hcju/isabelleps huggingface dataset:
inj_on_imageI2: fixes f' :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f"   and A :: "'d set" assumes "inj_on (f' \<circ> f) A" shows "inj_on f A"
assms_3: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "subterms f \<subseteq> Mapping.keys M"
completenessForSAT: fixes F0 :: "Literal list list"   and state0 :: "State"   and state :: "State"   and decisionVars :: "nat set" assumes "satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "\<not> formulaFalse F0 (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state))"
Fun_ex_if_subterm: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "t \<sqsubset> s" shows "\<exists>f T. Fun f T \<sqsubseteq> s \<and> t \<in> set T"
\<gamma>_def: fixes i :: "nat" shows "\<gamma> i = (- 1) ^ \<sigma> i * pow_int (f (i - 1)) (1 - int (\<delta> (i - 1))) * (\<Prod>l\<leftarrow>[3..<Suc i]. (\<beta> l / \<alpha> l) ^ (n (l - 1) - n (i - 1) + 1) * f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)))"
R18_4: shows "pmf (sds R18) c = 1 - pmf (sds R14) a"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Drop of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2) = f2"
assms_1: shows "AE x in M. f x = ennreal (g x)"
\<T>o_e_ide_1: fixes e :: "'s"   and b :: "'c" assumes "e \<in> F.SET a"   and "C.ide b" shows "\<guillemotleft>\<T>o e b : Y a b \<rightarrow>\<^sub>S F b\<guillemotright>"
weak_group_morphism_inv_into: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set"   and a :: "'a" assumes "weak_group_morphism f H G"   and "a \<in> carrier G" obtains h :: "'a" and h' :: "'a" where "h \<in> H" and "inv_into (carrier G) f (f a) = h \<otimes> a" and "h' \<in> H" and "inv_into (carrier G) f (f a) = a \<otimes> h'"
that: fixes x :: "(real, 'n) vec" assumes "x \<in> B"   and "x \<noteq> 0"   and "\<And>i. x $ i \<in> \<int>" shows "thesis"
deg_pm_plus: fixes s :: "'b \<Rightarrow>\<^sub>0 'a"   and t :: "'b \<Rightarrow>\<^sub>0 'a" shows "deg_pm (s + t) = deg_pm s + deg_pm t"
list_map_eq: fixes vl :: "nat"   and xs :: "nat list" assumes "vl = length xs" shows "map (\<lambda>i. xs ! (i - 1)) [Suc 0..<Suc vl] = xs"
fimageI: fixes x :: "'a"   and A :: "'a fset"   and f :: "'a \<Rightarrow> 'b" assumes "x |\<in>| A" shows "f x |\<in>| f |`| A"
type_update_select_length3: fixes cts :: "ct list"   and ctm :: "ct list" assumes "type_update_select (TopType cts) = TopType ctm"   and "3 \<le> length cts" shows "\<exists>cts' ct1 ct2 ct3. cts = cts' @ [ct1, ct2, ct3] \<and> ct_eq ct3 (TSome T_i32)"
cod_char: fixes f :: "'c" shows "Cop.cod f = C.dom f"
val_Bool_simp2_1: shows "\<lbrakk>TT\<rbrakk> = True"
convex_improper: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "\<not> proper f" shows "\<forall>x\<in>rel_interior (domain f). f x = - \<infinity>"
ge_tpoly_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and q :: "('a pvar, hmultiset) tpoly"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym q p = (\<forall>A. tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A \<longrightarrow> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of q))"
D_1: shows "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>) Var"
y_axis: fixes x :: "complex" assumes "is_imag x"   and "0 < Im x"   and "Im x < 1" shows "P (of_complex x)"
pmat_extension_def: fixes m :: "'a mat" shows "st.pmat_extension m = st.ptensor_mat m (1\<^sub>m st.d2)"
