hcju/isabelleps huggingface dataset:
abs_le_zero_iff: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) zero = (a = zero)"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and c :: "'a"   and ty :: "ty" assumes "\<And>c. P c One"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (Fun x1 x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (Syntax.Sum x1 x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (Syntax.Prod x1 x2)"   and "\<And>x1 x2 c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (Mu x1 x2)"   and "\<And>x c. P c (Alpha x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (AuthT x)" shows "P c ty"
simps: fixes t :: "'b trm" shows "subterms_of t = {s. occurs_in s t}"
sets_PiM_generator: shows "sets (Pi\<^sub>M I M) = sigma_sets (space (Pi\<^sub>M I M)) generator"
fds_eq_fds_iff: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "(fds f = fds g) = (\<forall>n>0. f n = g n)"
rep_abs_c: fixes y :: "'cl" assumes "y \<in> \<UU>C" shows "y = Rep_c (Abs_c y)"
cases: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol"   and a :: "'s Trace" assumes "jpTracesp jp a" obtains s :: "'s" where "a = tInit sa__" and "sa__ \<in> set envInit"   | t :: "'s Trace" and eact :: "'eAct" and aact :: "'a \<Rightarrow> 'aAct" and s :: "'s" where "a = t \<leadsto> s" and "jpTracesp jp t" and "eact \<in> set (envAction (tLast t))" and "\<And>a. aact a \<in> set (actJP jp t a)" and "s = envTrans eact aact (tLast t)"
pos_horner: fixes x :: "float" shows "0 < (let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in if y \<le> 0 then Float 1 (- 2) else y)"
neg_numeral_less_neg_one_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus m)) (uminus one) = (m \<noteq> num.One)"
ccInf_mono: fixes B :: "'a set"   and A :: "'a set" assumes "countable B"   and "countable A"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b" shows "Inf A \<le> Inf B"
assms_4: shows "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
returns_result_eq: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and y :: "'o"   and y' :: "'o" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
InitSucState_Target_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> Target TS"   and "A \<in> the (CompFun (HA ST) S)" shows "InitState A \<in> StepConf (HA ST) (Conf ST) TS"
seq_step_invariant_ctermI: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('f \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and I :: "'g seq_action \<Rightarrow> bool"   and P :: "('f \<times> 'h) \<times> 'g seq_action \<times> 'f \<times> 'h \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "simple_labels \<Gamma>"   and "automaton.trans A = seqp_sos \<Gamma>"   and "\<And>p pp l \<xi> a q l' \<xi>'. \<lbrakk>p \<in> cterms \<Gamma>; l \<in> labels \<Gamma> p; ((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>; ((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A; l' \<in> labels \<Gamma> q; (\<xi>, pp) \<in> reachable A I; p \<in> sterms \<Gamma> pp; (\<xi>', q) \<in> reachable A I; I a\<rbrakk> \<Longrightarrow> P ((\<xi>, l), a, \<xi>', l')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"
hyps_2: assumes "possible_steps e s r l i = {|(d, t)|}" shows "ffilter (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (origin, dest) \<Rightarrow> \<lambda>t. origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)) b) e = {|((s, d), t)|}"
preserves_reflects_arr: fixes f :: "'c \<times> 'c" shows "C.VV.arr (C.VV.map f) = C.VV.arr f"
us_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "us x \<le> us y"
fin_succ_hd_w: shows "finite (succs (hd w))"
