hcju/isabelleps huggingface dataset:
max_0_1_5: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq one (numeral.numeral one plus x) = numeral.numeral one plus x"
parent_upper_bound: fixes l :: "int"   and c :: "int" shows "(parent l c < c) = (l \<le> c)"
indep_vars_subset: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> real \<Rightarrow> 'c"   and I :: "'b set"   and J :: "'b set" assumes "\<mu>.indep_vars M' X I"   and "J \<subseteq> I" shows "\<mu>.indep_vars M' X J"
coreflexive_comp_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c"   and j :: "'c" assumes "matrix_idempotent_semiring.coreflexive f" shows "(f \<odot> g) (i, j) = f (i, i) \<sqinter> g (i, j)"
Diff_insert: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A - insert a B = A - B - {a}"
Cons_lift_append: fixes P :: "'b com"   and Q :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and ys :: "('b com option \<times> 'b) list" shows "(Some (Seq P Q), s) # map (lift Q) xs @ ys = map (lift Q) ((Some P, s) # xs) @ ys"
option_bind_Some: fixes s :: "'d"   and f :: "'d \<Rightarrow> 'c option" shows "option_bind (Some s) f = f s"
trans_equiv_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Equiv"   and "cmp b c = Less" shows "cmp a c = Less"
closed_except_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c"   and S :: "name fset" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees t S = (frees t |\<subseteq>| S)"
cong_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = (modulo b a = modulo c a)"
is_orthogonal_closure_cspan: fixes X :: "'b set"   and Y :: "'b set"   and x :: "'b"   and y :: "'b" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> Y\<rbrakk> \<Longrightarrow> is_orthogonal x y"   and "x \<in> closure (cspan X)"   and "y \<in> closure (cspan Y)" shows "is_orthogonal x y"
F: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_vector_derivative f x) (at x within {a..b})"
abs_eq: fixes x :: "\<kappa> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" shows "IsProperInX (make\<o> \<circ> x \<circ> id) = (\<forall>x v. (\<exists>a. \<nu>\<upsilon> a = \<nu>\<upsilon> x \<and> x (a\<^sup>P) dj v) = x (x\<^sup>P) dj v)"
countable_basis_openI: fixes B :: "'d set set" assumes "countable B"   and "topological_basis B" shows "countable_basis open B"
in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
E_ProjRcd: fixes fs :: "(char list \<times> trm) list"   and l :: "char list"   and v :: "trm" assumes "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>v\<rfloor>"   and "v \<in> value" shows "Rcd fs..l \<longmapsto>\<^sub>c v"
new_vars_private_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "gen_refine.new_vars_private \<R> \<equiv> \<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<forall>v\<^sub>C. (mem\<^sub>1\<^sub>C' v\<^sub>C \<noteq> mem\<^sub>1\<^sub>C v\<^sub>C \<or> dma mem\<^sub>1\<^sub>C' v\<^sub>C < dma mem\<^sub>1\<^sub>C v\<^sub>C) \<and> v\<^sub>C \<notin> range id \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C' AsmNoReadOrWrite) \<and> mds\<^sub>C AsmNoReadOrWrite - range id \<subseteq> mds\<^sub>C' AsmNoReadOrWrite - range id)"
flowsto_trans: fixes A :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and S :: "real set"   and B :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and C :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and D :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and E :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto A S B C"   and "flowsto C T D E" shows "flowsto A {s + t |s t. s \<in> S \<and> t \<in> T} (B \<union> D \<union> C) E"