pres_wf_exec: fixes G :: "nat graph_rec"   and ss :: "nat list list"   and w :: "nat list" assumes "dom_sl G"   and "dom_sl.wf_dom G ss w"   and "w \<noteq> []" shows "\<forall>(q, \<tau>)\<in>set (cfg_doms.exec G (hd w) (ss ! hd w)). \<tau> \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))"
head_subst_db: fixes i :: "nat"   and x :: "'a"   and s :: "('s, 'a) tm" shows "head (subst_db i x s) = head (subst (raw_db_subst i x) s)"
hyps_1: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "xs = smods p q"   and "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "cindex_poly a b q p = changes_itv_smods a b p q"
Procedure_defs_3: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('c, 'd) chan"   and p :: "PrIds" shows "Calling ch p \<equiv> cbit<ch!p> \<noteq> rbit<ch!p>"
subset_code_1: fixes xs :: "'c list"   and B :: "'c set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
weak_ranking_index_unique': fixes xs :: "'b set list"   and i :: "nat"   and x :: "'b" assumes "is_weak_ranking xs"   and "i < length xs"   and "x \<in> xs ! i" shows "i = find_index ((\<in>) x) xs"
assms: shows "prefix\<cdot>(xs :@ ys)\<cdot>zs = TT"
exp_times_arg_commute: fixes A :: "'a" shows "exp A * A = A * exp A"
imageEquivalence: fixes f :: "('a \<times> 'b set) set"   and x :: "'a" assumes "runiq f"   and "x \<in> Domain f" shows "f ,, x = f ,,, x"
FvarsT_psubstT: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)" shows "FvarsT (psubstT r txs) = FvarsT r - snd ` set txs \<union> \<Union> {if x \<in> FvarsT r then FvarsT t else {} |t x. (t, x) \<in> set txs}"
field_char_0_hom_axioms: shows "field_char_0_hom of_real"
superset_imp_sort_leq: fixes s2 :: "String.literal set"   and s1 :: "String.literal set" assumes "sort_ex cs s2"   and "s2 \<subseteq> s1" shows "sort_leq cs s1 s2"
FNCALL: fixes Xs :: "freeExp list"   and Xs' :: "freeExp list"   and F :: "nat" assumes "(Xs, Xs') \<in> listrel exprel" shows "FNCALL F Xs \<sim> FNCALL F Xs'"
fk'_2: fixes t :: "real" assumes "0 < t" shows "frechet_derivative (\<lambda>x. nth_derivative k f x 1) (at t) 1 = rp t * (exp (- inverse t) / t ^ (2 * k + 2))"
txs_trm: shows "fst ` set txs \<subseteq> trm"
reflp_onI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a. a \<in> A \<Longrightarrow> P a a" shows "reflp_on P A"
mk_fun_def: fixes Ts :: "Unary_PCF.type list"   and T :: "Unary_PCF.type" shows "Ts \<rightarrow>\<rightarrow> T = fold (\<rightarrow>) (rev Ts) T"
nf_invar'_axioms: shows "nf_invar' c src dst PRED C N d"
poly_subst: fixes \<alpha> :: "'e \<Rightarrow> 'd"   and \<sigma> :: "'f \<Rightarrow> ('e monom \<times> 'd) list"   and p :: "('f monom \<times> 'd) list" shows "eval_poly \<alpha> (poly_subst \<sigma> p) = eval_poly (\<lambda>v. eval_poly \<alpha> (\<sigma> v)) p"
bounded_iff: fixes b :: "'k"   and c :: "'k"   and a :: "'k" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
cases: fixes E :: "(nat \<times> nat) set"   and a1 :: "nat list"   and a2 :: "nat" assumes "path_entry E a1 a2" obtains "a1 = []" and "a2 = 0"   | u :: "nat" and v :: "nat" and l :: "nat list" where "a1 = u # l" and "a2 = v" and "(u, v) \<in> E" and "path_entry E l u"
matrix_matrix_mul_diag_matr: fixes A :: "(('d, 'e) vec, 'f) vec"   and f :: "'e \<Rightarrow> 'd" shows "A ** diag_mat f = (\<chi>i j. A $ i $ j * f j)"
cases: fixes p :: "'node"   and S :: "'node set"   and a :: "'node" assumes "reachable_through p S a" obtains "a = p"   | p' :: "'node" and Sl :: "'node set" and p'' :: "'node" where "a = p''" and "reachable_through p S p'" and "p' \<in> W" and "Sl \<in> slices p'" and "Sl \<subseteq> S" and "p'' \<in> Sl"
