hcju/isabelleps huggingface dataset:
module_hom_scale_left: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "module scale" shows "module_hom (*) scale (\<lambda>r. scale r x)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f" obtains "\<And>x. d_OUT f x \<le> weight \<Gamma> x" and "\<And>x. d_IN f x \<le> weight \<Gamma> x" and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT f x \<le> d_IN f x" and "\<And>a. a \<in> A \<Gamma> \<Longrightarrow> d_IN f a = 0" and "\<And>b. b \<in> B \<Gamma> \<Longrightarrow> d_OUT f b = 0" and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<Longrightarrow> f e = 0"
matches_in_names: fixes v :: "'c list"   and \<phi> :: "'c MFOTL.formula"   and x :: "char list \<times> 'c list" assumes "MFOTL.matches v \<phi> x" shows "fst x \<in> names \<phi>"
combine_with_key_code: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "combine_with_key f (AList_Mapping.Mapping xs) (AList_Mapping.Mapping ys) = Mapping.tabulate (remdups (map fst xs @ map fst ys)) (\<lambda>x. the (combine_options (f x) (map_of xs x) (map_of ys x)))"
alw_ev_scons: fixes P :: "'b stream \<Rightarrow> bool"   and a :: "'b"   and w :: "'b stream" shows "alw (ev P) (a ## w) = alw (ev P) w"
obtain_ideal_generated_pair: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<in> ideal_generated {a, b}" obtains p :: "'a" and q :: "'a" where "p * a + q * b = c"
sum_bounds_lt_plus1: fixes f :: "nat \<Rightarrow> 'a"   and mm :: "nat" shows "(\<Sum>k<mm. f (Suc k)) = sum f {1..mm}"
strip_iter_widen: fixes f :: "'a acom \<Rightarrow> 'a acom"   and C :: "'a acom"   and C' :: "'a acom" assumes "\<forall>C. strip (f C) = strip C"   and "iter_widen f C = Some C'" shows "strip C' = strip C"
boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "max b c \<le> a" obtains "b \<le> a" and "c \<le> a"
integrable_p: fixes y :: "'s"   and n :: "nat" shows "integrable (measure_pmf N) (\<lambda>x. p x y n)"
up_repr_le: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "deg R p \<le> n"   and "p \<in> carrier P" shows "(\<Oplus>\<^bsub>P\<^esub>i\<in>{..n}. monom P (coeff P p i) i) = p"
oinvariant_weakenD: fixes A :: "('e \<times> 'f, 'g) automaton"   and S' :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U' :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<times> 'f \<Rightarrow> bool"   and \<sigma> :: "'e"   and p :: "'f"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "A \<Turnstile> (S', U' \<rightarrow>) P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"   and "\<And>s s'. U s s' \<Longrightarrow> U' s s'" shows "P (\<sigma>, p)"
if\<^sub>C_cond_cong: fixes f :: "'a control_state_scheme \<Rightarrow> bool"   and \<sigma> :: "'a control_state_scheme"   and g :: "'a control_state_scheme \<Rightarrow> bool"   and c :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and d :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option" assumes "f \<sigma> = g \<sigma>" shows "(_if_SECLEAN f c d) \<sigma> = (_if_SECLEAN g c d) \<sigma>"
distinct_17: fixes x61 :: "'b formula"   and x62 :: "'b formula" shows "\<bottom> \<noteq> (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
plus_monotone_strict_left: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and u :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm_strict s t" shows "lex_pm_strict (u + s) (u + t)"
space_measure_spmf: fixes p :: "'h spmf" shows "space (measure_spmf p) = UNIV"
un: fixes as :: "'a list"   and bs :: "'b" shows "set_of (un as bs) = set as \<union> set_of bs"
