hcju/isabelleps huggingface dataset:
local_cE_def: shows "cE \<equiv> edges rg \<inter> (- S) \<times> S"
valid_states_def: fixes prob :: "(('d, 'f) fmap \<times> ('d, 'g) fmap) set" shows "valid_states prob \<equiv> {s. fmdom' s = prob_dom prob}"
p_t_stop: fixes a :: "real"   and v :: "real"   and s0 :: "real" shows "movement.p a v s0 (movement.t_stop a v) = movement.p_max a v s0"
WF_alt: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> WF(A)_v = (\<box>\<diamond>\<not> Enabled \<langle>A\<rangle>_v \<or> \<box>\<diamond>\<langle>A\<rangle>_v)"
out_degree_add_arc_iff: fixes aa :: "'a \<times> nat \<times> 'a"   and a :: "'a \<times> nat \<times> 'a"   and u :: "'a" shows "out_degree (H.add_arc aa a) u = (if tail G a = u \<and> a \<notin> arcs (add_arc aa) then out_degree (add_arc aa) u + 1 else out_degree (add_arc aa) u)"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>xs. P xs []"   and "\<And>v va. P [] (v # va)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
monom_mult_dist_right_minus: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult c t (p - q) = monom_mult c t p - monom_mult c t q"
is_span: shows "C.span gf.leg0 gf.leg1"
semiring_norm_86: fixes m :: "num" shows "(num.Bit1 m = num.One) = False"
tl_subset: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<noteq> []"   and "set xs \<subseteq> A" shows "set (tl xs) \<subseteq> A"
strict_mono_less_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x \<le> f y) = less_eq x y"
split_llist_first: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset xs" shows "\<exists>ys zs. xs = lappend ys (LCons x zs) \<and> lfinite ys \<and> x \<notin> lset ys"
closed: fixes A :: "'d set" assumes "closed_csubspace A" shows "closed A"
StableT: fixes P :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> $P \<and> A \<longrightarrow> P$" shows "\<turnstile> \<box>A \<longrightarrow> stable P"
subst_subst_range: fixes x :: "'b"   and t :: "('a, 'b) Term.term" shows "subst_range (subst x t) = (if t = Var x then {} else {t})"
bij_image_Collect_eq: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "bij f" shows "f ` Collect P = {y. P (inv f y)}"
bind_pmf_cong: fixes p :: "'a pmf"   and q :: "'a pmf"   and f :: "'a \<Rightarrow> 'b pmf"   and g :: "'a \<Rightarrow> 'b pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "bind_pmf p f = bind_pmf q g"
simps_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
periodN_D1: fixes w :: "'b list"   and n :: "nat" assumes "periodN w n" shows "w \<noteq> \<epsilon>"
bp_bins_finite: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P :: "'b set set" assumes "BinPacking U w c S L"   and "BinPacking.bp U w c P" shows "\<forall>B\<in>P. finite B"
Markov_Decision_Process_Invariant_axioms_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" shows "Markov_Decision_Process_Invariant_axioms K S \<equiv> \<forall>s D. s \<in> S \<longrightarrow> D \<in> K s \<longrightarrow> (\<forall>s'\<in>set_pmf D. s' \<in> S)"
obtain_frontier_elem: fixes M :: "'b zmultiset"   and t :: "'b" assumes "0 < zcount M t" obtains u :: "'b" where "u \<in>\<^sub>A frontier M" and "u \<le> t"
doesnt_have_mode: fixes x :: "'Var\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and m :: "Mode" shows "(x \<notin> mds\<^sub>A_of mds\<^sub>C m) = (var\<^sub>C_of x \<notin> mds\<^sub>C m)"
\<pi>_subset_elem_trans: fixes Y :: "item set"   and k :: "nat"   and T :: "(symbol \<times> character list) set"   and X :: "item set"   and z :: "item" assumes "Y \<subseteq> \<pi> k T X"   and "z \<in> \<pi> k T Y" shows "z \<in> \<pi> k T X"
finite: shows "\<integral>\<^sup>+ xa. ennreal (eval_cexpr f (case_nat x \<rho>) xa) \<partial>stock_measure t' \<noteq> \<infinity>"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" shows "src\<^sub>B \<mu> = src\<^sub>B f"
comp_mwb_lens: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'e \<Longrightarrow> 'f" assumes "mwb_lens x"   and "mwb_lens y" shows "mwb_lens (x ;\<^sub>L y)"
