hcju/isabelleps huggingface dataset:
arcsin_less_arcsin: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x < y"   and "y \<le> 1" shows "arcsin x < arcsin y"
fin_dim_def: shows "W.fin_dim = (\<exists>A. finite A \<and> A \<subseteq> carrier W \<and> W.module.gen_set A)"
no_loops_in_apath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and a :: "'a \<times> 'a" assumes "apath u p v"   and "a \<in> set p" shows "fst a \<noteq> snd a"
Max_in: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"
oalist_inv_raw_ConsD1: fixes x :: "'a \<times> 'e"   and xs :: "('a \<times> 'e) list" assumes "oalist_inv_raw (x # xs)" shows "oalist_inv_raw xs"
tendsto_iff: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" shows "(f \<longlongrightarrow> l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
ideal_prod_commute: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "I \<diamondsuit>\<^sub>r J = J \<diamondsuit>\<^sub>r I"
component_Join2: fixes G :: "'c program"   and F :: "'c program" shows "G \<le> F \<squnion> G"
has_derivativeI_sandwich: fixes e :: "real"   and f' :: "'d \<Rightarrow> 'e"   and s :: "'d set"   and x :: "'d"   and f :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> real" assumes "0 < e"   and "bounded_linear f'"   and "\<And>y. \<lbrakk>y \<in> s; y \<noteq> x; dist y x < e\<rbrakk> \<Longrightarrow> norm (f y - f x - f' (y - x)) / norm (y - x) \<le> H y"   and "(H \<longlongrightarrow> 0) (at x within s)" shows "(f has_derivative f') (at x within s)"
changes_R_smods_def: fixes p :: "real poly"   and q :: "real poly" shows "changes_R_smods p q = (let ps = smods p q in changes_poly_neg_inf ps - changes_poly_pos_inf ps)"
preserves_cod: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd"   and \<tau> :: "'b \<Rightarrow> 'd"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.cod B (\<tau> f) = G (partial_magma.cod A f)"
r_subset_y: shows "set_of r \<subset> set_of y"
crsp_step_dec_b_e_pre': fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and n :: "nat"   and la :: "cell list"   and ra :: "cell list"   and ires :: "cell list"   and e :: "nat"   and f :: "nat \<Rightarrow> (nat \<times> cell list \<times> cell list) \<times> nat \<times> nat"   and P :: "(nat \<times> cell list \<times> cell list) \<times> nat \<times> nat \<Rightarrow> bool"   and Q :: "(nat \<times> cell list \<times> cell list) \<times> nat \<times> nat \<Rightarrow> bool" assumes "ly = layout_of ap"   and "inv_locate_b (as, lm) (n, la, ra) ires"   and "abc_fetch as ap = Some (Dec n e)"   and "abc_lm_v lm n = 0"   and "f = (\<lambda>stp. (steps (Suc (start_of ly as) + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp, start_of ly as, n))"   and "P = (\<lambda>((s, l, r), ss, x). s = start_of ly e)"   and "Q = (\<lambda>((s, l, r), ss, x). dec_inv_1 ly x e (as, lm) (s, l, r) ires)" shows "\<exists>stp. P (f stp) \<and> Q (f stp)"
expand: fixes llist :: "'a llist"   and llist' :: "'a llist" assumes "lnull llist = lnull llist'"   and "\<lbrakk>\<not> lnull llist; \<not> lnull llist'\<rbrakk> \<Longrightarrow> lhd llist = lhd llist' \<and> ltl llist = ltl llist'" shows "llist = llist'"
path_edges: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v" shows "set p \<subseteq> edges g"
connected_space_topspace_empty: fixes X :: "'b topology" assumes "topspace X = {}" shows "connected_space X"
ce_rel_lm_26: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "Domain r \<in> ce_sets"
ideD_1: fixes a :: "'a" assumes "A.ide a" shows "A.arr a"
many_strongly_connected_iff_3: fixes x :: "'a" shows "many_strongly_connected x = (x \<le> x\<^sup>T\<^sup>\<star>)"
ijk: fixes i :: "nat"   and k :: "nat"   and j :: "nat" shows "i + k < Suc (i + j + k)"
qFreshInp_def: fixes xs :: "'varSort"   and x :: "'var"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" shows "qFreshInp xs x qinp \<equiv> liftAll (qFresh xs x) qinp"
