hcju/isabelleps huggingface dataset:
of_nat_0_less_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (semiring_1.of_nat one plus zero n) = (0 < n)"
mem_in_image: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and a :: "'h" assumes "f \<in> A \<rightarrow> B"   and "a \<in> A" shows "f a \<in> f ` A"
sup: fixes x :: "'a" shows "(SUP i. U i x) = u x"
IH: fixes \<delta> :: "V" assumes "\<delta> \<in> elts (\<omega>_sum \<alpha>s)" shows "\<exists>k<length \<alpha>s. \<exists>\<gamma>\<in>elts (\<omega> \<up> \<alpha>s ! k). \<delta> = \<omega>_sum (take k \<alpha>s) + \<gamma>"
l2_inv4'_trans: shows "{l2_inv4'} TS.trans l2 {> l2_inv4'}"
steps'_set: fixes qs :: "'b list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'b list" assumes "length qs = length as"   and "n \<le> length as" shows "set (steps' s qs as n) = set s"
a: shows " M \<longrightarrow>\<^sub>1 M'"
simps_4: fixes uv :: "'sets set" shows "arity (Abs uv) = 0"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "f x \<in> carrier C"
f_def: shows "f \<equiv> JVM_SemiType.sup P mxs mxl"
frag_cmul_sum: fixes a :: "int"   and b :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and I :: "'c set" shows "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"
image: fixes A :: "'a set"   and B :: "'a set" shows "(\<lambda>(x, y). x - y) ` (A \<times> B) = {x - y |x y. x \<in> A \<and> y \<in> B}"
poincare_mapsto_def: fixes P :: "'d set"   and X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and S :: "'d set"   and CX :: "'d set"   and Y :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" shows "poincare_mapsto P X0 S CX Y = (\<forall>(x, d)\<in>X0. returns_to P x \<and> fst ` X0 \<subseteq> S \<and> return_time P differentiable at x within S \<and> (\<exists>D. (poincare_map P has_derivative blinfun_apply D) (at x within S) \<and> (poincare_map P x, D o\<^sub>L d) \<in> Y) \<and> (\<forall>t\<in>{0<..<return_time P x}. flow0 x t \<in> CX))"
in_Under_above: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<in> Field r" shows "a \<in> Under r (above r a)"
i: fixes i :: "nat" shows "mkdisjoint A i \<in> measurable_sets M"
rng_f: shows "range \<F> \<subseteq> \<H>"
stream_pred_snth: fixes P :: "'d \<Rightarrow> bool"   and w :: "'d stream" shows "pred_stream P w = (\<forall>i. P (w !! i))"
lprefix_llength_eq_imp_eq: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys"   and "llength xs = llength ys" shows "xs = ys"
assms: shows "max_Var_floatarith ea \<le> i"
hfcomp: fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and RR' :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and h :: "'e \<Rightarrow> 'f nres"   and Q :: "'e \<Rightarrow> bool"   and T :: "('c \<times> 'e) set"   and U :: "('d \<times> 'f) set" assumes "(f, g) \<in> [P]\<^sub>a RR' \<rightarrow> S"   and "(g, h) \<in> [Q]\<^sub>f T \<rightarrow> \<langle>U\<rangle>nres_rel" shows "(f, h) \<in> [\<lambda>a. Q a \<and> (\<forall>a'. (a', a) \<in> T \<longrightarrow> P a')]\<^sub>a hrp_comp RR' T \<rightarrow> hr_comp S U"
class_add_wt_start: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and mxl :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "wt_start P C\<^sub>0 b Ts mxl \<tau>s"   and "\<not> is_class P C" shows "wt_start (class_add P (C, cdec)) C\<^sub>0 b Ts mxl \<tau>s"
queue_rotate1_def: shows "queue_rotate1 = (\<lambda>(x, y). queue_enqueue x y) \<circ> queue_dequeue"
assoc: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and c :: "('a \<times> 'a) set" shows "a ; b ; c = a ; (b ; c)"
l11_57: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and A' :: "'a"   and B :: "'a"   and B' :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet A A' B B'"   and "Tarski_neutral_dimensionless.Per Bet Cong B A A'"   and "Tarski_neutral_dimensionless.Per Bet Cong B' A' A"   and "Tarski_neutral_dimensionless.OS Bet A A' C C'"   and "Tarski_neutral_dimensionless.Per Bet Cong C A A'"   and "Tarski_neutral_dimensionless.Per Bet Cong C' A' A" shows "Tarski_neutral_dimensionless.CongA Bet Cong B A C B' A' C'"
