hcju/isabelleps huggingface dataset:
lookup_other_ev_None: fixes ev :: "'b"   and k :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "lookup_other_ev ev k A = None"   and "i < k" shows "A $$ (i, i) = ev"
leadsTo_Diff: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "F \<in> A - B \<longmapsto> C"   and "F \<in> B \<longmapsto> C" shows "F \<in> A \<longmapsto> C"
choose_1: fixes S :: "'b set" assumes "card S = 1" shows "\<exists>x. S = {x}"
Subst_preserves_finite_rhs: fixes rhs :: "'b trm set"   and yrhs :: "'b trm set"   and Y :: "'b list set" assumes "finite rhs"   and "finite yrhs" shows "finite (Subst rhs Y yrhs)"
enat_ord_simps_1: fixes m :: "nat"   and n :: "nat" shows "(enat m \<le> enat n) = (m \<le> n)"
nat_power_power_left: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and k :: "nat"   and n :: "'b" assumes "class.nat_power one times plus zero nat_power"   and "0 < m" shows "nat_power (m ^ k) n = power.power one times (nat_power m n) k"
distinct_10: fixes x61 :: "nat"   and x62 :: "recf" shows "Mn x61 x62 \<noteq> Z"
le_def: fixes c :: "int"   and t :: "Ferrack.num" shows "le c t = (if c = 0 then Le t else if 0 < c then Le (CN 0 c t) else Ge (CN 0 (- c) (Neg t)))"
sfoldl_sappend: fixes f :: "'s \<rightarrow> 'r \<rightarrow> 's"   and z :: "'s"   and xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "f\<cdot>\<bottom> = \<bottom>" shows "sfoldl\<cdot>f\<cdot>z\<cdot>(xs :@ ys) = sfoldl\<cdot>f\<cdot>(sfoldl\<cdot>f\<cdot>z\<cdot>xs)\<cdot>ys"
intros_3: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" assumes "CK f"   and "CK g" shows "CK (f \<circ> g)"
fun_type: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "typed_model arity public Ana \<Gamma>"   and "0 < arity f" shows "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)"
toy_wf: shows "wellformed \<Gamma>\<^sub>T\<^sub>O\<^sub>Y"
w_addrs_def: fixes vs :: "'a \<times> addr_loc \<Rightarrow> 'a val set" shows "w_addrs vs = {a. \<exists>adal. Addr a \<in> vs adal}"
nprv_addLemmaE: fixes \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "prv \<phi>"   and "nprv (insert \<phi> F) \<psi>"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F \<psi>"
consistent_def: shows "consistent \<equiv> \<not> {} \<turnstile> Fls"
conjugate_transpose_rank_le: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "vec_space.rank nc A\<^sup>H \<le> vec_space.rank n A"
reachable_pair_induct: fixes \<xi> :: "'e"   and p :: "'f"   and A :: "('e \<times> 'f, 'g) automaton"   and I :: "'g \<Rightarrow> bool"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "(\<xi>, p) \<in> reachable A I"   and "\<And>\<xi> p. (\<xi>, p) \<in> init A \<Longrightarrow> P \<xi> p"   and "\<And>\<xi> p \<xi>' p' a. \<lbrakk>(\<xi>, p) \<in> reachable A I; P \<xi> p; ((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P \<xi>' p'" shows "P \<xi> p"
semiring_mat: fixes n :: "nat"   and b :: "'c" shows "semiring (ring_mat TYPE('b) n b)"
wf_subst_support_iff_mgt: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "(\<theta> supports \<delta>) = (\<theta> \<preceq>\<^sub>\<circ> \<delta>)"
mult_mod_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "times c (modulo a b) = modulo (times c a) (times c b)"
cipher_queriesI: fixes ose :: "'d \<Rightarrow> ('a \<times> 'b \<times> 'c list) option"   and n :: "'d"   and ek :: "'a"   and dk :: "'b"   and ciphers :: "'c list"   and x :: "'c" assumes "ose n = Some (ek, dk, ciphers)"   and "x \<in> set ciphers" shows "x \<in> ind_cca2.cipher_queries ose"
iso_is_arr: fixes f :: "'a" assumes "S.iso f" shows "S.arr f"
assms_3: shows "R\<^sup>+\<^sup>+ y z"
True: fixes F :: "('f \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