pts: shows "q = p @ ts \<and> indexlt k' u' k u \<and> is_continuation k' u' p ts"
lm106: fixes Z :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "Z \<subseteq> X \<union> Y" shows "sum (X <| Y) Z = sum (chi X) Z"
mergeSL_def: fixes r :: "('c \<times> 'c) set"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and L :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and i :: "'c" assumes "wo_rel r" shows "wo_rel.mergeSL r S L f i \<equiv> if isSucc r i then S (pred r i) (f (pred r i)) else L f i"
rev_all_inA: shows "rev [0..<n] \<in> A"
interval_cbox: fixes a :: "'a"   and b :: "'a" shows "{a..b} = cbox a b"
p_max: fixes a :: "real"   and v :: "real"   and s0 :: "real"   and x :: "real" assumes "braking_movement a v" shows "movement.p a v s0 x \<le> movement.p_max a v s0"
cases: fixes x :: "('f, 'event, 'g, 'h, 'i, 'j, 'k) rest_scheme \<times> ('s_core \<times> 'f) \<times> (('iadv_core + 'g) + 'iusr_core + 'h)" obtains rest :: "('f, 'event, 'g, 'h, 'i, 'j, 'k) rest_scheme" and state :: "'s_core \<times> 'f" and iadv_core :: "'iadv_core" where "x = (resta__, state__, Inl (Inl iadv_core))"   | rest :: "('f, 'event, 'g, 'h, 'i, 'j, 'k) rest_scheme" and state :: "'s_core \<times> 'f" and iadv_rest :: "'g" where "x = (restb__, statea__, Inl (Inr iadv_rest))"   | rest :: "('f, 'event, 'g, 'h, 'i, 'j, 'k) rest_scheme" and state :: "'s_core \<times> 'f" and iusr_core :: "'iusr_core" where "x = (restc__, stateb__, Inr (Inl iusr_core))"   | rest :: "('f, 'event, 'g, 'h, 'i, 'j, 'k) rest_scheme" and state :: "'s_core \<times> 'f" and iusr_rest :: "'h" where "x = (rest, state, Inr (Inr iusr_rest))"
int_of_integer_of_nat: fixes n :: "nat" shows "int_of_integer (of_nat n) = int n"
gfinprod_cong: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "A = B"   and "f \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "gfinprod G f A = gfinprod G g B"
infinite_Ioc_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "infinite (ord.greaterThanAtMost less_eq less a b) = less a b"
is_pure_iff_pure_assn: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "is_pure P = (\<forall>x x'. is_pure_assn (P x x'))"
splitAt_simpB: fixes ram :: "'c"   and a :: "'c"   and b :: "'c list" assumes "ram \<noteq> a" shows "fst (splitAt ram (a # b)) = a # fst (splitAt ram b)"
triangle_right: shows "(k \<star> \<tau>) \<cdot> (\<sigma> \<star> k) = k"
p_def: shows "p \<equiv> \<lambda>(x, y). (x, \<lambda>i. y i + f' x i)"
gf: fixes z :: "'a \<times> real" shows "g (f z) = z"
tensor_lookup_cong: fixes ds :: "nat list"   and e\<^sub>1 :: "nat list \<Rightarrow> 'b"   and e\<^sub>2 :: "nat list \<Rightarrow> 'b"   and "is" :: "nat list" assumes "tensor_from_lookup ds e\<^sub>1 = tensor_from_lookup ds e\<^sub>2"   and "is \<lhd> ds" shows "e\<^sub>1 is = e\<^sub>2 is"
lhs_pivot_eq: fixes x\<^sub>j :: "nat"   and eq :: "nat \<times> linear_poly" assumes "x\<^sub>j \<in> rvars_eq eq"   and "lhs eq \<notin> rvars_eq eq" shows "lhs (pivot_eq eq x\<^sub>j) = x\<^sub>j"
sum_distrib_left: fixes r :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
l8_9: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C"   and "Col A B C" shows "A = B \<or> C = B"
INVIMP: fixes s :: "'a" assumes "I s" shows "I' s"
edge_succ_pred: fixes a :: "'b" shows "edge_succ HM (edge_pred HM a) = a"
transformation_group_def: fixes G :: "('c \<Rightarrow> 'c) set"   and S :: "'c set" shows "transformation_group G S \<equiv> subgroup G (monoid.Units (S \<rightarrow>\<^sub>E S) (compose S) (identity S)) (compose S) (identity S)"
valid_graph_rep_ex1: fixes l :: "('c \<times> 'd \<times> nat) list" assumes "valid_graph_rep l" shows "\<exists>!w. \<forall>u v d. (w (u, v) = enat d) = ((u, v, d) \<in> set l)"
is_ground_comp_subst: fixes \<sigma> :: "'s"   and \<tau> :: "'s" assumes "is_ground_subst \<sigma>" shows "is_ground_subst (\<tau> \<odot> \<sigma>)"
endpoint_in_S_swap: fixes a :: "proj2"   and b :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S" shows "endpoint_in_S a b \<noteq> endpoint_in_S b a"
