hcju/isabelleps huggingface dataset:
dom_eq_singleton_conv: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c" shows "(dom f = {x}) = (\<exists>v. f = [x \<mapsto> v])"
\<psi>_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "\<psi> x \<le> \<psi> y"
c: shows "\<bar>c\<bar> \<noteq> \<infinity>"
finabc: shows "finite ({a, b, c} \<union> (S \<union> T))"
\<delta>_eq_\<epsilon>: shows "\<delta>\<^sub>h = \<epsilon>\<^sub>h"
interchange: fixes h :: "'a"   and g :: "'a"   and h' :: "'a"   and g' :: "'a" assumes "seq h g"   and "seq h' g'" shows "(h \<otimes> h') \<cdot> (g \<otimes> g') = h \<cdot> g \<otimes> h' \<cdot> g'"
fps_mult_right_const_nth: fixes f :: "'a fps"   and c :: "'a"   and n :: "nat" shows "fps_nth (f * fps_const c) n = fps_nth f n * c"
norm_not_less_zero: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<not> norm x < 0"
pcompose_idR: fixes p :: "'a poly" shows "p \<circ>\<^sub>p [:0::'a, 1::'a:] = p"
matches_simps_1: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b"   and m' :: "'a match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m m') a p = matches \<gamma> m' a p"
length_word_rsplit_size: fixes n :: "nat"   and w :: "'d word"   and m :: "nat" assumes "n = LENGTH('c)" shows "(length (word_rsplit w) \<le> m) = (size w \<le> m * n)"
irreducible_cong: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b" shows "irreducible a = irreducible b"
simps_2: fixes f :: "'a" shows "EVAL \<^bold>\<langle>f\<^bold>\<rangle> = (if B.arr f then f else B.null)"
v1_in_Q: shows "\<exists>Q_hit\<in>Q. v1 \<in> set Q_hit"
heap_is_wellformed_one_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and ptr' :: "(_) object_ptr"   and children' :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children'"   and "set children \<inter> set children' \<noteq> {}" shows "ptr = ptr'"
lt_set_iff: fixes t :: "'x \<Rightarrow>\<^sub>0 nat"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "(t \<in> punit.lt_set F) = (\<exists>f\<in>F. f \<noteq> 0 \<and> lpp f = t)"
atom_map_inf_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<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 x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot (inf x y) = bounded_lattice.atom_map less_eq less bot x \<inter> bounded_lattice.atom_map less_eq less bot y"
sigma_finite_measure_def: fixes M :: "'c measure" shows "sigma_finite_measure M \<equiv> \<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
cname_of_def: fixes h :: "'heap"   and a :: "'addr" shows "cname_of h a = the_Class (ty_of_htype (the (typeof_addr h a)))"
match_term_list_complete: fixes tus :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option" assumes "match_term_list tus \<sigma> = None"   and "extends_subst \<sigma> \<tau>"   and "dom \<tau> = (\<Union>(t, u)\<in>set tus. vars_term t) \<union> dom \<sigma>" shows "\<exists>(t, u)\<in>set tus. subst_atm_abbrev t (subst_of_map Var \<tau>) \<noteq> u"
comm_neutral: fixes a :: "'k" shows "a * (1::'k) = a"
elims_2: fixes x :: "atom" assumes "depends\<^sub>d\<^sub>l\<^sub>o x" obtains i :: "nat" and j :: "nat" where "x = Eq i__ j__" and "i__ = 0 \<or> j__ = 0"   | i :: "nat" and j :: "nat" where "x = Less i j" and "i = 0 \<or> j = 0"
minus_eq: fixes x :: "'a list"   and y :: "'a list" shows "x \<ominus>\<^bsub>K [X]\<^esub> y = x \<oplus>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> y"
