hcju/isabelleps huggingface dataset:
vsubstEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var"   and y1 :: "'var" assumes "goodEnv rho"   and "freshEnv zs z rho"   and "zs \<noteq> ys \<or> z \<notin> {y, y1}" shows "freshEnv zs z (rho &[y1 // y]_ys)"
exhaust: fixes y :: "base" obtains x1 :: "char list" where "y = Repeats x1"   | x2 :: "char list" where "y = Shares x2"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<pi>"
Cons_eqvt: fixes p :: "perm"   and x :: "'b"   and xs :: "'b list" shows "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
exhaust: fixes y :: "'e option" obtains "y = None"   | x2 :: "'e" where "y = Some x2"
simps_8: fixes x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "fos (All1 x \<phi>) = remove1 x (fos \<phi>)"
Nle0_def: shows "0\<ge>\<^sub>N = (\<lambda>(a, b). a \<le> 0)"
A: shows "\<forall>n m. n \<noteq> m \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj n A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj m A)) Sec"
e: shows "rel = Leq_Rel \<and> e < 0 \<or> rel = Lt_Rel \<and> e \<le> 0"
prems: fixes i :: "nat" assumes "i < length (x # xs)" shows "length ((x # xs) ! i) = length ((y # ys) ! i)"
get_put: fixes m :: "'m" shows "get (\<lambda>s. put s m) = m"
elims_3: fixes x :: "'a list" assumes "\<not> sorted x" obtains x :: "'a" and ys :: "'a list" where "xb__ = x # ys" and "\<not> (Ball (set ys) ((\<le>) x) \<and> sorted ys)"
inf_import_p: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "inf x (uminus (inf x y)) = inf x (uminus y)"
ring_hom_mult: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
invc_upd_2: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "invc t" shows "invc2 (upd x y t)"
summable_LIMSEQ_zero: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "f \<longlonglongrightarrow> (0::'a)"
in_homE: fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and a :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and b :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" obtains "S.arr f" and "S.dom f = a" and "S.cod f = b"
alg_equiv_inv_auto_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : TBase" obtains p :: "trm" and q :: "trm" where "s \<Down> p" and "t \<Down> q" and "\<Gamma> \<turnstile> p \<leftrightarrow> q : TBase"
\<A>'_2: shows "\<I> \<Turnstile> \<langle>unlabel \<A>'\<rangle>"
f7: shows "4294967296 + - 1 * uint w + - 4294967296 = - 1 * uint w"
Real_Alg_Invariant_inverse: fixes y :: "real_alg_2" assumes "y \<in> Collect invariant_2" shows "rep_real_alg_3 (Real_Alg_Invariant y) = y"
hidesD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G\<turnstile> new hides old" shows "declclass new \<noteq> Object \<and> is_static new \<and> msig new = msig old \<and> G\<turnstile>declclass new\<prec>\<^sub>C declclass old \<and> G\<turnstile>Method new declared_in declclass new \<and> G\<turnstile>Method old declared_in declclass old"
actions_ok1_ex_actions_ok2: fixes s1 :: "('a, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'b wait_set_status option) \<times> 't set"   and t :: "'t"   and ta1 :: "('a, 't, 'x1, 'm1, 'b, 'c) thread_action"   and ta2 :: "('a, 't, 'x2, 'm2, 'b, 'c) thread_action" assumes "r1.actions_ok s1 t ta1"   and "ta1 \<sim>m ta2" obtains s2 :: "('a, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'b wait_set_status option) \<times> 't set" where "r2.actions_ok s2 t ta2"
exec: shows "exec_meth ci P (ins @ ins') (xt @ shift (length ins) xt') t h (stk, loc, length ins + pc, xcp) ta h' (stk', loc', pc', xcp')"
tAct_4: shows "\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)"
eq: fixes x :: "'a"   and y :: "'b" assumes "x \<in> set_pmf p"   and "y \<in> set_pmf q"   and "R x y" shows "measure_pmf.prob p {x. R x y} = measure_pmf.prob q {y. R x y}"
simps_2: fixes n :: "nat" shows "fac (Suc n) = Suc n * fac n"