hcmod_hcomplex_of_complex: fixes x :: "complex" shows "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
hyps_3: obtains p' :: "'t \<Rightarrow>\<^sub>0 'b" and q' :: "'t \<Rightarrow>\<^sub>0 'b" where "(red F)\<^sup>*\<^sup>* p p'" and "(red F)\<^sup>*\<^sup>* q q'" and "y = p' - q'"
scale_zero_left: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "module scale" shows "scale (0::'c) x = (0::'d)"
assms_3: shows "rk {B', C', P, Q} = 4"
lVs: shows "{l v \<cdot>\<^sub>v v |v. v \<in> Vs} \<subseteq> carrier_vec n"
assms_2: shows "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
IH: fixes p1 :: "'c"   and p2 :: "'d" assumes "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path AB (ws || r1s || r2s) (p1, p2)"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path A (ws || r1s) p1 \<and> path B (ws || r2s) p2 \<or> target (ws || r1s || r2s) (p1, p2) = FAIL \<and> 0 < length ws \<and> path A (butlast (ws || r1s)) p1 \<and> path B (butlast (ws || r2s)) p2 \<and> succ A (last ws) (target (butlast (ws || r1s)) p1) = {} \<and> succ B (last ws) (target (butlast (ws || r2s)) p2) \<noteq> {}"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.range x = range (lookup x) - {0::'c}"
conf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
inverse_1: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse one = one"
at_prm_rev_eq: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list" assumes "at TYPE('a)" shows " rev pi1 \<triangleq> rev pi2  =  pi1 \<triangleq> pi2 "
Pd_Xd: fixes x :: "'x"   and y :: "'x"   and d :: "'d" assumes "(x, y) \<in> Pd d" shows "Xd x = d \<and> Xd y = d"
ucast_le_ucast: fixes x :: "'a word"   and y :: "'a word" assumes "LENGTH('a) \<le> LENGTH('b)" shows "(UCAST('a \<rightarrow> 'b) x \<le> UCAST('a \<rightarrow> 'b) y) = (x \<le> y)"
take_all_iff: fixes n :: "nat"   and xs :: "'e list" shows "(take n xs = xs) = (length xs \<le> n)"
meas: shows "(\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set vs) (set vs') (\<sigma>, \<rho>)) \<delta>)) \<in> borel_measurable (state_measure (set vs) \<Gamma>)"
irreducible_m_def: fixes f :: "int poly" shows "p.irreducible_m f = (p.Mp f \<noteq> p.Mp 0 \<and> \<not> p.dvdm f 1 \<and> (\<forall>a b. p.eq_m f (a * b) \<longrightarrow> p.dvdm a 1 \<or> p.dvdm b 1))"
key_gen_def: fixes \<eta> :: "nat" shows "key_gen \<eta> = spmf_of_set (nlists UNIV (len \<eta>))"
uniform_limit_theorem: fixes A :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and F :: "'c filter"   and l :: "'d \<Rightarrow> 'e" assumes "\<forall>\<^sub>F n in F. continuous_on A (f n)"   and "uniform_limit A f l F"   and "F \<noteq> bot" shows "continuous_on A l"
strict_implies_not_eq: fixes b :: "'b word"   and a :: "'b word" assumes "b <s a" shows "a \<noteq> b"
has_field_derivative_zero_constant: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "convex s"   and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (0::'a)) (at x within s)" shows "\<exists>c. \<forall>x\<in>s. f x = c"
mgu_gives_MGU: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<theta>" shows "MGU \<theta> s t"
rreq_rrep_fresh_1: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "rreq_rrep_fresh crt (Rreq hops rreqid dip dsn dsk oip osn sip) = (sip \<noteq> oip \<longrightarrow> oip \<in> kD crt \<and> (osn < sqn crt oip \<or> sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val))"
toplevel_summands_lang_UN: fixes s :: "'b rexp" shows "lang s = \<Union> (lang ` toplevel_summands s)"
inf_dual: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice_with_dual inf less_eq less sup dual" shows "dual (inf x y) = sup (dual x) (dual y)"
list_update_overwrite: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a"   and y :: "'a" shows "xs[i := x, i := y] = xs[i := y]"
properfilters_def: shows "properfilters = {F \<in> filters. F \<noteq> UNIV}"
sample_bind_pmf: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 'p pmf"   and g :: "'p \<Rightarrow> 'm" shows "sample (p \<bind> f) g = sample p (\<lambda>x. sample (f x) g)"
