hcju/isabelleps huggingface dataset:
abs_eq: fixes x :: "'c \<Rightarrow> 'd"   and xa :: "real" assumes "eq_onp (\<lambda>x. x \<in> bcontfun) x x" shows "xa *\<^sub>R Bcontfun x = Bcontfun (\<lambda>xb. xa *\<^sub>R x xb)"
prefixed_Imagep_imp_empty: fixes X :: "'a set" assumes "X \<subseteq> ((\<sqsubset>) ``` X) \<inter> A" shows "X = {}"
proots_count_uminus: fixes p :: "'a poly"   and s :: "'a set" shows "proots_count (- p) s = proots_count p s"
inf_aforms_refine: fixes xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and d :: "nat"   and optns :: "nat" assumes "(xi, x) \<in> aforms_rel"   and "length xi = d" shows "(RETURN (inf_aforms optns xi), Inf_specs d x) \<in> \<langle>rl_rel\<rangle>nres_rel"
adjugate_def: fixes A :: "('a, 'b) sq_matrix" shows "Square_Matrix.adjugate A = Square_Matrix.transpose (Square_Matrix.cofactor A)"
prj_in_hom_2: shows "\<guillemotleft>fg.prj\<^sub>0 : f.leg0 \<down>\<down> g.leg1 \<rightarrow>\<^sub>C g.apex\<guillemotright>"
RealLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "real"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (RealLiteral c) : \<tau>" obtains "\<tau> = Real[1]"
Y_simp: fixes f :: "'c" assumes "C.arr f" shows "Y f = (\<lambda>g. Hom.map (g, f))"
set_map_fst_auxlist: fixes t :: "nat" assumes "t \<in> set (map fst auxlist)" shows "t \<le> nt"
density_RN_deriv: fixes N :: "'a measure" assumes "absolutely_continuous P N"   and "sets N = sets P" shows "density P (RN_deriv P N) = N"
monad_right_unit: fixes m :: "'d\<cdot>'e" shows "m \<bind> return = m"
ssubst_Eats: fixes p :: "perm"   and Vs :: "name set"   and V :: "name set"   and t :: "tm"   and u :: "tm" assumes "quote_perm p Vs"   and "finite V" shows "ssubst (Eats t u) V (make_F Vs p) = Eats (ssubst t V (make_F Vs p)) (ssubst u V (make_F Vs p))"
assms: shows "G\<^sub>1 \<simeq> G\<^sub>2"
fls_base_factor_X_power_decompose_2: fixes f :: "'a fls" shows "f = fls_X_intpow (fls_subdegree f) * fls_base_factor f"
Sigma_interval_disjoint: fixes A :: "'a set"   and v :: "'a \<Rightarrow> 'b"   and w :: "'b" shows "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A. {v i<..w}) = {}"
hyps: shows "ffact n a = (\<Sum>k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)"
r_result_converg: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and v :: "nat" assumes "i = encode f"   and "recfn (e_length x) f"   and "eval f (Partial_Recursive.list_decode x) \<down>= v" shows "\<exists>t. (\<forall>t'\<ge>t. eval r_result [t', i, x] \<down>= Suc v) \<and> (\<forall>t'<t. eval r_result [t', i, x] \<down>= 0)"
hypnat_of_nat_eq: fixes m :: "nat" shows "hypnat_of_nat m = star_n (\<lambda>n. m)"
fold_fusion_def: fixes g :: "('e, 'f) generator"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'f" shows "fold_fusion g f s = fold f (list.unfoldr g s)"
select_convs_1: fixes c_sinvar :: "'a graph \<Rightarrow> bool"   and c_offending_flows :: "'a graph \<Rightarrow> ('a \<times> 'a) set set"   and c_isIFS :: "bool"   and more :: "'b" shows "c_sinvar \<lparr>c_sinvar = c_sinvar, c_offending_flows = c_offending_flows, c_isIFS = c_isIFS, \<dots> = more\<rparr> = c_sinvar"
assms: shows "\<forall>i. r (f (Suc i)) (f i)"
l6_3_1: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Out A B" shows "A \<noteq> P \<and> B \<noteq> P \<and> (\<exists>C. C \<noteq> P \<and> Bet A P C \<and> Bet B P C)"
dim_span: fixes S :: "'c set" shows "vs2.dim (vs2.span S) = vs2.dim S"
prefixes_snoc: fixes xs :: "'b list"   and x :: "'b" shows "prefixes (xs @ [x]) = prefixes xs @ [xs @ [x]]"
simps: fixes x :: "'g"   and xs :: "'g list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
word_mult_less_dest: fixes i :: "'b word"   and k :: "'b word"   and j :: "'b word" assumes "i * k < j * k"   and "unat i * unat k < 2 ^ LENGTH('b)"   and "unat j * unat k < 2 ^ LENGTH('b)" shows "i < j"
