlekamp_resulting_mat u)"
rsd_def: fixes x :: "'a" shows "rsd x = (x \<cdot> \<top> \<le> x)"
extensible_refl: fixes R :: "('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "graph_homomorphism R G f" shows "extensible (R, R) G f"
assms_4: shows "a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
A_def: shows "A = {q \<in> B. card (snd q) = Suc m}"
Empty: shows "Var \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set"
h_plus_sum_finite: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat" assumes "flow_network \<Delta> g" shows "integral\<^sup>N (count_space UNIV) (flow_network.h_plus \<Delta> g i) \<noteq> \<top>"
d_type: shows "\<forall>P. (\<forall>x. x = d x \<longrightarrow> P x) = (\<forall>x. P (d x))"
rotate_Suc: fixes n :: "nat"   and xs :: "'b list" shows "rotate (Suc n) xs = rotate1 (rotate n xs)"
JN_insert: fixes a :: "'d"   and I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "JOIN (insert a I) F = F a \<squnion> JOIN I F"
split_at_conv_split_at_it'_1: fixes ts :: "'b list"   and ds :: "'b list"   and n :: "nat"   and xs :: "'b list"   and ts' :: "'b list"   and ds' :: "'b list"   and acc :: "'b list" assumes "(ts, ds) = split_at n xs"   and "(ts', ds') = split_at_it' acc n xs" shows "rev acc @ ts = ts'"
oprod_Partial_order: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Partial_order r"   and "Partial_order r'" shows "Partial_order (r *o r')"
rk2_increment_def: fixes x :: "real"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "rk2_increment x f = rk_increment f 2 (\<lambda>i j. if i = 2 \<and> j = 1 then x else 0) (\<lambda>i. if i = 1 then 1 - 1 / (2 * x) else 1 / (2 * x)) (\<lambda>i. if i = 2 then x else 0)"
equals_rsp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and xa :: "'e"   and xb :: "'e"   and ya :: "'e"   and yb :: "'e" assumes "Quotient3 R Abs Rep"   and "R xa xb"   and "R ya yb" shows "R xa ya = R xb yb"
mod_ring_plus_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (plus_p_integer pp) (+)"
Join_absorb: fixes F :: "'d program" shows "F \<squnion> F = F"
finite_dimensional_vector_space_on_axioms: shows "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1"
loc2_def: fixes s :: "(nat \<times> form) list"   and f :: "nat \<Rightarrow> nat \<times> (nat \<times> form) list"   and mo :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)" shows "loc2 s f mo \<equiv> loc1 s f \<and> loc2_axioms s mo"
add_cancel_right_left: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_ring_hom hom" shows "idom_hom hom"
dual_additive_def: fixes f :: "'a \<Rightarrow> 'a" shows "lattice.dual_additive (\<squnion>) (\<sqinter>) f \<equiv> \<forall>x y. f (x \<sqinter> y) = f x \<squnion> f y"
that: fixes U :: "'a set" assumes "open U"   and "B \<subseteq> U"   and "emeasure lborel (U - B) < ennreal e" shows "thesis"
list_all2_map2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and as :: "'d list"   and f :: "'f \<Rightarrow> 'e"   and bs :: "'f list" shows "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
simps_5: fixes ux :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and uy :: "'b"   and uz :: "'b list"   and va :: "'b list" shows "[] \<simeq> {uy # uz, va, ux} = False"
coeff_pCons: fixes x :: "'a"   and xa :: "'a poly" shows "poly.coeff (pCons x xa) = case_nat x (poly.coeff xa)"
not_greater_Max: fixes A :: "'a set"   and k :: "'a" assumes "finite A"   and "Max A < k" shows "k \<notin> A"
le_cube: fixes m :: "nat" shows "m \<le> m * (m * m)"
filter_a1_map_def: fixes pos :: "bool"   and tp :: "nat"   and a1_map :: "('a option list, nat) mapping" shows "filter_a1_map pos tp a1_map = {xs \<in> Mapping.keys a1_map. case Mapping.lookup a1_map xs of Some tp' \<Rightarrow> (pos \<longrightarrow> tp' \<le> tp) \<and> (\<not> pos \<longrightarrow> tp \<le> tp')}"
m_spec: shows "get_spec ((m_impl, m_spec) # M) = m_spec # get_spec M"
intro: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<subseteq> carrier G"   and "\<And>x y. \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H"   and "\<one>\<^bsub>G\<^esub> \<in> H" shows "submonoid H G"