wp1alternative_RS1: fixes p :: "('b, 'c) Combinators list" assumes "DenyAll \<in> set p" shows "wellformed_policy1_strong (removeShadowRules1 p)"
meta_spec2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> prop"   and x :: "'d"   and y :: "'e" assumes "\<And>x y. PROP P x y" shows "PROP P x y"
fbd_set: fixes f :: "'d \<Rightarrow> 'c set"   and X :: "'d set" shows "bd\<^sub>\<F> f X = {y. \<exists>x. s2p (f x) y \<and> s2p X x}"
max_MInf2: fixes x :: "ereal" shows "max x (- \<infinity>) = x"
assms_5: shows "\<not> p - Suc 0 \<le> lo"
sat_fv_cong: fixes v :: "'b list"   and v' :: "'b list"   and \<sigma> :: "'a trace"   and i :: "nat" assumes "\<And>x. x \<in> fv \<Longrightarrow> v ! x = v' ! x" shows "sat \<sigma> v i = sat \<sigma> v' i"
impr_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a r\<rightarrow> b \<equiv> if second a = (1::'c) \<and> second b = (1::'c) then Abs_OrdSum (first a r\<rightarrow> first b, 1::'c) else if first a = (1::'b) \<and> first b = (1::'b) then Abs_OrdSum (1::'b, second a r\<rightarrow> second b) else if first a = (1::'b) \<and> second b = (1::'c) then b else 1"
epiE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "CC.epi f"   and "CC.seq g f"   and "CC.seq g' f"   and "CC.comp g f = CC.comp g' f" shows "g = g'"
ExcessNotAt_eq: fixes g :: "graph" assumes "minGraphProps g"   and "final g" shows "\<exists>V. ExcessNotAt g None = ListSum V (ExcessAt g) \<and> separated g (set V) \<and> set V \<subseteq> \<V> g \<and> distinct V"
cong: fixes rep_key_order :: "'c \<Rightarrow> 'd key_order" shows "oalist_raw.lex_ord_raw rep_key_order \<equiv> oalist_raw.lex_ord_raw rep_key_order"
tendsto_0_if_Limsup_eq_0_ennreal: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> ennreal" assumes "Limsup F f = 0" shows "(f \<longlongrightarrow> 0) F"
residue_def: fixes n :: "int"   and m :: "int" shows "residue n m = m mod n"
rectpath_def: fixes a1 :: "complex"   and a3 :: "complex" shows "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3) in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
r_def: shows "r = (if fps_conv_radius f = \<infinity> then norm x + 1 else (norm x + real_of_ereal (fps_conv_radius f)) / 2)"
simps_5: fixes s :: "'a rexp"   and t :: "'a rexp" shows "rexp_ewp (s \<cdot>\<^sub>r t) = (rexp_ewp s \<and> rexp_ewp t)"
prems_1: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and n' :: "nat" assumes "is_path \<pi>'"   and "\<pi>' 0 = x"   and "\<pi>' n' = local.return"   and "\<forall>i>0. \<pi>' i \<noteq> x" shows "thesis"
Id_onI: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
\<Phi>_def: shows "\<Phi> \<equiv> \<lambda>g i r k. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> (r \<circ> k)) n x - g x) < e"
vmc_path_ldropn: fixes n :: "nat" assumes "enat n < llength (ldropn n__ P)" shows "vmc_path G (ldropn n (ldropn n__ P)) (ldropn n__ P $ n) p \<sigma>"
that: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p = rep_list r"   and "r \<in> sig_inv_set' (length fs)" shows "thesis"
tm_append_second_step_eq: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and B :: "(action \<times> nat) list"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and off :: "nat"   and A :: "(action \<times> nat) list" assumes "step0 (s, l, r) B = (s', l', r')"   and "s' \<noteq> 0"   and "off = length A div 2"   and "is_even (length A)" shows "step0 (s + off, l, r) (A @ shift B off) = (s' + off, l', r')"
matrix'_id_eq_matrix_change_of_basis: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "matrix' X Y id = matrix_change_of_basis X Y"
indets_empty_iff_poly_deg_zero: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(indets p = {}) = (poly_deg p = 0)"
coeff_sum: fixes f :: "'a \<Rightarrow> linear_poly"   and "is" :: "'a set"   and x :: "nat" shows "coeff (sum f is) x = (\<Sum>i\<in>is. coeff (f i) x)"
assms_7: shows "\<rho> U W s = \<rho> V W t"
nn_integral_measure_pmf: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = (\<Sum>\<^sup>+ x. ennreal (pmf p x) * f x)"