simps: fixes a1 :: "'a com"   and a2 :: "'a"   and a3 :: "'a" shows "Sem a1 a2 a3 = ((\<exists>f s. a1 = Basic f \<and> a2 = s \<and> a3 = f s) \<or> (\<exists>c1 s s'' c2 s'. a1 = Seq c1 c2 \<and> a2 = s \<and> a3 = s' \<and> Sem c1 s s'' \<and> Sem c2 s'' s') \<or> (\<exists>s b c1 s' c2. a1 = Cond b c1 c2 \<and> a2 = s \<and> a3 = s' \<and> s \<in> b \<and> Sem c1 s s') \<or> (\<exists>s b c2 s' c1. a1 = Cond b c1 c2 \<and> a2 = s \<and> a3 = s' \<and> s \<notin> b \<and> Sem c2 s s') \<or> (\<exists>s b x y c. a1 = While b x y c \<and> a2 = s \<and> a3 = s \<and> s \<notin> b) \<or> (\<exists>s b c s'' x y s'. a1 = While b x y c \<and> a2 = s \<and> a3 = s' \<and> s \<in> b \<and> Sem c s s'' \<and> Sem (While b x y c) s'' s'))"
8: shows "r\<^sup>T * - (1::'a) * r \<le> bot"
assms_2: shows "(\<lambda>x. f x \<bullet> j) piecewise_C1_differentiable_on s"
set_eqvt: fixes p :: "perm"   and xs :: "'e list" shows "p \<bullet> set xs = set (p \<bullet> xs)"
iso_points_ss_card: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and ps :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "ps \<subseteq> \<V>" shows "card ps = card (bij_map ` ps)"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray" shows "fi_state (f, mu, d) = get_nth_i f"
eq: shows "{x \<in> topspace X. f x \<in> topspace X' \<and> g (f x) \<in> U''} = {x \<in> topspace X. (g \<circ> f) x \<in> U''}"
qFreshInp_preserves_alphaInp_aux: fixes qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qinp' :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and xs :: "'varSort"   and x :: "'var" assumes "qGoodInp qinp \<or> qGoodInp qinp'"   and "(qinp, qinp') \<in> alphaInp"   and "qFreshInp xs x qinp" shows "qFreshInp xs x qinp'"
drop_pre_inv: fixes p :: "('d, 'c \<times> 'b) uexpr" assumes "out\<alpha> \<sharp> p" shows "\<lceil>\<lfloor>p\<rfloor>\<^sub><\<rceil>\<^sub>< = p"
chain_subset_trans_Union: fixes R :: "('a \<times> 'a) set set" assumes "chain\<^sub>\<subseteq> R"   and "\<forall>r\<in>R. trans r" shows "trans (\<Union> R)"
rep_eq: fixes x :: "nat"   and xa :: "nat \<Rightarrow> 'a" shows "Rep_vec (vec x xa) = (x, mk_vec x xa)"
leg0_in_hom_2: shows "\<guillemotleft>f\<^sub>0gh\<^sub>1.p\<^sub>0 : f\<^sub>0gh\<^sub>1.p\<^sub>0 \<rightarrow> f\<^sub>0gh\<^sub>1.p\<^sub>0\<guillemotright>"
simps_1: fixes f :: "'a \<Rightarrow> 'b" shows "ListUpdateE f [] = f"
LeftDerives1_implies_Derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.LeftDerives1 \<NN> \<TT> \<RR> a i r b" shows "CFG.Derives1 \<NN> \<TT> \<RR> a i r b"
union_fold_mset_add_mset: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A + B = fold_mset msins A B"
polyfun_tensor_deep_model: fixes "is" :: "nat list" assumes "is \<lhd> input_sizes (deep_model_l rs)" shows "polyfun {..<weight_space_dim} (\<lambda>f. Tensor.lookup (tensors_from_net (insert_weights shared_weights (deep_model_l rs) f) $ y) is)"
abs_add_nlex: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "coll 0 a b"   and "Counterclockwise_2D_Arbitrary.lex a 0"   and "Counterclockwise_2D_Arbitrary.lex b 0" shows "\<bar>a + b\<bar> = \<bar>a\<bar> + \<bar>b\<bar>"
is_iso_classI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C f" shows "category.is_iso_class C (category.iso_class C f)"
integral_has_vector_derivative_continuous_at': fixes s :: "real set"   and f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "real" assumes "finite s"   and "f integrable_on {a..b}"   and "x \<in> {a..b} - s"   and "continuous (at x within {a..b} - s) f" shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b} - s)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_inv_3: fixes l :: "'d strand_label"   and a :: "('e, 'f) stateful_strand_step"   and k :: "'d strand_label"   and b :: "('e, 'f) stateful_strand_step"   and t :: "('e, 'f) Term.term" assumes "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, a) = (k, b)"   and "a = send\<langle>t\<rangle>" shows "b = receive\<langle>t\<rangle>"