lim: fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))" shows "emeasure PF.lim (PF.emb UNIV J X) = emeasure (CI J) X"
fmeasurable_UN_bound: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "\<Union> (A ` I) \<in> fmeasurable M"
isoI: fixes f :: "'a"   and g :: "'a" assumes "Right.inverse_arrows f g" shows "Right.iso f"
weakBisimTransitiveCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "eqvt X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(bisim O X O bisim)> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
closed_halfspace_ge: fixes b :: "real"   and a :: "'b" shows "closed {x. b \<le> a \<bullet> x}"
\<beta>_ne_m: shows "\<beta> \<noteq> m"
assms: shows "[\<^bold>\<forall>F. \<lparr>F,a\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,b\<^sup>P\<rparr> in v]"
DynT_propI: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and a' :: "val"   and statT :: "ref_ty"   and mode :: "inv_mode" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"   and "wf_prog G"   and "mode = IntVir \<longrightarrow> a' \<noteq> Null" shows "G\<turnstile>mode\<rightarrow>invocation_class mode s a' statT\<preceq>statT"
one_reducible_lm_4: fixes A :: "nat set" assumes "A \<in> ce_sets" shows "one_reducible_to A univ_ce"
det_rank_iff: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A \<noteq> (0::'b)) = (vec_space.rank n A = n)"
i_set_contp_i_set_cont_eq: shows "i_set_contp = (\<lambda>x. x \<in> i_set_cont)"
has_dist_on_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and V :: "nat set" shows "has_dist_on G m n V = (\<exists>p. is_shortest_path_on G m n p V)"
cobounded2: fixes a :: "'c"   and b :: "'c" shows "min a b \<le> b"
execute_parallel_plan_precondition_cons_i: fixes op :: "'a strips_operator"   and op' :: "'a strips_operator"   and s :: "'a \<Rightarrow> bool option" assumes "\<not> are_operators_interfering op op'"   and "is_operator_applicable_in s op"   and "is_operator_applicable_in s op'" shows "is_operator_applicable_in (s ++ map_of (effect_to_assignments op)) op'"
Arg2pi_cnj: fixes z :: "complex" shows "Arg2pi (cnj z) = (if z \<in> \<real> then Arg2pi z else 2 * pi - Arg2pi z)"
case_2: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and x2 :: "'h" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
sublist_EQNS_1: fixes l :: "'c list" shows "subseq [] l = True"
qfree_map_fm: fixes f :: "'b \<Rightarrow> 'a"   and \<phi> :: "'b fm" shows "qfree (map\<^sub>f\<^sub>m f \<phi>) = qfree \<phi>"
spp_of_vec_of: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp" shows "spp_of (vec_of sp) = sp"
coinduct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a stream \<Rightarrow> bool"   and x :: "'a stream" assumes "class.order less_eq less"   and "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>a b w. x = a ## b ## w \<and> less_eq b a \<and> (X (b ## w) \<or> order.sdescending less_eq (b ## w))" shows "order.sdescending less_eq x"
derivation_suc: fixes P :: "'a set"   and p :: "'a" assumes "derivation P"   and "extreme P (\<sqsubseteq>) p" shows "derivation (P \<union> {f p})"
cap'_def: shows "cap' = (\<lambda>(x, y). if edge \<Gamma> x y then g (x, y) else if edge \<Gamma> y x then 1 else 0)"
linear_axioms: fixes A :: "(('b, 'c) vec, 'd) vec" shows "linear (*s) (*s) ((*v) A)"
downset_prop: fixes x :: "'a" shows "\<down> x = {y. y \<le> x}"
ordering_def: fixes f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set" shows "TernaryOrdering.ordering f ord X \<equiv> (\<forall>n. (finite X \<longrightarrow> n < card X) \<longrightarrow> f n \<in> X) \<and> (\<forall>x\<in>X. \<exists>n. (finite X \<longrightarrow> n < card X) \<and> f n = x) \<and> (\<forall>n n' n''. (finite X \<longrightarrow> n'' < card X) \<and> n < n' \<and> n' < n'' \<longrightarrow> ord (f n) (f n') (f n''))"
