hcju/isabelleps huggingface dataset:
distributed_measurable: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal" assumes "distributed M N X f" shows "X \<in> M \<rightarrow>\<^sub>M N"
eq_right_lang_def: shows "Collapse.eq_right_lang \<equiv> {(u, v). u \<in> states Collapse_dfa \<and> v \<in> states Collapse_dfa \<and> Collapse.right_lang u = Collapse.right_lang v}"
higher_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t"   and v :: "'t" shows "higher (higher p u) v = higher p (ord_term_lin.max u v)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) ((\<mu> \<star> \<nu>) \<star> \<pi>)"
prems: shows "s @ t = trace_hide (a # ta) (ev ` A)"
nth_ucast: fixes w :: "'b word"   and n :: "nat" shows "UCAST('b \<rightarrow> 'a) w !! n = (w !! n \<and> n < LENGTH('a))"
fresh_finite_union: fixes S :: "'b set"   and T :: "'b set"   and a :: "atom" assumes "finite S"   and "finite T" shows "a \<sharp> S \<union> T = (a \<sharp> S \<and> a \<sharp> T)"
assms_3: shows "g \<in> space\<^sub>N (\<LL> q M)"
eNorm_Norm': fixes x :: "'c"   and N :: "'c quasinorm" assumes "x \<notin> space\<^sub>N N" shows "Norm N x = 0"
assms: fixes i :: "'a"   and v :: "'b" assumes "inp i = Some v" shows "phi v = phi' v"
fixedaction_swap: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and a :: "action"   and m1 :: "'c match_expr list"   and m2 :: "'c match_expr list"   and s :: "state" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) (m1 @ m2)) s = approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) (m2 @ m1)) s"
len_n2_lt_n1_y_ys: shows "length (takeWhile ((\<noteq>) n2) (y # ys)) < length (takeWhile ((\<noteq>) n1) (y # ys))"
action_loc_aux_sim_action: fixes a :: "('a, 'b) obs_event action"   and a' :: "('a, 'b) obs_event action"   and P :: "'c prog" assumes "a \<approx> a'" shows "action_loc_aux P a = action_loc_aux P a'"
FWI_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat" shows "FWI M n k \<equiv> fwi M n k n n"
stable_img: fixes t :: "'s" assumes "terminal t" shows "t \<in> some_img ` hom unity t"
bit_xor_iff: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (xor a b) n = (bit a n \<noteq> bit b n)"
sinvar_mono_imp_negative_mono: fixes N :: "'v set"   and E :: "('v \<times> 'v) set"   and E' :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<not> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP" shows "\<not> sinvar \<lparr>nodes = N, edges = E\<rparr> nP"
divide_less_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide b c) a = (if less zero c then less b (times a c) else if less c zero then less (times a c) b else less zero a)"
mk_poly2_sylvester_upper: fixes i :: "nat"   and q :: "'b poly"   and p :: "'b poly" assumes "i < degree q" shows "mk_poly2 (sylvester_mat p q) $ i = monom (1::'b) (degree q - Suc i) * p"
Always_ConstrainsI: fixes F :: "'b program"   and INV :: "'b set"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> Always INV"   and "F \<in> INV \<inter> A Co A'" shows "F \<in> A Co A'"
spanE: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> MPoly_Type_Class.punit.pmdl B" obtains A :: "('b \<Rightarrow>\<^sub>0 'c) set" and q :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. MPoly_Type_Class.punit.mult_scalar (q b) b)"