symrun_interp_expansion: fixes \<Gamma>\<^sub>1 :: "'a constr list"   and \<Gamma>\<^sub>2 :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
Ag_carrier: shows "Ag \<in> carrier_mat dB dA"
r_inv_add_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (\<ominus> r |\<^bsub>rel\<^esub> s) = \<zero>\<^bsub>rec_rng_of_frac\<^esub>"
not_in_bounds: fixes x :: "'a"   and v :: "'a \<Rightarrow> 'b"   and lb :: "'a \<Rightarrow> 'b option"   and ub :: "'a \<Rightarrow> 'b option" shows "in_bounds x v (lb, ub) \<noteq> (v x <\<^sub>l\<^sub>b lb x \<or> v x >\<^sub>u\<^sub>b ub x)"
ring_hom_trans: fixes f :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and g :: "'d \<Rightarrow> 'g"   and T :: "('g, 'h) ring_scheme" assumes "f \<in> ring_hom R S"   and "g \<in> ring_hom S T" shows "g \<circ> f \<in> ring_hom R T"
assms_1: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and s :: "'sort" assumes "X \<in> set (XLF s)" shows "wls s X"
timpl_closure'_inv''_2: fixes f :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and g :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set"   and i :: "nat" assumes "(Fun f S, Fun g T) \<in> timpl_closure' TI"   and "i < length T" shows "(S ! i, T ! i) \<in> timpl_closure' TI"
7: shows "prim_E w v e\<^sup>T * top * prim_E w v e \<le> w\<^sup>+ \<squnion> w\<^sup>T\<^sup>+ \<squnion> (1::'a)"
that_2: fixes n :: "nat" assumes "x = Suc n"   and "y = n" shows "thesis"
transrec: fixes H :: "(V \<Rightarrow> V) \<Rightarrow> V \<Rightarrow> V"   and a :: "V" shows "transrec H a = H (restrict (transrec H) (elts a)) a"
cart2_append1: fixes p :: "proj2" assumes "z_non_zero p" shows "vector2_append1 (cart2_pt p) = cart2_append1 p"
unit_r_inv: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one"   and "Dmonoid.unit prod one x" shows "prod x (Dmonoid.inv prod one x) = one"
mcp_prefix_of_suffix: fixes vs :: "'d list"   and xs :: "'d list"   and V :: "'d list set"   and xs' :: "'d list" assumes "mcp (vs @ xs) V vs"   and "prefix xs' xs" shows "mcp (vs @ xs') V vs"
x_lin_x: fixes V :: "'c set"   and x :: "'c" assumes "vectorspace V"   and "x \<in> V" shows "x \<in> lin x"
floor_less_neg_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lfloor>x\<rfloor> < - numeral v) = (x < - numeral v)"
borel_measurable_sgn: shows "sgn \<in> borel_measurable borel"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "construct B f (b1 + b2) = construct B f b1 + construct B f b2"
evaln_InsInitE: fixes G :: "prog"   and s :: "st"   and c :: "stmt"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>In1l (InsInitE c e)\<succ>\<midarrow>n\<rightarrow> (v, s') = False"
strategy_attracts_via_SucE: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy_attracts_via G p \<sigma> v0 A W"   and "v0 \<notin> W" shows "\<exists>n. enat (Suc n) < llength P \<and> P $ Suc n \<in> W \<and> lset (ltake (enat (Suc n)) P) \<subseteq> A"
multpw_single: fixes x :: "'b"   and y :: "'b"   and ns :: "('b \<times> 'b) set" assumes "(x, y) \<in> ns" shows "({#x#}, {#y#}) \<in> multpw ns"
paths_are_winning_for_one_player: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and p :: "Player" assumes "ParityGame G"   and "Digraph.valid_path G P" shows "ParityGame.winning_path G p P = (\<not> ParityGame.winning_path G p** P)"
below_meet: fixes x :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> z" shows "x \<sqinter> z = x"
meval_elim: fixes cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.meval_abv (cms, mem) k (cms', mem')" obtains cm' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)" where "cms' = cms[k := cm']" and "(cms ! k, mem) \<leadsto>\<^sub>C (cm', mem')" and "k < length cms"
lift2_Err: fixes x :: "'c err"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b err" shows "x \<squnion>\<^bsub>lift2 f\<^esub> Err = Err"
test_1: fixes x :: "'a"   and y :: "'a" assumes "is_test x" shows "x ; 1 \<cdot> y = x ; y"