poly_mult_replicate_zero_1: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and n :: "nat" assumes "domain R"   and "set p \<subseteq> carrier R" shows "ring.poly_mult R (replicate n \<zero>\<^bsub>R\<^esub>) p = []"
GEN_ALGO_tagI: fixes P :: "bool" assumes "P" shows "GEN_ALGO_tag P"
hyps_3: assumes "(a, b) \<in> Rel ^^ m" shows "(a, b) \<in> Rel ^^ i"
div_minus_minus: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "divide (uminus a) (uminus b) = divide a b"
mem_Sigma_mset_iff: fixes a :: "'b"   and b :: "'c"   and A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" shows "((a, b) \<in># Sigma_mset A B) = (a \<in># A \<and> b \<in># B a)"
smooth_on_inverse: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "(0::'d) \<notin> f ` S"   and "open S" shows "k-smooth_on S (\<lambda>x. inverse (f x))"
ih: shows "(s \<star> rs) \<star> rs' = s \<star> append rs' rs"
dim_eq_0: fixes S :: "'b set" shows "(local.dim S = 0) = (S \<subseteq> {0::'b})"
diff_union_swap2: fixes y :: "'a"   and M :: "'a multiset"   and x :: "'a" assumes "y \<in># M" shows "add_mset x M - {#y#} = add_mset x (M - {#y#})"
noIrrelevantChannels_def: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "noIrrelevantChannels i chSet \<equiv> \<forall>x\<in>chSet. systemIN x i \<longrightarrow> (\<exists>Z\<in>minSetOfComponents i chSet. x \<in> IN Z)"
intros_3: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F, G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow>\<^sub>n"
shows_list_char_def: fixes cs :: "char list" shows "shows_list cs = shows_string cs"
akra_bazzi_asymptotics: fixes bs :: "real list"   and hb :: "real"   and e :: "real"   and p :: "real" assumes "\<And>b. b \<in> set bs \<Longrightarrow> b \<in> {0<..<1}"   and "0 < hb"   and "0 < e" shows "\<forall>\<^sub>F x in at_top. \<forall>b\<in>set bs. akra_bazzi_asymptotics b hb e p x"
assms_3: shows "i \<inter> - b \<subseteq> q"
axioms_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP" shows "U.Struct wtFsym wtPsym arOf parOf D intF intP"
mset_le_subtract_right: fixes A :: "'c multiset"   and B :: "'c multiset"   and X :: "'c multiset" assumes "A + B \<subseteq># X" shows "A \<subseteq># X - B \<and> B \<subseteq># X"
invol_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "dual \<circ> dual = id"
summable: shows "summable (\<lambda>n. diffs c n * z ^ n)"
atLeastLessThan_rev_at_least_Suc_atMost: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "prod g {n..<m} = (\<Prod>i = Suc n..m. g (m + n - i))"
excessAtType_def: fixes t :: "nat"   and q :: "nat"   and e :: "nat" shows "excessAtType t q e \<equiv> if e = 0 then if 7 < t + q then squanderTarget else \<b> t q - t * d3_const - q * d4_const else if t + q + e \<noteq> 6 then 0 else if t = 5 then \<a> else squanderTarget"
fun_evaluate_equiv': fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" shows "Evaluate_Single.evaluate env s e = map_prod id (map_result hd id) (fun_evaluate s env [e])"
Pxyz_nn: fixes x :: "'c \<times> 'd \<times> 'e" assumes "x \<in> space (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)" shows "0 \<le> Pxyz x"
pure_eq_iff: fixes h :: "'x"   and f :: "('x, 'z, 'y) prog" shows "(\<forall>h' x. h \<turnstile> f \<rightarrow>\<^sub>r x \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h' \<longrightarrow> h = h') = pure f h"
degree_poly_add_rat: fixes r :: "rat"   and p :: "int poly" shows "degree (poly_add_rat r p) = degree p"
simps: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and a1 :: "ty"   and a2 :: "ty" shows "P \<turnstile> a1 \<le> a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D) \<or> (\<exists>C. a1 = NT \<and> a2 = Class C))"
scalar_product_assoc: fixes nr :: "nat"   and nc :: "nat"   and m :: "'a list list"   and r :: "'a list"   and c :: "'a list" assumes "mat nr nc m"   and "vec nr r"   and "vec nc c" shows "scalar_prod (map (\<lambda>k. scalar_prod r (col m k)) [0..<nc]) c = scalar_prod r (map (\<lambda>k. scalar_prod (row m k) c) [0..<nr])"