surjective: fixes r :: "'a tg_scheme" shows "r = \<lparr>tg_mut = tg_mut r, tg_t = tg_t r, \<dots> = tg.more r\<rparr>"
v_dom: fixes i :: "nat" shows "vertices (S i) = Domain (g i)"
try_swap_eq: fixes p :: "'person"   and c\<^sub>1 :: "'color"   and c\<^sub>2 :: "'color"   and w :: "'person \<Rightarrow> 'color"   and x :: "'person"   and w' :: "'person \<Rightarrow> 'color" shows "(local.try_swap p c\<^sub>1 c\<^sub>2 w x = local.try_swap p c\<^sub>1 c\<^sub>2 w' x) = (w x = w' x)"
zero_closed: shows "0\<^sub>v n \<in> Rn"
has_derivative_cnj: fixes g :: "'b \<Rightarrow> complex"   and g' :: "'b \<Rightarrow> complex"   and F :: "'b filter" assumes "(g has_derivative g') F" shows "((\<lambda>x. cnj (g x)) has_derivative (\<lambda>x. cnj (g' x))) F"
prems_2: shows "\<forall>m'\<in>set (a # ms). m' x = Some y \<or> m' x = None"
OclOr_true1: fixes Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(true or Y) = true"
vproduct_vsubset_VPow: fixes I :: "V"   and A :: "V \<Rightarrow> V" shows "vproduct I A \<subseteq>\<^sub>\<circ> VPow (I \<times>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I))"
llist_of_eq_LCons_conv: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c llist" shows "(llist_of xs = LCons y ys) = (\<exists>xs'. xs = y # xs' \<and> ys = llist_of xs')"
not_Cons_self2: fixes x :: "'c"   and xs :: "'c list" shows "x # xs \<noteq> xs"
master3: fixes p' :: "real" assumes "g \<in> \<Theta>(\<lambda>x. real x powr p')"   and "(\<Sum>i<k. as ! i * bs ! i powr p') < 1" shows "f \<in> \<Theta>(\<lambda>x. real x powr p')"
commute_sym: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "commute R S" shows "commute S R"
add_mset_lt_right_lt: fixes A :: "'b multiset"   and B :: "'b multiset"   and a :: "'b" assumes "A < B" shows "add_mset a A < add_mset a B"
finfun_upd_apply: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a"   and a' :: "'b" shows "(f(a $:= b)) $ a' = (if a = a' then b else f $ a')"
Farkas'_Lemma_set_sum: fixes cs :: "rat le_constraint set" assumes "lec_rel ` cs \<subseteq> {Leq_Rel}"   and "finite cs" shows "(\<nexists>v. \<forall>c\<in>cs. v \<Turnstile>\<^sub>l\<^sub>e c) = (\<exists>C const. (\<forall>c\<in>cs. 0 \<le> C c) \<and> (\<Sum>c\<in>cs. Le_Constraint Leq_Rel (C c *R lec_poly c) (C c *R lec_const c)) = Le_Constraint Leq_Rel 0 const \<and> const < 0)"
blue_dfs_def: fixes G :: "('a, 'b) b_graph_rec_scheme" shows "blue_dfs G \<equiv> NDFS_SI_Statistics.start_nres \<bind> (\<lambda>_. FOREACH\<^sub>C (g_V0 G) (\<lambda>(uu_, uu_, cyc). cyc = NO_CYC) (\<lambda>v0 (blues, reds, uu_). if v0 \<notin> blues then REC\<^sub>T (\<lambda>D (blues, reds, onstack, s). let blues = insert s blues; onstack = insert s onstack; s_acc = s \<in> bg_F G in NDFS_SI_Statistics.vis_blue_nres \<bind> (\<lambda>_. FOREACH\<^sub>C (g_E G `` {s}) (\<lambda>(uu_, uu_, uu_, cyc). cyc = NO_CYC) (\<lambda>t (blues, reds, onstack, cyc). if t \<in> onstack \<and> (s_acc \<or> t \<in> bg_F G) then RETURN (blues, reds, onstack, init_wit_blue_early s t) else if t \<notin> blues then D (blues, reds, onstack, t) \<bind> (\<lambda>(blues, reds, onstack, cyc). RETURN (blues, reds, onstack, prep_wit_blue s cyc)) else NDFS_SI_Statistics.match_blue_nres \<bind> (\<lambda>_. RETURN (blues, reds, onstack, cyc))) (blues, reds, onstack, NO_CYC) \<bind> (\<lambda>(blues, reds, onstack, cyc). (if cyc = NO_CYC \<and> s_acc then red_dfs (g_E G) onstack reds s \<bind> (\<lambda>(reds, rcyc). RETURN (reds, init_wit_blue s rcyc)) else RETURN (reds, cyc)) \<bind> (\<lambda>(reds, cyc). let onstack = onstack - {s} in RETURN (blues, reds, onstack, cyc))))) (blues, reds, {}, v0) \<bind> (\<lambda>(blues, reds, uu_, cyc). RETURN (blues, reds, cyc)) else RETURN (blues, reds, NO_CYC)) ({}, {}, NO_CYC) \<bind> (\<lambda>(uu_, uu_, cyc). NDFS_SI_Statistics.stop_nres \<bind> (\<lambda>_. RETURN (extract_res cyc))))"