UP_subring_pderiv_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "pderiv g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
productF_path_reverse_ob: fixes w :: "('d \<times> 'e) list"   and r1 :: "'f list"   and r2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and p1 :: "'f"   and p2 :: "'g" assumes "length w = length r1"   and "length r1 = length r2"   and "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path AB (w || r1 || r2) (p1, p2)"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" obtains r2' :: "'g list" where "path B (w || r2') p2 \<and> length w = length r2'"
li_Cons: fixes u :: "'a"   and K :: "'a set"   and Us :: "'a list" assumes "u \<in> carrier R"   and "u \<notin> Span K Us"   and "local.independent K Us" shows "local.independent K (u # Us)"
linear_surjective_imp_injective: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f"   and "surj f"   and "vec.dim UNIV = vec.dim UNIV" shows "inj f"
transcendental_Ln: fixes x :: "complex" assumes "algebraic x"   and "x \<noteq> 0"   and "x \<noteq> 1" shows "\<not> algebraic (Ln x)"
is_ring_iso_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme" shows "R \<simeq> S = (ring_iso R S \<noteq> {})"
rev_fBexI: fixes x :: "'b"   and A :: "'b fset"   and P :: "'b \<Rightarrow> bool" assumes "x |\<in>| A"   and "P x" shows "fBex A P"
simps_2: fixes c :: "color"   and lt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
prems_2: shows "s' \<in> set_pmf (action (absc cfg))"
selection_char: fixes s :: "'a"   and x :: "'a" shows "selection s x = (\<exists>y. s = - y \<sqinter> x)"
Limit_vcpower_in_VsetI: fixes \<alpha> :: "V"   and n :: "V"   and A :: "V" assumes "Limit \<alpha>"   and "n \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "vfinite n" shows "A ^\<^sub>\<times> n \<in>\<^sub>\<circ> Vset \<alpha>"
underS_Field: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "underS r a \<subseteq> Field r"
assms_1: shows "safe_distance_5r < s\<^sub>o - s\<^sub>e"
continuous_on_closed: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on S f = (\<forall>T. closedin (top_of_set (f ` S)) T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
eq2: shows "(\<lambda>i. term_of_pair (0::'a, i)) ` {0..<length bs} = (\<Union>i\<in>{0..<length bs}. {term_of_pair (0::'a, i)})"
rep_eq: fixes x :: "'b prm" shows "($) x = preprm_apply (rep_prm x)"
simps: fixes x :: "'b extT" shows "- x = (if x = extT.Val bot then extT.Top else extT.Val bot)"
normalized_n_primitive_alist_and': fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and P :: "'c \<Rightarrow> bool"   and as :: "'b negation_type list" shows "normalized_n_primitive disc_sel P (alist_and' as) = (\<forall>a\<in>set as. normalized_n_primitive disc_sel P (negation_type_to_match_expr a))"
finite_set_plus: fixes s :: "'c set"   and t :: "'c set" assumes "finite s"   and "finite t" shows "finite (s + t)"
that_1: shows "(l, g, \<mu>) \<in> PTA.trans_of A"
seq_UINF_distl': fixes P :: "('b, 'd) urel"   and F :: "'e \<Rightarrow> ('d, 'c) urel" shows "P ;; (\<Sqinter> Q \<bullet> F Q) = (\<Sqinter> Q \<bullet> P ;; F Q)"
remove1_mset_add_mset_If: fixes L' :: "'b"   and C :: "'b multiset"   and L :: "'b" shows "remove1_mset L (add_mset L' C) = (if L = L' then C else remove1_mset L C + {#L'#})"