rec_def: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a" shows "rec e r x = (THE y. Rec e r x y)"
g0_def: shows "PROP g0 \<equiv> (\<And>kvs. Q 0 kvs)"
zeroth_homology_group: fixes X :: "'b topology" shows "homology_group 0 X \<cong> free_Abelian_group (path_components_of X)"
NatTransExt_def: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "NatTransExt \<eta> \<equiv> ($$) \<eta> \<in> extensional (obj\<^bsub>NTCatDom \<eta>\<^esub>)"
Basis_le_infnorm: fixes b :: "'b"   and x :: "'b" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> infnorm x"
CFG_axioms_def: fixes \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'h set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'h set" shows "CFG_axioms \<alpha>n invar defs uses \<equiv> ((\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> uses g n = {}) \<and> (\<forall>g n. finite (defs g n))) \<and> (\<forall>v g n. v \<in> uses g n \<longrightarrow> n \<in> set (\<alpha>n g)) \<and> (\<forall>g n. finite (uses g n)) \<and> (\<forall>g. invar g)"
upd_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and x :: "nat"   and y :: "nat"   and v :: "'c" shows "upd m x y v = m(x := (m x)(y := v))"
gcd: shows "gcd (nat (int e - int e') mod order \<G>) (order \<G>) = 1"
intro\<^sub>2: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<R>\<^sub>2p \<Gamma> x xa xaa xaaa xaaaa xaaaaa) xa) lc'" shows "\<R>p \<Gamma> lc lc'"
head_if: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "prod g {m..n} = (if n < m then 1::'b else prod g {m..<n} * g n)"
replicate_length_same: fixes xs :: "'c list"   and x :: "'c" assumes "\<forall>y\<in>set xs. y = x" shows "replicate (length xs) x = xs"
direct_arc_def: fixes uv :: "'a \<times> 'a" shows "direct_arc uv \<equiv> SOME e. {fst uv, snd uv} = {fst e, snd e}"
tab_in_vhom': shows "\<guillemotleft>\<rho> : g \<Rightarrow> r \<star> f\<guillemotright>"
sq_norm_poly_eq_0: fixes p :: "'a poly" shows "(\<parallel>p\<parallel>\<^sup>2 = (0::'a)) = (p = 0)"
set_plus_mono_b: fixes C :: "'c set"   and D :: "'c set"   and x :: "'c"   and a :: "'c" assumes "C \<subseteq> D"   and "x \<in> a +o C" shows "x \<in> a +o D"
lsided_eq_var2: fixes x :: "'b \<times> 'c" shows "prod.lsd x = (x = prod.bres top_prod x)"
dim_eq_card_independent: fixes B :: "'b set" assumes "independent B" shows "dim B = card B"
bilinear_form_scale_m: fixes v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex"   and k :: "complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "bilinear_form v1 v2 (k *\<^sub>s\<^sub>m H) = k * bilinear_form v1 v2 H"
yun_2: fixes a :: "complex poly"   and i :: "nat" assumes "(a, i) \<in> set pis" shows "square_free a \<and> degree a \<noteq> 0"
flip_altsublist_adjacent_in_Q': fixes as :: "'w list"   and s :: "'w"   and t :: "'w"   and bs :: "'w list" assumes "Abs_freelist (as @ pair_relator_halflist s t @ bs) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>" shows "Abs_freelist (as @ pair_relator_halflist t s @ bs) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
pairs_def: shows "stone_phi.pairs = {(x, y). \<exists>z. y = stone_phi (- x) \<squnion> up_filter z}"
intvs_singleton_1: fixes i :: "int" shows "{i..<i + 1} = {i}"
vlrestriction_vconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<circ> \<restriction>\<^sup>l\<^sub>\<circ> A = (r \<restriction>\<^sup>r\<^sub>\<circ> A)\<inverse>\<^sub>\<circ>"
fresh_5: fixes a :: "name"   and x1 :: "trm" shows "a \<sharp> [x1] = a \<sharp> x1"
