hcju/isabelleps huggingface dataset:
Cond1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and e\<^sub>1 :: "'addr expr1"   and e\<^sub>2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>if (Throw a) e\<^sub>1 else e\<^sub>2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
path_connected_sphere_eq: fixes a :: "'a"   and r :: "real" shows "path_connected (sphere a r) = (2 \<le> DIM('a) \<or> r \<le> 0)"
expands_to_imp_eventually_neg: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "trimmed_neg F" shows "\<forall>\<^sub>F x in at_top. f x < 0"
dg_op_dg_op_dg: shows "op_dg (op_dg \<AA>) = \<AA>"
global_assms_simps: fixes a :: "'form"   and i :: "'var itself" shows "(a |\<in>| global_assms i) = (\<exists>v p. a = labelAtOut v (Reg p) \<and> v |\<in>| vertices \<and> nodeOf v = Assumption p)"
hyps: fixes pns :: "'a list"   and Ts :: "'b list"   and E :: "'a \<Rightarrow> 'b option" assumes "length pns = length Ts"   and "distinct pns"   and "n < length Ts" shows "(E(pns [\<mapsto>] Ts)) (pns ! n) = \<lfloor>Ts ! n\<rfloor>"
eval_Z': fixes xs :: "nat list" assumes "length xs = 1" shows "eval Z xs \<down>= 0"
hom_0: fixes x :: "int poly" assumes "of_int_poly x = 0" shows "x = 0"
HEndPhase2_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv5_inner s q" shows "HInv5_inner s' q"
stalk_eq: shows "\<forall>U\<in>I. \<FF>1 U = \<FF>2 U \<and> add_str1 U = add_str2 U \<and> mult_str1 U = mult_str2 U \<and> zero_str1 U = zero_str2 U \<and> one_str1 U = one_str2 U"
IH: assumes "set Vs \<subseteq> carrier_vec n" shows "lincomb_list (\<lambda>i. a * c i) Vs = a \<cdot>\<^sub>v lincomb_list c Vs"
isPartition_wrt_trans: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" assumes "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"   and "isPartition_wrt R xs lo hi p" shows "\<forall>i j. lo \<le> i \<and> i < p \<and> p < j \<and> j \<le> hi \<longrightarrow> R (xs ! i) (xs ! j)"
simps_4: fixes Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "pred (Q\<hookleftarrow>\<^bsub>p\<^esub>f) (cf # cfs) = (Q cf \<and> cfs \<noteq> [])"
of_int_1_less_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less one (ring_1.of_int one plus zero minus z) = (1 < z)"
adv_P1_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd \<times> 'e) spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and Out1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and m1 :: "'a"   and m2 :: "'b"   and D :: "'c \<times> 'd \<times> 'e \<Rightarrow> bool spmf" shows "sim_non_det_def.adv_P1 R1 S1 Out1 funct m1 m2 D \<equiv> \<bar>spmf (R1 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). sim_non_det_def.Ideal1 S1 Out1 m1 m2 o1 \<bind> D)) True\<bar>"
put_eq: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and k :: "'a"   and u :: "'a"   and \<rho> :: "'b"   and v :: "'a" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
group_action_def: fixes G :: "('d, 'e) monoid_scheme"   and E :: "'f set"   and \<phi> :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "group_action G E \<phi> \<equiv> group_hom G (BijGroup E) \<phi>"
43: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> - y) \<sqinter> y = x \<sqinter> y"
code_gen_free_hull: fixes C :: "'b list set" assumes "code C" shows "\<langle>C\<rangle>\<^sub>F = \<langle>C\<rangle>"
elims: fixes x :: "fml option"   and y :: "game option" assumes "Testo x = y" obtains \<phi> :: "fml" where "x = Afml \<phi>" and "y = Agame (? \<phi>)"   | "x = undeff" and "y = undefg"
padd_correct: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b \<times> 'b \<times> 'b"   and p\<^sub>2 :: "'b \<times> 'b \<times> 'b" assumes "on_curvep a b p\<^sub>1"   and "on_curvep a b p\<^sub>2" shows "make_affine (padd a p\<^sub>1 p\<^sub>2) = ell_field_class.add a (make_affine p\<^sub>1) (make_affine p\<^sub>2)"
