hcju/isabelleps huggingface dataset:
triv_div_mult: fixes vSF :: "real"   and L :: "real"   and K :: "real"   and VS :: "real"   and V :: "real" assumes "vSF \<noteq> 1"   and "L = (K - vSF) / (1 - vSF)"   and "L \<noteq> 1" shows "VS / (1 - vSF) * V / (1 - L) = VS * V / (1 - K)"
Sem_no_zero_height_derivs: fixes s :: "Var \<Rightarrow> Val"   and c :: "IMP"   and t :: "Var \<Rightarrow> Val" assumes " s , c \<rightarrow>\<^sub>0  t " shows "False"
lve_Exit_edge: fixes e :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and X :: "'g"   and valid_edge :: "'l \<Rightarrow> bool"   and src :: "'l \<Rightarrow> 'g"   and trg :: "'l \<Rightarrow> 'g"   and knd :: "'l \<Rightarrow> ('h, 'i, 'j, 'k) edge_kind"   and E :: "'g" assumes "e = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)" shows "lift_valid_edge valid_edge src trg knd E X e"
ddep_slice1: fixes n'' :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
completeTA_def: shows "completeTA \<equiv> \<lparr>ta_initial = Some ` Qi, ta_rules = \<delta>complete\<rparr>"
convex_hull_bound: fixes X :: "'b vec set"   and Bnd :: "'b"   and n :: "nat" assumes "X \<subseteq> Bounded_vec Bnd"   and "X \<subseteq> carrier_vec n" shows "gram_schmidt.convex_hull n X \<subseteq> Bounded_vec Bnd"
invalidate_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" shows "invalidate rt dests \<equiv> \<lambda>ip. case (rt ip, dests ip) of (None, x) \<Rightarrow> None | (Some (a, ab, ad, af, bb), None) \<Rightarrow> Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) \<Rightarrow> Some (rsn, ab, Aodv_Basic.inv, af, bb)"
insert_ident: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "\<not> x \<in> A"   and "\<not> x \<in> B" shows "(insert x A = insert x B) = (A = B)"
inj_map_strong: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c"   and fa :: "'a \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>z \<in> range x; za \<in> range xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "f \<circ> x = fa \<circ> xa" shows "x = xa"
sbelow_rel_internal: fixes A :: "('c \<times> 'd) set" shows "sbelow_rel A = \<langle>A\<rangle>sctn_rel O br sbelow_halfspace top"
simps: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and base :: "'d"   and elements :: "'c list"   and n :: "nat"   and head :: "'c"   and tail :: "'c iarray"   and prevs :: "(nat \<times> 'd) list" shows "my_subseqs.invariant f base elements n (head, tail, prevs) = (if elements = [] then prevs = [] else head = hd elements \<and> tail = IArray (tl elements) \<and> set prevs = my_subseqs.set_prevs f base (tl elements) n)"
ESem_fresh_cong_subset: fixes e :: "exp"   and S :: "Vars.var set"   and \<rho> :: "Vars.var \<Rightarrow> Value"   and \<rho>' :: "Vars.var \<Rightarrow> Value" assumes "fv e \<subseteq> S"   and "\<rho> f|` S = \<rho>' f|` S" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
product_in_first_factor: fixes x :: "i"   and y :: "i" assumes "O x y" shows "P (x \<otimes> y) x"
star_left_induct_mult: fixes x :: "'a"   and y :: "'a" assumes "x * y \<le> y" shows "x\<^sup>\<star> * y \<le> y"
not_p2: fixes t :: "real" assumes "0 < t"   and "t \<le> h" shows "flow0 x0 t \<notin> p2"
conv_abscissa_PInf_iff: fixes f :: "'a fds" shows "(conv_abscissa f = \<infinity>) = (\<forall>s. \<not> fds_converges f s)"
derived_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "derived G H = generate G (derived_set G H)"
sum: fixes f :: "'q \<Rightarrow> 'r"   and g :: "'s \<Rightarrow> 'q"   and A :: "'s set" assumes "additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
dX_range: fixes X :: "'a set" shows "\<forall>d. ManiPref'.dX X d \<subseteq> {x. Xd x = d}"
that: fixes as' :: "'edge list" assumes "preds (slice_kinds S as') [cf]"   and "\<forall>V\<in>Use m'. state_val (transfers (slice_kinds S as') [cf]) V = state_val (transfers (kinds as) [cf]) V"   and "slice_edges S [] as = slice_edges S [] as'"   and "transfers (kinds as) [cf] \<noteq> []"   and "m -as'\<rightarrow>\<^sub>\<surd>* m'" shows "thesis"