mxl_def: shows "mxl \<equiv> (case b of Static \<Rightarrow> 0 | NonStatic \<Rightarrow> 1) + length Ts + mxl\<^sub>0"
D'_def: shows "D' = abstr (inv_of A l') (\<lambda>i j. \<infinity>) v"
step_3: shows "\<forall>n'>n'. n' \<le> n \<longrightarrow> (\<exists>nid'. (n', nid') \<in> his t n nid) \<longrightarrow> (\<exists>!nid'. (n', nid') \<in> his t n nid) \<and> ((\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid) \<longrightarrow> (\<exists>x. (\<exists>na. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' na) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))) \<and> (hisPred t n nid n', SOME nid'. (hisPred t n nid n', nid') \<in> his t n nid) = (SOME x. (\<exists>na. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' na) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))))"
IP: fixes \<sigma> :: "'a"   and k :: "'x"   and it :: "'x set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. j \<le> k"   and "\<forall>j\<in>\<alpha> s - it. k \<le> j"   and "it \<subseteq> \<alpha> s"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
divide_self: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "a / a = (1::'a)"
boxNotInitD: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<not> Init F \<equiv> \<box>\<not> F"
1: shows "p = (\<Sum>x | x \<in> keys (mapping_of p) \<and> x' \<in> keys x. MPoly_Type.monom x (MPoly_Type.coeff p x)) + (\<Sum>x | x \<in> keys (mapping_of p) \<and> x' \<notin> keys x. MPoly_Type.monom x (MPoly_Type.coeff p x))"
pol_ct: fixes \<sigma>1 :: "'tp"   and P :: "'psym"   and \<sigma>2 :: "'tp" shows "pol \<sigma>1 P = pol \<sigma>2 P"
refTE: fixes T :: "ty" assumes "is_refT T" obtains "T = NT"   | C :: "char list" where "T = Class C"
child_estimate_child: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and l :: "nat"   and i :: "int"   and p' :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p"   and "l \<le> lv p d"   and "ix p d < (i + 1) * 2 ^ (lv p d - l) \<and> (i - 1) * 2 ^ (lv p d - l) < ix p d"   and "p' = child p dir d" shows "ix p' d < (i + 1) * 2 ^ (lv p' d - l) \<and> (i - 1) * 2 ^ (lv p' d - l) < ix p' d"
nth_zip: fixes i :: "nat"   and xs :: "'g list"   and ys :: "'h list" assumes "i < length xs"   and "i < length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
interior_convex_hull_eq_empty: fixes S :: "'a set" assumes "card S = Suc DIM('a)" shows "(interior (convex hull S) = {}) = affine_dependent S"
simps_6: fixes x2 :: "('c, 'd) Term.term"   and x1 :: "('c, 'd) Term.term" shows "receive\<langle>x2\<rangle>\<^sub>s\<^sub>t \<noteq> send\<langle>x1\<rangle>\<^sub>s\<^sub>t"
floor_diff_numeral: fixes x :: "'b"   and v :: "num" shows "\<lfloor>x - numeral v\<rfloor> = \<lfloor>x\<rfloor> - numeral v"
sol: shows "(sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<psi>}"
wf_Int1: fixes r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set" assumes "wf r" shows "wf (r \<inter> r')"
linorder_sorted_wrt_list_of_set_1: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and B :: "'b set" assumes "linorder_on A R"   and "finite B"   and "B \<subseteq> A" shows "set (sorted_wrt_list_of_set R B) = B"
succ_in_Field: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "succ i \<in> Field r"
apply_infmatrix_closed: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c matrix" assumes "f (0::'c) = (0::'b)" shows "Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
frac_from_carrier_local: fixes X :: "('a \<times> 'a) set" assumes "X \<in> pi.carrier_local_ring_at" obtains r :: "'a" and s :: "'a" where "r \<in> R" and "s \<in> R" and "s \<notin> \<pp>" and "X = local.frac r s"
p_not_zero: shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
wpo_with_assms_axioms: shows "wpo_with_assms S NS prc prl \<sigma>\<sigma> ssimple large"