elims: fixes x :: "variable"   and xa :: "trm option"   and y :: "game option" assumes "Assigno x xa = y" obtains xb :: "variable" and \<theta> :: "trm" where "x = xb__" and "xa = Aterm \<theta>" and "y = Agame (xb__ := \<theta>)"   | xb :: "variable" where "x = xb" and "xa = undeft" and "y = undefg"
h1: shows "set (characterize_consistent_signs_at_roots_copr p [q]) \<subseteq> {[1], [- 1]}"
carrier_not_empty: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G" shows "carrier G \<noteq> {}"
strategy_attracts_irrelevant_override: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "G'.strategy_attracts p \<sigma> A W"   and "G'.strategy p \<sigma>"   and "G'.strategy p \<sigma>'" shows "G'.strategy_attracts p (override_on \<sigma>' \<sigma> (A - W)) A W"
invertible_mat_one: fixes n :: "nat" shows "invertible_mat (1\<^sub>m n)"
simps_14: fixes v :: "fm"   and va :: "fm" shows "not (Imp v va) = NOT (Imp v va)"
rsp: shows "eq_onp (\<lambda>m. finite (dom m)) Map.empty Map.empty"
le_refl: fixes n :: "nat" shows "(n::nat) \<le> n"
prems_6: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
cfid: shows "f\<inverse> O d \<subseteq> s \<union> ov \<union> d"
lm067: fixes f :: "('b \<times> 'a) set" shows "runiq (f\<inverse>) = (f \<in> converse ` runiqs)"
nat_term_compare_splus: fixes to :: "'a nat_term_order"   and t :: "'a"   and u :: "'a"   and v :: "'a" shows "nat_term_compare to (splus t u) (splus t v) = nat_term_compare to u v"
update_simps_5: fixes x :: "'h"   and y :: "'h"   and ys :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "update (x # y # ys) opt (Env b es) = Env b (es(x := case es x of None \<Rightarrow> None | Some e \<Rightarrow> Some (update (y # ys) opt e)))"
sound_isTrue: fixes \<phi> :: "fm" assumes "\<phi> \<in> UNIV"   and "Fvars \<phi> = {}"   and "{} \<turnstile> \<phi>" shows "eval_fm e0 \<phi>"
assms_4: shows "confs prb (root (red prb)) = init_conf prb"
Edka_Impl_axioms_def: fixes c :: "nat \<times> nat \<Rightarrow> int"   and N :: "nat" shows "Edka_Impl_axioms c N \<equiv> Graph.V c \<subseteq> {0..<N}"
left_neutral: fixes a :: "'e" shows "(0::'e) + a = a"
preserves_reflects_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "C.arr (D.map f) = J.arr f"
minus_float_round_up_eq: fixes prec :: "nat"   and x :: "float" shows "- float_round_up prec x = float_round_down prec (- x)"
plus_current_simps: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and e :: "'k \<times> 'k" shows "plus_current f g e = f e + g e"
mat_of_cblinfun_adj: fixes F :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "mat_of_cblinfun (F*) = mat_adjoint (mat_of_cblinfun F)"
strict_trans1: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "b \<unrhd> a"   and "c \<rhd> b" shows "c \<rhd> a"
sum0: shows "(\<Sum>x\<in>X. of_int (\<beta> x) * exp x) = 0"
diff_closed: fixes R :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Ring1 R"   and "g \<in> R"   and "h \<in> R" shows "g - h \<in> R"
final_notin_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and T :: "('e, 'f) xstate set" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T = (\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> t \<notin> T)"
inf_idem: fixes x :: "'g multiset" shows "x \<inter># x = x"
iteration_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<infinity> \<le> y\<^sup>\<infinity>"
\<delta>: shows "Unifier \<delta> (Fun f X) t"
not_finals_ConsI: fixes e :: "'a exp"   and es :: "'a exp list" assumes "\<not> final e" shows "\<not> finals (e # es)"
final_def: fixes cs :: "com \<times> (char list \<Rightarrow> int)" shows "final cs = (\<nexists>cs'. cs \<rightarrow> cs')"