TS_prefix: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) u = 0"
eval_append_aux: fixes p :: "'a list"   and b :: "'a"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "b \<in> carrier R"   and "a \<in> carrier R" shows "local.eval (p @ [b]) a = local.eval p a \<otimes> a \<oplus> b"
swap_fresh_fresh: fixes a :: "atom"   and x :: "'e"   and b :: "atom" assumes "a \<sharp> x"   and "b \<sharp> x" shows "(a \<rightleftharpoons> b) \<bullet> x = x"
gf_h: fixes x :: "'V'" assumes "x \<in> vertices G'" shows "(g \<circ> f) x = h x"
int_cases2: fixes z :: "int" obtains n :: "nat" where "z = int nb__"   | n :: "nat" where "z = - int n"
setbydecision_setbydecision_all_Allow: fixes iface :: "iface"   and rs :: "'b common_primitive rule list" shows "setbydecision TYPE('c) iface rs FinalAllow - setbydecision_all TYPE('c) iface rs FinalDeny = setbydecision TYPE('c) iface rs FinalAllow"
p'_def: shows "p' = p div gcd p (pderiv p)"
mult_integer_group: fixes x :: "int"   and y :: "int" shows "x \<otimes>\<^bsub>integer_group\<^esub> y = x + y"
eval_generalVS'': fixes L :: "atomUni list"   and x :: "real" shows "(\<exists>x. evalUni (list_conj_Uni (map AtomUni L)) x) = evalUni (generalVS_DNF L) x"
Q_implies_path: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "I \<subseteq> A"   and "Q I a b" shows "b \<in> A \<and> a \<in> A"
contD: fixes luba :: "'h set \<Rightarrow> 'h"   and orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and lubb :: "'i set \<Rightarrow> 'i"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i"   and Y :: "'h set" assumes "cont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
step: fixes \<sigma> :: "'a"   and p :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and p' :: "'b" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P \<sigma>"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P \<sigma>'"
H: shows "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) = fact m"
is_processT3_ST_pref: fixes t :: "'c event list"   and P :: "'c process"   and s :: "'c event list" assumes "t \<in> T P"   and "s \<le> t" shows "s \<in> T P"
asIMOD_igSubstAllIPresIGWlsAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstAllIPresIGWlsAll (asIMOD SEM)"
Abs_fm_cases: fixes x :: "fm" obtains y :: "fm_raw set" where "x = Abs_fm y" and "y \<in> {c. \<exists>x. alpha_fm_raw x x \<and> c = Collect (alpha_fm_raw x)}"
group_hom_axioms: shows "group_hom \<F>\<^bsub>{()}\<^esub> int_group (int.lift f)"
locked_in_def: fixes s :: "v_state"   and r :: "nat"   and v :: "val" shows "locked_in s r v = locked_in_vf (votes s r) v"
ring_hom_memI: fixes R :: "('c, 'd) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>" shows "h \<in> ring_hom R S"
chars_of_token_simp: fixes a :: "symbol"   and b :: "character list" shows "chars_of_token (a, b) = b"
case_eq_if: fixes f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c"   and enat :: "enat" shows "(case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = (if enat = 0 then f1 else f2 (epred enat))"
Infinitesimal_add_right_cancel: fixes y :: "'a star"   and x :: "'a star"   and z :: "'a star" assumes "y \<in> Infinitesimal"   and "x \<approx> z + y" shows "x \<approx> z"
elims: fixes x :: "nat"   and xa :: "atom"   and y :: "atom fm" assumes "substNegInfinity x xa = y" obtains var :: "nat" and p :: "real mpoly" where "x = vara__" and "xa = Eq p__" and "y = allZero p__ vara__"   | var :: "nat" and p :: "real mpoly" where "x = varb__" and "xa = Less pa__" and "y = alternateNegInfinity pa__ varb__"   | var :: "nat" and p :: "real mpoly" where "x = varc__" and "xa = Leq pb__" and "y = Or (alternateNegInfinity pb__ varc__) (allZero pb__ varc__)"   | var :: "nat" and p :: "real mpoly" where "x = var" and "xa = Neq p" and "y = Neg (allZero p var)"