less_restrictive_def: fixes R1 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and R2 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" shows "less_restrictive R1 R2 = (\<forall>P1 P2 C. R2 P1 P2 C \<longrightarrow> R1 P1 P2 C \<or> R1 P2 P1 C)"
reducedly_statically_complete: fixes B :: "'a"   and N :: "'a set" assumes "B \<in> Bot"   and "reduc_saturated N"   and "N \<Turnstile> {B}" shows "\<exists>B'\<in>Bot. B' \<in> N"
safe_sblit_def: fixes a :: "'b array"   and s :: "nat"   and d :: "nat"   and l :: "nat" shows "safe_sblit a s d l \<equiv> if d < s then sblit a s d l else srblit a s d l"
x_prop: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> F \<^bold>= K) in dw]"
step: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r x y"   and "star r y z" shows "star r x z"
subst_ground_ident_compose_2: fixes t :: "('d, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv (t \<cdot> \<sigma>) = {}" shows "t \<cdot> \<sigma> \<circ>\<^sub>s \<theta> = t \<cdot> \<sigma>"
assms: fixes i :: "'a" shows "((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
Suc: shows "jordan_block n a ^\<^sub>m r = mat n n (\<lambda>b. case b of (i, j) \<Rightarrow> if i \<le> j then of_nat (r choose (j - i)) * a ^ (r + i - j) else (0::'a))"
iterates_above_le_f: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "a \<in> Field leq" shows "(x, f x) \<in> leq"
ternop_type_det: fixes op :: "string_ternop + collection_ternop"   and k :: "call_kind"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<upsilon>\<^sub>1 :: "'a OCL_Types.type"   and \<upsilon>\<^sub>2 :: "'a OCL_Types.type" assumes "ternop_type op k \<tau> \<sigma> \<rho> \<upsilon>\<^sub>1"   and "ternop_type op k \<tau> \<sigma> \<rho> \<upsilon>\<^sub>2" shows "\<upsilon>\<^sub>1 = \<upsilon>\<^sub>2"
a2: shows "\<forall>f\<in>pmdl G. f \<noteq> 0 \<longrightarrow> is_red G f"
lemma_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R"   and "incid A (line B' C') \<or> incid C' (line A B)"   and "incid C (line A' B') \<or> incid B' (line A C)"   and "incid B (line A' C') \<or> incid A' (line B C)" shows "col M N P \<or> triangle_circumscribes_triangle A B C A' B' C' \<or> triangle_circumscribes_triangle A' B' C' A B C"
Am_simp3': fixes c :: "'b"   and b :: "'b"   and rr :: "'b tree"   and l :: "'b tree"   and rl :: "'b tree" assumes "c < b"   and "bst rr"   and "rr \<noteq> \<langle>\<rangle>" shows "Am \<langle>l, c, \<langle>rl, b, rr\<rangle>\<rangle> = (case splay_max rr of \<langle>rrl, x, rrr\<rangle> \<Rightarrow> Am rr + local.\<phi> rrl \<langle>l, c, rl\<rangle> + local.\<phi> l rl - local.\<phi> rl rr - local.\<phi> rrl rrr + 1)"
prod_constant: fixes y :: "'b"   and A :: "'c set" shows "(\<Prod>x\<in>A. y) = y ^ card A"
igWlsAbs_termMOD: shows "igWlsAbs termMOD = wlsAbs"
ahm_rel_sv: fixes Rk :: "('a \<times> 'b) set"   and Rv :: "('c \<times> 'd) set"   and bhc :: "nat \<Rightarrow> 'a \<Rightarrow> nat" assumes "single_valued Rk"   and "single_valued Rv" shows "single_valued (\<langle>Rk, Rv\<rangle>Impl_Array_Hash_Map.ahm_rel bhc)"
inter_sym: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X Inter A B C D" shows "X Inter C D A B"
jonsson4: shows "(\<forall>x y. x \<lhd> y = x \<cdot> ((1::'a) \<lhd> y)) = (\<forall>x y. x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y)"
pigeonhole_CCs: fixes E1 :: "'edge set"   and E2 :: "'edge set" assumes "finite V"   and "card (CCs E1) < card (CCs E2)" shows "\<exists>u v. u \<in> V \<and> v \<in> V \<and> CC E1 u = CC E1 v \<and> CC E2 u \<noteq> CC E2 v"
blinfun_apply_inverse: fixes x :: "'b \<Rightarrow>\<^sub>L 'c" shows "Blinfun (blinfun_apply x) = x"
nrows_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "nrows (Finite_Cartesian_Product.transpose A) = ncols A"
scale_shift_ms_aux_MSLNil: fixes x :: "'a \<times> real" shows "scale_shift_ms_aux x MSLNil = MSLNil"
mn: shows "length (map of_int_hom.vec_hom fs) \<le> n"