r2_def: shows "r2 = map_prod (map_prod id Link) (map_prod id Link) ` inv_image {(l, l'). pointsTo l' = Link l} snd"
delayedR_correct_aux: fixes u :: "'c \<Rightarrow> real"   and I :: "'c \<Rightarrow> Regions.intv"   and r :: "('c \<times> 'c) set"   and t :: "real" assumes "u \<in> Regions.region \<X> I r"   and "Regions.valid_region \<X> k I r"   and "\<forall>c\<in>\<X>. \<not> Regions.isConst (I c)"   and "\<forall>c\<in>\<X>. Regions.isIntv (I c) \<longrightarrow> (u \<oplus> t) c < real (intv_const (I c) + 1)"   and "0 \<le> t" shows "u \<oplus> t \<in> Regions.region \<X> I r"
simps_1: fixes I :: "'f \<Rightarrow> 'g option"   and P :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and t :: "'f"   and x :: "'h"   and m :: "'i" shows "upd_inv I P (NewThread t x m) = I(t \<mapsto> SOME i. P i t x m)"
natural_isomorphism_\<rr>: shows "natural_isomorphism (\<cdot>) (\<cdot>) R local.map \<rr>"
arr_char: fixes f :: "'a" shows "hom.arr f = \<guillemotleft>f : trg e \<rightarrow> trg (trg f)\<guillemotright>"
is_RB_in_insertI: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_RB_in d rword G u"   and "u \<prec>\<^sub>t lt g" shows "is_RB_in d rword (insert g G) u"
assms_7: shows "\<forall>x\<in>S. finite (cl_ecl x)"
integration_by_parts_interior_strong: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'b"   and y :: "'c" assumes "bounded_bilinear prod"   and "finite s"   and "a \<le> b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "\<And>x. x \<in> {a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"   and "((\<lambda>x. prod (f x) (g' x)) has_integral prod (f b) (g b) - prod (f a) (g a) - y) {a..b}" shows "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
rep_eq: fixes x :: "uint32"   and xa :: "nat"   and xb :: "bool" shows "Rep_uint32 (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (Rep_uint32 x) xa xb"
sr: shows "s \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"
env_restr_belowD: fixes S :: "'c set"   and m\<^sub>1 :: "'c \<Rightarrow> 'd"   and m\<^sub>2 :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "m\<^sub>1 f|` S \<sqsubseteq> m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x \<sqsubseteq> m\<^sub>2 x"
q2_def: shows "q2 \<equiv> [:1, 1:]"
dim_mono: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and V :: "'e set"   and W :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "V \<subseteq> module.span scale W" shows "vector_space.dim scale V \<le> vector_space.dim scale W"
ivl_subset: fixes i :: "'c"   and j :: "'c"   and m :: "'c"   and n :: "'c" shows "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> n)"
gso_dim: fixes i :: "nat" assumes "i < length fs" shows "dim_vec (gso i) = n"
igVarIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igVarIPresIGWls MOD \<equiv> \<forall>xs x. igWls MOD (asSort xs) (igVar MOD xs x)"
step: fixes l :: "'a llist"   and a :: "'a" assumes "l \<in> A\<^sup>\<clubsuit>"   and "P l"   and "a \<in> A" shows "P (a ## l)"
assms_5: shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M"
FieldI2: fixes i :: "'b"   and j :: "'b"   and R :: "('b \<times> 'b) set" assumes "(i, j) \<in> R" shows "j \<in> Field R"
OstrowskiTr4: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a"   and t :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "t \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a s)"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a t)" shows "0 < v (1\<^sub>r \<plusminus> -\<^sub>a s \<cdot>\<^sub>r t)"
FAssRedsThrow1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>"