MST_validn_def: fixes K :: "nat" shows "MST_validn K = (\<forall>C m par code l0 T MI Anno. mbody_is C m (par, code, l0) \<longrightarrow> MST\<down>(C, m) = Some (T, MI, Anno) \<longrightarrow> \<Turnstile>\<^sub>K \<lbrace> (\<lambda>s0 s. s = mkState s0) \<rbrace> C , m , l0 \<lbrace> mkPost T \<rbrace> mkInv MI )"
tl1_to_tl2_simps_5: fixes s2 :: "'s2"   and s1 :: "'s1"   and tl1 :: "'tl1"   and s1' :: "'s1"   and stls1' :: "('s1 \<times> 'tl1 \<times> 's1) llist" shows "tl1_to_tl2 s2 (LCons (s1, tl1, s1') stls1') = LCons (s2, SOME (tl2, s2'). s2 -2-tl2\<rightarrow> s2' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2) (tl1_to_tl2 (snd (SOME (tl2, s2'). s2 -2-tl2\<rightarrow> s2' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2)) stls1')"
stable_def: fixes r :: "('a \<times> 'a) set" shows "stable r \<equiv> \<forall>A F. |A| <o r \<and> (\<forall>a\<in>A. |F a| <o r) \<longrightarrow> |Sigma A F| <o r"
assms_4: shows "bijective (i * top) \<and> bijective (i[[top]])"
bdd_above_nat: fixes X :: "nat set" shows "bdd_above X = finite X"
inv_solve_right': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(b \<oplus> \<ominus> c = a) = (b = a \<oplus> c)"
lsumr_rsuml_left_gpv: fixes gpv :: "('h, 'i, 'l) gpv" shows "map_gpv' id lsumr rsuml (left_gpv gpv) = left_gpv (left_gpv gpv)"
drop_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list" shows "drop n (drop m xs) = drop (n + m) xs"
fmrel_on_fset_trans: fixes xm :: "'g \<rightharpoonup>\<^sub>f 'd"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and ym :: "'g \<rightharpoonup>\<^sub>f 'e"   and zm :: "'g \<rightharpoonup>\<^sub>f 'f" assumes "\<And>x y z. \<lbrakk>x \<in> fmran' xm; P x y; Q y z\<rbrakk> \<Longrightarrow> R x z"   and "subtuple P xm ym"   and "subtuple Q ym zm" shows "subtuple R xm zm"
Bfun_metric_def: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "Bfun f F = (\<exists>y K. 0 < K \<and> (\<forall>\<^sub>F x in F. dist (f x) y \<le> K))"
simps_5: fixes V :: "'c set"   and s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<langle>check: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t V S"
FI_RESULT_def: fixes M :: "(assn \<times> assn) list"   and UP :: "assn"   and UQ :: "assn"   and F :: "assn" shows "FI_RESULT M UP UQ F \<equiv> fi_m_match M \<longrightarrow> (fi_m_fst M * UP \<Longrightarrow>\<^sub>A fi_m_snd M * UQ * F)"
lens_indep_sym: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'g \<Longrightarrow> 'f" assumes "x \<bowtie> y" shows "y \<bowtie> x"
Suspend: fixes w :: "'b" shows "Suspend w \<notin> set (wa # was)"
assms_1: shows "sorted_sublist_map R h xs lo (lo' - Suc 0)"
LID\<^sub>GE: fixes l :: "'d"   and s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option" assumes "l \<in> LID\<^sub>G s" obtains r :: "'e" and ls :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" where "s r = Some ls" and "l \<in> LID\<^sub>L ls"
compact_UN: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> compact (B x)" shows "compact (\<Union> (B ` A))"
slow_select_def: fixes k :: "nat"   and xs :: "'a list" shows "slow_select k xs = isort xs ! k"
orthonormal_basis_subspace: fixes S :: "'b set" assumes "subspace S" obtains B :: "'b set" where "B \<subseteq> S" and "pairwise orthogonal B" and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B" and "card B = dim S" and "span B = S"