ord_spmf_eqD_pmf_None: fixes Y :: "'c spmf set"   and x :: "'d spmf"   and y :: "'d spmf" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "ord_spmf (=) x y" shows "pmf y None \<le> pmf x None"
simps: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and r :: "'d \<Rightarrow> nat option"   and \<nu> :: "'c"   and q :: "'d" shows "semi_mojmir_def.pre_ranks \<Sigma> \<delta> q\<^sub>0 r \<nu> q = {i. \<exists>q'. r q' = Some i \<and> q = \<delta> q' \<nu>} \<union> (if q = q\<^sub>0 then {semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0} else {})"
Jmp: fixes G :: "prog"   and A :: "'b triple set"   and j :: "jump"   and P :: "'b assn" shows "G,A\<turnstile>{Normal (abupd (\<lambda>a. Some (Jump j)) .; P\<leftarrow>\<diamondsuit>)} .Jmp j. {P}"
length_update: fixes b :: "'c array"   and i :: "nat"   and v :: "'c"   and h :: "heap" shows "Array.length (Array.update b i v h) = Array.length h"
cols_mat_mul: fixes A :: "'b mat"   and B :: "'b mat" shows "set (cols (A * B)) = (*\<^sub>v) A ` set (cols B)"
res_one_eq: fixes m :: "int" assumes "residues m" shows "\<one>\<^bsub>residue_ring m\<^esub> = 1"
before_in_index2: fixes l :: "'b list"   and x :: "'b"   and y :: "'b" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 1 else 0) = index l y"
det_I: shows "Determinants.det (Finite_Cartesian_Product.mat (1::'b)) = (1::'b)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Sum_Type.Inl x \<Rightarrow> f1 x | Sum_Type.Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Sum_Type.Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Sum_Type.Inr x2 \<longrightarrow> P (f2 x2)))"
sum_mult_indicator: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "finite A" shows "(\<Sum>x\<in>A. f x * indicator B x) = sum f (A \<inter> B)"
simple_bochner_integrable_compose2: fixes p :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and M :: "'f measure"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" assumes "p (0::'d) (0::'e) = (0::'c)"   and "simple_bochner_integrable M f"   and "simple_bochner_integrable M g" shows "simple_bochner_integrable M (\<lambda>x. p (f x) (g x))"
marked_imp_black_or_grey_1: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and r :: "'g" assumes "marked r s" shows "black r s \<or> grey r s"
valid_cons: fixes \<Gamma> :: "binding list"   and x :: "vrs"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "x \<sharp> trm_dom \<Gamma>"   and "T closed_in \<Gamma>" shows "\<turnstile> (VarB x T # \<Gamma>) ok"
R\<^sub>N_def: fixes p :: "real"   and n :: "nat"   and u :: "nat"   and l :: "nat" shows "random_skip_list.R\<^sub>N p n u l = random_skip_list.R p {..<n} u l"
f_base2: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "fb2 \<le> f x"
inv_into_image_cancel: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and S :: "'a set" assumes "inj_on f A"   and "S \<subseteq> A" shows "inv_into A f ` f ` S = S"
cong_3_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Cong3 Cong A' B' C' A B C"
funpow_cycle_mult: fixes k :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and m :: "nat" assumes "(f ^^ k) x = x" shows "(f ^^ (m * k)) x = x"
ts_prop_2: fixes k :: "cnf" shows "\<parallel>the_singleton\<parallel>\<^bsub>k\<^esub>"
IrrFinGroupRepresentation_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" shows "IrrFinGroupRepresentation G smult V \<equiv> FinGroupRepresentation G smult V \<and> IrrFinGroupRepresentation_axioms G smult V"
set_bd_1: fixes x :: "('d, 'e, 'f) F1" shows "|F1set1 x| \<le>o bd_F1"
power_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times one n = one"
assms_5: shows "(r, Store e2 mo2 r) \<in> set (node_deliver_messages xs)"
Map_memb_in_hhom: fixes F :: "('a, 'a set) arr"   and A :: "('a, 'a set) arr"   and B :: "('a, 'a set) arr"   and f :: "'a" assumes "\<guillemotleft>F : A \<rightarrow> B\<guillemotright>"   and "f \<in> Map F" shows "\<guillemotleft>f : local.Dom A \<rightarrow>\<^sub>B local.Dom B\<guillemotright>"
