hcju/isabelleps huggingface dataset:
int_def: fixes idx_interval :: "'a idx_interval" shows "idx_interval.int idx_interval = (case idx_interval of IdxInterval x1 x2 \<Rightarrow> x1)"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (inf_class.inf a b = b)"
exec_step_input_StepID: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and i :: "instr" assumes "exec_step_input P C M pc ics = StepI i" shows "(ics = Called [] \<or> ics = No_ics) \<and> instrs_of P C M ! pc = i"
mcalc_iff: fixes \<sigma> :: "'tp"   and c :: "('a, 'b) lit list" shows "(\<sigma> \<turnstile> c) = (infTp \<sigma> \<or> (\<forall>x\<in>nvC c. tpOfV x \<noteq> \<sigma>))"
ss: shows "{b. \<exists>a\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r} \<subseteq> vsnd ` elts r"
word_split_bin': fixes w :: "'d word" shows "word_split w = (ucast (drop_bit LENGTH('c) w), ucast w)"
exhaustive_fusion_def: fixes g :: "('a, 'c) generator"   and s :: "'c" shows "exhaustive_fusion g s = exhaustive (list.unfoldr g s)"
hseqI': fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "hseq \<nu> \<mu>"
lessThan_strict_subset_iff: fixes m :: "'b"   and n :: "'b" shows "({..<m} \<subset> {..<n}) = (m < n)"
assms: shows "f \<in> borel_measurable (F (Suc n))"
find_max_def: fixes P :: "'b \<Rightarrow> bool"   and S :: "'b set" shows "find_max P S \<equiv> FOREACH\<^sub>O\<^sub>C\<^bsup>\<lambda>x y. y \<le> x,find_max_invar P S\<^esup> S (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x _. RETURN (if P x then Some x else None)) None"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) A'.map"
yy': shows "{y EQ y'} \<turnstile> P x' y IFF P x' y'"
error_free_halloc: fixes G :: "prog"   and s0 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1"   and "error_free s0" shows "error_free s1"
utp_defs_7: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<longleftrightarrow> Q \<equiv> \<lambda>A. P A = Q A"
neg_minus_divideC_le_eq: fixes c :: "complex"   and b :: "'b"   and a :: "'b" assumes "c < 0" shows "(- (b /\<^sub>C c) \<le> a) = (c *\<^sub>C a \<le> - b)"
update_accessor_accessor_eqE: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a"   and v :: "'b"   and f :: "'a \<Rightarrow> 'a"   and v' :: "'b"   and x :: "'a" assumes "Record.iso_tuple_update_accessor_eq_assist upd ac v f v' x" shows "ac v = x"
prod_mset_Un: fixes A :: "'a multiset"   and B :: "'a multiset" shows "\<Prod>\<^sub># (A + B) = \<Prod>\<^sub># A * \<Prod>\<^sub># B"
elims_2: fixes x :: "'v"   and xa :: "'v list" assumes "adj_path x xa" obtains v :: "'v" where "x = v__" and "xa = []" and "v__ \<in> V"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "adjacent v u \<and> adj_path u us"
monoid_axioms: shows "Group_Theory.monoid stk2.carrier_stalk stk2.add_stalk (stk2.zero_stalk U)"
degree_npolyhCN: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh (CN c n p) n0" shows "Polynomial_Expression.degree c = 0"
L_deltas: fixes w :: "'a list"   and s :: "'s" assumes "wf_word n w"   and "wf_state s" shows "L (fold delta w s) = fold lQuot w (L s)"
real_root_gt_0_iff: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "(0 < root n y) = (0 < y)"
from_nat_into_to_nat_on: fixes A :: "'b set"   and a :: "'b" assumes "countable A"   and "a \<in> A" shows "from_nat_into A (to_nat_on A a) = a"
reflpI: fixes r :: "'x \<Rightarrow> 'x \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
fun_ord_def: fixes ord :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
\<phi>_in_hom: fixes y :: "'c"   and f :: "'b"   and x :: "'b" assumes "C.ide y"   and "\<guillemotleft>f : F' y \<rightarrow>\<^sub>B x\<guillemotright>" shows "\<guillemotleft>\<phi>' y f : y \<rightarrow>\<^sub>C G' x\<guillemotright>"