cexpr_sem_Mult: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL"   and "\<sigma> \<in> space (state_measure V \<Gamma>)"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V" shows "extract_real (cexpr_sem \<sigma> (e1 *\<^sub>c e2)) = extract_real (cexpr_sem \<sigma> e1) * extract_real (cexpr_sem \<sigma> e2)"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_index: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and i :: "nat" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"   and "i < length F" shows "\<exists>d\<in>set D. G ! i = (pair (F ! i), pair d)"
assms_2: shows "h \<circ> r = r' \<circ> h"
n0_1: shows "n\<^sub>0 \<noteq> 0"
monic_i_def: fixes ops :: "'b arith_ops_record"   and pp :: "'b list" shows "monic_i ops pp = (lead_coeff_i ops pp = arith_ops_record.one ops)"
inverse_renaming: fixes A :: "'at set"   and L :: "'at Literal" shows "rename_literal A (rename_literal A L) = L"
ht_hash_def: fixes l :: "('c \<times> 'd) list list" shows "ht_hash l \<equiv> \<forall>i<length l. \<forall>x\<in>set (l ! i). bounded_hashcode_nat (length l) (fst x) = i"
finsum_Suc: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> {..Suc n} \<rightarrow> carrier R" shows "finsum R f {..Suc n} = f (Suc n) \<oplus> finsum R f {..n}"
surjective_mk_program: fixes F :: "'c program" shows "mk_program (Init F, Acts F, AllowedActs F) = F"
B: fixes x :: "real" assumes "norm x < 1" shows "summable (\<lambda>n. arctan_coeffs n * x ^ n)"
input2state_Some: fixes i :: "value list"   and x :: "nat" shows "(\<exists>v. input2state i $ x = Some v) = (x < length i)"
is_Vcomp_def: fixes "term" :: "'b local.term" shows "is_Vcomp term = (\<exists>x41 x42. term = x41 \<^bold>\<cdot> x42)"
simps_3: fixes uv :: "'b"   and t :: "'c tree" shows "U (Insert uv) [t] = 2 * \<phi> t + 5 / 2"
strategyproofness_1: shows "(pmf (sds R32) d < pmf (sds R28) a \<or> pmf (sds R32) d + pmf (sds R32) b < pmf (sds R28) a + pmf (sds R28) c) \<or> pmf (sds R32) d = pmf (sds R28) a \<and> pmf (sds R32) d + pmf (sds R32) b = pmf (sds R28) a + pmf (sds R28) c"
numer_cf5_mono: fixes y :: "real"   and x :: "real" assumes "y \<le> x" shows "numer_cf5 y \<le> numer_cf5 x"
run_asset_i_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "case run_asset_i n s of (s', S, x) \<Rightarrow> (Asset n, Crypt (Auth_ShaKey n) (PriKey S)) \<in> s'"
splitFace_holds_faces_distinct: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and n :: "nat" assumes "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"   and "minGraphProps g'" shows "faces_distinct (snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n])))"
del_vert_simps_3: fixes u :: "'a" shows "tail (del_vert u) = tail G"
singleton_iff: fixes b :: "'e"   and a :: "'e" shows "(b \<in> {a}) = (b = a)"
mat_mult_plus_distrib_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and ncc :: "nat"   and m2 :: "'c list list"   and m3 :: "'c list list" assumes "class.semiring_0 plus zero times"   and "mat nr nc m1"   and "mat nc ncc m2"   and "mat nc ncc m3" shows "mat_multI zero plus times nr m1 (mat_plusI plus m2 m3) = mat_plusI plus (mat_multI zero plus times nr m1 m2) (mat_multI zero plus times nr m1 m3)"
cprank_max_prod_order1: fixes A :: "'b tensor"   and n :: "nat"   and B :: "'b tensor" assumes "order A = 1"   and "cprank_max n B" shows "cprank_max n (A \<otimes> B)"
assms_3: shows "set ps = ps\<^sub>L \<union> ps\<^sub>R"
E_ss_VxV: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "Graph.E c \<subseteq> Graph.V c \<times> Graph.V c"
degree_linear_factors: fixes f :: "'b \<Rightarrow> 'a"   and as :: "'b list" shows "degree (\<Prod>a\<leftarrow>as. [:f a, 1::'a:]) = length as"
elims_1: fixes x :: "'b nres"   and xa :: "'b nres"   and y :: "bool" assumes "(x \<le> xa) = y" obtains uu_ :: "'b nres" where "x = uu_" and "xa = FAILi" and "y"   | a :: "'b set" and b :: "'b set" where "x = RES a" and "xa = RES b" and "y = (a \<subseteq> b)"   | uv_ :: "'b set" where "x = FAILi" and "xa = RES uv_" and "\<not> y"