chain_reparam_weak_chain_def: fixes one_chain1 :: "('a \<times> (real \<Rightarrow> 'b)) set"   and one_chain2 :: "('a \<times> (real \<Rightarrow> 'b)) set" shows "chain_reparam_weak_chain one_chain1 one_chain2 \<equiv> \<exists>f. bij f \<and> f ` one_chain1 = one_chain2 \<and> (\<forall>(k, \<gamma>)\<in>one_chain1. if k = fst (f (k, \<gamma>)) then reparam_weak \<gamma> (snd (f (k, \<gamma>))) else reparam_weak \<gamma> (reversepath (snd (f (k, \<gamma>)))))"
F5: shows "(f (lub H cl), lub H cl) \<in> r"
lift_type: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type"   and i :: "nat"   and U :: "LambdaType.type" assumes "e \<turnstile> t : T" shows "e\<langle>i:U\<rangle> \<turnstile> lift t i : T"
sub: shows "\<Union> \<D>0 \<subseteq> cbox a b"
favorites_def: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'b" shows "favorites R i = Max_wrt (R i)"
sdrop_sconst: fixes n :: "nat"   and s :: "'d stream"   and x :: "'d"   and m :: "nat" assumes "sdrop n s = sconst x"   and "n \<le> m" shows "s !! m = x"
inverse_squares_sums: shows "(\<lambda>n. 1 / real ((n + 1)\<^sup>2)) sums (pi\<^sup>2 / 6)"
types_snoc: fixes e :: "nat \<Rightarrow> LambdaType.type"   and ts :: "dB list"   and Ts :: "LambdaType.type list"   and t :: "dB"   and T :: "LambdaType.type" assumes "e \<tturnstile> ts : Ts"   and "e \<turnstile> t : T" shows "e \<tturnstile> ts @ [t] : Ts @ [T]"
hyps: assumes "strict_mono_on g {0..<n}" shows "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]"
dom_def_var: fixes x :: "'a" shows "d x = (x \<cdot> U \<sqinter> 1\<^sub>\<pi>) \<parallel> 1\<^sub>\<sigma>"
Cons: shows "p \<noteq> [] \<longrightarrow> (\<forall>x s. s \<in> set p \<and> x \<in> dom (Cp A) \<longrightarrow> x \<notin> dom (Cp s)) \<longrightarrow> y \<in> dom (Cp A) \<longrightarrow> y \<notin> dom (Cp (list2FWpolicy p))"
Z: shows "Esigma (2 ^ k * p * q) = Esigma (2 ^ k * r)"
lset_intersect_lnth: fixes xs :: "'c llist"   and A :: "'c set" assumes "lset xs \<inter> A \<noteq> {}" shows "\<exists>n. enat n < llength xs \<and> xs $ n \<in> A"
lmeasurable_iff_integrable_on: fixes S :: "'a set" shows "(S \<in> lmeasurable) = ((\<lambda>x. 1) integrable_on S)"
simps_1: fixes proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a" shows "proper_intrvl.exhaustive_above proper_interval x [] = (\<not> proper_interval (Some x) None)"
P: shows "{x \<in> space M. \<not> P x} \<subseteq> A"
j_def: shows "j \<equiv> LEAST k. \<forall>a. coprime a p \<longrightarrow> [a ^ (2 ^ k * m) = 1] (mod p)"
BJ: shows "ennreal (1 - d) \<le> emeasure MJ {x \<in> space MJ. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - delta l * real l < uJ n x - uJ (n - l) ((TJ ^^ l) x)}}"
not_neutral_contains_not_neutral: fixes g :: "'b \<Rightarrow> bool"   and A :: "'b set" assumes "bool.F g A \<noteq> True" obtains a :: "'b" where "a \<in> A" and "g a \<noteq> True"
\<Phi>o_def: shows "\<Phi>o = (\<lambda>a\<in>Collect S.ide. S'.mkIde (\<phi> ` S.set a))"
connected_spaceD: fixes X :: "'c topology"   and U :: "'c set"   and V :: "'c set" assumes "connected_space X"   and "openin X U"   and "openin X V"   and "topspace X \<subseteq> U \<union> V"   and "U \<inter> V = {}"   and "U \<noteq> {}"   and "V \<noteq> {}" shows "False"