treap_NodeI: fixes l :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree"   and k :: "'a"   and p :: "'b" assumes "treap l"   and "treap r"   and "\<forall>k'\<in>keys l. k' < k"   and "\<forall>k'\<in>keys r. k < k'"   and "\<forall>p'\<in>prios l. p \<le> p'"   and "\<forall>p'\<in>prios r. p \<le> p'" shows "treap \<langle>l, (k, p), r\<rangle>"
perm_simps_2: fixes p :: "perm"   and fmla1 :: "fmla"   and fmla2 :: "fmla" shows "p \<bullet> fmla1 OR fmla2 = (p \<bullet> fmla1) OR (p \<bullet> fmla2)"
closed_Collect_all: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.topological_space open"   and "\<And>i. topological_space.closed open {x. P i x}" shows "topological_space.closed open {x. \<forall>i. P i x}"
simps_7: fixes e :: "char list exp"   and F :: "char list"   and D :: "char list" shows "fv (e\<bullet>F{D}) = fv e"
power_Suc: fixes a :: "'a"   and n :: "nat" shows "a ^ Suc n = a \<cdot> a ^ n"
smult_conv_map_poly: fixes c :: "'a"   and p :: "'a poly" shows "smult c p = map_poly ((*) c) p"
tagged_fun_relD_rhs: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set"   and x :: "'c"   and x' :: "'e" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, Let x' f') \<in> B"
dsafe_Fun: fixes args :: "'a \<Rightarrow> ('b, 'a) trm"   and i :: "'b" assumes "\<And>i. dsafe (args i)" shows "dsafe ($f i args)"
last_suffixes: fixes xs :: "'c list" shows "last (suffixes xs) = xs"
concat_equal_length_map: fixes a :: "nat"   and f :: "nat \<Rightarrow> 'b list"   and d :: "nat" assumes "\<And>i. i < a \<Longrightarrow> length (f i) = d" shows "length (concat (map f [0..<a])) = a * d"
init_def: shows "init = integral\<^sup>L N (discounted_value r (\<lambda>m. der) matur)"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_end1 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "0 < n \<and> (l, r) = ([Bk], Oc # Bk \<up> n @ Oc \<up> n)"
vrestr_comp: fixes g :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"   and A :: "(nat \<Rightarrow> 'a) set" assumes "local.P.random_variable P g" shows "vimage_restr (f \<circ> g) A = vimage_restr g (vimage_restr f A)"
take_bit_Suc_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit (Suc n) one = one"
assms_3: shows "set xvec \<subseteq> supp T"
iso_runit: fixes a :: "'a" assumes "ide a" shows "local.iso (MC.runit a)"
tl_enc: fixes x :: "nat + nat set"   and I :: "(nat + nat set) list"   and w :: "'a list" assumes "\<forall>x\<in>set (x # I). case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "SAMEQUOT (any, replicate (length I) False) (map \<pi> ` enc (w, x # I)) = enc (w, I)"
hyperb_ineq: fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) - deltaG TYPE('a) \<le> Gromov_product_at e x z"
fv_less_nfv: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and x :: "nat" assumes "fo_spec nfv fv sat"   and "x \<in> fv" shows "x < nfv"
surj_plus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "surj (plus a)"
basic_trans_rules_28: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b < c" shows "a < c"
cases: fixes x :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('a \<times> 'c) list \<times> ('a \<times> 'c) list" obtains f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and k :: "'a" and v :: "'c" and as :: "('a \<times> 'c) list" and k' :: "'a" and v' :: "'c" and bs :: "('a \<times> 'c) list" where "x = (f__, (k, v) # as__, (k', v') # bs__)"   | f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and bs :: "('a \<times> 'c) list" where "x = (fa__, [], bs)"   | f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and as :: "('a \<times> 'c) list" where "x = (f, as, [])"
P'RelQ': shows "(\<Psi>, P', Q') \<in> Rel"
CondFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and n :: "nat"   and t :: "('d, 'f) xstate"   and c\<^sub>1 :: "('d, 'e, 'f) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
continuous_on_open_UN: fixes S :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>s. s \<in> S \<Longrightarrow> open (A s)"   and "\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f" shows "continuous_on (\<Union> (A ` S)) f"
cut_ge_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<ge> t = {}) = (\<forall>x\<in>I. x < t)"
left_commute: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" shows "gcp b (gcp a c) = gcp a (gcp b c)"
Diff_Diff_Int: fixes A :: "'q set"   and B :: "'q set" shows "A - (A - B) = A \<inter> B"
ennreal_mult_strict_right_mono: fixes a :: "ennreal"   and c :: "ennreal"   and b :: "ennreal" assumes "a < c"   and "0 < b"   and "b < top" shows "a * b < c * b"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "S.seq (S.UP g) (S.UP f)"
map_le_member: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(f \<subseteq>\<^sub>m g) = (\<forall>x y. (x, y) \<in>\<^sub>m f \<longrightarrow> (x, y) \<in>\<^sub>m g)"
fls_left_inverse: fixes x :: "'a"   and f :: "'a fls" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)" shows "fls_left_inverse f x * f = 1"
mergeFrameSimps: fixes A\<^sub>F :: "name list"   and \<Psi> :: "'b"   and \<Psi>\<^sub>F :: "'b" assumes "A\<^sub>F \<sharp>* \<Psi>" shows "(\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) \<otimes>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle> = \<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>\<rangle>"
simps: fixes s :: "name"   and t :: "tm"   and u :: "tm"   and k :: "name" assumes "atom s \<sharp> (t, u, k)"   and "atom k \<sharp> (t, u)" shows "QuoteP t u = SyntaxN.Ex s (SyntaxN.Ex k (SeqQuoteP t u (Var s) (Var k)))"
affS: shows "aff_dim S = int (dim ((+) (- a) ` S))"
id_f: shows "id_func AA \<^bsub>\<a>\<^esub> f = f"
borel_measurable_has_bochner_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "f \<in> borel_measurable M"
hensel_result: shows "linear_hensel_binary p n C D H = (D', H')"
retractions_compose: fixes e :: "'a"   and e' :: "'a" assumes "retraction e"   and "retraction e'"   and "seq e' e" shows "retraction (e' \<cdot> e)"
assms_3: shows "a = insertion (nth_default 0 (xs' @ y # xs)) A"
1: shows "f * fps_deriv R + fps_deriv f * R = 0"
word_set_nth_iff: fixes w :: "'a word"   and n :: "nat"   and b :: "bool" shows "(set_bit_class.set_bit w n b = w) = (w !! n = b \<or> size w \<le> n)"
respectfulness: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'b" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "f \<phi> \<nu> \<equiv>\<^sub>P f \<psi> \<nu>"
comb_2: fixes arg :: "Term_Class.term"   and t :: "sterm" assumes "arg \<in> set args"   and "is_value t" shows "match arg t = smatch' (mk_pat arg) t"
alpha_Tree_induct': fixes P :: "('a, 'b, 'c) Formula.Tree \<Rightarrow> ('a, 'b, 'c) Formula.Tree \<Rightarrow> bool"   and a0 :: "('a, 'b, 'c) Formula.Tree"   and a1 :: "('a, 'b, 'c) Formula.Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (Formula.Tree.tConj tset1) (Formula.Tree.tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (Formula.Tree.tNot t1) (Formula.Tree.tNot t2)"   and "\<And>\<phi>1 \<phi>2. P (Formula.Tree.tPred \<phi>1) (Formula.Tree.tPred \<phi>2)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (Formula.Tree.tAct \<alpha>1 t1) (Formula.Tree.tAct \<alpha>2 t2)"   and "\<And>v va. P (Formula.Tree.tNot v) (Formula.Tree.tConj va)"   and "\<And>v va. P (Formula.Tree.tNot v) (Formula.Tree.tPred va)"   and "\<And>v va vb. P (Formula.Tree.tNot v) (Formula.Tree.tAct va vb)"   and "\<And>v va. P (Formula.Tree.tPred v) (Formula.Tree.tConj