modular_lattice_axioms_def: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.modular_lattice_axioms inf less_eq sup \<equiv> \<forall>x y z. less_eq x y \<longrightarrow> sup x (inf y z) = inf y (sup x z)"
is_extensional: fixes f :: "'a" assumes "\<not> hom'.arr f" shows "\<psi>.map f = hom'.null"
dy_fake_chan_mono_chan: fixes C :: "chan set"   and C' :: "chan set"   and bad :: "agent set"   and T :: "msg set" assumes "C \<subseteq> C'" shows "dy_fake_chan bad T C \<subseteq> dy_fake_chan bad T C'"
lessThan_iff: fixes i :: "'a word"   and k :: "'a word" shows "(i \<in> signed.lessThan k) = (i <s k)"
borel_measurable_mono_on_fnc: fixes f :: "real \<Rightarrow> real"   and A :: "real set" assumes "mono_on f A" shows "f \<in> borel_measurable (restrict_space borel A)"
v: fixes r :: "rat"   and c :: "constraint" assumes "(r, c) \<in> set C" shows "v \<Turnstile>\<^sub>l\<^sub>e lec_of_constraint (r *R c)"
that: shows "is_normal_Float TYPE(('e, 'f) IEEE.float) x"
morphism_presheaves_of_rings_fam_cong: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and fam_morphisms' :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "\<And>U x. \<lbrakk>is_open U; x \<in> \<FF> U\<rbrakk> \<Longrightarrow> fam_morphisms' U x = fam_morphisms U x"   and "\<And>U. is_open U \<Longrightarrow> fam_morphisms' U \<in> extensional (\<FF> U)" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms'"
\<alpha>w_def: fixes g :: "'m"   and e :: "'v set" shows "\<alpha>w g e \<equiv> case epair e of None \<Rightarrow> 0 | Some (u, v) \<Rightarrow> case M_lookup g u of None \<Rightarrow> 0 | Some xs \<Rightarrow> the_default 0 (Map.map_of xs v)"
R: shows "(impl, mf) \<in> [Q]\<^sub>a R \<rightarrow> S"
mat_of_row_dim_1: fixes y :: "'a Matrix.vec" shows "dim_row (mat_of_row y) = 1"
chain_tllist_ord_tSup: fixes A :: "('c, 'b) tllist set"   and xs :: "('c, 'b) tllist" assumes "Complete_Partial_Order.chain local.tllist_ord A"   and "xs \<in> A" shows "local.tllist_ord xs (local.tSup A)"
bind_fmap: fixes f :: "'h \<rightarrow> 'g"   and xs :: "'h\<cdot>'f"   and k :: "'g \<rightarrow> 'e\<cdot>'f" shows "fmap\<cdot>f\<cdot>xs \<bind> k = xs \<bind> (\<Lambda> x. k\<cdot>(f\<cdot>x))"
inv_plus_ivl_def: fixes iv :: "ivl"   and iv1 :: "ivl"   and iv2 :: "ivl" shows "inv_plus_ivl iv iv1 iv2 = (iv1 \<sqinter> (iv - iv2), iv2 \<sqinter> (iv - iv1))"
seqE: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.seq g f" obtains "C.VV.arr f" and "C.VV.arr g" and "C.VV.dom g = C.VV.cod f"
length_mtf: fixes x :: "'a"   and xs :: "'a list" shows "length (mtf x xs) = length xs"
nn_integral_suminf: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M = (\<Sum>i. integral\<^sup>N M (f i))"
conf_s0: shows "s0\<Colon>\<preceq>(G, L)"
hom_add_eq_zero: fixes x :: "'a poly"   and y :: "'a poly" assumes "x + y = 0" shows "pCons (0::'a) x + pCons (0::'a) y = 0"
size_mset: fixes xs :: "'k list" shows "size (mset xs) = length xs"
prems_1: fixes i :: "nat" assumes "i < num_args t"   and "args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at (a # p) d t) ! i"   and "\<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args (emb_step_at (a # p) d t) ! j" shows "thesis"
nonzero_Solutions_iff: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Solutions" shows "nonzero x = nonzero y"
assms_3: shows "q \<in> hom_deg_set z A"
a1: shows "store_word_mem s addr data byte_mask asi = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
Q: shows "Q = (\<lambda>((as, lm), ss, arity). mn_ind_inv (as, lm) (length fap) x rsx anything xs)"