inject_1: fixes x1 :: "int \<times> int"   and y1 :: "int \<times> int" shows "(C x1 = C y1) = (x1 = y1)"
diff_empty: shows "\<forall>i<length cmds. differing_vars_lists mem\<^sub>1 mem\<^sub>2 (replicate (length cmds) (mem\<^sub>1, mem\<^sub>2)) i = {}"
l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.to_tree_order"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "ManiPref.blocking_on ds X h X''" shows "\<not> ManiPref.stable_on ds X"
zero_less_measure_iff: fixes M :: "'d measure"   and A :: "'d set" shows "(0 < Sigma_Algebra.measure M A) = (Sigma_Algebra.measure M A \<noteq> 0)"
finite_Un: fixes F :: "'x set"   and G :: "'x set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
subst_cls_empty_iff: fixes C :: "'a literal multiset"   and \<eta> :: "'s" shows "(C \<cdot> \<eta> = {#}) = (C = {#})"
funion_commute: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<union>| B = B |\<union>| A"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (L a)"
h: shows "abc_steps_l (as, am) ap (Suc stp) = (bs, bm)"
qiter_fun_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "prod.qiter_fun x y = sup_prod x \<circ> times_prod' y"
hSuc_not_zero: fixes m :: "nat star" shows "hSuc m \<noteq> 0"
2: shows "loop_super_orientation (x \<squnion> (1::'a)) y"
WTrt_elim_cases_12: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and F :: "String.literal"   and D :: "String.literal"   and T :: "ty" assumes "P,E,h \<turnstile> e\<bullet>F{D} : T" obtains U :: "ty" and C :: "String.literal" and fm :: "fmod" where "P,E,h \<turnstile> e : U" and "class_type_of' U = \<lfloor>C\<rfloor>" and "P \<turnstile> C has F:T (fm) in D"   | "P,E,h \<turnstile> e : NT"
HPhase1or2ReadThen_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d q"   and "HInv4b s p" shows "HInv4b s' p"
strict_runit': fixes f :: "'a" assumes "ide f" shows "\<r>\<^sup>-\<^sup>1[f] = f"
e2eenreal_enn2ereal_diff: fixes x :: "ennreal"   and y :: "ennreal" shows "e2ennreal (enn2ereal x - enn2ereal y) = x - y"
coretract_inverse: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and y :: "'e" assumes "coretract G"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y) = y"
join_subpaths_middle: fixes p :: "real \<Rightarrow> 'a" shows "subpath 0 (1 / 2) p +++ subpath (1 / 2) 1 p = p"
Cauchy_theorem_global: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "open S"   and "f holomorphic_on S"   and "valid_path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "path_image \<gamma> \<subseteq> S"   and "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" shows "(f has_contour_integral 0) \<gamma>"
Sigma_mset_empty1: fixes B :: "'a \<Rightarrow> 'b multiset" shows "Sigma_mset {#} B = {#}"
ennreal_inverse_eq_0_iff: fixes a :: "ennreal" shows "(inverse a = 0) = (a = top)"
strictly_preserves_T: fixes f :: "'c"   and g :: "'c" assumes "C.arr f"   and "C.arr g" shows "F (f \<otimes>\<^sub>C g) = F f \<otimes>\<^sub>D F g"
fmrel_on_fset_acyclic': fixes ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "acyclicP_on (fmran' ym) R"   and "subtuple R\<^sup>+\<^sup>+ xm ym"   and "subtuple R ym xm" shows "xm = ym"
Osum_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "trans r"   and "trans r'" shows "trans (r \<union>o r')"
assms_3: shows "R \<subseteq> {v. v \<turnstile> g}"
2: fixes r :: "nat"   and c :: "nat"   and ts :: "((nat \<times> nat) \<times> int) list"   and v :: "int"   and x :: "nat \<Rightarrow> int"   and i :: "nat" assumes "(r, c) \<notin> fst ` set ts \<and> distinct (map fst ts)" shows "local.pr (\<alpha> (((r, c), v) # ts)) x i = local.pr (\<alpha> [((r, c), v)]) x i + local.pr (\<alpha> ts) x i"