simps_5: fixes v :: "trm"   and va :: "trm" shows "get_and_list (Less v va) = [Less v va]"
set_rel_sv: fixes R :: "('b \<times> 'c) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>set_rel)"
sees_method_decl_above: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
Pi_coprime: fixes i :: "nat"   and j :: "nat" assumes "i \<noteq> j" shows "coprime (Pi i) (Pi j)"
star_ax: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x\<^sup>\<star> + x \<le> x\<^sup>\<star>"
healthy_wp_Seq: fixes t :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy (wp t)"   and "healthy (wp u)" shows "healthy (wp (t ;; u))"
amalgamation_V01_total: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> V\<^sub>0\<^sub>1" shows "total1 (amalgamation (the (f 0)) (the (f 1)))"
adjust_not_in_span: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "w \<notin> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us)" shows "adjuster n w us + w \<notin> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us)"
simps_3: fixes \<phi> :: "fmla"   and \<psi> :: "fmla" shows "max_depth (Conj \<phi> \<psi>) = Suc (max (max_depth \<phi>) (max_depth \<psi>))"
tp_len_tp_unfold: shows "[tp - len..<tp] = (tp - len) # [tp - (len - 1)..<tp]"
strict_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a <lex b"   and "b <lex c" shows "a <lex c"
deg_pp_alt: fixes s :: "('c, 'b) pp" shows "deg_pp s = sum (lookup_pp s) (keys_pp s)"
atom_not_fresh_eq: fixes a :: "'a"   and x :: "'a" assumes "\<not> atom a \<sharp> x" shows "a = x"
distinct_sorted_list_of_set: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
degree_of_difference_diff_degree: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "deg R (p \<ominus>\<^bsub>P\<^esub> q) = deg R p"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b tree \<Rightarrow> 'b \<Rightarrow> 'b tree \<Rightarrow> 'a"   and tree :: "'b tree" shows "(case tree of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = (if tree = \<langle>\<rangle> then f1 else f2 (un_Node1 tree) (value tree) (un_Node3 tree))"
FDERIV_floatarith_def: fixes fa :: "floatarith"   and xs :: "nat list"   and d :: "floatarith list" shows "FDERIV_floatarith fa xs d = inner_floatariths (map (\<lambda>x. fold_const_fa (DERIV_floatarith x fa)) xs) d"
accessible_imp_states: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M"   and "q \<in> dfa.accessible M" shows "q \<in> dfa.states M"
ncoplanar_perm_10: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B D A C"
\<tau>Execsr1step: 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 s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_moves ci P t es h (stk', loc', pc', xcp') \<lbrace>\<rbrace> h s'"   and "\<tau>moves2 P h stk' es pc' xcp'" shows "\<tau>Exec_movesr ci P t es h (stk', loc', pc', xcp') s'"
isomorphic_implies_ide_2: fixes f :: "'b"   and g :: "'b" assumes "D.isomorphic f g" shows "D.ide g"
the_apartment_iso_apartment_chamber_map: fixes Xa :: "'b set set"   and \<A> :: "'b set set set"   and X :: "'b set set"   and Y :: "'b set set"   and Ca :: "'b set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem Xa \<A>"   and "X \<in> \<A>"   and "Y \<in> \<A>"   and "SimplicialComplex.maxsimp Xa Ca"   and "Ca \<in> X \<inter> Y"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp Y (ChamberComplexWithApartmentSystem.the_apartment_iso X Y ` C)"
to_from_view\<^sub>m: fixes x :: "view\<^sub>m" shows "view_merkle.to_view\<^sub>m (view_merkle.from_view\<^sub>m x) = x"
arr_char: fixes Arr :: "'a set"   and f :: "(bool, 'a list) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f = (f = quiver.Zero \<or> f = quiver.One \<or> f \<in> quiver.fromArr Arr ` Arr)"
equals_minim: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> (a, b) \<in> r" shows "a = minim B"