iso_arcs_eq_iff: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and e1 :: "'b"   and e2 :: "'b" assumes "digraph_isomorphism hom"   and "e1 \<in> arcs G"   and "e2 \<in> arcs G" shows "(iso_arcs hom e1 = iso_arcs hom e2) = (e1 = e2)"
semilattice_order_axioms: shows "semilattice_order (\<sqinter>) (\<preceq>) (\<prec>)"
mat_mult_assoc: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<otimes> (g \<otimes> h) = f \<otimes> g \<otimes> h"
add_less_le_mono: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset"   and d :: "'d multiset" assumes "a \<subset># b"   and "c \<subseteq># d" shows "a + c \<subset># b + d"
comp_in_homE: fixes g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and a :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and c :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "\<guillemotleft>S.comp g f : a \<rightarrow>\<^sub>S c\<guillemotright>" obtains b :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" where "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>S c\<guillemotright>"
sel_2: fixes x1 :: "'d \<Rightarrow> 'e list \<Rightarrow> nat \<Rightarrow> 'd list \<times> 'c"   and x2 :: "'c \<Rightarrow> 'd list \<times> 'c" shows "next_subseqs_foldr (Sublists_Foldr_Impl x1 x2) = x2"
simps_1: fixes x11 :: "String.literal"   and x12 :: "typ list"   and y11 :: "String.literal"   and y12 :: "typ list" shows "(Ty x11 x12 = Ty y11 y12) = (x11 = y11 \<and> x12 = y12)"
srtc_transitive: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<leftrightarrow>\<^sup>* b"   and "b \<leftrightarrow>\<^sup>* c" shows "a \<leftrightarrow>\<^sup>* c"
simps_7: fixes jmps :: "jump set"   and j :: "jump" shows "jumpNestingOkS jmps (Jmp j) = (j \<in> jmps)"
hyps: assumes "\<forall>s\<in>set ss. s + s = (0::'a)" shows "snd (signed_list_lconjaction ss x) = (if gcd_nat.greater_eq (count_list (lconjseq (rev ss)) (fst x)) 2 then snd x else \<not> snd x)"
rel_converter_Grp: fixes f :: "'e \<Rightarrow> 'a"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'f"   and C :: "'c set"   and h :: "'c \<Rightarrow> 'g"   and k :: "'h \<Rightarrow> 'd" shows "rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp B g) (BNF_Def.Grp C h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse> = BNF_Def.Grp {conv. results_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> B \<and> outs_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> C} (map_converter f g h k)"
inject_6: fixes x61 :: "Unary_PCF.type"   and x62 :: "expr"   and y61 :: "Unary_PCF.type"   and y62 :: "expr" shows "(\<Lambda>\<langle>x61\<rangle> x62 = \<Lambda>\<langle>y61\<rangle> y62) = (x61 = y61 \<and> x62 = y62)"
alw: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "\<phi> xs"   and "alw \<phi> (stl xs)" shows "alw \<phi> xs"
simps: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.swap_cols_rows_block i j A = (if i < j then ??.Jordan_Normal_Form_Existence.swap_cols_rows_block (Suc i) j (swap_cols_rows i j A) else A)"
vsnd_def: fixes p :: "V" shows "vsnd p \<equiv> THE y. \<exists>x. p = \<langle>x, y\<rangle>"
h: fixes i :: "'a" assumes "i \<in> I" shows "continuous_map (prod_topology (top_of_set {0..1}) (X i)) (Y i) (h i)"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "prod g A = prod h B"
UniverseSubset: fixes u :: "ZF"   and v :: "ZF" assumes "HOLZF.subset u v"   and "Elem v U" shows "Elem u U"