select_convs_4: fixes dwa_cond :: "'a \<Rightarrow> bool"   and dwa_step :: "'a \<Rightarrow> 'a"   and dwa_initial :: "'a"   and dwa_invar :: "'a set"   and more :: "'b" shows "dwa_invar \<lparr>dwa_cond = dwa_cond, dwa_step = dwa_step, dwa_initial = dwa_initial, dwa_invar = dwa_invar, \<dots> = more\<rparr> = dwa_invar"
compatible_tail: fixes G :: "('a, 'b) pre_digraph"   and H :: "('a, 'b) pre_digraph" assumes "compatible G H" shows "tail G = tail H"
l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog" shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node \<equiv> get_component = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component get_root_node to_tree_order \<and> is_strongly_dom_component_safe = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order \<and> is_weakly_dom_component_safe = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order"
isomorphic_implies_hpar_2: fixes f :: "'b"   and f' :: "'b" assumes "D.isomorphic f f'" shows "D.ide f'"
case_2: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and x2 :: "'g" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) \<Pi>o\<Delta>.map"
lebesgue_measurable_vimage_borel: fixes f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "f \<in> borel_measurable lebesgue"   and "T \<in> sets borel" shows "{x. f x \<in> T} \<in> sets lebesgue"
Dom_in_Obj: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" shows "concrete_category.Dom f \<in> Obj"
simps_2: fixes t :: "symbol \<times> character list"   and ts :: "(symbol \<times> character list) list" shows "chars (t # ts) = chars_of_token t @ chars ts"
assms_7: fixes x' :: "'a option list" assumes "wf_tuple n A' x'"   and "P x'"   and "Q' x'" shows "Q (restrict A x') \<and> x' ! y = Some (f (restrict A x'))"
power_zero_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and k :: "num" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (numeral k) = zero"
select_convs_2: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "E\<^bsub>\<lparr>Graph.verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = arcs"
ale_antisym: fixes x :: "ant"   and y :: "ant" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
zero_less_mult_pos2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times b a)"   and "less zero a" shows "less zero b"
set_lub_spmf: shows "set_spmf local.lub_spmf = \<Union> (set_spmf ` Y)"
hext_allocate: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and hT :: "htype" assumes "(h', a) \<in> allocate h hT" shows "h \<unlhd> h'"
that: fixes C :: "'a set" assumes "card ({x \<in> A. P x} - C) = card {x \<in> A. Q x}"   and "C \<noteq> {}"   and "C \<subseteq> {x \<in> A. P x}" shows "thesis"
generalise_output_context_check_directly_subsumes_original: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat"   and v :: "value"   and s :: "nat"   and s' :: "nat"   and e1 :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and e2 :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" assumes "stored_reused t' t = Some (r, p)"   and "Outputs t ! p = L v"   and "generalise_output_context_check v r s s' e1 e2" shows "directly_subsumes (tm e1) (tm e2) s s' t' t"
MakeFormP_Neg: fixes x :: "tm"   and y :: "tm" shows "{} \<turnstile> MakeFormP (Q_Neg x) x y"
sorted_distinct_set_unique: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'f list"   and ys :: "'f list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
tt_cons1: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "(\<exists>q'. \<lparr>p\<rparr>x\<lparr>q'\<rparr> \<and> q' \<le> q) = \<lparr>p\<rparr>x\<lparr>q\<rparr>"
comm_ring_hom_axioms: shows "comm_ring_hom of_real"