dom_subid_aux2'': fixes x :: "'a"   and y :: "'a" shows "x \<cdot> d y \<le> x"
2: fixes n :: "nat" shows "Sigma_Algebra.measure lebesgue (\<Union> (S ` {..n})) \<le> B"
one_V_def: shows "1 \<equiv> ZFC_in_HOL.succ []\<^sub>\<circ>"
sum_moebius_mu_aux1: shows "(\<lambda>x. M x / x - H x / (x * ln x)) \<in> O(\<lambda>x. 1 / ln x)"
euclid_B_cart2_common_line: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r"   and "B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r" shows "\<exists>l. proj2_incident p l \<and> proj2_incident q l \<and> proj2_incident r l"
cod_char: fixes f :: "'a \<times> 'a" shows "VxV.cod f = (if arr (fst f) \<and> arr (snd f) then (cod (fst f), cod (snd f)) else (null, null))"
iso_runit: fixes f :: "'a" assumes "B.ide f" shows "B.iso \<r>[f]"
int_of_nat: fixes m :: "nat" shows "of_nat m = int_of_nat m"
intros_17: fixes c' :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and b :: "expr" assumes "c' \<turnstile> n -et\<rightarrow> (_Exit_)"   and "n \<noteq> (_Entry_)" shows "while (b) c' \<turnstile> n \<oplus> 2 -et\<rightarrow> (_ 0 _)"
openin_open_trans: fixes T :: "'c set"   and S :: "'c set" assumes "openin (top_of_set T) S"   and "open T" shows "open S"
to_gbg_ext_def: fixes m :: "'b" shows "to_gbg_ext m \<equiv> \<lparr>g_V = V, g_E = E, g_V0 = V0, gbg_F = F, \<dots> = m\<rparr>"
append_id_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i1 :: "'a"   and i2 :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "Deliver (i1, Append i2 e) \<in> set xs" shows "i1 = i2"
rep_eq: fixes x :: "'a"   and xa :: "nat" shows "poly.coeff (Polynomial.monom x xa) = (\<lambda>n. if xa = n then x else (0::'a))"
sup_ty_opt_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty err" shows "P \<turnstile> T \<le>\<^sub>\<top> T"
closed_Collect_neg: fixes P :: "'a \<Rightarrow> bool" assumes "open {x. P x}" shows "closed {x. \<not> P x}"
subset_maxchain_max: fixes A :: "'c set set"   and C :: "'c set set"   and X :: "'c set" assumes "subset.maxchain A C"   and "X \<in> A"   and "\<Union> C \<subseteq> X" shows "\<Union> C = X"
that: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set ops"   and "op' = \<phi>\<^sub>O \<Psi> op" shows "thesis"
simps: fixes a1 :: "'c"   and a2 :: "'d"   and r :: "('c \<times> 'e) set"   and s :: "('e \<times> 'd) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
continuous_map_pairwise: fixes Z :: "'d topology"   and X :: "'e topology"   and Y :: "'f topology"   and f :: "'d \<Rightarrow> 'e \<times> 'f" shows "continuous_map Z (prod_topology X Y) f = (continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f))"
symD: fixes r :: "('b \<times> 'b) set"   and b :: "'b"   and a :: "'b" assumes "sym r"   and "(b, a) \<in> r" shows "(a, b) \<in> r"
coset_assoc: fixes x :: "'a set"   and y :: "'a set"   and H :: "'a set set" assumes "x \<in> carrier (G Mod N)"   and "y \<in> carrier (G Mod N)"   and "H \<subseteq> carrier (G Mod N)" shows "x <#\<^bsub>G Mod N\<^esub> (H #>\<^bsub>G Mod N\<^esub> y) = x <#\<^bsub>G Mod N\<^esub> H #>\<^bsub>G Mod N\<^esub> y"
