hcju/isabelleps huggingface dataset:
extended_Gromov_product_at_topological: fixes e :: "'b"   and x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_Gromov_product_at e x y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) |u v. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x \<and> (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
maintained_def: fixes R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph" shows "maintained R G \<equiv> \<forall>f. graph_homomorphism (fst R) G f \<longrightarrow> extensible R G f"
redT_updTs_preserves_wset_thread_ok: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and nts :: "('a, 'c, 'e) new_thread_action list" assumes "wset_thread_ok ws ts" shows "wset_thread_ok ws (redT_updTs ts nts)"
cond_def: fixes g :: "'g" shows "uninst_code.cond uses phis g \<equiv> \<lambda>(u, p). uninst.redundant ((usesOf \<circ> uses)(g := u)) ((\<lambda>g. Mapping.lookup (phis g))(g := p)) g"
inext_predicate_change_exists2: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P a"   and "P b" shows "\<exists>n\<in>I. a \<le> n \<and> n < b \<and> \<not> P n \<and> (\<forall>k\<in>I. n < k \<and> k \<le> b \<longrightarrow> P k)"
nat_pow_mult: fixes x :: "'a vec"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier_vec n" shows "[n] \<cdot>\<^bsub>V\<^esub> x + [m] \<cdot>\<^bsub>V\<^esub> x = [(n + m)] \<cdot>\<^bsub>V\<^esub> x"
weak_arrow_of_homs_def: fixes C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's" shows "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F \<equiv> (horizontal_homs C src\<^sub>C trg\<^sub>C \<and> horizontal_homs D src\<^sub>D trg\<^sub>D) \<and> functor C D F \<and> weak_arrow_of_homs_axioms C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"
fofu_I_II: assumes "isMaxFlow f" shows "\<not> Ex isAugmentingPath"
enum_def: fixes i :: "nat"   and j :: "nat" shows "s.enum i j = (if j \<in> u_s ` {..<i} then Suc (b_s j) else b_s j)"
S1_nS2: fixes s :: "'s" assumes "s \<in> S1" shows "s \<notin> S2"
lred_imp_sred: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>l t" shows "s \<rightarrow>\<^sub>s t"
eq_iff: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "(a = b) = (b \<preceq> a \<and> a \<preceq> b)"
unit_not_fst: fixes P :: "'b trm" shows "Unit \<noteq> Fst P"
token_run_intial_state: fixes x :: "nat" shows "\<MM>.token_run x x = Abs \<phi>"
repeat_conv_replicate: fixes n :: "nat"   and xs :: "'a list" shows "repeat n xs = concat (replicate n xs)"
lfp_fixpoint: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "f (\<nu> f) = \<nu> f"
cases: fixes x :: "('b \<times> nat) tree" obtains t1 :: "('b \<times> nat) tree" and b :: "'b" and lvb :: "nat" and t2 :: "('b \<times> nat) tree" and a :: "'b" and lva :: "nat" and t3 :: "('b \<times> nat) tree" where "x = \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle>"   | "x = \<langle>\<rangle>"   | va :: "'b \<times> nat" and vb :: "('b \<times> nat) tree" where "x = \<langle>\<langle>\<rangle>, va, vb\<rangle>"
assms_1: assumes "b" shows "m1 \<le> m1'"
in_dom_g: fixes x :: "'a" shows "(x \<in> dom (\<lambda>x. if P x then Some x else None)) = P x"
length_coeffs: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "length (coeffs p) = degree p + 1"
valid_R: shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c R,B"
reflect_poly_const: fixes a :: "'a" shows "reflect_poly [:a:] = [:a:]"
Compl_UN: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "- \<Union> (B ` A) = (\<Inter>x\<in>A. - B x)"
get_dom_component_def: fixes ptr :: "(_) object_ptr" shows "get_dom_component ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
hyps_2: shows "\<Gamma>\<turnstile> (r', s') \<rightarrow> (r'', s'')"
return_nondet_def: fixes x :: "'a" shows "return_nondet x = NondetT (return (single x))"
T_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "tensor f g = f \<otimes> g"