inj_on_verts_sccs: shows "inj_on verts G.sccs"
A_1: fixes x :: "'a" assumes "x \<in> space (restr_to_subalg M F) - A" shows "P x"
vcomp_ide_dom: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B F \<tau> = \<tau>"
refl: fixes a :: "'b" shows "(a::'b) \<le> a"
split_comb_seq_asm: fixes P :: "'b \<Rightarrow> bool"   and i :: "nat"   and \<omega> :: "nat \<Rightarrow> 'b"   and \<omega>' :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "P (comb_seq i \<omega> \<omega>' j) = (\<not> (j < i \<and> \<not> P (\<omega> j) \<or> (\<exists>k. j = i + k \<and> \<not> P (\<omega>' k))))"
degree_monom_eq: fixes a :: "'a"   and n :: "nat" assumes "a \<noteq> (0::'a)" shows "Polynomial.degree (Polynomial.monom a n) = n"
replace_f: fixes m' :: "'a match_expr"   and a :: "'b negation_type" assumes "m' \<in> set (normalize_match (replace_f a))" shows "normalized_n_primitive (disc2, sel2) f m'"
Cy: shows "\<forall>z\<in>C. z \<sim> y"
finite_Fvars: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and \<phi> :: "'c" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "\<phi> \<in> fmla" shows "finite (Fvars \<phi>)"
pred_def: fixes P :: "bexp set" shows "A.pred P \<equiv> \<lambda>mem. \<forall>p\<in>P. ev\<^sub>B mem p"
A: shows "A \<in> sets (Pi\<^sub>F {J} M)"
finite_image_set2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite {x. P x}"   and "finite {y. Q y}" shows "finite {f x y |x y. P x \<and> Q y}"
ordLeq_cprod2: fixes p1 :: "('c \<times> 'c) set"   and p2 :: "('d \<times> 'd) set" assumes "(p1, czero) \<notin> ordIso \<and> Card_order p1"   and "Card_order p2" shows "p2 \<le>o p1 *c p2"
not_less: fixes x :: "'e Word.word"   and y :: "'e Word.word" shows "(\<not> x <s y) = (y \<le>s x)"
antisym: fixes b :: "'a list"   and a :: "'a list" assumes "??.Sublist.prefix b a"   and "??.Sublist.prefix a b" shows "a = b"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>n. P (Bound n)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Add p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Sub p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Mul p q)"   and "\<And>p. P p \<Longrightarrow> P (Neg p)"   and "\<And>p n. P p \<Longrightarrow> P (Pw p n)"   and "\<And>c n p. P (Add c (Mul (Bound n) p)) \<Longrightarrow> P (CN c n p)"   and "\<And>c. P (c)\<^sub>p" shows "P a0"
slim_paths_def: shows "slim_paths \<equiv> (\<lambda>e. (fst e, choose_iapath (fst e) (snd e), snd e)) ` parcs (gen_contr_graph (with_proj G) (verts3 (with_proj G)))"
invertible_qmpt_def: shows "I.invertible_qmpt \<equiv> bij natural_extension_map \<and> I.random_variable natural_extension_measure (inv natural_extension_map)"
term_ok'_bind_fv: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and v :: "variable"   and \<tau> :: "typ" assumes "term_ok' \<Sigma> t" shows "term_ok' \<Sigma> (bind_fv (v, \<tau>) t)"
symeval_def: fixes P :: "(char list \<Rightarrow> int option) \<times> 'a \<Rightarrow> bool"   and e :: "aexp"   and v :: "int" shows "symeval P e v \<equiv> \<forall>s n. P (s, n) \<longrightarrow> paval' e s = Some v"
card_UNIV_sum_def: shows "card_UNIV_class.card_UNIV = Phantom('a + 'b) (let ca = of_phantom card_UNIV_class.card_UNIV; cb = of_phantom card_UNIV_class.card_UNIV in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