parse_nodes_help: fixes s :: "char list" shows "parse_nodes_dom s \<and> (\<forall>x r. parse_nodes s = Inr (x, r) \<longrightarrow> length r \<le> length s)"
pdivides_imp_roots_incl: fixes p :: "'a list"   and q :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "q \<in> carrier (poly_ring R)"   and "q \<noteq> []"   and "p pdivides q" shows "roots p \<subseteq># roots q"
all_in_list_sublist: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list"   and P :: "('a, 'b) Combinators \<Rightarrow> bool" assumes "singleCombinators xs"   and "all_in_list xs l" shows "all_in_list (qsort (filter P xs) l) l"
mult2_alt_add: fixes X1 :: "'a multiset"   and Y1 :: "'a multiset"   and b1 :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X2 :: "'a multiset"   and Y2 :: "'a multiset"   and b2 :: "bool" assumes "(X1, Y1) \<in> mult2_alt b1 ns s"   and "(X2, Y2) \<in> mult2_alt b2 ns s" shows "(X1 + X2, Y1 + Y2) \<in> mult2_alt (b1 \<and> b2) ns s"
that: fixes R :: "real" assumes "0 < R"   and "R < ub"   and "f analytic_on cball s R" shows "thesis"
Og_def: shows "Og = (\<lambda>n. {x \<in> space M. \<forall>l\<in>{1..n}. - delta l * real l < u n ((Tinv ^^ n) x) - u (n - l) ((Tinv ^^ (n - l)) x)})"
hyp2_in_middle: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "p \<in> S"   and "q \<in> S"   and "r \<in> hyp2 \<union> S"   and "proj2_Col p q r"   and "p \<noteq> q" shows "B\<^sub>\<real> cart2_pt p cart2_pt r cart2_pt q"
set_pmf_J: fixes x :: "'a" shows "set_pmf (J x) = I x"
exhaust: fixes y :: "'a option" obtains "y = \<bottom>"   | x2 :: "'a" where "y = \<lfloor>x2\<rfloor>"
commute: fixes a :: "'a"   and b :: "'a" shows "f a b = f b a"
assms_4: shows "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"
hyps_2: shows "awalk_cyc_decomp p = (q, r, s)"
primroot_of_root: fixes x :: "'b list"   and u :: "'b list"   and r :: "'b list" assumes "x \<noteq> \<epsilon>"   and "x \<in> u*"   and "u \<in>\<^sub>p r *" shows "x \<in>\<^sub>p r *"
IH_3: assumes "avl r" shows "avl (AVL.insert x r)"
2: fixes u :: "('a, 'b) pp"   and i :: "nat" shows "snd (rep_nat_term (u, i)) = i"
P: shows "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"
member_filter: fixes x :: "'e"   and P :: "'e \<Rightarrow> bool"   and A :: "'e set" shows "(x \<in> Set.filter P A) = (x \<in> A \<and> P x)"
out_in_arcs_reflexive: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a"   and e :: "'b" assumes "sym_digraph G"   and "v \<in> verts G"   and "e \<in> out_arcs G v" shows "\<exists>e'. e' \<in> in_arcs G v \<and> tail G e' = head G e"
assms_1: shows "atoms_formula S = {X. \<exists>I<N. X = A I}"
D_def: shows "D = {b \<in> B. y \<notin> b \<and> f -` b \<inter> space M \<in> null_sets M}"
consistent_def: fixes M :: "('a, 'b) message set" shows "consistent M \<equiv> inj_on I (insert_messages M) \<and> \<Union> (deps ` M) \<subseteq> I ` insert_messages M \<and> wfP (depends_on (insert_messages M)) \<and> (\<exists>a. a_conditions (insert_messages M) a)"
intros_5: fixes u :: "'d \<Rightarrow> 'c"   and r :: "'d"   and c :: "'d"   and d :: "'c" assumes "u r - u c \<le> d" shows "dbm_entry_val u (Some r) (Some c) (Le d)"
agree_on_def: fixes G :: "('e, 'f) labeled_graph"   and f\<^sub>1 :: "('f \<times> 'g) set"   and f\<^sub>2 :: "('f \<times> 'g) set" shows "agree_on G f\<^sub>1 f\<^sub>2 \<equiv> \<forall>v\<in>vertices G. f\<^sub>1 `` {v} = f\<^sub>2 `` {v}"
sum_prime_factorization_conv_sum_primepow_factors: fixes n :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "n \<noteq> (0::'a)" shows "(\<Sum>q\<in>primepow_factors n. f (aprimedivisor q)) = \<Sum>\<^sub># (image_mset f (prime_factorization n))"
divisors_hd_coeffs1: fixes as :: "atom list"   and a :: "atom" assumes "\<forall>a\<in>set as. divisor a \<noteq> 0"   and "\<forall>a\<in>set as. hd_coeff a \<noteq> 0"   and "a \<in> set (hd_coeffs1 as)" shows "divisor a \<noteq> 0"
assms: shows "\<S>\<bullet>\<C> \<turnstile> es @ es' : ts _> ts'"