g': fixes x :: "complex" assumes "x \<in> ball z r" shows "(g has_field_derivative g'__ x) (at x)"
next_free_stutter_invariant: fixes \<phi> :: "'b pltl" assumes "next_free \<phi>" shows "stutter_invariant \<phi>"
RECT_gfp_def: fixes B :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" shows "REC\<^sub>T B x = (if trimono B then gfp B x else top)"
assms_1: fixes r :: "'a" shows "Case_Labeling.CTXT (Suc OC1) ((''bind'', Suc OC1, [Case_Labeling.VAR r]) # CT) OC (ovalidNF (R r) (g r) Q)"
countable_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
g_non_zero_ind_mod: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "dcharacter n \<chi>"   and "dcharacter.gauss_sum n \<chi> k \<noteq> 0"   and "\<not> coprime k n"   and "0 < k" shows "dcharacter.induced_modulus n \<chi> (n div gcd k n)"
mergeFrames: fixes A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>G :: "'b"   and \<Psi>\<^sub>F :: "'b" assumes "A\<^sub>F \<sharp>* A\<^sub>G"   and "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"   and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F" shows "(\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) \<otimes>\<^sub>F \<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle> = \<langle>(A\<^sub>F @ A\<^sub>G), \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>G\<rangle>"
eqvt: fixes x1 :: "trm"   and x2 :: "trm"   and pi :: "(name \<times> name) list" assumes "x1 \<mapsto> x2" shows "pi \<bullet> x1 \<mapsto> pi \<bullet> x2"
content_0_subset: fixes a :: "'a"   and b :: "'a"   and s :: "'a set" assumes "content (cbox a b) = 0"   and "s \<subseteq> cbox a b" shows "content s = 0"
normal: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x'm' :: "'x \<times> 'm" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ln = no_wait_locks"   and "t \<turnstile> (x, shr s) -ta\<rightarrow> x'm'"   and "actions_ok s t ta" shows "t \<in> active_threads s"
weak_Sup_distl: fixes Y :: "'a set"   and x :: "'a" assumes "Y \<noteq> {}" shows "x \<cdot> \<Squnion>Y = \<Squnion>{x \<cdot> y |y. y \<in> Y}"
hh_1: shows "hermitean H1 \<and> H1 \<noteq> mat_zero"
typeof_unboxed_eq_const_2: fixes x :: "('a, 'b, 'c) unboxed" shows "(typeof x = Some Ubx1) = (\<exists>n. x = OpUbx1 n)"
unitary_exproj_psi_minus_1: shows "unitary (2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d)"
analytic: shows "(\<lambda>z. if z = 1 then a else newman z + deriv zeta z + c * zeta z) analytic_on {s. 1 \<le> Re s}"
exhaust: fixes y :: "QDelta" obtains x1 :: "rat" and x2 :: "rat" where "y = QDelta x1 x2"
hcomp_simps_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.hseq \<nu> \<mu>" shows "trg\<^sub>C (\<nu> \<star>\<^sub>C \<mu>) = trg\<^sub>C \<nu>"
NatTransComp_defs_6: fixes \<eta> :: "('g, 'j, 'h, 'k, 'i, 'l, 'm) NatTrans_scheme" shows "NTCatDom \<eta> \<equiv> CatDom (NTDom \<eta>)"
dns5: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> (d x + d y) = d x"
assms_3: shows "eoutKnowCorrect A (sKS m)"
subst_sends_strand_fv_to_img: fixes S :: "('b, 'a) strand_step list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> fv\<^sub>s\<^sub>t S \<union> range_vars \<delta>"
length_filter_map: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (filter P (map f xs)) = length (filter (P \<circ> f) xs)"
1: shows "\<exists>k. f (Suc k) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc k)\<^esub>"
A: shows "summable (\<lambda>n. (cmod z)\<^sup>2 * (inverse (real (Suc (Suc n))) * cmod z ^ n))"