l11_13: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and D' :: "'p" assumes "A B C CongA D E F"   and "Bet A B A'"   and "A' \<noteq> B"   and "Bet D E D'"   and "D' \<noteq> E" shows "A' B C CongA D' E F"
assms_2: shows "B\<^sub>2 \<subseteq> {u \<in> U. w u \<le> real c / 2}"
c_secure_ipurge_tr: fixes I :: "('e \<times> 'e) set"   and step :: "'f \<Rightarrow> 'g \<Rightarrow> 'f"   and out :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and s\<^sub>0 :: "'f"   and D :: "'g \<Rightarrow> 'e"   and x :: "'g"   and xs :: "'g list"   and ys :: "'g list" assumes "refl I"   and "c_secure step out s\<^sub>0 I D" shows "ipurge_tr I (c_dom D) (D x) (c_tr step out (step (foldl step s\<^sub>0 xs) x) ys) = ipurge_tr I (c_dom D) (D x) (c_tr step out (foldl step s\<^sub>0 xs) ys)"
weak_composition_axioms: shows "weak_composition (\<cdot>) (\<star>)"
rec_less_def: shows "rec_less = Cn 2 rec_sg [Cn 2 rec_minus [recf.id 2 1, recf.id 2 0]]"
can_take_transition_def: fixes t :: "'a transition_scheme"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "can_take_transition t i r = can_take (Arity t) (Guards t) i r"
strict_coboundedI2: fixes b :: "'t"   and c :: "'t"   and a :: "'t" assumes "b \<prec>\<^sub>t c" shows "ord_term_lin.min a b \<prec>\<^sub>t c"
assms: fixes xs :: "real list" shows "eval (clearQuantifiers \<phi>) xs = eval \<phi> xs"
s0: shows "\<forall>i. i \<noteq> j \<longrightarrow> s i = (0::'a)"
compatible_iff_seq: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "(VV.domains g \<inter> VV.codomains f \<noteq> {}) = VV.seq g f"
Seq1: fixes c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> int" shows "(SKIP;; c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
similar_def: fixes x :: "Value"   and y :: "CValue" shows "(x \<triangleleft>\<triangleright> y) = (\<forall>n. \<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>y)"
less_1: fixes y :: "nat"   and i :: "nat"   and i' :: "nat" assumes "y < k"   and "obsp \<sigma> i \<noteq> obsp \<sigma>' i' \<or> cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'"   and "is_kth_obs \<pi> y i"   and "is_kth_obs \<pi>' y i'" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> \<pi> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> \<pi>' k' \<in> dom local.att"
effect_def: fixes c :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" shows "effect c h h' r = (execute c h = Some (r, h'))"
sestoftUnGCStack: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and e :: "exp"   and r :: "Vars.var list" assumes "heap_upds_ok (\<Gamma>, S)" obtains \<Gamma>' :: "(Vars.var \<times> exp) list" and S' :: "stack_elem list" where "(\<Gamma>, e, S) \<Rightarrow>\<^sup>* (\<Gamma>', e, S')" and "to_gc_conf r (\<Gamma>, e, S) = to_gc_conf r (\<Gamma>', e, S')" and "\<not> isVal e \<or> safe_hd S' = safe_hd (restr_stack (- set r) S')"
apx_def: fixes Z :: "('a \<Rightarrow> real) set" shows "Approx\<^sub>\<beta> Z \<equiv> \<Inter> {S. \<exists>U M. S = \<Union> U \<and> U \<subseteq> \<R> \<and> Z \<subseteq> S \<and> (S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n) \<and> normalized M}"
map_add_subsumed2: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "f \<subseteq>\<^sub>m g" shows "g ++ f = g"
deg_pm_zero: shows "deg_pm 0 = (0::'c)"