va)"   and "\<And>v va. P (Formula.Tree.tPred v) (Formula.Tree.tNot va)"   and "\<And>v va vb. P (Formula.Tree.tPred v) (Formula.Tree.tAct va vb)"   and "\<And>v va vb. P (Formula.Tree.tAct v va) (Formula.Tree.tConj vb)"   and "\<And>v va vb. P (Formula.Tree.tAct v va) (Formula.Tree.tNot vb)"   and "\<And>v va vb. P (Formula.Tree.tAct v va) (Formula.Tree.tPred vb)"   and "\<And>va v. P (Formula.Tree.tConj va) (Formula.Tree.tNot v)"   and "\<And>va v. P (Formula.Tree.tPred va) (Formula.Tree.tNot v)"   and "\<And>va vb v. P (Formula.Tree.tAct va vb) (Formula.Tree.tNot v)"   and "\<And>va v. P (Formula.Tree.tConj va) (Formula.Tree.tPred v)"   and "\<And>va v. P (Formula.Tree.tNot va) (Formula.Tree.tPred v)"   and "\<And>va vb v. P (Formula.Tree.tAct va vb) (Formula.Tree.tPred v)"   and "\<And>vb v va. P (Formula.Tree.tConj vb) (Formula.Tree.tAct v va)"   and "\<And>vb v va. P (Formula.Tree.tNot vb) (Formula.Tree.tAct v va)"   and "\<And>vb v va. P (Formula.Tree.tPred vb) (Formula.Tree.tAct v va)" shows "P a0 a1"
modify_def: fixes f :: "'b \<Rightarrow> 'b" shows "modify f \<equiv> do s \<leftarrow> get; put (f s) od"
cSup_union_distrib: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq A"   and "B \<noteq> {}"   and "preorder.bdd_above less_eq B" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
ceiling_def: fixes x :: "'b" shows "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>"
assms_1: shows "LeftDerivationFix (\<delta> @ \<alpha>) i D j \<gamma>"
realizable_Append_matched: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'" shows "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
set_permutation_the_inv_restrict1: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "bij_betw f A A" shows "the_inv (restrict1 f A) = restrict1 (the_inv_into A f) A"
natural_isomorphism_\<rr>: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "right_hom_with_unit V H \<a> \<iota> a" shows "natural_isomorphism (subcategory.comp V (weak_composition.right V H a)) (subcategory.comp V (weak_composition.right V H a)) (weak_composition.H\<^sub>R H a) (identity_functor.map (subcategory.comp V (weak_composition.right V H a))) (\<lambda>\<mu>. if partial_magma.arr (subcategory.comp V (weak_composition.right V H a)) \<mu> then subcategory.comp V (weak_composition.right V H a) \<mu> (right_hom_with_unit.runit V H \<a> \<iota> a (partial_magma.dom (subcategory.comp V (weak_composition.right V H a)) \<mu>)) else partial_magma.null V)"
cases: fixes x :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<times> 'a list" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = (P__, [])"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xa :: "'a" and ys :: "'a list" where "x = (P, xa # ys)"
ports_to_set: fixes pts :: "(16 word \<times> 16 word) list" shows "ports_to_set pts = \<Union> {{s..e} |s e. (s, e) \<in> set pts}"
simps_3: fixes G :: "('c, 'b) labeled_graph"   and a :: "'c allegorical_term" shows ":G:\<lbrakk>A_Cnv a\<rbrakk> = (:G:\<lbrakk>a\<rbrakk>)\<inverse>"
fresh_ps_smallstep_P: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term"   and x :: "Syntax.var" assumes "\<lless>\<pi>, e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   and "atom x \<sharp> e"   and "atom x \<sharp> \<pi>" shows "atom x \<sharp> \<pi>'"
rel_optionT_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'e" shows "rel_optionT (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_optionT x \<subseteq> A} (map_optionT f)"
echelon_foldl_condition2: fixes ma :: "'d"   and k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" assumes "\<not> is_zero_row_upt_k ma k A"   and "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'d). A $ m $ mod_type_class.from_nat k = (0::'e)" shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) = (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) A)"