simps: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf"   and a1 :: "'a spmf"   and a2 :: "'g spmf" shows "trace_core_closure core1 core2 E IA IU p q a1 a2 = (\<exists>tr. a1 = trace_core_aux core1 p tr \<and> a2 = trace_core_aux core2 q tr \<and> set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV)"
assms: fixes x :: "'a" shows "f \<le> SPEC (P x)"
resChainAppend: fixes xvec :: "name list"   and yvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "\<lparr>\<nu>*(xvec @ yvec)\<rparr>P = \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>*yvec\<rparr>P"
runit_agreement: fixes a :: "'a" assumes "ide a" shows "runit a = C.runit a"
arr_char: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F = (bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Dom F) \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Dom F) \<and> bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Cod F) \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B (concrete_category.Cod F) \<and> bicategorical_language.Src src\<^sub>B (concrete_category.Dom F) = bicategorical_language.Src src\<^sub>B (concrete_category.Cod F) \<and> bicategorical_language.Trg trg\<^sub>B (concrete_category.Dom F) = bicategorical_language.Trg trg\<^sub>B (concrete_category.Cod F) \<and> partial_magma.in_hom V\<^sub>B (concrete_category.Map F) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Dom F)) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Cod F)) \<and> F \<noteq> concrete_category.Null)"
c_le: shows "c < prod_list (nths (dims (dematricize rmodes A ds)) (- rmodes))"
N_vector_pp: fixes x :: "'a" shows "N (x * top) = - (x * top) '"
index_of_correct: fixes x :: "'c"   and y :: "'c list" assumes "x \<in> set y" shows "y ! index_of x y = x"
vsv_axioms: fixes a :: "V"   and b :: "V" shows "vsv (set {\<langle>a, b\<rangle>})"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "inf b (inf a c) = inf a (inf b c)"
FreeGroupD_transfer: fixes xs :: "('b \<times> bool) list"   and S :: "'b set" assumes "proper_signed_list xs"   and "Abs_freeword xs \<in> FreeGroup S" shows "order.greater_eq S (fst ` set xs)"
ran: shows "range f = {i'. \<forall>j'>i'. j' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i'}"
keys_homogenizeE_alt: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys (homogenize x p)" obtains q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" and t' :: "'b \<Rightarrow>\<^sub>0 nat" where "q \<in> hom_components p" and "t' \<in> keys q" and "t = monomial (poly_deg p - poly_deg q) x + t'"
assms: shows "w1 @ w2 \<in> LS M q"
is_at_def: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and v' :: "nat" shows "is_at p i branch v v' \<equiv> \<exists>ps. branch !. v = Some (ps, i) \<and> ps !. v' = Some p"
projection_on_union: fixes l :: "'a list"   and Y :: "'a set"   and X :: "'a set" assumes "l \<upharpoonleft> Y = []" shows "l \<upharpoonleft> (X \<union> Y) = l \<upharpoonleft> X"
axioms_4: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme" assumes "bisimulation R A B" shows "simulation (R\<inverse>) B A"
wp_under_wlp_def: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "wp_under_wlp prog \<equiv> \<forall>P. unitary P \<longrightarrow> wp prog P \<tturnstile> wlp prog P"
cycleD: fixes as :: "atom list"   and "is" :: "nat list"   and xs :: "'b list" assumes "cycle as is" shows "\<not> DLO.I (list_conj (map Atom as)) xs"
residue_add: fixes p :: "int"   and k :: "nat"   and x :: "int"   and y :: "int" assumes "padic_integers p" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = (x + y) mod p ^ k"
at_eq_sup_left_right: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder_topology less_eq less open" shows "topological_space.at_within open x UNIV = sup (topological_space.at_within open x (ord.lessThan less x)) (topological_space.at_within open x (ord.greaterThan less x))"