swap_def: fixes p :: "'b \<times> 'a" shows "prod.swap p = (snd p, fst p)"
lookup_empty: fixes ko :: "'c nat_term_order"   and k :: "'c" shows "OAlist_lookup_ntm (OAlist_empty_ntm ko) k = (0::'b)"
set_lbounds: fixes as :: "atom list" shows "set (lbounds as) = {(r / c, (- 1 / c) *\<^sub>s cs) |r c cs. Less r (c # cs) \<in> set as \<and> 0 < c}"
eval_norm_eq_id: fixes q :: "('c multiset \<Rightarrow> 'a) list"   and a :: "'c multiset \<Rightarrow> 'a" assumes "q \<in> carrier (poly_ring L)"   and "0 < degree q"   and "a \<in> carrier L" shows "(eval_pmod q \<circ> L.poly_of_const) a = a"
merge: fixes idx :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and ws :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "idx_sequence idx"   and "n \<in> {idx i..<idx (Suc i)}" shows "merge ws idx n = ws i n"
abs_eq: fixes xb :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'a \<Rightarrow> 'b option"   and x :: "'a \<Rightarrow> 'b option" shows "Mapping.combine xb (Mapping xa) (Mapping x) = Mapping (\<lambda>xc. combine_options xb (xa xc) (x xc))"
case_prod_conv: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and a :: "'d"   and b :: "'e" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
face_im: fixes w :: "'a set"   and y :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "w \<in> X"   and "order.greater_eq (f ` w) y" shows "y \<in> f \<turnstile> X"
lower: fixes a :: "ereal" assumes "a < l" shows "\<forall>\<^sub>F n in sequentially. a < v n"
Healthy_def: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and H :: "('c \<times> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<times> 'c \<Rightarrow> bool" shows "P is H healthy \<equiv> P = H P"
accept: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and p :: "'c" assumes "matches \<gamma> m Accept p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m Accept], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
filter_rev_alt: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "filter_rev P l = filter P (rev l)"
rel_intros: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list" assumes "list_all2 R x y" shows "rel_iarray R (IArray x) (IArray y)"
simps_3: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and n :: "'c \<Rightarrow> bool"   and P :: "'b" shows "normalized_n_primitive (disc, sel) n (MatchNot (Match P)) = (if disc P then False else True)"
encode_operator_effect_valid: fixes sv :: "sat_plan_variable"   and t :: "nat"   and op :: "(nat \<times> nat) strips_operator" assumes "sv \<in> atoms (encode_operator_effect (\<phi> prob_with_noop abs_prob ) t op)"   and "t < h"   and "op \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)" shows "valid_state_var sv"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleEnd G smult V T" shows "FGModuleHom G smult V smult T"
simps: fixes s :: "'b"   and f :: "'e \<rightarrow> ('c, 'd) Stream"   and h :: "'b \<rightarrow> ('e, 'b) Step" assumes "s \<noteq> \<bottom>" shows "concatMapS\<cdot>f\<cdot>(Stream\<cdot>h\<cdot>s) = Stream\<cdot>(concatMapStep\<cdot>f\<cdot>h)\<cdot>(s :!: Nothing)"
bool_or_not_bool: fixes x :: "Value"   and y :: "CValue"   and b :: "bool discr" assumes "x \<triangleleft>\<triangleright> y" shows "(x = B\<cdot>b) = (y = CB\<cdot>b)"
lub: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "Semilat A r f"   and "x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
subst_fresh_noop: fixes x :: "Vars.var"   and e :: "exp"   and y :: "Vars.var" assumes "atom x \<sharp> e" shows "e[x::=y] = e"
inverse_arrows_def: fixes f :: "'a"   and g :: "'a" shows "Right.inverse_arrows f g \<equiv> Right.ide (Right.comp g f) \<and> Right.ide (Right.comp f g)"
divide_less_0_iff: 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 a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