communicating_def: fixes K :: "'a \<Rightarrow> 'a pmf" shows "MC_syntax.communicating K = (SIGMA x:UNIV. set_pmf (K x))\<^sup>* \<inter> ((SIGMA x:UNIV. set_pmf (K x))\<^sup>*)\<inverse>"
poly_reflect_poly_0: fixes p :: "'a poly" shows "poly (reflect_poly p) (0::'a) = lead_coeff p"
test_result_def: shows "test_result \<equiv> [''a'', ''b'', ''c'', ''0'']"
pivot_fun_def: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat" shows "pivot_fun A f nc \<equiv> let nr = dim_row A in \<forall>i<nr. f i \<le> nc \<and> (f i < nc \<longrightarrow> A $$ (i, f i) = (1::'c) \<and> (\<forall>i'<nr. i' \<noteq> i \<longrightarrow> A $$ (i', f i) = (0::'c))) \<and> (\<forall>j<f i. A $$ (i, j) = (0::'c)) \<and> (Suc i < nr \<longrightarrow> f i < f (Suc i) \<or> f (Suc i) = nc)"
Prim_Impl_Adts_axioms: shows "Prim_Impl_Adts G.\<alpha>w G.\<alpha>g G.invar G_adj G_empty G_add_edge RBT_Set.empty RBT_Map.update RBT_Map.delete lookup M.invar PST_RBT.empty PST_RBT.update PST_RBT.delete PM.invar lookup rbt_is_empty pst_getmin"
I_minWeightBasisD_1: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "indep T"
ereal_minus_5: shows "- \<infinity> - \<infinity> = - \<infinity>"
degree: shows "MPoly_Type.degree p x = 1 \<or> MPoly_Type.degree p x = 2"
vmc_path_lnull_ltl_no_deadend: assumes "\<not> lnull (ltl P)" shows "vmc_path_no_deadend G P v0 p \<sigma>"
\<alpha>n_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label list" shows "\<alpha>n fg e \<equiv> if e = [] then ({}, {}) else (mon_e fg (hd e), mon_w fg (tl e))"
LeastI: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "P k" shows "P (LEAST x. P x)"
d: shows "d \<in> ran (ve ++ map_of (map (\<lambda>(v, l). ((v, b''), \<A> (L l) \<beta>' ve)) ls))"
refl_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
lderivs_snoc: fixes ws :: "'a list"   and w :: "'a"   and r :: "'b rexp" shows "lderivs (ws @ [w]) r = lderiv w (lderivs ws r)"
refl: fixes a :: "('d, 'e) Term.term" shows "a \<unrhd> a"
FaultProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f" shows "\<Gamma>\<turnstile> (c # cs, css, Fault f) \<rightarrow> (cs, css, Fault f)"
collect_pair_finite': fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {(x, y). P x y}" shows "finite {(x, y). P x y \<and> R x y}"
bind_def: fixes f :: "'b Heap"   and g :: "'b \<Rightarrow> 'a Heap" shows "f \<bind> g = Heap (\<lambda>h. case execute f h of None \<Rightarrow> None | Some (x, h') \<Rightarrow> execute (g x) h')"
offs_next_zero_cons_neq: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and x :: "'c"   and ns :: "nat list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and xs :: "'c list"   and ub :: "nat" assumes "index key x (length ns) mi ma = i"   and "i < length ns"   and "0 < i"   and "offs_set_prev ns (x # xs) index key mi ma i = {}" shows "offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma 0 = (if 0 < offs_num (length ns) xs index key mi ma i then Suc (ns ! i) else offs_next ns ub (x # xs) index key mi ma i)"
plus_game_comm: fixes G :: "game"   and H :: "game" shows "plus_game G H = plus_game H G"
constr: fixes name :: "name"   and rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and ts :: "sterm list"   and us :: "value list" assumes "name |\<in>| C"   and "list_all2 (veval rs \<Gamma>) ts us" shows "rs, \<Gamma> \<turnstile>\<^sub>v name $$ ts \<down> Vconstr name us"
PiE_eq_subset: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and F' :: "'c \<Rightarrow> 'd set"   and i :: "'c" assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"   and "Pi\<^sub>E I F = Pi\<^sub>E I F'"   and "i \<in> I" shows "F i \<subseteq> F' i"
elims_2: fixes x :: "real_alg_2" assumes "invariant_2 x" obtains n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai" and "invariant_1_2 rai \<and> n = card {y. y \<le> real_of_1 rai \<and> poly (real_of_int_poly (poly_real_alg_1 rai)) y = 0}"   | r :: "rat" where "x = Rational r"
Semilat_def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "Semilat A r f \<equiv> semilat (A, r, f)"