adjacent_def: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.adjacent G v v' \<equiv> \<exists>w. (v, w, v') \<in> edges G"
OclSize_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "x->size\<^sub>S\<^sub>e\<^sub>t() = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> finite \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> then \<lfloor>\<lfloor>int (card \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil>)\<rfloor>\<rfloor> else \<bottom>)"
Gfun: fixes f :: "(val \<times> val) fset" shows "good (VFun f) = {BClos x e \<rho> |x e \<rho>. ffold (good_entry x e \<rho>) True (good_prod |`| f)}"
sup_None1: fixes x :: "'b option"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err" shows "x \<squnion>\<^bsub>Opt.sup f\<^esub> None = OK x"
redsp_reds_eq: fixes P :: "(char list list \<times> char list exp) prog" shows "redsp P = reds' P"
cbmi: shows "b O m\<inverse> \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
is_induced_map_iff: fixes w :: "'a" shows "is_induced_map w = (ide w \<and> (\<exists>\<nu> \<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (\<mu> \<star> r\<^sub>0) \<cdot> \<rho> = (s \<star> \<theta>) \<cdot> \<a>[s, s\<^sub>0, w] \<cdot> (\<sigma> \<star> w) \<cdot> \<nu>))"
hfrefI: fixes P :: "'b \<Rightarrow> bool"   and RS :: "('b \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('b \<Rightarrow> 'a \<Rightarrow> assn)"   and f :: "'a \<Rightarrow> 'c Heap"   and T :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and g :: "'b \<Rightarrow> 'd nres" assumes "\<And>c a. P a \<Longrightarrow> hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)" shows "(f, g) \<in> [P]\<^sub>a RS \<rightarrow> T"
eq: fixes x :: "'a" assumes "x \<in> \<Omega>" shows "emeasure M {x} = emeasure N {x}"
case_1: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "'a"   and x1 :: "vname" shows "(case VNam x1 of VNam x \<Rightarrow> f1 x | Res \<Rightarrow> f2) = f1 x1"
extended_Gromov_product_at_diff3: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and z :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y \<le> extended_Gromov_product_at e x z + extended_Gromov_distance y z"
hl_ax2: shows "\<forall>P x. P x \<longrightarrow> P (Eps P)"
coprime_absorb_right: fixes y :: "'a"   and x :: "'a" assumes "y dvd x" shows "coprime x y = is_unit y"
quot_set_one: fixes A :: "'a set" shows "A \<div> (1::'a) = A"
odd_numeral: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit1 n))"
E_Abs: fixes v\<^sub>2 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm" assumes "v\<^sub>2 \<in> value" shows "(\<lambda>:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet> v\<^sub>2 \<longmapsto>\<^sub>c t\<^sub>1\<^sub>2[0 \<mapsto> v\<^sub>2]"
binop_axioms: shows "binop \<int>\<^sub>\<circ> vint_mult"
SMP_MP_split: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> SMP M"   and "\<forall>m\<in>M. is_Fun m" shows "(\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>) \<or> t \<in> SMP (subterms\<^sub>s\<^sub>e\<^sub>t M \<union> \<Union> ((set \<circ> fst \<circ> Ana) ` M) - M)"
step: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "f y \<in> orbit f x"
pred_sets2: fixes A :: "'c set"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "A \<in> sets N"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "pred M (\<lambda>x. f x \<in> A)"
s0g_def: shows "s0g \<equiv> \<lparr>init = s0g_init, trans = s0g_trans, obs = id\<rparr>"
ord_isom_convert: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and x :: "'a"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "x \<in> carrier D"   and "a \<in> carrier D" shows "(\<forall>y\<in>carrier D. x \<prec> y \<longrightarrow> \<not> y \<prec> a) = (\<forall>z\<in>carrier E. f x \<prec>\<^bsub>E\<^esub> z \<longrightarrow> \<not> z \<prec>\<^bsub>E\<^esub> f a)"