range_eqI: fixes b :: "'k"   and f :: "'l \<Rightarrow> 'k"   and x :: "'l" assumes "b = f x" shows "b \<in> range f"
U_l: shows "\<forall>(t, n)\<in>set (uset (rlfm (simpfm p))). numbound0 t \<and> 0 < n"
10: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
wf_send_compose: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and X :: "('b, 'a) Term.term list"   and S' :: "('b, 'a) strand_step list"   and f :: "'b" shows "wf\<^sub>s\<^sub>t V (S @ map Send X @ S') = wf\<^sub>s\<^sub>t V (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S')"
assms_6: fixes k' :: "'a" assumes "k' \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)"   and "lt ko k' k" shows "f k' (lookup_raw xs k') (lookup_raw ys k') = Some Eq"
simps_5: shows "cospan_shape.Dom cospan_shape.BT = cospan_shape.BB"
rmult_0: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and r :: "'h" assumes "module R M"   and "r \<in> carrier R" shows "r \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
map_return_pmf: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
simps_4: fixes x51 :: "'a"   and x52 :: "('a, 'b) MWLfCom"   and y51 :: "'a"   and y52 :: "('a, 'b) MWLfCom" shows "(while x51 do x52 od = while y51 do y52 od) = (x51 = y51 \<and> x52 = y52)"
SC3: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "singleCombinators p" shows "singleCombinators (FWNormalisationCore.sort p l)"
ass: shows "\<forall>x y. (x, y) \<in> set xys \<longrightarrow> y < x"
Abs_type: fixes \<Gamma> :: "binding list"   and x :: "vrs"   and s :: "trm"   and S :: "ty"   and T :: "ty"   and U :: "ty"   and U' :: "ty" assumes "\<Gamma> \<turnstile> (\<lambda>x:S. s) : T"   and "\<Gamma>\<turnstile>T<:U \<rightarrow> U'"   and "x \<sharp> \<Gamma>" obtains S' :: "ty" where "\<Gamma>\<turnstile>U<:S" and "VarB x S # \<Gamma> \<turnstile> s : S'" and "\<Gamma>\<turnstile>S'<:U'"
add_less_cancel_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "contained.additive.Class a [\<cdot>] contained.additive.Class b = contained.additive.Class (a \<cdot> b)"
Inf_b_less: fixes i' :: "nat"   and j' :: "nat"   and i :: "nat"   and j :: "nat"   and k' :: "nat"   and k :: "nat" assumes "i' < j'"   and "i < j"   and "j' \<le> k'"   and "j \<le> k"   and "k' < k" shows "\<Sqinter> (b k' (j', i')) < \<Sqinter> (b k (j, i))"
assms_1: shows "m = WriterT\<cdot>(return\<cdot>\<bottom>)"
prod_mono_strict: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "finite A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less (f i) (g i)"   and "A \<noteq> {}" shows "less (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
cases: fixes x :: "'a" obtains z :: "int" where "x = of_int z" and "0 \<le> z" and "z < n"
HAInitValue_def: shows "HAInitValue \<equiv> fst \<circ> Rep_hierauto"
Disjoint_Eats_Imp: fixes x :: "tm"   and z :: "tm"   and y :: "tm" shows "{Disjoint x z, Neg (y IN z)} \<turnstile> Disjoint (Eats x y) z"
coeffs_to_int_poly: fixes ff_ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c mod_ring \<Rightarrow> bool"   and p :: "int"   and x :: "'c mod_ring poly" assumes "mod_ring_gen ff_ops R p" shows "coeffs (to_int_poly x) = map to_int_mod_ring (coeffs x)"
simps_2: fixes p :: "char list"   and ts :: "fterm list"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "Neg p ts \<cdot>\<^sub>l \<sigma> = Neg p (ts \<cdot>\<^sub>t\<^sub>s \<sigma>)"
elims: fixes x :: "lambda"   and y :: "val set" assumes "vals x = y" obtains uu_ :: "nat" and vs :: "(nat \<times> char list) list" and c :: "call" where "x = Lambda uu_ vs c" and "y = valsC c"
r_result_converg': fixes i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_univ [i, x] \<down>= v" shows "\<exists>t. (\<forall>t'\<ge>t. eval r_result [t', i, x] \<down>= Suc v) \<and> (\<forall>t'<t. eval r_result [t', i, x] \<down>= 0)"
H_star: fixes i :: "'a"   and x :: "'a" assumes "H i x i" shows "H i (x\<^sup>\<star>) i"