top_assn_def: shows "true = Abs_assn (Assn in_range_assn)"
Min_gr_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Min less_eq A) = (\<forall>a\<in>A. less x a)"
map_snd_pair_pmf: fixes A :: "'b pmf"   and B :: "'a pmf" shows "map_pmf snd (pair_pmf A B) = B"
cong: fixes I :: "('b \<times> 'c) spmf"   and range :: "'b \<Rightarrow> 'a set"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "ETP_base.R1 I range F \<equiv> ETP_base.R1 I range F"
lspasl_starr_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "h1,h2\<triangleright>h0"   and "\<not> (A \<and>* B) h0" shows "(h1,h2\<triangleright>h0) \<and> \<not> (A h1 \<or> (A \<and>* B) h0) \<and> starr_applied h1 h2 h0 (A \<and>* B) \<or> (h1,h2\<triangleright>h0) \<and> \<not> (B h2 \<or> (A \<and>* B) h0) \<and> starr_applied h1 h2 h0 (A \<and>* B)"
distinct_23: fixes x3 :: "'b formula"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "\<^bold>\<not> (x3::'b formula) \<noteq> (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
is_orthogonal_cspan: fixes S :: "'a set"   and a :: "'a"   and x :: "'a" assumes "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s"   and "x \<in> cspan S" shows "is_orthogonal a x"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q; P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (Eq a)) \<Longrightarrow> P (Dvd i a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (NEq a)) \<Longrightarrow> P (NDvd i a)"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Or p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Imp p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q); P (NOT p); P q\<rbrakk> \<Longrightarrow> P (NOT (Iff p q))"   and "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "\<And>a. P (Ge a) \<Longrightarrow> P (NOT (Lt a))"   and "\<And>a. P (Gt a) \<Longrightarrow> P (NOT (Le a))"   and "\<And>a. P (Le a) \<Longrightarrow> P (NOT (Gt a))"   and "\<And>a. P (Lt a) \<Longrightarrow> P (NOT (Ge a))"   and "\<And>a. P (NEq a) \<Longrightarrow> P (NOT (Eq a))"   and "\<And>a. P (Eq a) \<Longrightarrow> P (NOT (NEq a))"   and "\<And>i a. P (NDvd i a) \<Longrightarrow> P (NOT (Dvd i a))"   and "\<And>i a. P (Dvd i a) \<Longrightarrow> P (NOT (NDvd i a))"   and "\<And>Pa. P (NOT (Closed Pa))"   and "\<And>Pa. P (NOT (NClosed Pa))"   and "P T"   and "P F"   and "\<And>va. P (NOT (E va))"   and "\<And>va. P (NOT (A va))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
mkArr_restrict_eq: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "mkArr A B (restrict F A) = mkArr A B F"
emb_step_equiv': fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "(t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s) = ((\<exists>p. p \<noteq> [] \<and> emb_step_at' p t = s) \<and> t \<noteq> s)"
positive_add: fixes x :: "rat"   and y :: "rat" assumes "positive x"   and "positive y" shows "positive (x + y)"
CompFun_Int_disjoint: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<noteq> T"   and "S \<in> HAStates A"   and "T \<in> HAStates A" shows "the (CompFun A T) \<inter> the (CompFun A S) = {}"
real_sqrt_eq_zero_cancel_iff: fixes x :: "real" shows "(sqrt x = 0) = (x = 0)"
sign_changes_rat_def: fixes ps :: "rat poly list"   and x :: "rat" shows "sign_changes_rat ps x = length (remdups_adj (list_neq (map (\<lambda>p. sgn (poly p x)) ps) 0)) - 1"
non_volatile_owned_or_read_only_outstanding_refs: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb" shows "outstanding_refs (Not \<circ> is_volatile) sb \<subseteq> \<O> \<union> all_acquired sb \<union> read_only_reads \<O> sb"
numeral_Bit1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
IH: fixes q :: "'c" assumes "well_formed M"   and "path M p q" shows "set (map fst (map fst p)) \<subseteq> inputs M"