irreducible_altdef: fixes x :: "'c" shows "irreducible x = (x \<noteq> (0::'c) \<and> \<not> x dvd (1::'c) \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> b dvd (1::'c)))"
\<alpha>n_fst_snd: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and w :: "('c, 'f) label list" shows "fst (\<alpha>n fg w) \<union> snd (\<alpha>n fg w) = mon_w fg w"
\<Delta>\<Phi>_pass1_ub_pass1: fixes hs :: "'b tree" assumes "hs \<noteq> \<langle>\<rangle>" shows "\<Phi> (pass\<^sub>1 hs) - \<Phi> hs \<le> ub_pass\<^sub>1 hs"
exhaust: fixes y :: "cell" obtains "y = Bk"   | "y = Oc"
hyps_3: fixes w :: "'b wait_set_status" assumes "ws'' t = \<lfloor>w'\<rfloor>"   and "ws' t = \<lfloor>w\<rfloor>"   and "\<And>w. Suspend w \<notin> set was" shows "w' = w \<or> (\<exists>w'' w'''. w = InWS w'' \<and> w' = PostWS w''')"
tendsto_of_nat_iff: fixes f :: "'a \<Rightarrow> nat"   and c :: "nat"   and F :: "'a filter" shows "((\<lambda>n. of_nat (f n)) \<longlongrightarrow> of_nat c) F = (\<forall>\<^sub>F n in F. f n = c)"
zip_update: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a"   and ys :: "'b list"   and y :: "'b" shows "zip (xs[i := x]) (ys[i := y]) = (zip xs ys)[i := (x, y)]"
conga_trivial_1: 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"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> D" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B A C D C"
vbrelation_axioms: fixes U :: "V"   and f :: "V \<Rightarrow> V" shows "vbrelation (VLambda U f)"
pullback_iso_self: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.iso C f" shows "prj0 f f = prj1 f f"
degree_add_pdevs_le: fixes X :: "'a pdevs"   and n :: "nat"   and Y :: "'a pdevs" assumes "degree X \<le> n"   and "degree Y \<le> n" shows "degree (add_pdevs X Y) \<le> n"
minus_zero: shows "\<ominus>\<^bsub>F\<^esub> \<zero>\<^bsub>F\<^esub> = \<zero>\<^bsub>F\<^esub>"
simps_3: fixes v :: "'a" shows "Some v \<squnion> None = Some v"
domintros: fixes l :: "nat list"   and i :: "nat" assumes "l ! i \<noteq> i \<Longrightarrow> rep_of_dom (l, l ! i)" shows "rep_of_dom (l, i)"
step_1: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "c = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child_node"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "h \<turnstile> get_root_node_si parent \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
zero_in_bigo: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(\<lambda>_. 0::'b) \<in> O[F](f)"
hcomp_closed: fixes f :: "'a"   and g :: "'a" assumes "B.arr f \<and> src\<^sub>B f = a \<and> trg\<^sub>B f = a"   and "B.arr g \<and> src\<^sub>B g = a \<and> trg\<^sub>B g = a"   and "trg\<^sub>B f = src\<^sub>B g" shows "B.hseq g f \<and> src\<^sub>B (g \<star>\<^sub>B f) = a \<and> trg\<^sub>B (g \<star>\<^sub>B f) = a"
y': shows "\<forall>ya\<in>S. norm (v - y) \<le> norm (v - ya)"
assms: shows "insert (a EQ c) (insert (b EQ d) H) \<turnstile> A"
bf_ifex_rel_def: shows "bf_ifex_rel = {(a, b). (\<forall>ass. a ass = val_ifex b ass) \<and> ro_ifex b}"
minus_eq: fixes x :: "rat"   and y :: "rat" shows "x - y = x + - y"
drop_map_def: fixes i :: "nat" shows "drop_map i \<equiv> \<lambda>(g, uu). (g i, delete_map i g, uu)"
induced_function_integral_nonergodic_2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integral\<^sup>L M (induced_function A f) = set_lebesgue_integral M (\<Union>n. (T ^^ n) --` A) f"
L_While: fixes X :: "char list set"   and b :: "bexp"   and c :: "com" assumes "finite X" shows "L (WHILE b DO c) X = while (\<lambda>Y. vars b \<union> X \<union> L c Y \<noteq> Y) (\<lambda>Y. vars b \<union> X \<union> L c Y) {}"
norm_bounded: fixes f :: "'d \<Rightarrow>\<^sub>C 'c"   and x :: "'d" shows "norm (apply_bcontfun f x) \<le> norm f"