enumerator_last: fixes ls :: "nat list"   and m :: "nat"   and n :: "nat" assumes "ls \<in> set (enumerator m n)" shows "last ls = n - 1"
gs_3: shows "\<forall>x\<in>aezfun_setspan G. \<exists>rs. length rs = length gs \<and> x = sum_list (map2 (\<delta>\<delta>) rs gs)"
in_rel: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "('c, 'e, 'g) F1"   and b :: "('d, 'f, 'h) F1" shows "F1rel R1 R2 R3 a b = (\<exists>z. z \<in> F1in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F1map fst fst fst z = a \<and> F1map snd snd snd z = b)"
maxsimpchainD_adj: fixes xs :: "'a set list" assumes "maxsimpchain xs" shows "adjacentchain xs"
trans: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
itypeI: fixes c :: "'b"   and I :: "interface" shows "c ::\<^sub>i I"
Prim1_imp_R1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> Prim1" shows "f \<in> \<R>"
size_eq_0_iff_empty: fixes M :: "'k multiset" shows "(size M = 0) = (M = {#})"
covs: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>X VS. x \<in> X \<and> openin (top_of_set S) X \<and> \<Union> VS = c \<inter> p -` X \<and> (\<forall>u\<in>VS. openin (top_of_set c) u) \<and> disjoint VS \<and> (\<forall>u\<in>VS. \<exists>q. homeomorphism u X p q)"
blinfun_apply_integral: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c"   and s :: "'a set"   and x :: "'b" assumes "f integrable_on s" shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
case_ifexi_rule: fixes s :: "'s"   and ni :: "'ni"   and n :: "'a ifex"   and fti :: "'s \<Rightarrow> ('b \<times> 'c) option"   and ft :: "'d"   and Q :: "'s \<Rightarrow> ('b \<times> 'd) set"   and I' :: "'c \<Rightarrow> bool"   and ffi :: "'s \<Rightarrow> ('b \<times> 'c) option"   and ff :: "'d"   and fii :: "'a \<Rightarrow> 'ni \<Rightarrow> 'ni \<Rightarrow> 's \<Rightarrow> ('b \<times> 'c) option"   and fi :: "'a \<Rightarrow> 'a ifex \<Rightarrow> 'a ifex \<Rightarrow> 'd" assumes "I s"   and "(ni, n) \<in> R s"   and "n = Trueif \<Longrightarrow> ospec (fti s) (\<lambda>(r, s'). (r, ft) \<in> Q s \<and> I' s')"   and "n = Falseif \<Longrightarrow> ospec (ffi s) (\<lambda>(r, s'). (r, ff) \<in> Q s \<and> I' s')"   and "\<And>ti ei v t e. \<lbrakk>n = IF v t e; (ti, t) \<in> R s; (ei, e) \<in> R s\<rbrakk> \<Longrightarrow> ospec (fii v ti ei s) (\<lambda>(r, s'). (r, fi v t e) \<in> Q s \<and> I' s')" shows "ospec (case_ifexi fti ffi fii ni s) (\<lambda>(r, s'). (r, case n of Trueif \<Rightarrow> ft | Falseif \<Rightarrow> ff | IF x xa xb \<Rightarrow> fi x xa xb) \<in> Q s \<and> I' s')"
induct: fixes P :: "('d, 'e, 'f) cntxt \<Rightarrow> ('d, 'e, 'f) expr \<Rightarrow> bool"   and a0 :: "('d, 'e, 'f) cntxt"   and a1 :: "('d, 'e, 'f) expr" assumes "\<And>e. P \<box> e"   and "\<And>\<E> e1 e. P \<E> e \<Longrightarrow> P (ApplyL\<^sub>\<E> \<E> e1) e"   and "\<And>val \<E> e. P \<E> e \<Longrightarrow> P (ApplyR\<^sub>\<E> val \<E>) e"   and "\<And>\<E> e1 e2 e. P \<E> e \<Longrightarrow> P (Ite\<^sub>\<E> \<E> e1 e2) e"   and "\<And>\<E> e. P \<E> e \<Longrightarrow> P (Ref\<^sub>\<E> \<E>) e"   and "\<And>\<E> e. P \<E> e \<Longrightarrow> P (Read\<^sub>\<E> \<E>) e"   and "\<And>\<E> e1 e. P \<E> e \<Longrightarrow> P (AssignL\<^sub>\<E> \<E> e1) e"   and "\<And>l \<E> e. P \<E> e \<Longrightarrow> P (AssignR\<^sub>\<E> l \<E>) e"   and "\<And>\<E> e. P \<E> e \<Longrightarrow> P (Rjoin\<^sub>\<E> \<E>) e" shows "P a0 a1"