gen_g_impl_rel_ext_def: fixes Rm :: "('k \<times> 'm) set"   and Rv :: "('h \<times> 'l set) set"   and Re :: "('i \<times> ('l \<times> 'l) set) set"   and Rv0 :: "('j \<times> 'l set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
INF_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "(\<Sqinter> (f ` A)) x = (\<Sqinter>y\<in>A. f y x)"
fvs_simp_5: fixes A :: "'b trm"   and B :: "'b trm" shows "fvs (SimplyTyped.Pair A B) = fvs A \<union> fvs B"
invariance_of_domain_Euclidean_space_gen: fixes n :: "nat"   and m :: "nat"   and U :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "n \<le> m"   and "openin (Euclidean_space m) U"   and "continuous_map (subtopology (Euclidean_space m) U) (Euclidean_space n) f"   and "inj_on f U" shows "openin (Euclidean_space n) (f ` U)"
lappend_lnull1: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lnull xs" shows "xs @@ ys = ys"
inverse_incidence_sys_iso: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "incidence_system_isomorphism \<V>' \<B>' \<V> \<B> (inv_into \<V> bij_map)"
ngba_g_def: fixes A :: "('d, 'c) ngba" shows "ngba_g A \<equiv> \<lparr>g_V = UNIV, g_E = E_of_succ (successors A), g_V0 = initial A\<rparr>"
leD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
strict_trans: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "b \<prec> a"   and "c \<prec> b" shows "c \<prec> a"
writes_small_big: fixes SW :: "('l, 'm, 'n) prog set"   and setter :: "('l, 'm, 'o) prog"   and h :: "'l"   and h' :: "'l"   and P :: "'l \<Rightarrow> 'l \<Rightarrow> bool" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> P h h'"   and "reflp P"   and "transp P" shows "P h h'"
real_lepoll_ratrat: shows "UNIV \<lesssim> UNIV"
below_monic_chain_inv: fixes Y :: "nat \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" assumes "chain Y"   and "\<forall>i. \<exists>y. Y i = f y \<and> P y"   and "below_monic f" shows "\<exists>Y'. chain Y' \<and> (\<forall>i. Y i = f (Y' i) \<and> P (Y' i))"
assms_2: shows "(r', s') \<in> carrier R \<times> S"
comp_Im_same: fixes x :: "complex"   and y :: "complex" assumes "x \<le> y" shows "Im x = Im y"
msubstleneg_nb: fixes t :: "tm"   and a :: "poly"   and r :: "tm"   and c :: "poly" assumes "tmbound0 t"   and "islin (Le (CNP 0 a r))" shows "bound0 (msubstleneg c t a r)"
orient_lit_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and u :: "'b trm"   and v :: "'b trm"   and s :: "sign" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.orient_lit trm_ord L u v s = ((L = Pos (Eq u v) \<or> L = Pos (Eq v u)) \<and> (u, v) \<notin> trm_ord \<and> s = pos \<or> (L = Neg (Eq u v) \<or> L = Neg (Eq v u)) \<and> (u, v) \<notin> trm_ord \<and> s = neg)"
effect_nthE: fixes a :: "'b array"   and i :: "nat"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect (a.(i)) h h' r" obtains "i < Array.length h a" and "r = Array.get h a ! i" and "h' = h"
sams2_suma2__conga456: 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"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D' E' F'"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D' E' F' G H I" shows "Tarski_neutral_dimensionless.CongA Bet Cong D E F D' E' F'"
padic_val_ultrametric: fixes p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "prime p"   and "a \<in> carrier (padic_int p)"   and "b \<in> carrier (padic_int p)"   and "a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"   and "b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"   and "a \<oplus>\<^bsub>padic_int p\<^esub> b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>" shows "min (padic_val p a) (padic_val p b) \<le> padic_val p (a \<oplus>\<^bsub>padic_int p\<^esub> b)"