that: fixes k :: "int" assumes "real_of_int k = real_of_float (floor_fl f)" shows "thesis"
leg_simps_1: shows "C.arr HfHgh.cod.leg0"
m2ZFinj_on: fixes T :: "(ZF, ZF) Axioms" assumes "ZFAxioms T" shows "inj_on m2ZF {TermEquivCl' T A e B |A e B. True}"
hq: shows "\<forall>m\<in>monomials q. lookup m var \<le> MPoly_Type.degree q var"
i_not_ia: shows "column i (fst (QR_decomposition A)) \<noteq> column ia (fst (QR_decomposition A))"
param_rec_list: fixes Ra :: "('a \<times> 'c) set"   and Rb :: "('b \<times> 'd) set" shows "(rec_list, rec_list) \<in> Ra \<rightarrow> (Rb \<rightarrow> \<langle>Rb\<rangle>list_rel \<rightarrow> Ra \<rightarrow> Ra) \<rightarrow> \<langle>Rb\<rangle>list_rel \<rightarrow> Ra"
SUP_bot_conv_1: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
\<omega>_def: shows "\<omega> \<equiv> set (range ord_of_nat)"
simps_3: fixes s :: "dB"   and i :: "nat" shows "free (Abs s) i = free s (i + 1)"
upd_left_comm: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" shows "comp_fun_commute (\<lambda>a. (case a of (a, b) \<Rightarrow> \<lambda>c f. finfun_update f a (finfun_update (finfun_apply f a) b c)) (f a))"
span_superset: fixes S :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "S \<subseteq> pmdl S"
bigo_lesseq1: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(h)"   and "\<forall>x. \<bar>g x\<bar> \<le> \<bar>f x\<bar>" shows "g \<in> O(h)"
SN_rel_on_def: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "SN_rel_on R S \<equiv> SN_on (S\<^sup>* O R O S\<^sup>*)"
pol_deg_0: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "(deg R S X p = 0) = (p \<in> carrier S)"
subseteq_list_Union_mset: fixes Ci :: "'b multiset list"   and n :: "nat"   and CAi :: "'b multiset list" assumes "length Ci = n"   and "length CAi = n"   and "\<forall>i<n. Ci ! i \<subseteq># CAi ! i" shows "\<Sum>\<^sub># (mset Ci) \<subseteq># \<Sum>\<^sub># (mset CAi)"
rtranclp_reflclp_absorb: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "R\<^sup>*\<^sup>*\<^sup>=\<^sup>= = R\<^sup>*\<^sup>*"
END_def: fixes i :: "'b" shows "END i = {j |j. (j, i) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>}"
Gd12: fixes k :: "nat" assumes "k \<le> m" shows "rat_of_int (gs.Gramian_determinant fs k) = (\<Prod>j<k. \<parallel>fs.gs.gso j\<parallel>\<^sup>2)"
map_upd_triv: fixes t :: "'g \<Rightarrow> 'f option"   and k :: "'g"   and x :: "'f" assumes "t k = \<lfloor>x\<rfloor>" shows "t(k \<mapsto> x) = t"
inf_sup_aci_3: fixes 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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
invar_def: fixes ho :: "'b hp option" shows "invar ho = (case ho of None \<Rightarrow> True | Some h \<Rightarrow> php h)"
intros: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
ipresSubstAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresSubstAll h hA MOD \<equiv> ipresSubst h MOD \<and> ipresSubstAbs h hA MOD"
inject_2: fixes x2 :: "nat set"   and y2 :: "nat set" shows "(PubK x2 = PubK y2) = (x2 = y2)"
division_decomp: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (times b c)" shows "\<exists>b' c'. a = times b' c' \<and> dvd.dvd times b' b \<and> dvd.dvd times c' c"
2: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" assumes "A \<in> set ss" shows "phull.subspace A"
tl_generate_check_filter: fixes b :: "nat list"   and a :: "nat list"   and aa :: "nat list"   and ba :: "nat list" assumes "suffs (cond1 b (zeroes (length b))) a (zeroes (length a), 0)"   and "suffs (cond2 aa ba) b (zeroes (length b), 0)" shows "tl (c12_generate_check aa ba a b) = filter (\<lambda>(xs, ys). suffs (cond1 b (fst ys)) a xs \<and> suffs (cond2 aa ba) b ys) (tl (alls2 (Max (set b)) (Max (set aa)) a b))"
orbit_list_not_nil: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "orbit_list f x \<noteq> []"
true_cls_mset_add_mset: fixes I :: "'a set"   and C :: "'a literal multiset"   and CC :: "'a literal multiset multiset" shows "(I \<Turnstile>m add_mset C CC) = (I \<Turnstile> C \<and> I \<Turnstile>m CC)"