bind_ask: fixes f :: "'b \<Rightarrow> 'm"   and g :: "'a fset \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
prod_prime_factors': fixes n :: "nat" assumes "0 < n" shows "totient n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n - p ^ (multiplicity p n - 1))"
cdf_unique': fixes M1 :: "real measure"   and M2 :: "real measure" assumes "finite_borel_measure M1"   and "finite_borel_measure M2"   and "cdf M1 = cdf M2" shows "M1 = M2"
compl_top_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus top = bot"
simps: fixes lm :: "nat list"   and n :: "nat" shows "abc_lm_v lm n = (if n < length lm then lm ! n else 0)"
dgrad_p_set_le_trdsp: fixes d :: "'a \<Rightarrow> nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" assumes "dickson_grading d" shows "dgrad_p_set_le d {trdsp bs (p, q)} (insert (fst p) (insert (fst q) (set bs)))"
Sync_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (sync\<^bsub>V\<^esub> (e) e2, xs) (sync\<^bsub>V\<^esub> (e') e2, xs')"
Interleaves_interleaves: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<cong> {ys, zs, P}" shows "xs \<simeq> {ys, zs, P}"
power_increasing_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less_eq (power.power one times b x) (power.power one times b y) = (x \<le> y)"
sa1tb1: shows "s * a1 + t * b1 = (1::'a)"
resr_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<le> z) = (y \<le> x \<rightarrow> z)"
simps_1: fixes p :: "perm"   and tset :: "('b, 'c, 'd, 'e) Tree set['b]" shows "p \<bullet> tConj tset = tConj (map_bset (permute p) tset)"
set_map_pmf: fixes f :: "'f \<Rightarrow> 'e"   and M :: "'f pmf" shows "set_pmf (map_pmf f M) = f ` set_pmf M"
subst_mgt_comp: fixes \<theta> :: "'f \<Rightarrow> ('d, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('d, 'e) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta> \<circ>\<^sub>s \<delta>"
elims_3: fixes x :: "'fsym trm" assumes "\<not> wt x" obtains f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "\<not> (wtFsym f \<and> list_all wt Tl \<and> arOf f = map tpOf Tl)"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "(to_fract x = to_fract y) = (x = y)"
field_axioms_def: fixes R :: "('g, 'h) ring_scheme" shows "field_axioms R \<equiv> Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}"
dghm_up_ObjMap_app: fixes x :: "V"   and \<CC> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = (\<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ObjMap\<rparr>\<lparr>x\<rparr>)"
to_from_Gromov_completion: fixes x :: "'c Gromov_completion" assumes "x \<notin> Gromov_boundary" shows "to_Gromov_completion (from_Gromov_completion x) = x"
cong: shows "exception_\<I> \<I>,exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>\<^sub>C cnv \<sim> cnv'"
nb\<^sub>1: fixes v :: "'a" assumes "(\<Pi>\<^sub>G) v \<noteq> None" shows "(trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1)) v = (\<Pi>\<^sub>G) v"
rel_spmfE: fixes P :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'d spmf" assumes "rel_spmf P p q" obtains pq :: "('b \<times> 'd) spmf" where "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y" and "p = map_spmf fst pq" and "q = map_spmf snd pq"
rotate_append: fixes l :: "'b list"   and q :: "'b list" shows "rotate \<^bold>|l\<^bold>| (l \<cdot> q) = q \<cdot> l"
subring_props_2: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "\<zero>\<^bsub>K [X]\<^esub> \<in> K"
distinct_1: fixes x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set"   and x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat" shows "Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<noteq> Read\<^sub>s\<^sub>b x21 x22 x23 x24"
drlex_pm_lin: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "drlex_pm s t \<or> drlex_pm t s"