get_get: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> ('v \<Rightarrow> int option) \<Rightarrow> 'm" shows "get (\<lambda>s. get (f s)) = get (\<lambda>s. f s s)"
union_le_diff_plus: fixes P :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset" assumes "P \<subseteq># M"   and "N < P" shows "M - P + N < M"
segment_le_mono: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (segment D a \<subseteq> segment D b)"
bk: shows "b \<in> mat_kernel (char_matrix A lam)"
ipresIGAbs_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'varSort, 'sort, 'n, 'o, 'h, 'j, 'p) model_scheme"   and MOD' :: "('q, 'r, 'varSort, 's, 't, 'o, 'i, 'k, 'u) model_scheme" shows "ipresIGAbs h hA MOD MOD' \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> hA (igAbs MOD xs x X) = igAbs MOD' xs x (h X)"
map_eq_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and g :: "'h \<Rightarrow> 'g" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
emp_ref: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists \<langle>G\<rangle>\<^sub>+"   and "Ref G us = \<epsilon>" shows "us = \<epsilon>"
apex_def: shows "g.apex \<equiv> C.dom g.leg0"
fct2: shows "(0::'capacity) < Flow.val cf s (augmentingFlow p)"
scopeFresh: fixes x :: "name"   and P :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>P \<sim>\<^sub>l P"
fls_divide_shift_denom: fixes f :: "'a fls"   and m :: "int"   and g :: "'a fls" shows "f / fls_shift m g = fls_shift (- m) (f / g)"
submodule_is_module: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and N :: "'i set" assumes "module R M"   and "LinearCombinations.submodule R N M" shows "module R (M\<lparr>carrier := N\<rparr>)"
ode_d1_eq: fixes x :: "'c"   and j :: "'c" shows "ode_d1 x = ode_d 0 x j"
A_def: shows "A = (SIGMA nm:UNIV. (\<lambda>x. (n (fst nm) (snd nm) x, m (fst nm) (snd nm) x)) ` set_pmf (p (fst nm) (snd nm)))"
ferrack01: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "(\<exists>x. Ifm (x # bs) (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = Ifm bs (ferrack01 p) \<and> qfree (ferrack01 p)"
lookup_pair_update_by_fun_gr_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k1 :: "'a"   and f :: "'b \<Rightarrow> 'b"   and k2 :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.lookup_pair comp (comparator.update_by_fun_gr_pair comp k1 f xs) k2 = (if k1 = k2 then f else id) (comparator.lookup_pair comp xs k2)"
eqinv: fixes X :: "complex mat" assumes "X \<in> carrier_mat d d" shows "f 0 + 1 / c \<cdot>\<^sub>m (c \<cdot>\<^sub>m (X - f 0)) = X"
polyn_expr_restrict1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (n, f)"   and "pol_coeff S (Suc (m + n), g)" shows "polyn_expr R X (m + n) (add_cf S (n, f) (m + n, g)) = polyn_expr R X (m + n) (m + n, snd (add_cf S (n, f) (Suc (m + n), g)))"
value_h_plus: fixes i :: "nat" shows "value_flow \<Delta> (h_plus i) = value_flow \<Delta> g"
Cauchy_contour_integral_circlepath: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex"   and k :: "nat" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc k) = complex_of_real (2 * pi) * \<i> * (deriv ^^ k) f w / fact k"
bind_UNION: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "Set.bind A f = \<mu> (\<P> f A)"
append_imp_prefix: fixes a :: "'d list"   and b :: "'d list" shows "prefix a (a @ b)"
33: shows "bprv (all yy' (imp (SS \<langle>\<chi>\<rangle> (Var yy')) (eql \<langle>\<psi>\<rangle> (Var yy'))))"
measure_eq_prod: fixes f :: "complex poly"   and g :: "complex poly" shows "mahler_measure_poly (f * g) = mahler_measure_poly f * mahler_measure_poly g"