sum_fun_image_sup: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and z :: "'b" assumes "finite A" shows "(\<Sum> (f ` A) \<le> z) = (\<forall>a\<in>A. f a \<le> z)"
NCut_def: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and s :: "nat"   and t :: "nat"   and k :: "nat set" shows "NCut c s t k \<equiv> Network c s t \<and> Cut c k \<and> NCut_axioms s t k"
pf_2: assumes "n \<noteq> 0" shows "prod_list (prime_factorization_nat n) = n"
defNode_2: fixes v :: "'val"   and g :: "'g" assumes "v \<in> uninst_code.ssa.allVars uses phis g" shows "v \<in> uninst_code.ssa.allDefs phis g (uninst_code.ssa.defNode phis g v)"
cs_length_one_2: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "length (IFC_def.cs prob \<pi> i) = 1" shows "\<forall>k. \<not> IFC_def.is_cdi prob i \<pi> k"
ideal_prod_def: shows "ideal_prod \<equiv> \<lambda>R I J. {x. ideal_prodp R (\<lambda>x. x \<in> I) (\<lambda>x. x \<in> J) x}"
Un_absorb: fixes A :: "'g set" shows "A \<union> A = A"
weakCongClosed: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "P \<doteq>\<^sub>c Q" shows "p \<bullet> P \<doteq>\<^sub>c p \<bullet> Q"
Goal: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"
monic_degree_m: fixes f :: "int poly" assumes "monic f" shows "degree_m f = degree f"
simps_2: fixes ea :: "floatarith"   and eas :: "floatarith list" shows "interpret_slp (ea # eas) = (\<lambda>xs. interpret_slp eas (interpret_floatarith ea xs # xs))"
simple_integral_completion: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "simple_function M f" shows "integral\<^sup>S (completion M) f = integral\<^sup>S M f"
arcs_ends_blocks: fixes x :: "'a"   and y :: "'a" assumes "{x, y} \<in># arcs_blocks" shows "x \<rightarrow> y \<and> y \<rightarrow> x"
simps_9: fixes v :: "Ferrack.num" shows "not (NEq v) = NOT (NEq v)"
Abs_dlist_inverse: fixes y :: "'b list" assumes "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('b) = None}" shows "list_of_dlist (Abs_dlist y) = y"
list_all2_append1: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list"   and zs :: "'e list" shows "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)"
factorial_monoid_imp_ufd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.idom times one plus zero minus uminus"   and "factorial_monoid \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>" shows "class.ufd times one plus zero minus uminus"
assms_5: shows "p' \<in> sterms \<Gamma> p"
closedin_closed_Int: fixes S :: "'a set"   and U :: "'a set" assumes "closed S" shows "closedin (top_of_set U) (U \<inter> S)"
remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
at_eq_bot_iff: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.topological_space open" shows "(topological_space.at_within open a UNIV = bot) = open {a}"
m_choose_component_def: fixes e :: "'a"   and v :: "'a" shows "m_choose_component e v \<equiv> if choose_component_input_condition e v then e * minarc v * top else bot"
ring_simprules_13: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
glb_in_lattice: fixes S :: "'a set" assumes "S \<subseteq> A" shows "glb S cl \<in> A"
ContractsWithUnilateralSubstitutes_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithUnilateralSubstitutes Xd Xh Pd Ch \<equiv> Contracts Xd Xh Pd Ch \<and> ContractsWithUnilateralSubstitutes_axioms Xd Ch"
TensorDiag_Prim: fixes t :: "'b local.term"   and f :: "'b" assumes "t \<noteq> \<^bold>\<I>" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> t = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> t"
vimage_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "A \<subseteq> B" shows "f -` A \<subseteq> f -` B"
transfer: shows "rel_fun cr_uint8 (rel_fun cr_uint8 cr_uint8) (+) (+)"
M: shows "\<Union> (subterms ` set fas) \<subseteq> Mapping.keys M \<and> Mapping.keys Mapping.empty \<subseteq> Mapping.keys M \<and> (\<forall>f\<in>Mapping.keys M. subterms f \<subseteq> Mapping.keys M \<and> the (Mapping.lookup M f) < length slp \<and> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs)"