bsoks_def: fixes es :: "'a expr1 list"   and n :: "nat" shows "bsoks es n \<equiv> \<B>s es n \<and> expr_lockss es = (\<lambda>ad. 0)"
iso_locally_ringed_spaces_axioms_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "iso_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> homeomorphism X is_open\<^sub>X Y is_open\<^sub>Y f \<and> iso_sheaves_of_rings Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y (im_sheaf.im_sheaf X \<O>\<^sub>X f) (im_sheaf.im_sheaf_morphisms X \<rho>\<^sub>X f) b (im_sheaf.add_im_sheaf X add_str\<^sub>X f) (im_sheaf.mult_im_sheaf X mult_str\<^sub>X f) (im_sheaf.zero_im_sheaf X zero_str\<^sub>X f) (im_sheaf.one_im_sheaf X one_str\<^sub>X f) \<phi>\<^sub>f"
cr_tllist_def: shows "cr_tllist \<equiv> \<lambda>(xs, b). (=) (tllist_of_llist b xs)"
Rep_less_n: fixes x :: "'a" shows "Rep x < int CARD('a)"
enum_char_def: shows "enum_class.enum = [CHR 0x00, CHR 0x01, CHR 0x02, CHR 0x03, CHR 0x04, CHR 0x05, CHR 0x06, CHR 0x07, CHR 0x08, CHR 0x09, CHR ''\<newline>'', CHR 0x0B, CHR 0x0C, CHR 0x0D, CHR 0x0E, CHR 0x0F, CHR 0x10, CHR 0x11, CHR 0x12, CHR 0x13, CHR 0x14, CHR 0x15, CHR 0x16, CHR 0x17, CHR 0x18, CHR 0x19, CHR 0x1A, CHR 0x1B, CHR 0x1C, CHR 0x1D, CHR 0x1E, CHR 0x1F, CHR '' '', CHR ''!'', CHR 0x22, CHR ''#'', CHR ''$'', CHR ''%'', CHR ''&'', CHR 0x27, CHR ''('', CHR '')'', CHR ''*'', CHR ''+'', CHR '','', CHR ''-'', CHR ''.'', CHR ''/'', CHR ''0'', CHR ''1'', CHR ''2'', CHR ''3'', CHR ''4'', CHR ''5'', CHR ''6'', CHR ''7'', CHR ''8'', CHR ''9'', CHR '':'', CHR '';'', CHR ''<'', CHR ''='', CHR ''>'', CHR ''?'', CHR ''@'', CHR ''A'', CHR ''B'', CHR ''C'', CHR ''D'', CHR ''E'', CHR ''F'', CHR ''G'', CHR ''H'', CHR ''I'', CHR ''J'', CHR ''K'', CHR ''L'', CHR ''M'', CHR ''N'', CHR ''O'', CHR ''P'', CHR ''Q'', CHR ''R'', CHR ''S'', CHR ''T'', CHR ''U'', CHR ''V'', CHR ''W'', CHR ''X'', CHR ''Y'', CHR ''Z'', CHR ''['', CHR 0x5C, CHR '']'', CHR ''^'', CHR ''_'', CHR 0x60, CHR ''a'', CHR ''b'', CHR ''c'', CHR ''d'', CHR ''e'', CHR ''f'', CHR ''g'', CHR ''h'', CHR ''i'', CHR ''j'', CHR ''k'', CHR ''l'', CHR ''m'', CHR ''n'', CHR ''o'', CHR ''p'', CHR ''q'', CHR ''r'', CHR ''s'', CHR ''t'', CHR ''u'', CHR ''v'', CHR ''w'', CHR ''x'', CHR ''y'', CHR ''z'', CHR ''{'', CHR ''|'', CHR ''}'', CHR ''~'', CHR 0x7F, CHR 0x80, CHR 0x81, CHR 0x82, CHR 0x83, CHR 0x84, CHR 0x85, CHR 0x86, CHR 0x87, CHR 0x88, CHR 0x89, CHR 0x8A, CHR 0x8B, CHR 0x8C, CHR 0x8D, CHR 0x8E, CHR 0x8F, CHR 0x90, CHR 0x91, CHR 0x92, CHR 0x93, CHR 0x94, CHR 0x95, CHR 0x96, CHR 0x97, CHR 0x98, CHR 0x99, CHR 0x9A, CHR 0x9B, CHR 0x9C, CHR 0x9D, CHR 0x9E, CHR 0x9F, CHR 0xA0, CHR 0xA1, CHR 0xA2, CHR 0xA3, CHR 0xA4, CHR 0xA5, CHR 0xA6, CHR 0xA7, CHR 0xA8, CHR 0xA9, CHR 0xAA, CHR 0xAB, CHR 0xAC, CHR 0xAD, CHR 0xAE, CHR 0xAF, CHR 0xB0, CHR 0xB1, CHR 0xB2, CHR 0xB3, CHR 0xB4, CHR 0xB5, CHR 0xB6, CHR 0xB7, CHR 0xB8, CHR 0xB9, CHR 0xBA, CHR 0xBB, CHR 0xBC, CHR 0xBD, CHR 0xBE, CHR 0xBF, CHR 0xC0, CHR 0xC1, CHR 0xC2, CHR 0xC3, CHR 0xC4, CHR 0xC5, CHR 0xC6, CHR 0xC7, CHR 0xC8, CHR 0xC9, CHR 0xCA, CHR 0xCB, CHR 0xCC, CHR 0xCD, CHR 0xCE, CHR 0xCF, CHR 0xD0, CHR 0xD1, CHR 0xD2, CHR 0xD3, CHR 0xD4, CHR 0xD5, CHR 0xD6, CHR 0xD7, CHR 0xD8, CHR 0xD9, CHR 0xDA, CHR 0xDB, CHR 0xDC, CHR 0xDD, CHR 0xDE, CHR 0xDF, CHR 0xE0, CHR 0xE1, CHR 0xE2, CHR 0xE3, CHR 0xE4, CHR 0xE5, CHR 0xE6, CHR 0xE7, CHR 0xE8, CHR 0xE9, CHR 0xEA, CHR 0xEB, CHR 0xEC, CHR 0xED, CHR 0xEE, CHR 0xEF, CHR 0xF0, CHR 0xF1, CHR 0xF2, CHR 0xF3, CHR 0xF4, CHR 0xF5, CHR 0xF6, CHR 0xF7, CHR 0xF8, CHR 0xF9, CHR 0xFA, CHR 0xFB, CHR 0xFC, CHR 0xFD, CHR 0xFE, CHR 0xFF]"