hcju/isabelleps huggingface dataset:
simps_6: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "usubstappt \<sigma> U (Times \<theta> \<eta>) = Timeso (usubstappt \<sigma> U \<theta>) (usubstappt \<sigma> U \<eta>)"
cone_cone: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n" shows "gram_schmidt.cone n (gram_schmidt.cone n Vs) = gram_schmidt.cone n Vs"
closed_Diff: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "open T" shows "topological_space.closed open (S - T)"
weakStatImpE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<lessapprox><Rel> Q" obtains Q' :: "('a, 'b, 'c) psi" and Q'' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'" and "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi>" and "\<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q''" and "(\<Psi> \<otimes> \<Psi>', P, Q'') \<in> Rel"
ag_pOp_add_l: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "c \<plusminus>\<^bsub>A\<^esub> a = c \<plusminus>\<^bsub>A\<^esub> b"
B: fixes s :: "'a event list"   and t :: "'a event set" assumes "s \<notin> D P"   and "s \<notin> D S" shows "(s \<noteq> [] \<and> ((s, t) \<in> F P \<or> (s, t) \<in> F S) \<longrightarrow> (s, t) \<in> F Q \<or> (s, t) \<in> F S) \<and> (s = [] \<and> tick \<notin> t \<and> ([tick] \<in> T P \<or> [tick] \<in> T S) \<longrightarrow> ([], t) \<in> F Q \<and> ([], t) \<in> F S \<or> [] \<in> D Q \<or> [tick] \<in> T Q \<or> [tick] \<in> T S)"
R2_1: shows "pmf (sds R2) a = 1 / 2"
nhop_update_changed: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "update rt dip (dsn, dsk, flg, hops, sip, {}) \<noteq> rt" shows "the (nhop (update rt dip (dsn, dsk, flg, hops, sip, {})) dip) = sip"
IH: fixes as_1 :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as_2 :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap" assumes "as_1 @ as_2 = as"   and "sat_precond_as s as" shows "sat_precond_as s as_1 \<and> sat_precond_as (exec_plan s as_1) as_2"
asm: shows "Max (norm ` {x. poly p x = 0}) + 1 \<le> norm x"
ord_\<Gamma>_saturated_upto_def: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist"   and N :: "'a literal multiset set" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}" shows "redundancy_criterion.saturated_upto (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) Rf (standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts))) N = (inference_system.inferences_from (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) (N - Rf N) \<subseteq> standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) N)"
trace_rest'_simps_2: fixes x1ca :: "('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme"   and x2ca :: "'f spmf"   and xa :: "'a"   and x :: "'b \<times> 'c list"   and xaa :: "('a \<times> 'b \<times> 'c list + 'd \<times> 'e \<times> 'c list) list" shows "trace_rest' x1ca x2ca (Inl (xa, x) # xaa) = trace_rest' x1ca (cond_spmf_fst (x2ca \<bind> (\<lambda>s. rfunc_adv x1ca s xa)) x) xaa"
UN_extend_simps_9: fixes C :: "'c \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'c set"   and A :: "'b set" shows "(\<Union>x\<in>A. \<Union> (C ` B x)) = \<Union> (C ` \<Union> (B ` A))"
sup_bot_right: fixes x :: "'a multiset" shows "x \<union># {#} = x"
independence1: fixes x :: "'a"   and y :: "'a" assumes "x ; y = 0" shows "x\<^sup>\<star> ; y = y"
that: fixes xs :: "'a list"   and ys :: "'a list" assumes "us = xs \<cdot> [u] \<cdot> ys"   and "xs \<cdot> ys \<noteq> \<epsilon>" shows "thesis"
orthogonal_matrix_fst_QR_decomposition': fixes A :: "((real, 'a) vec, 'a) vec" assumes "rank A = ncols A" shows "orthogonal_matrix (fst (QR_decomposition A))"
constant_image_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "prob_space M" shows "prob_space.constant_image M f = (if \<exists>c. \<forall>x\<in>space M. f x = c then SOME c. \<forall>x\<in>space M. f x = c else undefined)"
open_continuous_vimage': fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (s \<inter> f -` B)"