information_space_axioms: shows "information_space (point_measure keys (\<lambda>x. ennreal (K x))) b"
fun_def: fixes x :: "'a" shows "fun x = (if recfun_aux_dom x then recfun_aux x else b)"
vwalk_join_hd': fixes p :: "'a list"   and q :: "'a list" assumes "p \<noteq> []" shows "hd (p \<oplus> q) = hd p"
bound_on_all_plans_bounds_MPLS_thesis: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and k :: "nat"   and x :: "nat" assumes "finite PROB"   and "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < k)"   and "x \<in> MPLS PROB" shows "x < k"
add_new_table_vmsaux_def: shows "add_new_table_vmsaux = (\<lambda>args rel2 (cur, auxlist). (cur, case auxlist of [] \<Rightarrow> [(cur, rel2)] | (t, y) # ts \<Rightarrow> if t = cur then (t, y \<union> rel2) # ts else (cur, rel2) # auxlist))"
SN: shows "SN {(x, y). (0::'a) \<le> y \<and> x \<succ> y}"
FAcc1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and F :: "String.literal"   and D :: "String.literal"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<bullet>F{D},s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
sumlist_carrier: fixes xs :: "'a vec list" assumes "set xs \<subseteq> carrier_vec n" shows "M.sumlist xs \<in> carrier_vec n"
CallRec': fixes p :: "'d"   and Procs :: "'d set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and Q :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and A :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and F :: "'f set"   and Z :: "'g" assumes "p \<in> Procs"   and "Procs \<subseteq> dom \<Gamma>"   and "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>p\<in>Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)})\<turnstile>\<^bsub>/F \<^esub>(P p Z) the (\<Gamma> p) (Q p Z),(A p Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P p Z) Call p (Q p Z),(A p Z)"
arr_dom_iff_arr: fixes f :: "'a" shows "hom'.arr (hom'.dom f) = hom'.arr f"
mapl_FGco_def: fixes l1 :: "'o \<Rightarrow> 'ab" shows "mapl_FGco l1 = map_FGco l1 id id id id id id id id id id"
simps_7: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" shows "((w, I) \<Turnstile> FExists \<phi>) = (\<exists>p. (w, Inl p # I) \<Turnstile> \<phi>)"
dual_weak_order: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L" shows "weak_partial_order (inv_gorder L)"
assms_3: fixes r1 :: "'a"   and r2 :: "'a" assumes "r1 \<in> carrier R"   and "r2 \<in> carrier R" shows "P (r1 \<otimes> r2)"
wp_assigns_r: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and r :: "'a upred" shows "\<langle>\<sigma>\<rangle>\<^sub>a wp r = \<sigma> \<dagger> r"
impr_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a r\<rightarrow> b \<equiv> if second a = (1::'b) \<and> second b = (1::'b) then Abs_OrdSum (first a r\<rightarrow> first b, 1::'b) else if first a = (1::'a) \<and> first b = (1::'a) then Abs_OrdSum (1::'a, second a r\<rightarrow> second b) else if first a = (1::'a) \<and> second b = (1::'b) then b else 1"
special_hyperplane_span: fixes k :: "'a" assumes "k \<in> Basis" shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
emeasure_eq_1_AE: fixes M :: "'b measure"   and S :: "'b set" assumes "prob_space M"   and "S \<in> sets M"   and "AE x in M. x \<in> S" shows "emeasure M S = 1"
orthogonal_transformation_def: fixes f :: "'a \<Rightarrow> 'b" shows "orthogonal_transformation f = (linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w))"
distinct_1: fixes x1 :: "'a" shows "nSome x1 \<noteq> nNone"
pmf_return: fixes x :: "'a"   and y :: "'a" shows "pmf (return_pmf x) y = indicat_real {y} x"
rep_eq: fixes x :: "nat set" shows "lookup (monom_of_set x) = (\<lambda>xa. if finite x \<and> xa \<in> x then 1 else 0)"
rcosets_mult_eq: fixes M :: "'a set" assumes "M \<in> rcosets H" shows "H <#> M = M"
g_def: shows "g \<equiv> \<lambda>z. \<Sum>i\<in>Basis. (z \<bullet> basf i) *\<^sub>R i"