le_adm: fixes u :: "'b \<Rightarrow> 'c process"   and v :: "'b \<Rightarrow> 'c process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<le> v x)"
arcosh_real_has_field_derivative: fixes x :: "real"   and A :: "real set" assumes "1 < x" shows "(arcosh has_real_derivative 1 / sqrt (x\<^sup>2 - 1)) (at x within A)"
fun_relD1: fixes f :: "'g \<Rightarrow> 'h"   and f' :: "'i \<Rightarrow> 'j"   and Ra :: "('g \<times> 'i) set"   and Rr :: "('h \<times> 'j) set"   and x :: "'g"   and r :: "'h" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
true_cls_mset_union: fixes I :: "'b set"   and CC :: "'b literal multiset multiset"   and DD :: "'b literal multiset multiset" shows "(I \<Turnstile>m CC + DD) = (I \<Turnstile>m CC \<and> I \<Turnstile>m DD)"
K_cfg_map_absc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "K_cfg (absc cfg) = map_pmf absc (K_cfg cfg)"
Ex_other_prime_factor: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "'a"   and p :: "'a" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "n \<noteq> zero"   and "\<nexists>k. normalize n = power.power one times p k"   and "normalization_semidom.prime zero times one normalize p" shows "\<exists>q\<in>#factorial_semiring.prime_factorization zero times one normalize n. q \<noteq> p"
pa: fixes p :: "real \<Rightarrow> 'b" assumes "path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p" shows "\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)"
Aexp_Lam: fixes y :: "Vars.var"   and e :: "exp"   and n :: "Arity" shows "env_delete y (\<A>\<^bsub>pred\<cdot>n\<^esub> e) \<sqsubseteq> \<A>\<^bsub>n\<^esub> (Lam [y]. e)"
subgroup_intersect: fixes H :: "'a set"   and H' :: "'a set" assumes "subgroup H G"   and "subgroup H' G" shows "subgroup (H \<inter> H') G"
success_returnI: fixes x :: "'b"   and h :: "heap" shows "success (return x) h"
valid: shows "valid_unMultigraph (del_unEdge v3 w v2 G)"
sub_mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'b list list" shows "sub_mat nr nc m = map (sub_vec nr) (take nc m)"
IH_2: fixes a :: "atom"   and b :: "atom" shows "((a \<rightleftharpoons> b) \<bullet> t1 =\<^sub>\<alpha> t1) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t1)"
PreValid_def: fixes A :: "'a \<Rightarrow> bool" shows "|~ A \<equiv> \<forall>w. A w"
simple_group_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "simple_group_axioms G \<equiv> 1 < order G \<and> (\<forall>H. H \<lhd> G \<longrightarrow> H = carrier G \<or> H = {\<one>\<^bsub>G\<^esub>})"
quotient_hom_im: fixes G :: "'c set"   and T :: "'c \<Rightarrow> 'd"   and H :: "'c set"   and g :: "'c" assumes "GroupHom G T"   and "Group H \<and> order.greater_eq G H"   and "Group.normal G H"   and "order.greater_eq (ker T \<inter> G) H"   and "g \<in> G" shows "restrict0 (quotientfun T \<circ> the_inv_into (G // Restr (lcoset_rel H) G) (BinOpSetGroup.Abs_G_perm (G // Restr (lcoset_rel H) G) (+))) (BinOpSetGroup.Abs_G_perm (G // Restr (lcoset_rel H) G) (+) ` G // Restr (lcoset_rel H) G) (BinOpSetGroup.Abs_G_perm (G // Restr (lcoset_rel H) G) (+) (Restr (lcoset_rel H) G `` {g})) = T g"
homeomorphic_imp_homotopy_eqv: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "S homotopy_eqv T"
not_inf_impl_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<not> AlternativeSmallStep.inf \<Gamma> [c] [] s" shows "\<Gamma>\<turnstile>c \<down> s"
emeasure_PiM_emb: fixes J :: "'c set"   and X :: "'c \<Rightarrow> real set" assumes "J \<subseteq> UNIV"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> E.events" shows "emeasure (Pi\<^sub>M UNIV (\<lambda>_. exponential 1)) (P.emb UNIV J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (exponential 1) (X i))"