gauge_integral_Fubini_curve_bounded_region_x: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and s :: "('a \<times> 'b) set"   and g :: "'a \<times> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "integrable lborel f"   and "\<And>x. (\<lambda>y. f (x, y)) integrable_on UNIV"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) \<in> borel_measurable lborel"   and "f = (\<lambda>x. if x \<in> s then g x else (0::'c))"   and "s = {(x, y). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g1 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g2 x \<bullet> i)}" shows "integral s g = integral (cbox a b) (\<lambda>x. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (x, y)))"
right_idem: fixes a :: "nat"   and b :: "nat" shows "gcd (gcd a b) b = gcd a b"
asm: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s) (Abs_fv x \<tau>' t)"
of_nat_le_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
simps_1: fixes xs :: "'b list" shows "inorder2_tm \<langle>\<rangle> xs =1 return xs"
CLS_in_hom: fixes f :: "'a" assumes "is_left_adjoint f" shows "Maps.in_hom \<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>src f\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>trg f\<rbrakk>\<rbrakk>"
isTrue_implies_bprv_PPf: fixes n :: "'trm"   and \<phi> :: "'fmla" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PPf n \<langle>\<phi>\<rangle>)" shows "bprv (PPf n \<langle>\<phi>\<rangle>)"
map_uminus_order2: fixes ss :: "'a list" assumes "\<forall>s\<in>set ss. s + s = (0::'a)" shows "map uminus ss = ss"
assms_2: fixes x :: "'a" assumes "P' x" shows "f x = f' x"
le_iff_inf: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod x y = (inf_prod x y = x)"
mmm_def: shows "mmm = mahler_landau_graeffe_approximation kk dd g"
dom_char: fixes f :: "'a" shows "hhom\<^sub>C.dom f = (if hhom\<^sub>C.arr f then C.dom f else C.null)"
elims_2: fixes x :: "'b match_expr" assumes "has_primitive x" obtains a :: "'b" where "x = Match a"   | m :: "'b match_expr" where "x = MatchNot m" and "has_primitive m"   | m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = MatchAnd m1 m2" and "has_primitive m1 \<or> has_primitive m2"
LAss_\<tau>red1t: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (V:=e, xs) (V:=e', xs')"
P: shows "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'"
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "v"   and a5 :: "(pat \<times> exp0) list"   and a6 :: "v"   and a7 :: "'b state \<times> (v, v) result" assumes "evaluate_match a1 a2 a3 a4 a5 a6 a7" obtains ck :: "bool" and env :: "v sem_env" and v1 :: "v" and err_v :: "v" and s :: "'b state" where "a1 = cka__" and "a2 = enva__" and "a3 = sa__" and "a4 = v1__" and "a5 = []" and "a6 = err_va__" and "a7 = (sa__, Rerr (Rraise err_va__))"   | ck :: "bool" and env :: "v sem_env" and env' :: "(char list \<times> v) list" and v1 :: "v" and p :: "pat" and pes :: "(pat \<times> exp0) list" and e :: "exp0" and bv :: "'b state \<times> (v, v) result" and err_v :: "v" and s :: "'b state" where "a1 = ckb__" and "a2 = envb__" and "a3 = sb__" and "a4 = v1a__" and "a5 = (pa__, ea__) # pesb__" and "a6 = err_vb__" and "a7 = bva__" and "allDistinct (pat_bindings pa__ []) \<and> pmatch (c envb__) (refs sb__) pa__ v1a__ [] = Match env' \<and> evaluate ckb__ (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v envb__)) envb__) sb__ ea__ bva__"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and bv :: "'b state \<times> (v, v) result" and s :: "'b state" and err_v :: "v" where "a1 = ckc__" and "a2 = envc__" and "a3 = sc__" and "a4 = v1b__" and "a5 = (pb__, eb__) # pesc__" and "a6 = err_vc__" and "a7 = bv" and "allDistinct (pat_bindings pb__ []) \<and> pmatch (c envc__) (refs sc__) pb__ v1b__ [] = No_match \<and> evaluate_match ckc__ envc__ sc__ v1b__ pesc__ err_vc__ bv"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and s :: "'b state" and err_v :: "v" where "a1 = ckd__" and "a2 = envd__" and "a3 = sd__" and "a4 = v1c__" and "a5 = (pc__, ec__) # pesd__" and "a6 = err_vd__" and "a7 = (sd__, Rerr (Rabort Rtype_error))" and "pmatch (c envd__) (refs sd__) pc__ v1c__ [] = Match_type_error"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and s :: "'b state" and err_v :: "v" where "a1 = ck" and "a2 = env" and "a3 = s" and "a4 = v1" and "a5 = (p, e) # pes" and "a6 = err_v" and "a7 = (s, Rerr (Rabort Rtype_error))" and "\<not> allDistinct (pat_bindings p [])"