assms_3: shows "fst ` set txs \<subseteq> trm"
IH_1: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
add_param: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' s $ Fv x \<tau>') (Abs \<tau>' t $ Fv x \<tau>')"
simps_9: fixes v :: "'b poly"   and vc :: "nat"   and vb :: "'b poly" shows "head (CN v (Suc vc) vb) = CN v (Suc vc) vb"
asymp_equiv_at_top_transfer: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "f \<sim>[F] g"   and "filterlim f at_top F" shows "filterlim g at_top F"
mod_div_mult_eq: fixes a :: "'a"   and b :: "'a" shows "a mod b + a div b * b = a"
match_simplematcher_SrcDst_not_1: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "'a ipt_iprange"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (MatchNot (Match (Src X))) a p = (p_src p \<notin> ipt_iprange_to_set X)"
real_of_rat_list_neq: fixes xs :: "rat list" shows "list_neq (map real_of_rat xs) 0 = map real_of_rat (list_neq xs 0)"
fin: shows "finite {x. a \<le> x \<and> x \<le> b \<and> poly p x = 0}"
weakPsiCongCasePres: fixes CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b" assumes "\<And>\<phi> P. (\<phi>, P) mem CsP \<Longrightarrow> \<exists>Q. (\<phi>, Q) mem CsQ \<and> guarded Q \<and> (\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q)"   and "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> (\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q)" shows "\<Psi> \<rhd> Cases CsP \<doteq> Cases CsQ"
f3: shows "rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha> \<notin> game_sem (ii (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) \<alpha> (selectlike (FF (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) (rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V)))) \<or> rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha> \<in> game_sem (ii (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) \<alpha> (FF (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>)"
enabled: shows "\<forall>\<sigma>. P \<sigma> \<longrightarrow> cond \<sigma>"
fls_base_factor_X_power_decompose_1: fixes f :: "'a fls" shows "f = fls_base_factor f * fls_X_intpow (fls_subdegree f)"
mult_add_mem_1: fixes a :: "V"   and x :: "V"   and r :: "V"   and y :: "V" assumes "a * x + r \<in> elts (a * y)"   and "r \<sqsubset> a" shows "x \<in> elts y"
mono: shows "monotone_on A (\<sqsubseteq>) (\<sqsubseteq>) f"
log_eq_one: fixes a :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "log a a = 1"
conv_def: shows "conv \<equiv> \<lambda>ars. {(xa, x). convp (\<lambda>x xa xb. (x, xa, xb) \<in> ars) xa x}"
u_def: shows "u = (P * A * Q) $$ (0, 0)"
Min_eq_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(local.Min A = m) = (m \<in> A \<and> (\<forall>a\<in>A. le m a))"
cA: shows "char_matrix A ev ^\<^sub>m k \<in> carrier_mat n n"
map_id: fixes t :: "'a blindable\<^sub>h" shows "map_blindable\<^sub>h id t = t"
lprodr_simps: fixes a :: "'a"   and b :: "'b"   and c :: "'c" shows "lprodr (a, b, c) = ((a, b), c)"
lNact_least: fixes n :: "nat"   and n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. latestAct_cond c t n k)" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
WT_plus_oracle_iff: fixes \<I>l :: "('a, 'b) \<I>"   and \<I>r :: "('c, 'd) \<I>"   and s :: "'s" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd> = (\<I>l \<turnstile>c left s \<surd> \<and> \<I>r \<turnstile>c right s \<surd>)"
iAll_empty: fixes P :: "nat \<Rightarrow> bool" shows "\<box> t {}. P t"
inj_on_empty: fixes f :: "'a \<Rightarrow> 'b" shows "inj_on f {}"
summable_iff_convergent_prod: fixes p :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> p i" shows "summable p = convergent (\<lambda>n. \<Prod>i<n. p i + 1)"