elims: fixes x :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and y :: "nat list" assumes "lin_ts_mmuaux x = y" obtains tp :: "nat" and tss :: "nat queue" and len :: "nat" and maskL :: "bool list" and maskR :: "bool list" and a1_map :: "('a option list, nat) mapping" and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" and "done" :: "'a option list set list" and done_length :: "nat" where "x = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)" and "y = linearize tss"
VDMAx: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "A \<in> G" shows "G \<rhd> Call : A"
inv_less': fixes i1 :: "int"   and i2 :: "int"   and a1 :: "'av"   and a2 :: "'av"   and a\<^sub>1' :: "'av"   and a\<^sub>2' :: "'av" assumes "inv_less' (i1 < i2) a1 a2 = (a\<^sub>1', a\<^sub>2')"   and "i1 \<in> \<gamma> a1"   and "i2 \<in> \<gamma> a2" shows "i1 \<in> \<gamma> a\<^sub>1' \<and> i2 \<in> \<gamma> a\<^sub>2'"
fls_base_factor_nth: fixes f :: "'a fls"   and n :: "int" shows "fls_nth (fls_base_factor f) n = fls_nth f (n + fls_subdegree f)"
s0_secrecy_init: shows "init s0 \<subseteq> s0_secrecy"
atLeastatMost_empty_iff: fixes a :: "'b word"   and b :: "'b word" shows "(signed.atLeastAtMost a b = {}) = (\<not> a \<le>s b)"
that: fixes b :: "'a" assumes "b \<in> carrier R"   and "ring_irreducible b"   and "b divides a" shows "thesis"
map_add_dom_app_simps_2: fixes m :: "'c"   and l1 :: "'c \<Rightarrow> 'd option"   and l2 :: "'c \<Rightarrow> 'd option" assumes "m \<notin> dom l1" shows "(l1 ++ l2) m = l2 m"
D_set_def: fixes x :: "'b set"   and y :: "'b set" shows "D x y \<equiv> x \<inter> y = {}"
validn: shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
range_nat_of_char: shows "range of_char = {0..<256}"
inverse_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "trimmed_ms_aux F" shows "is_expansion_aux (inverse_ms_aux F) (\<lambda>x. inverse (f x)) basis"
carrier_nhd: fixes x :: "'a llist" assumes "x \<in> carrier" shows "carrier \<in> nhds x"
f: fixes i :: "'a" assumes "i \<in> I" shows "f i \<in> S i"
unfold: fixes x :: "rat list" assumes "x \<in> A" shows "x \<in> set (take_indices B (find_nonzeros_from_input_vec C))"
ordinal_less_plusL: fixes y :: "ordinal"   and x :: "ordinal" assumes "y < x * \<omega>" shows "y < x + y"
map_add_find_right: fixes n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and xx :: "'a"   and m :: "'b \<Rightarrow> 'a option" assumes "n k = \<lfloor>xx\<rfloor>" shows "(m ++ n) k = \<lfloor>xx\<rfloor>"
subset_imp_msubset_mset_set: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "finite B" shows "mset_set A \<subseteq># mset_set B"
set_def: fixes s' :: "'c" shows "State_Monad.set s' = State (\<lambda>_. ((), s'))"
simps: fixes l :: "'d strand_label"   and s :: "('e, 'f) stateful_strand_step"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "(l, s) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = (l, s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"
comp_ipresIGOp: fixes h :: "'l \<Rightarrow> 'm"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'n, 'l, 'o, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'q, 'm, 'r, 's) model_scheme"   and hA :: "'o \<Rightarrow> 'r"   and h' :: "'m \<Rightarrow> 't"   and hA' :: "'r \<Rightarrow> 'u"   and MOD'' :: "('index, 'bindex, 'v, 'w, 'opSym, 'x, 't, 'u, 'y) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGWlsAbs hA MOD MOD'"   and "ipresIGOp h hA MOD MOD'"   and "ipresIGOp h' hA' MOD' MOD''" shows "ipresIGOp (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
valid_alternative_with_GC: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<forall>ps n. P (ps, n) \<longrightarrow> (\<exists>ps' ps'' m e e'. (c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' + ps'' \<and> ps' ## ps'' \<and> n = e + e' + m \<and> Q (ps', e))" shows "\<forall>ps n. P (ps, n) \<longrightarrow> (\<exists>ps' m e. (c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' \<and> n = e + m \<and> (Q \<and>* (\<lambda>s. True)) (ps', e))"