vars_for_consts: fixes p :: "('a, 'b) form"   and xs :: "'a list" assumes "infinite (- params p)"   and "[] \<turnstile> p" shows "[] \<turnstile> vars_for_consts p xs"
wf_finite_greater: fixes C :: "'b set" assumes "finite C" shows "wf {(x, y). {x, y} \<subseteq> C \<and> y < x}"
2: shows "(\<Oplus>\<^bsub>M\<^esub>x\<in>insert a A. f x \<odot>\<^bsub>M\<^esub> v) = f a \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> (\<Oplus>\<^bsub>M\<^esub>x\<in>A. f x \<odot>\<^bsub>M\<^esub> v)"
cases: fixes x :: "tm" obtains k :: "name" and s :: "name" and y :: "tm" where "atom k \<sharp> (s, y)" and "atom s \<sharp> y" and "x = y"
step: shows "St' = (if \<exists>Ci\<in>set P \<union> set Q. fst Ci = [] then ([], [], remdups_clss P @ Q, n + length (remdups_clss P)) else case N of [] \<Rightarrow> case P of [] \<Rightarrow> (N, P, Q, n) | P0 # P' \<Rightarrow> let (C, i) = select_min_weight_clause P0 P'; N = map (\<lambda>D. (D, n)) (remdups_gen mset (resolve_rename C C @ concat (map (resolve_rename_either_way C \<circ> fst) Q))); P = filter (\<lambda>(D, j). mset D \<noteq> mset C) P; Q = (C, i) # Q; n = Suc n in (N, P, Q, n) | (C, i) # N \<Rightarrow> let C = reduce (map fst (P @ Q)) [] C in if C = [] then ([], [], [([], i)], Suc n) else if is_tautology C \<or> subsume (map fst (P @ Q)) C then (N, P, Q, n) else let P = reduce_all C P; (back_to_P, Q) = reduce_all2 C Q; P = back_to_P @ P; Q = filter (Not \<circ> strictly_subsume [C] \<circ> fst) Q; P = filter (Not \<circ> strictly_subsume [C] \<circ> fst) P; P = (C, i) # P in (N, P, Q, n))"
af_simps_3: fixes \<phi> :: "'a ltl"   and x :: "'a set"   and xs :: "'a set list" shows "af (X \<phi>) (x # xs) = af \<phi> xs"
Ord_Least: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "Ord (LEAST i. Ord i \<and> P i)"
lossless_spmf_def: fixes p :: "'c spmf" shows "lossless_spmf p = (weight_spmf p = 1)"
in_foldr_UnI: fixes x :: "'a"   and A :: "'a set"   and xs :: "'a set list" assumes "x \<in> A"   and "A \<in> set xs" shows "x \<in> foldr (\<union>) xs {}"
wordinterval_setminus_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_setminus r1 r2) = wordinterval_to_set r1 - wordinterval_to_set r2"
Abs_atMost_induct: fixes P :: "'c atMost \<Rightarrow> bool"   and x :: "'c atMost" assumes "\<And>y. y \<in> {..<LENGTH('c)} \<Longrightarrow> P (Abs_atMost y)" shows "P x"
horiz_vector_helper_pos_ind_R: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "((\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x) = (z_R I sign = 1)"
paint_rbt_sorted: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt"   and c :: "color" assumes "ord.rbt_sorted less t" shows "ord.rbt_sorted less (paint c t)"
left_commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ac_operator f" shows "f a (f b c) = f b (f a c)"
cf_of_augment_edge: fixes u :: "nat"   and v :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "(u, v) \<in> cfE_of f" shows "cf_of (augment_edge f (u, v) \<Delta>) = augment_edge_cf (cf_of f) (u, v) \<Delta>"
list_all2_alt: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list" shows "list_all2 P xs ys = list_all2_alt P xs ys"
MM10: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" assumes "|~ F = G" shows "\<turnstile> \<box>[F]_v = \<box>[G]_v"
inj_map_strong: fixes x :: "'b pmf"   and xa :: "'b pmf"   and f :: "'b \<Rightarrow> 'c"   and fa :: "'b \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>z \<in> set_pmf x; za \<in> set_pmf xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_pmf f x = map_pmf fa xa" shows "x = xa"
finite_cone_iff_cone_list: fixes Vs :: "'b vec set"   and n :: "nat"   and Vsl :: "'b vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl" shows "gram_schmidt.finite_cone n Vs = gram_schmidt.cone_list n Vsl"
fresh_set_empty: fixes a :: "atom" shows "a \<sharp> {}"