proper_it_mono_dres: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('i dres \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'i dres \<Rightarrow> 'i dres) \<Rightarrow> 'i dres \<Rightarrow> 'i dres"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'i dres"   and f' :: "'h \<Rightarrow> 'i \<Rightarrow> 'i dres"   and s :: "'f"   and c :: "'i \<Rightarrow> bool"   and \<sigma> :: "'i dres" assumes "proper_it' it it'"   and "\<And>kv x. f kv x \<le> f' kv x" shows "it' s (case_dres False False c) (\<lambda>kv s. s \<bind> f kv) \<sigma> \<le> it' s (case_dres False False c) (\<lambda>kv s. s \<bind> f' kv) \<sigma>"
subtract_entries_raw_def: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "subtract_entries_raw xs ys = foldr (\<lambda>(k, v). AList.map_entry k (\<lambda>v'. v' - v)) ys xs"
negative_cycle_dest_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<not> cycle_free M n" shows "\<exists>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<and> len M i i xs < \<one>"
empty_bind: fixes f :: "'d \<Rightarrow> 'c set" shows "{} \<bind> f = {}"
B_4: fixes S :: "'b set" assumes "open S"   and "snd x \<in> S" shows "\<exists>a\<in>B. a \<subseteq> S"
assms_2: fixes x :: "'b" assumes "x \<in> A" shows "prime (f x)"
countable_Int2: fixes B :: "'a set"   and A :: "'a set" assumes "countable B" shows "countable (A \<inter> B)"
foldseq_zero: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "f (0::'b) (0::'b) = (0::'b)"   and "\<forall>i\<le>n. s i = (0::'b)" shows "foldseq f s n = (0::'b)"
IH: assumes "subst_domain \<theta> \<inter> bvars\<^sub>s\<^sub>t A = {}" shows "trms\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>t \<theta>) = trms\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
redexE: fixes e :: "('a, 'b, 'c) expr" assumes "redex e" obtains x :: "'c" and ea :: "('a, 'b, 'c) expr" and v :: "('a, 'b, 'c) val" where "e = Apply (VE (Lambda x ea__)) (VE v__)"   | e1 :: "('a, 'b, 'c) expr" and e2 :: "('a, 'b, 'c) expr" where "e = Ite (VE (CV T)) e1__ e2__"   | e1 :: "('a, 'b, 'c) expr" and e2 :: "('a, 'b, 'c) expr" where "e = Ite (VE (CV F)) e1 e2"   | v :: "('a, 'b, 'c) val" where "e = Ref (VE va__)"   | l :: "'b" where "e = Read (VE (Loc l__))"   | l :: "'b" and v :: "('a, 'b, 'c) val" where "e = Assign (VE (Loc l)) (VE v)"   | ea :: "('a, 'b, 'c) expr" where "e = Rfork ea"   | r :: "'a" where "e = Rjoin (VE (Rid r))"
nat_to_bv_def: fixes n :: "nat" shows "nat_to_bv n = nat_to_bv_helper n []"
Suc_7: shows "x \<in> lvars (\<T> (l ! j))"
prob1_geq_zero: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "0 \<le> prob1 n v i"
method_typings_lookup: fixes M :: "methodDef list"   and f :: "methodDef \<Rightarrow> bool"   and mDef :: "methodDef"   and CT :: "nat \<Rightarrow> classDef option"   and C :: "nat" assumes "lookup M f = Some mDef"   and "CT \<turnstile>+ M OK IN C" shows "CT \<turnstile> mDef OK IN C"
IH: fixes t :: "'a trm"   and s :: "'a trm" assumes "congruence I"   and "I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"   and "subterm t q u"   and "replace_subterm t q v s" shows "I (t \<lhd> \<sigma>) (s \<lhd> \<sigma>)"
ideI: fixes a :: "'a" assumes "left (trg \<mu>) a"   and "ide a" shows "Left.ide a"
q_cons_rel: fixes Bot :: "'a set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "consequence_relation_family Bot Q entails_q" shows "\<forall>q\<in>Q. consequence_relation Bot (entails_q q)"
is_unit_diagonal: fixes U :: "(('a, 'b) vec, 'b) vec" assumes "upper_triangular U"   and "is_unit (det U)" shows "\<forall>i. is_unit (U $ i $ i)"