simps_8: fixes x31 :: "name"   and x32 :: "nterm"   and x1 :: "name" shows "(\<Lambda>\<^sub>n x31. x32) \<noteq> Nconst x1"
assms_2: shows "{b1, b2} \<subseteq> B"
grid_change_dim: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat"   and X :: "nat \<times> int" assumes "p \<in> grid b ds" shows "p[d := X] \<in> grid (b[d := X]) ds"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "cachedWeight (Node w t\<^sub>1 t\<^sub>2) = w"
leadsTo_cancel2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A' \<union> B"   and "F \<in> B \<longmapsto> B'" shows "F \<in> A \<longmapsto> A' \<union> B'"
matrix_to_iarray_Hermite_of_row_i: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" shows "matrix_to_iarray (Hermite_of_row_i ass res A i) = Hermite_of_row_i_iarray ass res (matrix_to_iarray A) (mod_type_class.to_nat i)"
prob_space_axioms: shows "prob_space (Pi\<^sub>M UNIV (\<lambda>n. exponential (R n)))"
fst_conv: fixes x1 :: "'f"   and x2 :: "'g" shows "fst (x1, x2) = x1"
insert_DiffM: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "add_mset x (M - {#x#}) = M"
delta_wf_state: fixes a :: "'a"   and s :: "'s" assumes "a \<in> set (\<sigma> n)"   and "wf_state s" shows "wf_state (delta a s)"
F2: shows "\<forall>x\<^sub>1. parts x\<^sub>1 \<union> synth (analz x\<^sub>1) = parts (synth (analz x\<^sub>1))"
commutative_squareE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" 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"
Says_Tgs_message_form: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "B \<noteq> Tgs \<and> servK \<notin> range shrK \<and> servK \<notin> authKeys evs \<and> servK \<in> symKeys \<and> servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace> \<and> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys"
SN_preserved: fixes t :: "trm"   and t' :: "trm" assumes "SN t"   and "t \<mapsto> t'" shows "SN t'"
verdicts_def: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and \<sigma> :: "'c trace" assumes "fo_spec nfv fv sat" shows "fo_spec.verdicts nfv fv sat \<sigma> = {(i, v). wf_tuple nfv fv v \<and> sat \<sigma> (map the v) i}"
gf: fixes i :: "'a"   and x :: "real" assumes "i \<in> Basis"   and "a \<bullet> i < x"   and "x < b \<bullet> i" shows "g i (f i x) = x"
invert_side_of_triangle_1: fixes h :: "'c"   and f :: "'c"   and g :: "'c" assumes "C.arr h"   and "f \<cdot>\<^sub>C g = h"   and "C.iso f" shows "C.seq (C.inv f) h \<and> g = C.inv f \<cdot>\<^sub>C h"
equation_minus_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (b = uminus a)"
diff_eint_def: fixes a :: "eint"   and b :: "eint" shows "a - b = (case a of eint x \<Rightarrow> case b of eint y \<Rightarrow> eint (x - y) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
cnf_of_negative_transition_frame_axiom: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and v :: "'b" shows "cnf (encode_negative_transition_frame_axiom \<Pi> k v) = {{(State k (index (\<Pi>\<^sub>\<V>) v))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}}"
even_minus: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_parity plus minus zero times one divide modulo uminus" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (uminus a) = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"
convex_hull_list_eq_set: fixes Vs :: "'a vec list"   and Ws :: "'a vec list" assumes "set Vs \<subseteq> carrier_vec n"   and "set Vs = set Ws" shows "convex_hull_list Vs = convex_hull_list Ws"
lossless_R2_14: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "lossless_spmf (R2_14 M C)"
nf_ACI_nFOR: fixes \<phi>s :: "('a, 'b) aformula list" assumes "sorted \<phi>s"   and "distinct \<phi>s"   and "list_all nf_ACI \<phi>s"   and "list_all (\<lambda>x. \<not> is_FOr x) \<phi>s" shows "nf_ACI (nFOR \<phi>s)"
Q_1: fixes X :: "complex multiset" assumes "X \<in> Roots_ms" shows "poly_roots (Ring_Hom_Poly.of_int_poly (Q X)) = R X"