fun_le: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "\<forall>v1 v2. (v1, v2) \<in> set t1 \<longrightarrow> (\<exists>v3 v4. (v3, v4) \<in> set t2 \<and> v1 \<sqsubseteq> v3 \<and> v3 \<sqsubseteq> v1 \<and> v2 \<sqsubseteq> v4 \<and> v4 \<sqsubseteq> v2)" shows "t1 \<lesssim> t2"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L S.map \<ll>.map"
lfinite_rev_induct: fixes xs :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "lfinite xs"   and "P <>"   and "\<And>x xs. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (xs $ <x>)" shows "P xs"
improper_iff: fixes f :: "'b \<Rightarrow> ereal" shows "(\<not> proper f) = ((\<exists>x. f x = - \<infinity>) \<or> (\<forall>x. f x = \<infinity>))"
formula: shows "2 ^ (multiplicity 2 m + 1) * (m div 2 ^ multiplicity 2 m) = (2 ^ (multiplicity 2 m + 1) - 1) * sigma (m div 2 ^ multiplicity 2 m)"
power_one: fixes n :: "nat" shows "(1::'a) ^ n = (1::'a)"
OrR1_in_SNa: fixes M :: "trm"   and a :: "coname"   and b :: "coname" assumes "SNa M" shows "SNa (OrR1 <a>.M b)"
step_lem2b: fixes list :: "'a set list"   and x1 :: "'a set"   and a :: "'a set"   and aa :: "'a attree"   and listb :: "'a attree list" assumes "0 = length list"   and "map (\<lambda>i. \<N>\<^bsub>((x1 # a # list) ! i, (a # list) ! i)\<^esub>) [0..<length list] @ [\<N>\<^bsub>((x1 # a # list) ! length list, (a # list) ! length list)\<^esub>] = aa # listb" shows "\<N>\<^bsub>(x1, a)\<^esub> = aa"
order: fixes one :: "'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.impl_lr_ord_algebra one imp_l imp_r less_eq less" shows "less_eq a b = (imp_l a b = one)"
assms_2: shows "ss \<equiv> split 0 X (lpp ` G)"
monom_mult_dist_right: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.monom_mult c t (p + q) = local.punit.monom_mult c t p + local.punit.monom_mult c t q"
one_aux: shows "\<chi>\<^sub>0 (Suc 0) = 1"
derangements_add_one: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set"   and x :: "nat"   and y :: "nat" assumes "d \<in> derangements S"   and "x \<notin> S"   and "y \<in> S" shows "d(x := y, inv d y := x) \<in> derangements (insert x S)"
mr_cos_h_stable1: fixes H :: "'a set"   and m :: "'a"   and h :: "'a" assumes "submodule R M H"   and "m \<in> carrier M"   and "h \<in> H" shows "(m \<plusminus> h) \<uplus>\<^bsub>M\<^esub> H = m \<uplus>\<^bsub>M\<^esub> H"
INT_extend_simps_9: fixes C :: "'c \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'c set"   and A :: "'b set" shows "(\<Inter>x\<in>A. \<Inter> (C ` B x)) = \<Inter> (C ` \<Union> (B ` A))"
inject_2: fixes x21 :: "nat"   and x22 :: "'a"   and y21 :: "nat"   and y22 :: "'a" shows "(Geq x21 x22 = Geq y21 y22) = (x21 = y21 \<and> x22 = y22)"
secure_implies_iu_condition: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" assumes "secure P I D" shows "future_consistent P D (rel_ipurge P I D)"
assms: shows "D x \<in> sinks I D u (xs @ [x])"
iia_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "iia_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. \<forall>a\<in>C. a \<in> f B \<longrightarrow> a \<in> f C)"
elims_2: fixes x :: "nat \<Rightarrow> nat"   and xa :: "'b memref list" assumes "volatile_reads_consistent x xa" obtains m :: "nat \<Rightarrow> nat" where "x = ma__" and "xa = []"   | m :: "nat \<Rightarrow> nat" and r :: "'b memref" and rs :: "'b memref list" where "x = m" and "xa = r # rs" and "case r of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> volatile_reads_consistent (m(a := v)) rs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> (volatile \<longrightarrow> v = m a) \<and> volatile_reads_consistent m rs | _ \<Rightarrow> volatile_reads_consistent m rs"
ideD_2: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.dom a = a"
exhaust: fixes y :: "finite_2" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"
ty_REL: shows "TYREL (\<langle>Id\<rangle>ahs.rel)"
map_scale_eq_times: fixes c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "c \<cdot> p = monomial c (0::'c) * p"