pop_commute: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<plusminus>\<^bsub>R\<^esub> b = b \<plusminus>\<^bsub>R\<^esub> a"
Ri_def: fixes N :: "'a literal multiset set" shows "Ri N = {\<gamma> \<in> \<Gamma>. concl_of \<gamma> \<in> N}"
P1: shows "Bet A' M'' B' \<and> A M B Cong3 A' M'' B'"
cnj_alpha_l: fixes l :: "nat" shows "conjugate (alpha_l l) = alpha_l l"
fps_binomial_power: fixes a :: "'a"   and n :: "nat" shows "fps_binomial a ^ n = fps_binomial (of_nat n * a)"
lossless_fast_dice_roll: fixes c :: "nat"   and v :: "nat" assumes "c < v"   and "v \<le> n" shows "lossless_spmf (local.fast_dice_roll v c)"
SAMEQUOT_def: fixes a :: "'b"   and A :: "'b list set" shows "SAMEQUOT a A = {fin_cut_same a x @ replicate m a |x m. x \<in> A}"
padd_Nil2: fixes p :: "'a list" shows "p +++ [] = p"
span_span: fixes A :: "'b set" shows "local.span (local.span A) = local.span A"
trivial_limit_at_left_real: fixes x :: "'a" shows "at_left x \<noteq> \<bottom>"
\<phi>h: shows "\<phi> (C.cod g, C.dom f) h \<in> local.set (C.cod g, C.dom f)"
assms_1: shows "summable (\<lambda>k. norm (a k))"
extends_subst_def: fixes \<sigma> :: "'a \<Rightarrow> 'b option"   and \<tau> :: "'a \<Rightarrow> 'b option" shows "extends_subst \<sigma> \<tau> = (\<forall>x\<in>dom \<sigma>. \<sigma> x = \<tau> x)"
accessible_fieldD: fixes G :: "prog"   and accC :: "qtname"   and membr :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> membr of C accessible_from accC"   and "is_field membr" shows "G \<turnstile> membr member_of C \<and> G \<turnstile> Class C accessible_in pid accC \<and> G \<turnstile> membr in C permits_acc_from accC"
star_inductr_var_eq: fixes y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" assumes "y O x = y" shows "y O x\<^sup>* \<subseteq> y"
WHILEI_le_WHILEIT: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and s :: "'a" shows "WHILE\<^bsup>I\<^esup> b f s \<le> WHILE\<^sub>T\<^bsup>I\<^esup> b f s"
pmf_of_set_code_aux: fixes A :: "'b set"   and xs :: "'b list" assumes "A \<noteq> {}"   and "set xs = A"   and "distinct xs" shows "mapping_of_pmf (pmf_of_set A) = Mapping.tabulate xs (\<lambda>_. 1 / real (length xs))"
wlog_non_Red_F: fixes DD0 :: "'f set"   and N :: "'f set"   and CC :: "'f set"   and E :: "'f"   and D :: "'f" assumes "DD0 \<subseteq> N"   and "DD0 \<union> CC \<Turnstile> {E}"   and "\<forall>D'\<in>DD0. D' < D" shows "\<exists>DD\<subseteq>N - Red_F N. DD \<union> CC \<Turnstile> {E} \<and> (\<forall>D'\<in>DD. D' < D)"
kernel_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" shows "kernel G H h = {x \<in> carrier G. h x = \<one>\<^bsub>H\<^esub>}"
simps_2: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" shows "split_sign_changes [p, q] x = [[p, q]]"
unfolding_prop_unfold_idem: fixes \<phi> :: "'b ltln" shows "Unf \<phi> \<sim>\<^sub>Q \<phi>"
G4_def: shows "G4 = sdiv_poly (pseudo_mod G2 G3) ((- (1::'b)) ^ (degree G2 - degree G3 + 1) * lead_coeff G2 * h2 ^ (degree G2 - degree G3))"
unitaryI: fixes P :: "'b \<Rightarrow> real" assumes "sound P"   and "bounded_by 1 P" shows "unitary P"
top_elim: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<le> x \<cdot> \<top>"
snd_half_is_partial_heap: fixes l :: "int"   and r :: "int"   and m :: "int"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a" assumes "(l + r) div 2 \<le> m" shows "is_partial_heap cmp heap l m r"
explicationGoodness3: shows "\<forall>c x xa. \<lfloor>\<^bold>\<diamond>\<^sub>px\<rfloor>\<^sub>c \<longrightarrow> ob (\<lambda>w. \<lfloor>Good xa x\<rfloor>\<^sup>D) (x c)"
Gromov_product_at_quasi_isometry_1: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "lambda C -quasi_isometry f" shows "Gromov_product_at x y z / lambda - 187 * lambda\<^sup>2 * (C + deltaG TYPE('c) + deltaG TYPE('d)) \<le> Gromov_product_at (f x) (f y) (f z)"