insert_remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "join.Sup_fin (insert x A) = (if A - {x} = {} then x else x + join.Sup_fin (A - {x}))"
Extra_single_not: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and str :: "char list"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (MatchNot (Match (Extra str))) a p = \<alpha> a p"
iterate_to_list_correct: fixes it :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list"   and S0 :: "'c set" assumes "set_iterator it S0" shows "set (iterate_to_list it) = S0 \<and> distinct (iterate_to_list it)"
orthogonal_comp_UNIV: shows "UNIV\<^sup>\<bottom> = {0::'b}"
right_unique_pcr_spmf_gpv: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_unique A" shows "right_unique (pcr_spmf_gpv A)"
assms_1: shows "\<forall>y. f y \<noteq> 0 \<longrightarrow> y \<noteq> x \<longrightarrow> v1 y = v2 y"
KL_divergence_def: fixes b :: "real"   and M :: "'d measure"   and N :: "'d measure" shows "KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)"
replace6: fixes oldfs :: "'a list"   and x :: "'a"   and oldF :: "'a"   and newfs :: "'a list" assumes "distinct oldfs" shows "(x \<in> set (replace oldF newfs oldfs)) = ((x \<noteq> oldF \<or> oldF \<in> set newfs) \<and> (oldF \<in> set oldfs \<and> x \<in> set newfs \<or> x \<in> set oldfs))"
prod_filter_assoc: fixes F :: "'d filter"   and G :: "'e filter"   and H :: "'f filter" shows "(F \<times>\<^sub>F G) \<times>\<^sub>F H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (F \<times>\<^sub>F G \<times>\<^sub>F H)"
idiom_ext_rule_def: shows "idiom_ext_rule = sup idiom_rule extra_rule"
minus_add_uminus_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A - B = A + - B"
AX1_def: shows "AX1 \<equiv> {x. AX1p x}"
fps_numeral_fps_const: fixes k :: "num" shows "numeral k = fps_const (numeral k)"
assms_23: shows "InvariantVarsF (getF state) F0 Vbl"
R5: fixes p\<^sub>1 :: "'a point"   and x\<^sub>1 :: "'a"   and y\<^sub>1 :: "'a"   and p\<^sub>2 :: "'a point"   and x\<^sub>2 :: "'a"   and y\<^sub>2 :: "'a"   and p\<^sub>3 :: "'a point"   and x\<^sub>3 :: "'a"   and y\<^sub>3 :: "'a"   and l :: "'a" assumes "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"   and "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"   and "p\<^sub>3 = Point x\<^sub>3 y\<^sub>3"   and "p\<^sub>3 = local.add a p\<^sub>1 p\<^sub>2"   and "x\<^sub>1 \<noteq> x\<^sub>2"   and "l = (y\<^sub>2 \<ominus> y\<^sub>1) \<oslash> (x\<^sub>2 \<ominus> x\<^sub>1)"   and "x\<^sub>3 = l [^] 2 \<ominus> x\<^sub>1 \<ominus> x\<^sub>2"   and "y\<^sub>3 = \<ominus> y\<^sub>1 \<ominus> l \<otimes> (x\<^sub>3 \<ominus> x\<^sub>1)" shows "P p\<^sub>1 p\<^sub>2 p\<^sub>3"
u: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi"
of_ocircline_ounit_circle: shows "of_ocircline ounit_circle = unit_circle"
eq_neutr_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<union># b = {#}) = (a = {#} \<and> b = {#})"
convex_inter_rel_interior_same_closure: fixes I :: "'a set set" assumes "\<forall>S\<in>I. convex S"   and "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}" shows "closure (\<Inter> {rel_interior S |S. S \<in> I}) = closure (\<Inter> I)"
flow_initial_time_if: fixes x0 :: "real \<times> real" shows "c.rev.flow0 x0 0 = (if 0 \<in> UNIV \<and> x0 \<in> UNIV then x0 else 0)"
bezout_coefficients_i_def: fixes ff_ops :: "'b arith_ops_record"   and f :: "'b list"   and g :: "'b list" shows "bezout_coefficients_i ff_ops f g = fst (euclid_ext_poly_i ff_ops f g)"
base_pair1: fixes x :: "state \<Rightarrow> 'a"   and y :: "state \<Rightarrow> 'b" assumes "basevars (x, y)" shows "basevars x"
mult_ac_3: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "b * (a * c) = a * (b * c)"