mat_of_rows_list_def: fixes nc :: "nat"   and rs :: "'a list list" shows "mat_of_rows_list nc rs = Matrix.mat (length rs) nc (\<lambda>(i, j). rs ! i ! j)"
assms_2: fixes i :: "nat" shows "coeff q i \<in> R"
list_all2': shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map') (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map')) (drop (length done) (update_until args rel1 rel2 nt auxlist)) (zip (linearize tss') [tp + 1 - (len + 1)..<tp + 1])"
intros_9: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow> t'"
2: shows "(\<Oplus>\<^bsub>M\<^esub>x\<in>insert a A. c \<odot>\<^bsub>M\<^esub> g x) = c \<odot>\<^bsub>M\<^esub> g a \<oplus>\<^bsub>M\<^esub> (\<Oplus>\<^bsub>M\<^esub>x\<in>A. c \<odot>\<^bsub>M\<^esub> g x)"
assms_3: shows "io_targets M q1 (vs @ xs) = {q3}"
singular_chain_minus: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_chain p X (- c) = singular_chain p X c"
semiring_simprules_12: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
simps_1: fixes uu :: "'k multiset" shows "repeat_mset 0 uu = {#}"
cases: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'f" assumes "(r OO s) a1 a2" obtains a :: "'d" and b :: "'e" and c :: "'f" where "a1 = a" and "a2 = c" and "r a b" and "s b c"
assms_1: shows "(minarc\<^sub>M f) e \<noteq> bot"
facesAt_eq_def: fixes g :: "graph" shows "facesAt_eq g \<equiv> \<forall>v\<in>\<V> g. set (facesAt g v) = {f \<in> \<F> g. v \<in> \<V> f}"
in_vars_addE: fixes x :: "nat"   and p :: "'a mpoly"   and q :: "'a mpoly" assumes "x \<in> vars (p + q)" obtains "x \<in> vars p"   | "x \<in> vars q"
simps_4: fixes n :: "nat"   and k :: "nat" shows "stirling (Suc n) (Suc k) = n * stirling n (Suc k) + stirling n k"
map_M_pure_I: fixes xs :: "'x list"   and f :: "'x \<Rightarrow> ('y, 'z, 'aa) prog"   and h :: "'y" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_M f xs) h"
primitive_iff_some_content_dvd_1: fixes f :: "'a poly" shows "primitive f = (some_gcd.listgcd (coeffs f) dvd (1::'a))"
simps_13: fixes i :: "int"   and a :: "Cooper.num" shows "simpfm (NDvd i a) = (if i = 0 then simpfm (NEq a) else if \<bar>i\<bar> = 1 then F else let a' = simpnum a in case a' of C v \<Rightarrow> if \<not> i dvd v then T else F | _ \<Rightarrow> NDvd i a')"
one_neq_iffs_1: fixes x :: "one" shows "(x \<noteq> notOneShot) = (x = oneShot)"
keys_Mapping: fixes t :: "('c, 'd) RBT.rbt" shows "Mapping.keys (RBT_Mapping.Mapping t) = set (RBT.keys t)"
mult_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d"   and "bot \<le> b"   and "bot \<le> c" shows "a \<sqinter> c \<le> b \<sqinter> d"
map_add_cancel: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m g" shows "f ++ (g -- f) = g"
"1_5": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "\<not> n < 0"   and "odd n"   and "\<not> a < 0"   and "a \<noteq> 0"   and "\<not> n \<le> a"   and "even a" shows "Jacobi (a div 2) n = jacobi_code (a div 2) n"
exp_eq_0_imp_not_bit: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n = zero" shows "\<not> bit a n"
assms: shows "\<not> (a = \<infinity> \<and> b = 0 \<or> a = 0 \<and> b = \<infinity>)"