szip_unfold: fixes a :: "'b"   and s1 :: "'b stream"   and b :: "'c"   and s2 :: "'c stream" shows "a ## s1 ||| b ## s2 = (a, b) ## (s1 ||| s2)"
list_induct2'': fixes P :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'c list" assumes "P [] []"   and "\<And>xs. P xs []"   and "\<And>ys. P [] ys"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
Call_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and M :: "String.literal"   and v :: "'addr val" assumes "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<bullet>M(es)) h (stk @ [v], loc, length (compE2 e) + pc, xcp) (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
powr_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and A :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X"   and "a \<in> aform_err e A"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "powr_aform_err p X A = Some Y" shows "x powr a \<in> aform_err e Y"
evalHeap_reorder_head: fixes x :: "'d"   and y :: "'d"   and e1 :: "'f"   and e2 :: "'f"   and \<Gamma> :: "('d \<times> 'f) list"   and eval :: "'f \<Rightarrow> 'e" assumes "x \<noteq> y" shows "evalHeap ((x, e1) # (y, e2) # \<Gamma>) eval = evalHeap ((y, e2) # (x, e1) # \<Gamma>) eval"
assms: fixes s :: "(char list \<Rightarrow> int option) \<times> nat" shows "A s \<longrightarrow> B s"
less_eq_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "(d1 \<sqsubseteq> d2) = (d1 = d2 \<or> d1 = Low)"
atLeastatMost_subset_iff: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word"   and d :: "'b word" shows "(signed.atLeastAtMost a b \<subseteq> signed.atLeastAtMost c d) = (\<not> a \<le>s b \<or> c \<le>s a \<and> b \<le>s d)"
IH2: assumes "q \<in> subterms p2"   and "r \<in> stermsl q" shows "r \<in> ctermsl p2"
gcd'_greatest: fixes gcd' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "'a"   and a :: "'a"   and b :: "'a" assumes "is_gcd gcd'"   and "l dvd a"   and "l dvd b" shows "l dvd gcd' a b"
sum_list_replicate: fixes n :: "nat"   and x :: "nat" shows "sum_list (replicate n x) = n * x"
aezfun_diff_minus: fixes a :: "('a, 'b) aezfun"   and b :: "('a, 'b) aezfun" shows "a - b = a + - b"
solves_ode_on_subset_domain: fixes x :: "real \<Rightarrow> 'a"   and S :: "real set"   and Y :: "'a set" assumes "(x solves_ode f) S Y"   and "x t = flow t0 x0 t"   and "t \<in> S"   and "is_interval S"   and "S \<subseteq> cball t et" shows "(x solves_ode f) S (cball (flow t0 x0 t) ex)"
subprob_emeasure_le_1: fixes M :: "'b measure"   and X :: "'b set" assumes "subprob_space M" shows "emeasure M X \<le> 1"
real_of_nat_ge_one_iff: fixes n :: "nat" shows "(1 \<le> real n) = (1 \<le> n)"
sum_iff: fixes u :: "V"   and A :: "V"   and B :: "V" shows "(u \<in> elts (A \<Uplus> B)) = ((\<exists>x. x \<in> elts A \<and> u = ZFC_Cardinals.Inl x) \<or> (\<exists>y. y \<in> elts B \<and> u = ZFC_Cardinals.Inr y))"
permute_minus_cancel_2: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "perm"   and x :: "'c" assumes "class.pt permute" shows "permute (- p) (permute p x) = x"
Fun_Range: fixes f :: "ZF"   and U :: "ZF"   and V :: "ZF" assumes "Elem f (Fun U V)" shows "HOLZF.subset (HOLZF.Range f) V"
subgroup_in_normalizer: fixes H :: "'a set" assumes "subgroup H G" shows "H \<lhd> G\<lparr>carrier := normalizer G H\<rparr>"
split_uv: shows "splits_at (v @ u) i \<alpha> N (\<beta> @ u)"