SdlHPPRc: shows "SdlHPPB d PP Rc"
ctr: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and s1 :: "'e stream"   and s2 :: "'f stream" shows "smap2 f s1 s2 = f (shd s1) (shd s2) ## smap2 f (stl s1) (stl s2)"
of_nat_eq_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = zero) = (m = 0)"
Ri_spec: fixes x :: "'b"   and y :: "'c" shows "x \<le> Ri x y y"
inter_image_empty_iff: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "(X \<inter> {p. f p = x} = {}) = (x \<notin> f ` X)"
optimal_paths: fixes paths' :: "'a list set"   and P_new' :: "'a list" assumes "ProofStepInduct G v0 v1 paths' P_new' sep_size" shows "H.distance (last P_new) v1 \<le> H.distance (last P_new') v1"
submanifold_atlasI: fixes c :: "('a, 'e) chart" assumes "c \<in> src.atlas" shows "restrict_chart S c \<in> sub.atlas"
m_inv_monoid_consistent: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> Units (G\<lparr>carrier := H\<rparr>)"   and "submonoid H G" shows "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x = inv x"
gs_gso: fixes i :: "nat" assumes "i < k" shows "gs ! i = gso i"
d2: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "d (p \<sqinter> x) \<le> p"
is_unit_right_imp_coprime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "algebraic_semidom.coprime times one a b"
of_match_field_typeset: fixes field_match :: "of_match_field" shows "field_match \<in> {IngressPort (case field_match of IngressPort s \<Rightarrow> s), EtherSrc (case field_match of EtherSrc s \<Rightarrow> s), EtherDst (case field_match of EtherDst s \<Rightarrow> s), EtherType (case field_match of EtherType s \<Rightarrow> s), VlanId (case field_match of VlanId s \<Rightarrow> s), VlanPriority (case field_match of VlanPriority s \<Rightarrow> s), IPv4Src (case field_match of IPv4Src s \<Rightarrow> s), IPv4Dst (case field_match of IPv4Dst s \<Rightarrow> s), IPv4Proto (case field_match of IPv4Proto s \<Rightarrow> s), L4Src (case field_match of L4Src p m \<Rightarrow> p) (case field_match of L4Src p m \<Rightarrow> m), L4Dst (case field_match of L4Dst p m \<Rightarrow> p) (case field_match of L4Dst p m \<Rightarrow> m)}"
abs_pred_nterm_def: fixes P :: "nterm \<Rightarrow> bool"   and t :: "nterm" shows "abs_pred P t = (\<forall>t' x. t = (\<Lambda>\<^sub>n x. t') \<longrightarrow> P t' \<longrightarrow> P t)"
right_neutral: fixes a :: "'c multiset" shows "a \<union># {#} = a"
atom_eqvt: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and atom :: "'a \<Rightarrow> atom"   and p :: "perm"   and a :: "'a" assumes "class.at_base permute atom" shows "p \<bullet> atom a = atom (permute p a)"
lset_ldropn_subset: fixes n :: "nat"   and xs :: "'c llist" shows "lset (ldropn n xs) \<subseteq> lset xs"
pp_comp_semi_commute: fixes x :: "'a"   and y :: "'a" shows "- - x * y \<le> - - (x * y)"
parallelI: fixes xs :: "'b list"   and ys :: "'b list" assumes "\<not> prefix xs ys"   and "\<not> prefix ys xs" shows "xs \<parallel> ys"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<longrightarrow>cbv x2"   and "\<And>v x t y. \<lbrakk>x \<sharp> y; val v; x \<sharp> v\<rbrakk> \<Longrightarrow> P y (APP LAM [x].t v) (t[x::=v])"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (APP t t2) (APP t' t2)"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (APP t2 t) (APP t2 t')"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y t -- t2 t' -- t2"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y t2 -- t t2 -- t'"   and "\<And>n1 n2 y. P y NUM n1 -- NUM