msub_sprod: fixes M :: "('e, 'a, 'f) Module_scheme"   and M1 :: "('e, 'a, 'f) Module_scheme"   and a :: "'a"   and x :: "'e" assumes "R module M"   and "R module M1"   and "msubmodule R M M1"   and "a \<in> carrier R"   and "x \<in> carrier M1" shows "a \<cdot>\<^sub>s\<^bsub>M1\<^esub> x = a \<cdot>\<^sub>s\<^bsub>M\<^esub> x"
hyps_finite: fixes p :: "'b pl"   and t :: "'b set" shows "finite (hyps p t)"
simps_18: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Br i) ts = (if i < length (label \<C>) then type_update ts (to_ct_list (label \<C> ! i)) (TopType []) else Bot)"
finite_leaves: fixes i :: "int"   and t :: "int tree" shows "finite (leaves i t)"
eq_iff: fixes a :: "'d list"   and b :: "'d list" shows "(a = b) = (prefix b a \<and> prefix a b)"
shift_simps_2: fixes xs :: "'d list"   and s :: "'d stream" shows "stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"
ide_in_hom: fixes a :: "'a" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
equivp_symp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "equivp R"   and "R x y" shows "R y x"
execp_exec_eq: fixes eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set" shows "execp (\<lambda>x xa xb xc. ((x, xa), xb, xc) \<in> eval) = (\<lambda>x xa. exec_rel (x, xa) eval)"
lincomb_append_right: fixes rs :: "'r list"   and ms :: "'m list"   and ns :: "'m list" shows "rs \<bullet>\<cdot> (ms @ ns) = rs \<bullet>\<cdot> ms + drop (length ms) rs \<bullet>\<cdot> ns"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>c. P (C c)"   and "P (Bound 0)"   and "\<And>v. P (Bound (Suc v))"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>a. P a \<Longrightarrow> P (Floor a)"   and "\<And>c a. P a \<Longrightarrow> P (CN 0 c a)"   and "\<And>v c a. P a \<Longrightarrow> P (CN (Suc v) c a)"   and "\<And>c s t. \<lbrakk>P s; P t\<rbrakk> \<Longrightarrow> P (CF c s t)" shows "P a0"
pp_1: shows "set (pivot_positions A) = {(i, p i) |i. i < n \<and> p i \<noteq> n}"
rep_eq: fixes x :: "'a fset" shows "fcard x = card (fset x)"
d_def: fixes x :: "'a" shows "d x = d1 x \<inter> d3 x"
lspasl_iu_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 zero" shows "heap_sep_algebra.tern_rel plus sep_disj zero zero zero \<and> h1 = zero \<and> h2 = zero"
sg_i_closed: fixes H :: "'a set"   and x :: "'a" assumes "G \<guillemotright> H "   and "x \<in> H" shows "\<rho> x \<in> H"
hpk: fixes z :: "real \<times> real" assumes "z \<in> {0..1} \<times> {0..1}" shows "h z = p (k z)"
edges_undef_def: fixes l :: "('g \<times> 'h \<times> 'i) list"   and w :: "'g \<times> 'h \<Rightarrow> 'j" shows "edges_undef l w \<equiv> \<forall>u v d. (u, v, d) \<in> set l \<longrightarrow> w (u, v) = \<infinity>"
filterlim_principal: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'b set"   and F :: "'a filter" shows "filterlim f (principal S) F = (\<forall>\<^sub>F x in F. f x \<in> S)"
one_dim_iso_adjoint_complex: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "one_dim_iso (A*) = cnj (one_dim_iso A)"
integrable_bound: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "integrable M f"   and "g \<in> borel_measurable M"   and "AE x in M. norm (g x) \<le> norm (f x)" shows "integrable M g"
eq: shows "t - punit.lt (rep_list (bs ! i)) + punit.lt (rep_list (bs ! i)) = t"
dom_char: fixes f :: "('o, 'a) arr" shows "local.dom f = (if arr f then MkIde (Dom f) else null)"
ind_True: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n"   and "0 < n" shows "ind P n = (1::'b)"
as1: shows "\<forall>s\<in>T P. tickFree s \<longrightarrow> (s, UNIV) \<notin> F P"
rep_eq: fixes x :: "\<nu> \<Rightarrow> \<o>" shows "eval\<o> (forall\<^sub>\<nu> x) = (\<lambda>s w. \<forall>xa. eval\<o> (x xa) s w)"
distinct_15: fixes x51 :: "'a formula"   and x52 :: "'a formula" shows "\<bottom> \<noteq> x51 \<^bold>\<or> x52"
invertible_inverse_inverse: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "- (- u) = u"
to_pair_unit_def: fixes x :: "'a" shows "to_pair_unit x = (x, ())"