a1: shows "low_equal s1 s2 \<and> t1 = s1\<lparr>state_var := new_state_var\<rparr> \<and> t2 = s2\<lparr>state_var := new_state_var\<rparr>"
exhaust: fixes y :: "fresh_t" obtains x1 :: "fid_t" and x2 :: "nat" where "y = x1 $ x2"
prems_7: shows "is_tangent p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3)"
perp2_left_comm: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "P Perp2 A B C D" shows "P Perp2 B A C D"
type_less_x_OclVoid: fixes \<tau> :: "'a basic_type"   and P :: "bool" assumes "\<tau> < OclVoid" shows "P"
map_transfer: fixes R1b :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S1d :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R2b :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and S2d :: "'i \<Rightarrow> 'm \<Rightarrow> bool"   and R3b :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and S3d :: "'k \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_fun (rel_fun R1b S1d) (rel_fun (rel_fun R2b S2d) (rel_fun (rel_fun R3b S3d) (rel_fun (F1rel R1b R2b R3b) (F1rel S1d S2d S3d)))) F1map F1map"
mult_le_cancel_iff1: fixes z :: "'b"   and x :: "'b"   and y :: "'b" assumes "(0::'b) < z" shows "(x * z \<le> y * z) = (x \<le> y)"
seqSubs_def: fixes P :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" shows "P[<\<sigma>>] \<equiv> foldl (\<lambda>Q (xvec, Tvec). subs Q xvec Tvec) P \<sigma>"
insort_is_Cons: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'h list"   and f :: "'h \<Rightarrow> 'g"   and a :: "'h" assumes "class.linorder less_eq less"   and "\<forall>x\<in>set xs. less_eq (f a) (f x)" shows "linorder.insort_key less_eq f a xs = a # xs"
FG_advice_subfrmlsn: fixes \<phi> :: "'b ltln"   and Y :: "'b ltln set" shows "subfrmlsn (\<phi>[Y]\<^sub>\<mu>) \<subseteq> {\<psi>[Y]\<^sub>\<mu> |\<psi>. \<psi> \<in> subfrmlsn \<phi>}"
comp_char: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "A1xA2.comp g f = (if A1.seq (fst g) (fst f) \<and> A2.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>A\<^sub>1 fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f) else (A1.null, A2.null))"
select_convs_1: fixes Obj :: "'d set"   and Mor :: "'e set"   and Dom :: "'e \<Rightarrow> 'd"   and Cod :: "'e \<Rightarrow> 'd"   and Id :: "'d \<Rightarrow> 'e"   and Comp :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f" shows "obj\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Obj"
sets_bind_measurable: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and B :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets B"
eventually_elim2: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter"   and Q :: "'d \<Rightarrow> bool"   and R :: "'d \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F"   and "\<And>i. \<lbrakk>P i; Q i\<rbrakk> \<Longrightarrow> R i" shows "eventually R F"
ind_add_str_def: fixes V :: "'a set" shows "ind_add_str V \<equiv> +\<^bsub>U \<inter> V\<^esub>"
sel_1: fixes x1 :: "'f" shows "terminal (TNil x1) = x1"
dynimethd_def: fixes G :: "prog"   and I :: "qtname"   and dynC :: "qtname" shows "dynimethd G I dynC = (\<lambda>sig. if DeclConcepts.imethds G I sig \<noteq> {} then methd G dynC sig else dynmethd G Object dynC sig)"
pointwise_natUnion_swap: fixes f :: "'a set \<Rightarrow> 'b set"   and G :: "nat \<Rightarrow> 'a set" assumes "pointwise f" shows "f (natUnion G) = natUnion (\<lambda>u. f (G u))"
viable_market_def: fixes Mkt :: "(bool stream, 'b) discrete_market" shows "viable_market Mkt = (\<forall>p. stock_portfolio Mkt p \<longrightarrow> \<not> arbitrage_process Mkt p)"
sum_shift_lb_Suc0_0_upt: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "class.comm_monoid_add plus zero"   and "f 0 = zero" shows "comm_monoid_add.sum plus zero f {Suc 0..<k} = comm_monoid_add.sum plus zero f {0..<k}"
to_option_Some: fixes r :: "('m, 'l) result"   and x :: "'l" shows "(Result.to_option r = Some x) = (r = Ok x)"