n2 (NUM (n1 - n2))"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y t ++ t2 t' ++ t2"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y t2 ++ t t2 ++ t'"   and "\<And>n1 n2 y. P y NUM n1 ++ NUM n2 (NUM (n1 + n2))"   and "\<And>t t' t1 t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (IF t t1 t2) (IF t' t1 t2)"   and "\<And>t1 t2 y. P y (IF TRUE t1 t2) t1"   and "\<And>t1 t2 y. P y (IF FALSE t1 t2) t2"   and "\<And>x t y. x \<sharp> y \<Longrightarrow> P y FIX [x].t (t[x::=FIX [x].t])"   and "\<And>t t' y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (ZET t) (ZET t')"   and "\<And>y. P y (ZET (NUM 0)) TRUE"   and "\<And>n y. 0 < n \<Longrightarrow> P y (ZET (NUM n)) FALSE"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (EQI t t2) (EQI t' t2)"   and "\<And>t t' t2 y. \<lbrakk>t \<longrightarrow>cbv t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (EQI t2 t) (EQI t2 t')"   and "\<And>n y. P y (EQI (NUM n) (NUM n)) TRUE"   and "\<And>n1 n2 y. n1 \<noteq> n2 \<Longrightarrow> P y (EQI (NUM n1) (NUM n2)) FALSE" shows "P z x1 x2"
assms_1: shows "tensor_from_lookup ds e\<^sub>1 = tensor_from_lookup ds e\<^sub>2"
dims_matricize_2: fixes rmodes :: "nat set"   and T :: "'m tensor" shows "dim_col (matricize rmodes T) = prod_list (nths (dims T) (- rmodes))"
g: shows "split_block (four_block_mat A B C D) (dim_row A) (dim_col A) = (A1, A2, A3, A4)"
obj_simps_1: fixes a :: "'a" assumes "obj a" shows "arr a"
Exit_source: fixes a :: "'a" assumes "valid_edge a"   and "sourcenode a = Exit" shows "False"
square2_Suc: fixes n :: "nat"   and i :: "nat"   and j :: "nat" shows "square2 (Suc n) i j = square2 n i j \<union> square2 n (2 ^ n + i) j \<union> square2 n i (2 ^ n + j) \<union> square2 n (2 ^ n + i) (2 ^ n + j)"
assms_4: shows "geodesic_segment_between G (f a) (f b)"
k5: shows "K\<^bsub>5\<^esub> (with_proj G)"
canceling_indep: fixes i :: "nat"   and l :: "(bool \<times> 'c) list"   and a :: "(bool \<times> 'c) list"   and j :: "nat"   and b :: "(bool \<times> 'c) list" assumes "cancels_to_1_at i l a"   and "cancels_to_1_at j l b"   and "Suc i < j" obtains c :: "(bool \<times> 'c) list" where "cancels_to_1_at (j - 2) a c" and "cancels_to_1_at i b c"
bg_in1: shows "BG \<in> \<G>_F B"
simps_1: shows "andfold_MatchExp [] = MatchAny"
map\<^sub>0_simp: fixes a :: "'b" assumes "B.obj a" shows "map\<^sub>0 a = a"
poly_square_subst_def: fixes f :: "'c poly" shows "poly_square_subst f = poly_of_list (drop_half (coeffs f))"
the_inv_into_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "the_inv_into A f = (\<lambda>x. THE y. y \<in> A \<and> f y = x)"
suffix_unsnoc: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys"   and "xs \<noteq> []" obtains x :: "'c" where "xs = butlast xs @ [x]" and "ys = butlast ys @ [x]"
read_buffer_def: shows "read_buffer \<equiv> (Skip@[control_var +=\<^sub>m AsmNoWrite]) ;; (Skip@[temp +=\<^sub>m AsmNoReadOrWrite]) ;; (temp \<leftarrow> Load buffer) ;; Stmt.If (Eq control_var 0) (low_var \<leftarrow> Load temp) (high_var \<leftarrow> Load temp) ;; (temp \<leftarrow> Const 0) ;; Skip@[temp -=\<^sub>m AsmNoReadOrWrite]"
bitval_bin_last: fixes i :: "int" shows "of_bool (odd i) = i AND 1"
of_hypnat_def: shows "of_hypnat = *f* of_nat"
tpart1_su: fixes V :: "Preliminaries.var set" assumes "tinfinite V" shows "tpart1 V \<subseteq> V"
rec: fixes f1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "old.rec_prod f1 (a, b) = f1 a b"