assms_1: fixes t :: "real" assumes "t \<in> {x..y}" shows "(f has_real_derivative f' t) (at t)"
pairwise_disjnt_imp_sunflower: fixes F :: "'b set set" assumes "pairwise disjnt F" shows "sunflower F"
image_mset_is_empty_iff: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset" shows "(image_mset f M = {#}) = (M = {#})"
Rep_mod: fixes x :: "'b bit1" shows "Rep_bit1 x mod int CARD('b bit1) = Rep_bit1 x"
finfun_dom_conv: fixes f :: "('a, 'b) finfun"   and x :: "'a" shows "finfun_apply (finfun_dom f) x = (finfun_apply f x \<noteq> finfun_default f)"
assms_3: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "a\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
More_def: shows "More \<equiv> \<Lambda> app. resT_abs\<cdot>(sinr\<cdot>(up\<cdot>app))"
assms_2: fixes m :: "'b set" assumes "m open\<^bsub>T\<^esub>" shows "carrier \<inter> f -` m open"
Inf_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.finite_lattice_complete Inf Sup inf less_eq less sup bot top" shows "Inf A = Finite_Set.fold inf top A"
kozen_induct_l: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y + z \<le> y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
LLL_many_factorization_def: fixes f :: "int poly" shows "LLL_many_factorization f = (let p = suitable_prime_bz f; (uu_, fs) = finite_field_factorization_int p f; n = degree f; no = \<parallel>f\<parallel>\<^sup>2; B = sqrt_int_ceiling (2 ^ (5 * (n div 2) * (n div 2)) * no ^ (2 * (n div 2))); l = find_exponent p B; us = hensel_lifting p l f fs; pl = p ^ l in LLL_implementation.LLL_many_reconstruction p pl f us)"
digraph_isomorphism_def: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" shows "H.digraph_isomorphism hom \<equiv> wf_digraph H \<and> inj_on (iso_verts hom) (verts H) \<and> inj_on (iso_arcs hom) (arcs H) \<and> (\<forall>a\<in>arcs H. iso_verts hom (tail H a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head H a) = iso_head hom (iso_arcs hom a))"
size_mset: fixes xs :: "'c list" shows "size (mset xs) = length xs"
rep_eq: fixes x :: "'c set \<Rightarrow> 'b set"   and xa :: "'c trace" shows "Rep_trace (map_\<Gamma> x xa) = smap (\<lambda>(xa, i). (x xa, i)) (Rep_trace xa)"
zero_Integer_neq_bottom: shows "0 \<noteq> \<bottom>"
invertible_fst_row_add_iterate_PA: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and P :: "(('a, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'b" assumes "n < nrows A"   and "invertible P" shows "invertible (fst (row_add_iterate_PA (P, A) n i j))"
RelStay: fixes P :: "pi"   and Q :: "pi" assumes "(P \<parallel> !P, Q) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim" shows "(!P, Q) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim"
Regions'_def: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d" shows "Regions' X v n not_in_X \<equiv> (finite X \<and> (\<forall>c\<in>X. 0 < v c) \<and> (\<forall>c. c \<notin> X \<longrightarrow> n < v c)) \<and> bij_betw v X {1..n} \<and> X \<noteq> {} \<and> not_in_X \<notin> X"
propa_ss1: shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). ss ! q = \<tau> \<squnion>\<^bsub>f\<^esub> a ! q"
eSubst_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and eY :: "'c withERR"   and y :: "'var"   and eX :: "'c withERR"   and Z :: "'c" assumes "eSubst MOD ys eY y eX = OK Z" shows "\<exists>s X Y. eY = OK Y \<and> eX = OK X \<and> igWls MOD s X \<and> igWls MOD (asSort ys) Y \<and> Z = igSubst MOD ys Y y X"