ch2ch_Rep_cfunL: fixes F :: "nat \<Rightarrow> 'a \<rightarrow> 'b"   and x :: "'a" assumes "chain F" shows "chain (\<lambda>i. F i\<cdot>x)"
codomains_comp: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "D.codomains (g \<cdot>\<^sub>D f) = D.codomains g"
simps_3: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet" shows "tensors_from_net (Pool m1 m2) = component_mult (tensors_from_net m1) (tensors_from_net m2)"
elem: fixes x :: "'d"   and xs :: "'d list" shows "ListMem x (x # xs)"
permute_Value_def: fixes p :: "perm"   and v :: "Value" shows "p \<bullet> v = v"
take_eq_Nil: fixes n :: "nat"   and xs :: "'k list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
intro: fixes g_ops :: "('a, 'b, 'c, 'd) graph_ops_scheme"   and mr_ops :: "('a, ('a \<times> 'b \<times> 'a) list \<times> 'b, 'e, 'f) map_ops_scheme"   and qw_ops :: "('a, 'b infty, 'g, 'h) uprio_ops_scheme" assumes "StdGraph g_ops"   and "StdMap mr_ops"   and "StdUprio qw_ops" shows "dijkstraC g_ops mr_ops qw_ops"
prems_2: shows "valid fg ({#s#} + c1 + c2)"
assms_1: shows "returns_to {x \<in> S. s x = 0} x"
cinner_commute: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x y = cnj (cinner y x)"
mtx_nonzero_def: fixes m :: "nat \<times> nat \<Rightarrow> 'a" shows "mtx_nonzero m \<equiv> {(i, j). m (i, j) \<noteq> (0::'a)}"
kripke_tc_rels_worlds: fixes w :: "'e"   and w' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and as :: "'f set" assumes "(w, w') \<in> (\<Union> (relations M ` as))\<^sup>+"   and "kripke M" shows "w \<in> worlds M \<and> w' \<in> worlds M"
ind_ax_def: shows "ind_ax = A(decode_Var v::=Zero) IMP SyntaxN.All (decode_Var v) (SyntaxN.All (decode_Var w) (A IMP Aw IMP Ae)) IMP SyntaxN.All (decode_Var v) A"
sq_norm_poly_pos: fixes p :: "'d poly" shows "((0::'d) < \<parallel>p\<parallel>\<^sup>2) = (p \<noteq> 0)"
vwb_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "vwb_lens x \<equiv> wb_lens x \<and> mwb_lens x"
that: fixes R :: "(real \<times> real pdevs) \<times> real" assumes "approx_floatarith p fa XS = Some R" shows "thesis"
union'_nodes_finite: fixes A :: "('e, 'd) dca"   and B :: "('e, 'f) dca" assumes "finite (DCA.nodes A)"   and "finite (DCA.nodes B)" shows "finite (DGCA.nodes (union' A B))"
aug_inf_noninf_is_z: fixes x :: "ant" assumes "x \<in> Z\<^sub>\<infinity>"   and "x \<noteq> \<infinity>" shows "\<exists>z. x = ant z"
subterms_subst_subset': fixes \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and M :: "('a, 'c) Term.term set" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma> \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma>)"
put_option_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure"   and T :: "nat"   and K :: "real" assumes "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G" shows "CRR_market.put_option Mkt stk T K = (\<lambda>w. max (K - prices Mkt stk T w) 0)"
f_the_inv_into_f_bij_betw: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and x :: "'b" assumes "bij_betw f A B"   and "bij_betw f A B \<Longrightarrow> x \<in> B" shows "f (the_inv_into A f x) = x"
vangle_le_pi: fixes u :: "'b"   and v :: "'b" shows "vangle u v \<le> pi"
transfers_intra_slice_kinds_slice_edges: fixes as :: "'edge list"   and cs :: "'edge list"   and S :: "'node SDG_node set"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "\<forall>a\<in>set as. intra_kind (kind a)"   and "\<forall>c\<in>set cs. sourcenode c \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "transfers (slice_kinds S (slice_edges S cs as)) s = transfers (slice_kinds S as) s"