wf_inst_eq_aux: fixes x :: "variable"   and T :: "PDDL_STRIPS_Semantics.type" assumes "Q x = Some T" shows "objT (f x) \<noteq> None"
valid_index_length: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "length is = length ds"
le_nat_iff: fixes k :: "int"   and n :: "nat" assumes "0 \<le> k" shows "(n \<le> nat k) = (int n \<le> k)"
proves_eq_elim_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq A B"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> A \<longmapsto> B"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<sqinter> b = b)"
deleteMin_def: fixes bq :: "('c, 'd) BsSkewBinomialTree list" shows "deleteMin bq = (let min = getMinTree bq in insertList (filter (\<lambda>t. rank t = 0) (children min)) (meld (rev (filter (\<lambda>t. 0 < rank t) (children min))) (remove1Prio (prio min) bq)))"
word_not_not: fixes x :: "'b word" shows "~~ ~~ x = x"
rtranclp_def: shows "rtranclp \<equiv> \<lambda>r. lfp (\<lambda>p x1 x2. (\<exists>a. x1 = a \<and> x2 = a) \<or> (\<exists>a b c. x1 = a \<and> x2 = c \<and> p a b \<and> r b c))"
monomial_d: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat" assumes "pol_coeff S c" shows "polyn_expr R X d (ext_cf S d c) = snd c 0 \<cdot>\<^sub>r X^\<^bsup>R d\<^esup>"
wf_sees_method_fun: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list"   and mthd' :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C has least M = mthd via Cs"   and "P \<turnstile> C has least M = mthd' via Cs'"   and "wf_prog wf_md P" shows "mthd = mthd' \<and> Cs = Cs'"
Act\<^sub>\<alpha>_eq_iff: fixes \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c) Tree\<^sub>\<alpha>"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" shows "(Act\<^sub>\<alpha> \<alpha>1 t1 = Act\<^sub>\<alpha> \<alpha>2 t2) = (tAct \<alpha>1 (rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> tAct \<alpha>2 (rep_Tree\<^sub>\<alpha> t2))"
VectorSpaceIso_\<phi>: shows "VectorSpaceIso (\<star>\<currency>) (HRepHomSet (\<star>) W) (\<star>\<cdot>) \<phi> (GRepHomSet (\<star>) W)"
substT_substT: fixes s :: "'trm"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "s \<in> atrm"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "y \<notin> FvarsT s" shows "substT (substT s (Var y) x) t y = substT s t x"
qSwap_ident: fixes zs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "X #[[x \<and> x]]_zs = X"
Areflassump: shows "Arefl R0 \<inter> Arefl R \<subseteq> {[]}"
C_def: shows "C = (\<lambda>p. real_of_int (l ^ (n * p)) * (2 * R) ^ (n * p - 1))"
no_overlaps_defeq: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and fe :: "('d, 'f) flow_entry_match list"   and p :: "'e" assumes "no_overlaps \<gamma> fe" shows "OF_same_priority_match2 \<gamma> fe p = OF_priority_match \<gamma> fe p"
minsky_reaching_s_not_t_def: fixes s :: "nat"   and t :: "nat" shows "minsky_reaching_s_not_t s t = {uu_. \<exists>M m vs vs'. uu_ = m \<and> (M, vs) = decode_minsky_state m \<and> ((2, vs), s, vs') \<in> (step M)\<^sup>* \<and> ((2, vs), t, vs') \<notin> (step M)\<^sup>*}"
ZObisL_update: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and c' :: "('test, 'atom, 'choice) cmd"   and d' :: "('test, 'atom, 'choice) cmd"   and n :: "nat" assumes "ZObisL cl dl"   and "c' \<approx>01 d'" shows "ZObisL (cl[n := c']) (dl[n := d'])"
abstract_filter_def: fixes f :: "unit \<Rightarrow> 'a filter" shows "Filter.abstract_filter f = f ()"
nlex_pdevs_def: shows "nlex_pdevs \<equiv> map_fun pdevs_apply Abs_pdev (\<lambda>x n. if Counterclockwise_2D_Arbitrary.lex 0 (x n) then - x n else x n)"