subst_pp_trivial: fixes t :: "'d \<Rightarrow>\<^sub>0 nat" shows "subst_pp (\<lambda>_. 0) t = (if t = 0 then 1 else 0)"
assms_2: shows "B \<in> carrier_mat (m - 1) n"
normal_vector_7: fixes W :: "'a vec set"   and Bnd :: "'a" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n"   and "W \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec Bnd" shows "normal_vector W \<in> \<int>\<^sub>v \<inter> Bounded_vec (det_bound (n - 1) Bnd)"
finsum_cong': fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'a"   and f :: "'e \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier R"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum R f A = finsum R g B"
MGT\<^sub>e_def: fixes e :: "expr"   and Z :: "state" shows "MGT\<^sub>e e Z \<equiv> ((=) Z, e, \<lambda>v t. \<exists>n. Z -e\<succ>v-n\<rightarrow> t)"
take_hd_drop: fixes n :: "nat"   and xs :: "'a list" assumes "n < |xs|" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
Inum_Mul: fixes c :: "nat"   and t :: "Reflection_Examples.num"   and vs :: "nat list" shows "Inum (num.Mul c t) vs = c * Inum t vs"
not_one_le_zero: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
contour_integrable_neg_iff: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" shows "((\<lambda>x. - f x) contour_integrable_on g) = (f contour_integrable_on g)"
the_coext_prop_1: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' y f" shows "\<guillemotleft>the_coext x' f : x' \<rightarrow>\<^sub>D x\<guillemotright>"
Rats_induct: fixes q :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "q \<in> \<rat>"   and "\<And>r. P (of_rat r)" shows "P q"
add_implies_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero"   and "plus c b = a" shows "c = minus a b"
eq_iff: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "(a = b) = (less_eq_prod b a \<and> less_eq_prod a b)"
nat_pow_mult: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier G" shows "x [^] n \<otimes> x [^] m = x [^] (n + m)"
local_lipschitz_def: fixes T :: "'d set"   and X :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "local_lipschitz T X f \<equiv> \<forall>x\<in>X. \<forall>t\<in>T. \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
nonneg_matpow: fixes n :: "nat"   and i :: "'n"   and j :: "'n" shows "(0::'a) \<le> matpow A n $h i $h j"
prod_le_def: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "prod_le P1 P2 = (\<lambda>(p1, p2) (q1, q2). P1 p1 q1 \<and> P2 p2 q2)"
d2: fixes d2 :: "nat"   and e2 :: "nat"   and x :: "nat"   and b :: "nat"   and r :: "nat"   and s1 :: "nat"   and y :: "nat"   and s2 :: "nat" assumes "d2 = ((e2 + q - x) mod q * b + (q - r)) mod q"   and "s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q"   and "s2 = (x * y + (q - s1)) mod q"   and "x < q"   and "y < q"   and "r < q"   and "b < q"   and "e2 < q" shows "d2 = (e2 * b + (q - s2)) mod q"
entropy_density_def: fixes b :: "real"   and M :: "'d measure"   and N :: "'d measure" shows "entropy_density b M N = log b \<circ> enn2real \<circ> RN_deriv M N"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union> A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)"
monofunE: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "monofun f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
viewR_def: fixes t :: "('c, 'd) FingerTree" shows "viewR t \<equiv> extract_viewres (viewR_aux t)"
cet_less_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : INTEG"   and "\<Gamma> \<turnstile>\<^sub>c e2 : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c e1 <\<^sub>c e2 : BOOL"
simps_9: fixes x101 :: "stmt"   and x102 :: "stmt"   and y101 :: "stmt"   and y102 :: "stmt" shows "(x101 Finally x102 = y101 Finally y102) = (x101 = y101 \<and> x102 = y102)"
divide_eq_0_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = zero) = (a = zero \<or> b = zero)"