obs_a_conv: fixes ls :: "'b lock_actions"   and nts :: "('c, 'd, 'e) new_thread_action list"   and js :: "'c conditional_action list"   and wss :: "('c, 'f) wait_set_action list"   and "is" :: "'c interrupt_action list"   and obs :: "'a list" shows "\<lbrace>(ls, nts, js, wss, is, obs)\<rbrace>\<^bsub>o\<^esub> = obs"
ranking_index_le_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(weak_ranking_index y \<le> weak_ranking_index x) = le x y"
shift_bounds_Suc_ivl: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = (\<Sum>i = m..<n. g (Suc i))"
is_natural_2: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "\<Phi> (C.VV.cod f) \<cdot>\<^sub>D H\<^sub>DoFF.map f = \<Phi> f"
polyhedron_Inter: fixes F :: "'b set set" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> polyhedron S" shows "polyhedron (\<Inter> F)"
add_numeral_left: fixes v :: "num"   and w :: "num"   and z :: "'d" shows "numeral v + (numeral w + z) = numeral (v + w) + z"
has_field_derivative_powser: fixes z :: "'a"   and f :: "nat \<Rightarrow> 'a"   and A :: "'a set" assumes "ereal (norm z) < conv_radius f" shows "((\<lambda>z. \<Sum>n. f n * z ^ n) has_field_derivative (\<Sum>n. diffs f n * z ^ n)) (at z within A)"
mbuf2t_take_eqD_2: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and nts' :: "nat list"   and i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat" assumes "mbuf2t_take f z buf nts = (z', buf', nts')"   and "wf_mbuf2 i ja jb P Q buf"   and "list_all2 R [i..<j] nts"   and "ja \<le> j"   and "jb \<le> j" shows "list_all2 R [min ja jb..<j] nts'"
case_2: fixes f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
Part_def: fixes A :: "'b set"   and h :: "'c \<Rightarrow> 'b" shows "Part A h = A \<inter> {x. \<exists>z. x = h z}"
findMin_def: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "findMin bq = (let min = getMinTree bq in (val min, prio min))"
scc_root_unique_is_scc: fixes v :: "'v"   and scc :: "'v set" assumes "scc_root s v scc" shows "scc_root s v (scc_of E v)"
det_minor_row: fixes B :: "('a, 'b) sq_matrix"   and i :: "'b"   and j :: "'b" shows "Square_Matrix.det (minor B i j) = Square_Matrix.det (upd_row B i (axis j (1::'a)))"
prod_1_id_def: fixes p :: "'e \<Rightarrow> 'f decision option"   and q :: "'e \<Rightarrow> 'g decision option" shows "p \<Otimes>\<^sub>1\<^sub>I q = p \<Otimes>\<^sub>1 q \<circ> (\<lambda>x. (x, x))"
wls_swap_Op_simp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "Op delta inp binp #[x1 \<and> x2]_xs = Op delta (inp %[x1 \<and> x2]_xs) (binp %%[x1 \<and> x2]_xs)"
assms_7: fixes x :: "'a"   and ws :: "'a list"   and s :: "'b"   and r :: "'b" assumes "I x"   and "\<forall>x\<in>set ws. I x"   and "S s"   and "P ws (f x s) r" shows "P (x # ws) s r"
swapAbs_substAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "A $[Y / y]_ys $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
seq_progress: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t"
sequential_def: fixes P :: "'c option process" shows "sequential P \<equiv> (\<forall>xs\<in>traces P. None \<notin> set (butlast xs)) \<and> (\<forall>xs\<in>sentences P. next_events P xs = {None})"
not_gr_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
NotOrD: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "\<not> [\<phi> \<^bold>\<or> \<psi> in v]" shows "\<not> [\<phi> in v] \<and> \<not> [\<psi> in v]"
Gromov_completion_dist_comparison_3: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "min (esqrt (extended_Gromov_distance x y)) (eexp (ereal (- epsilonG TYPE('a)) * extended_Gromov_product_at basepoint x y)) \<le> 2 * ereal (dist x y)"
CompDiag_preserves_Diag_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Cod t"