IH: fixes gpv :: "('c, 'a, 'b) gpv" assumes "colossless_gpv \<I> gpv" shows "expectation_gpv' gpv = expectation_gpv'' gpv"
intro: assumes "\<And>x y. PP x y = (P x y \<and> \<not> P y x)"   and "\<And>x y. \<lbrakk>P x y; P y x\<rbrakk> \<Longrightarrow> x = y" shows "M1_axioms"
wf_table_iff: fixes n :: "nat"   and A :: "nat set"   and Q :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set" shows "wf_table n A Q X = (\<forall>x. (x \<in> X) = (Q x \<and> wf_tuple n A x))"
mes: fixes D :: "'a set" assumes "D \<in> sets M" shows "(\<lambda>x. f x * indicator D x) \<in> borel_measurable M"
soundnessForSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "getConflictFlag state = False"   and "\<not> formulaFalse (getF state) (elements (getM state))"   and "decisionVars \<subseteq> vars (elements (getM state))" shows "model (elements (getM state)) F0"
orientable_down_closed: fixes x :: "'a"   and y :: "'a" assumes "local.comp_inf.symmetric x"   and "x \<le> y"   and "bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id y" shows "bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id x"
autoref_monadicI1_2: fixes b :: "'c nres"   and a :: "'d nres"   and R :: "('c \<times> 'd) set"   and c :: "'c" assumes "(b, a) \<in> \<langle>R\<rangle>nres_rel"   and "RETURN c \<le> b" shows "RETURN c \<le> \<Down> R a"
dfs_append: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree option list"   and ys :: "('d, 'e) bintree option list" shows "dfs f (xs @ ys) = dfs f xs @ dfs f ys"
ct_suffix_less: fixes xs :: "ct list"   and xs' :: "ct list"   and ys :: "ct list" assumes "ct_suffix (xs @ xs') ys" shows "ct_suffix xs' ys"
igSubstIPresIGWls_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstIPresIGWls MOD \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> igWls MOD s (igSubst MOD ys Y y X)"
expr_type_Some_iff: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type" shows "(expr_type \<Gamma> e = Some t) = (\<Gamma> \<turnstile> e : t)"
monoid_axioms: shows "monoid M (\<cdot>) \<one>"
rep_eq: fixes x :: "uint64"   and xa :: "nat"   and xb :: "bool" shows "Rep_uint64 (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (Rep_uint64 x) xa xb"
jD: shows "j \<in> {1..c * \<delta> p}"
insertion_isovarspars_free: fixes L :: "real list"   and var :: "nat"   and x :: "real"   and p :: "real mpoly"   and i :: "nat"   and y :: "real" shows "insertion (nth_default 0 (L[var := x])) (isolate_variable_sparse p var i) = insertion (nth_default 0 (L[var := y])) (isolate_variable_sparse p var i)"
Equiv_def: fixes P :: "('a, 'b, 'c) formula"   and Q :: "('a, 'b, 'c) formula" shows "(P \<leftrightarrow> Q) = (P && Q || ! P && ! Q)"
assms_1: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}"
rel_refl_strong: fixes x :: "'a stream"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> sset x \<Longrightarrow> Ra z z" shows "stream_all2 Ra x x"
ih_5: shows "InvariantClCurrentLevel (getCl state') (getM state')"
fps_conv_radius_const: fixes c :: "'a" shows "fps_conv_radius (fps_const c) = \<infinity>"
pow_cong: fixes x :: "int"   and n :: "nat" shows "(x mod int p) [^]\<^bsub>residue_ring (int p)\<^esub> n = x ^ n mod int p"
correct_count: shows "\<forall>t. np - maxfaults \<le> count (\<lambda>p. correct p t) np"
tjs_union: fixes u :: "'c"   and tjs :: "'c list" assumes "u \<in> set tjs" shows "set tjs = set (tl (dropWhile ((\<noteq>) u) tjs)) \<union> insert u (set (takeWhile ((\<noteq>) u) tjs))"
gcd'_dvd2: fixes gcd' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "is_gcd gcd'" shows "gcd' a b dvd b"
score_state_pair_def: fixes strat :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and s2 :: "nat" shows "score_state_pair strat e s1 s2 = (let T1 = Inference.outgoing_transitions s1 e; T2 = Inference.outgoing_transitions s2 e in fSum ((\<lambda>((uu_, uu_, t1), uu_, uu_, t2). strat t1 t2 e) |`| (T1 |\<times>| T2)))"