SUP_inf: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd"   and B :: "'e set"   and a :: "'d" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup (f ` B)) a = Sup ((\<lambda>b. inf (f b) a) ` B)"
pairwise_disjoint_path_components_of: fixes X :: "'a topology" shows "disjoint (path_components_of X)"
when_2: fixes P :: "bool"   and a :: "'a" assumes "\<not> P" shows "(a when P) = (0::'a)"
assms_3: shows "eval c t t' n (\<circle>\<^sub>b\<gamma>)"
\<alpha>_is_\<gamma>c_i\<delta>: shows "\<alpha> = \<gamma> @ [c_i] @ \<delta>"
replace_coeff_def: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c mpoly" shows "replace_coeff f p = MPoly (Abs_poly_mapping (\<lambda>m. f (lookup (mapping_of p) m)))"
Suc_1: fixes p :: "'a list" assumes "n = degree p"   and "p \<in> carrier (poly_ring R)" shows "finite (Collect (is_root p))"
ap: shows "arc_pos (A $$ (0, 0) * B $$ (0, 0))"
algebra_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "algebra \<Omega> M \<equiv> ring_of_sets \<Omega> M \<and> algebra_axioms \<Omega> M"
real_mon_conv_outgrow: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and z :: "real" assumes "x\<up>y"   and "z < y" shows "\<exists>n. \<forall>m\<ge>n. z < x m"
edom_evalHeap_subset: fixes h :: "('a \<times> 'c) list"   and eval :: "'c \<Rightarrow> 'b" shows "edom (evalHeap h eval) \<subseteq> domA h"
assms_3: shows "t' \<otimes>\<otimes> t = t'"
cfun_eqI: fixes f :: "'g \<rightarrow> 'h"   and g :: "'g \<rightarrow> 'h" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
exhaust: fixes y :: "('d, 'e) stateT" obtains x :: "'d \<Rightarrow> 'e" where "y = StateT x"
product_assoc: fixes A :: "'b fset set"   and B :: "'b fset set"   and C :: "'b fset set" shows "(A \<otimes> B) \<otimes> C = A \<otimes> B \<otimes> C"
Read_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and d :: "'e \<Rightarrow> 'f list"   and c :: "'d \<Rightarrow> 'f list"   and a :: "'f \<Rightarrow> 'c"   and c_update :: "('f list \<Rightarrow> 'f list) \<Rightarrow> 'd \<Rightarrow> 'd"   and d_update :: "('f list \<Rightarrow> 'f list) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "P \<preceq>S Q"   and "\<And>A. d A = c (S A)" shows "read a (c, c_update) P \<preceq>S read a (d, d_update) Q"
abs_eq: fixes x :: "int" shows "- integer_of_int x = integer_of_int (- x)"
preimage_included_then_almost_invariant: fixes A :: "'a set" assumes "A \<in> sets M"   and "T --` A \<subseteq> A" shows "A \<Delta> T --` A \<in> null_sets M"
assms_2: shows "\<forall>j<i. \<not> P (xs ! j)"
state_to_strips_state_effect_consistent: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and v :: "'a"   and a :: "'b"   and s :: "'a \<Rightarrow> 'b option"   and a' :: "'b" assumes "is_valid_problem_sas_plus \<Psi>"   and "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)"   and "(v, a') \<in> dom (\<phi>\<^sub>S \<Psi> s)"   and "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some True"   and "(\<phi>\<^sub>S \<Psi> s) (v, a') = Some True" shows "(v, a) = (v, a')"
polynomial_function: fixes as :: "'a list" assumes "set as \<subseteq> carrier R" shows "polynomial as \<in> carrier (function_ring (carrier R) R)"
sv01_in_P1: shows "s\<^bsub>01\<^esub> \<in> \<P>"
Collect_split_mem: fixes Q :: "('b \<times> 'c) set" shows "{(x, y). (x, y) \<in> Q} = Q"
map_invar_rule3: fixes s :: "'a"   and ii :: "'b"   and ti :: "'b"   and ei :: "'b"   and ri :: "'b" assumes "I s" shows "R (U s (ii, ti, ei) ri) = R s"
that: fixes p' :: "'a list" assumes "polynomial K p'"   and "p' \<noteq> []"   and "p = p' @ [\<zero>]" shows "thesis"
nth_map2: fixes bl :: "'a list"   and al :: "'b list"   and i :: "nat"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "length bl = length al"   and "i < length al" shows "map2 f al bl ! i = f (al ! i) (bl ! i)"
sel: fixes x :: "'c \<Rightarrow> 'd" shows "run_env (EnvT x) = x"