from_nat_neq_rows: fixes i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "i < nrows A"   and "k < nrows A"   and "i \<noteq> k" shows "from_nat i \<noteq> from_nat k"
has_densityD_2: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M N f" shows "M = density N f"
kth_obs_stable: fixes \<pi> :: "nat \<Rightarrow> 'n"   and l :: "nat"   and j :: "nat"   and k :: "nat" assumes "is_kth_obs \<pi> l j"   and "k < l" shows "\<exists>i. is_kth_obs \<pi> k i"
pair_memberid_def: fixes p :: "'b \<times> 'c" shows "memberid p = memberid (snd p)"
assms_5: shows "p = 2 ^ (k - 1) * 3 - 1"
linear_sum: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and g :: "'e \<Rightarrow> ('b, 'c) vec"   and S :: "'e set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
indeps_2: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a" assumes "myss x y" shows "y \<bowtie> x"
Rep_uint16_numeral: fixes n :: "num" shows "Rep_uint16 (numeral n) = numeral n"
meet_imp_conversion: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>\<up> \<subseteq> r\<^sup>\<leftrightarrow>\<^sup>*"
com_rel_elim_cases_10: fixes c :: "('a, 'b, 'c) com"   and P :: "bool" assumes "(c, Throw) \<in> com_rel" shows "P"
N_dist: shows "\<forall>n\<ge>N. dist (norm (f (Suc n)) / norm (f n)) c < (1 - c) / 2"
numeral_times_minus_swap: fixes w :: "num"   and x :: "'a" shows "numeral w * - x = x * - numeral w"
elims: fixes x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree"   and y :: "nat" assumes "T_merge x xa = y" obtains t :: "('b \<times> nat) tree" where "x = \<langle>\<rangle>" and "xa = t" and "y = 1"   | v :: "('b \<times> nat) tree" and va :: "'b \<times> nat" and vb :: "('b \<times> nat) tree" where "x = \<langle>v, va, vb\<rangle>" and "xa = \<langle>\<rangle>" and "y = 1"   | l1 :: "('b \<times> nat) tree" and a1 :: "'b" and n1 :: "nat" and r1 :: "('b \<times> nat) tree" and l2 :: "('b \<times> nat) tree" and a2 :: "'b" and n2 :: "nat" and r2 :: "('b \<times> nat) tree" where "x = \<langle>l1, (a1, n1), r1\<rangle>" and "xa = \<langle>l2, (a2, n2), r2\<rangle>" and "y = (let t2 = \<langle>l2, (a2, n2), r2\<rangle>; t1 = \<langle>l1, (a1, n1), r1\<rangle> in (if a1 \<le> a2 then T_merge r1 t2 else T_merge t1 r2) + 1)"
P_nonzero_entry: fixes j :: "nat" assumes "j < n" shows "\<exists>i<n. P $$ (i, j) \<noteq> 0"
fmupd_alt_def: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) fmap" shows "m(k $$:= v) = m ++\<^sub>f fmap_of_list [(k, v)]"
start_fin_in_steps_atom: fixes a :: "'b"   and w :: "'b list" shows "((start (atom a), [False]) \<in> steps (atom a) w) = (w = [a])"
WTrtWhile: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and c :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : Boolean"   and "P,E,h,sh \<turnstile> c : T" shows "P,E,h,sh \<turnstile> while (e) c : Void"
ord_of_nat_in_Vset: fixes \<alpha> :: "V"   and a :: "nat" assumes "\<Z> \<alpha>" shows "a\<^sub>\<nat> \<in>\<^sub>\<circ> Vset \<alpha>"
star_one_def: shows "1 \<equiv> star_of (1::'d)"
Pd_linear: shows "\<forall>d. Linear_order (Pd_shuffle_to_top ds' Y d)"
Inf_FL_to_Inf_F: fixes \<iota>\<^sub>F\<^sub>L :: "('f \<times> 'l) inference" assumes "\<iota>\<^sub>F\<^sub>L \<in> Inf_FL" shows "Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L)) \<in> Inf_F"
simps_6: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; delete\<langle>t,s\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. \<lbrakk>M; D - {(t, s) \<cdot>\<^sub>p \<I>}; S\<rbrakk>\<^sub>s \<I>)"
Obj_Src: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Obj (E.Src t)"
iso_assoc: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "category.iso V (\<a> f g h)"
commutative_squareE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "category C"   and "category.commutative_square C f g h k" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.arr C h" and "partial_magma.arr C k" and "partial_magma.cod C f = partial_magma.cod C g" and "partial_magma.dom C h = partial_magma.dom C k" and "partial_magma.dom C f = partial_magma.cod C h" and "partial_magma.dom C g = partial_magma.cod C k" and "C f h = C g k"