Units_inv_Units: fixes x :: "'a" assumes "x \<in> Units G" shows "inv x \<in> Units G"
fix_in_H: fixes H :: "'a set"   and x :: "'a" assumes "H = {x. (x, f x) \<in> r \<and> x \<in> A}"   and "x \<in> P" shows "x \<in> H"
iii: shows "({#x \<lhd> \<sigma>. x \<in># mset_lit L#}, {#x \<lhd> \<sigma>. x \<in># mset_lit M#}) \<in> mult trm_ord"
inv_mult_group: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)" shows "inv\<^bsub>DirProds G I\<^esub> (x \<otimes>\<^bsub>DirProds G I\<^esub> y) = inv\<^bsub>DirProds G I\<^esub> y \<otimes>\<^bsub>DirProds G I\<^esub> inv\<^bsub>DirProds G I\<^esub> x"
le_type_eq_is_bound_typ_instance: fixes t :: "typ"   and sch :: "type_scheme" shows "(mk_scheme t \<le> sch) = t <| sch"
0: shows "p0 \<cdot> induced_arrow (C.dom p0') (D.mkCone p0' p1') = p0'"
that: fixes L :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" assumes "algebraic_closure L (indexed_const ` carrier R)"   and "indexed_const \<in> ring_hom R L" shows "thesis"
uminus_mult_left_mat: fixes A :: "'c mat"   and B :: "'c mat" assumes "dim_col A = dim_row B" shows "- A * B = - (A * B)"
quasi_inverses_def: fixes f :: "'a"   and g :: "'a" shows "B.quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. B.internal_equivalence f g \<phi> \<psi>"
case_distrib: fixes h :: "'y \<Rightarrow> 'x"   and f1 :: "'y"   and f2 :: "'z \<Rightarrow> 'y"   and option :: "'z option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
coinduct: fixes X :: "('c, 'd) resource \<Rightarrow> bool"   and x :: "('c, 'd) resource"   and \<I> :: "('c, 'd) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>res. x = res \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> lossless_spmf (run_resource res x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (run_resource res x) \<longrightarrow> X xb \<or> lossless_resource \<I> xb)" shows "lossless_resource \<I> x"
zeroI: fixes c :: "'c"   and u :: "'c \<Rightarrow> real" assumes "c \<in> \<X>"   and "u \<in> V"   and "u c = 0" shows "zero c ([u]\<^sub>\<R>)"
impl_model_check_def: fixes cfg :: "'cfg_l2b \<times> 'cfg_int \<times> 'cfg_ce"   and sys :: "'sai"   and \<phi> :: "'prop ltlc" shows "impl_model_check cfg sys \<phi> \<equiv> let ba = ltl_to_gba_impl (cfg_l2b cfg) (not\<^sub>c \<phi>); (G, map_q) = inter_impl (cfg_int cfg) sys ba; ce = find_ce_impl (cfg_ce cfg) G in case ce of None \<Rightarrow> None | Some None \<Rightarrow> Some None | Some (Some ce) \<Rightarrow> Some (Some (map_run_impl map_q ce))"
Image_empty1: fixes X :: "'c set" shows "{} `` X = {}"
obj_trg: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.obj (Span.trg \<mu>)"
eval_rs: fixes x :: "nat" shows "s x = eval r_s [x]"
rep_eq: fixes x :: "'a scene"   and xa :: "'a scene" shows "Rep_scene (x \<squnion>\<^sub>S xa) = (if \<forall>s\<^sub>1 s\<^sub>2. Rep_scene xa (Rep_scene x s\<^sub>1 s\<^sub>2) s\<^sub>2 = Rep_scene x (Rep_scene xa s\<^sub>1 s\<^sub>2) s\<^sub>2 then \<lambda>s\<^sub>1 s\<^sub>2. Rep_scene xa (Rep_scene x s\<^sub>1 s\<^sub>2) s\<^sub>2 else (\<lambda>s\<^sub>1 s\<^sub>2. s\<^sub>1))"
base: fixes bn :: "'a poly"   and cn :: "'a poly" assumes "bn = 1" shows "P bn cn"
equivp_reflp2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "equivp R" shows "reflp R"
plus_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x + y \<equiv> pnat_min x y"
bij_betw_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"