divmod: shows "x = 2 * (x div 2) + x mod 2"
exists_split_characterisations_4: shows "(\<exists>x. linear_orderable_1 x) = transitively_orientable (- (1::'a))"
simps_3: fixes p :: "perm"   and \<phi> :: "'c" shows "p \<bullet> tPred \<phi> = tPred (p \<bullet> \<phi>)"
init_locs_Nat_add: fixes s :: "state" shows "init_locs Nat add s = s"
vprod_size_inj: fixes A :: "(val \<times> val) fset" shows "inj_on vprod_size (fset A)"
mono: fixes c :: "'t"   and a :: "'t"   and d :: "'t"   and b :: "'t" assumes "c \<preceq>\<^sub>t a"   and "d \<preceq>\<^sub>t b" shows "ord_term_lin.max c d \<preceq>\<^sub>t ord_term_lin.max a b"
subst_apply_pairs_pset_subst: fixes F :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) list"   and \<theta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "set (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = set F \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
r_le_rtrancl: fixes S :: "('a \<times> 'a) set" shows "S \<subseteq> S\<^sup>*"
4: shows "(\<lambda>i. LBINT x. indicat_real {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> C"
bv_extend_rem_initial: fixes w :: "bit list"   and b :: "bit" shows "bv_extend (length w) b (rem_initial b w) = w"
assms_1: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
coherent_Lunit'_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "coherent \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[a\<^bold>]"
tabulate_alt: fixes f :: "nat \<Rightarrow> 'b"   and x :: "nat"   and n :: "nat" shows "tabulate f x n = map f [x..<x + n]"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<sim>\<^sub>l \<zero>"
bind_option_def: fixes fail :: "'c"   and x :: "'d option"   and f :: "'d \<Rightarrow> 'c" shows "monad.bind_option fail x f = (case x of None \<Rightarrow> fail | Some x' \<Rightarrow> f x')"
sorted2: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" shows "sorted (x # y # zs) = (x \<le> y \<and> sorted (y # zs))"
Gen_Shleg_0_left: fixes n :: "nat"   and x :: "real" shows "Gen_Shleg n 0 x = x ^ n * (1 - x) ^ n"
quotient_eq: shows "{f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // range_permutation A B = {F \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B. univ (\<lambda>f. inj_on f A) F}"
higher_differentiable_on_Pair': fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'd"   and k :: "nat"   and g :: "'c \<Rightarrow> 'e" assumes "open S"   and "open T"   and "higher_differentiable_on S f k"   and "higher_differentiable_on T g k" shows "higher_differentiable_on (S \<times> T) (\<lambda>x. (f (fst x), g (snd x))) k"
rep_eq: fixes x :: "nat"   and xa :: "float" shows "real_of_float (float_round_up x xa) = truncate_up x (real_of_float xa)"
gt_embedding_step_property: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "t >\<^sub>t s"
right_mult_zero_mat': fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "dim_col A = n" shows "A * 0\<^sub>m n nc = 0\<^sub>m (dim_row A) nc"
elims: fixes x :: "char"   and xa :: "char list"   and y :: "char list" assumes "string_cons x xa = y" obtains c :: "char" and s :: "char list" where "x = c" and "xa = s" and "y = (if MAX_STR \<le> length s then s else c # s)"
lossless_sample_primes: shows "lossless_spmf sample_primes"
finite_subset_chain: fixes Y :: "nat \<Rightarrow> 'a set"   and S :: "'a set" assumes "chain Y"   and "S \<subseteq> \<Union> (range Y)"   and "finite S" shows "\<exists>i. S \<subseteq> Y i"
mono_bfilter: fixes S :: "'a Abs_State.st option"   and S' :: "'a Abs_State.st option"   and b :: "bexp"   and r :: "bool" assumes "S \<sqsubseteq> S'" shows "bfilter b r S \<sqsubseteq> bfilter b r S'"