hcju/isabelleps huggingface dataset:
cSUP_upper: 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 x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> A"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (f x) (Sup (f ` A))"
timpl_closure_is_timpl_closure': fixes s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "(s \<in> timpl_closure t TI) = ((t, s) \<in> timpl_closure' TI)"
suminf_scaleR_right: fixes X :: "nat \<Rightarrow> 'a"   and r :: "real" assumes "summable X" shows "r *\<^sub>R (\<Sum>n. X n) = (\<Sum>n. r *\<^sub>R X n)"
value_in_aug_inf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "v x \<in> Z\<^sub>\<infinity>"
v2: shows "atom v2 \<sharp> (t, z, X0, v1, X3)"
iso_tuple_update_swap_fst_snd: fixes h :: "'f \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'f"   and j :: "'g \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'g" shows "(iso_tuple_snd_update isom \<circ> h) f \<circ> (iso_tuple_fst_update isom \<circ> j) g = (iso_tuple_fst_update isom \<circ> j) g \<circ> (iso_tuple_snd_update isom \<circ> h) f"
subset': shows "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> A"
Lstar: fixes M :: "nat rexp set" shows "L (starS M) = star (L M)"
enc_reflects_pred_iff_source_target_rel_reflects_pred_1: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_reflects_pred Rel Pred)"
one_closed: shows "\<one>\<^bsub>DirProds Hs J\<^esub> \<in> carrier (DirProds Hs J)"
relImage_def: fixes R :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'b" shows "relImage R f \<equiv> {(f a1, f a2) |a1 a2. (a1, a2) \<in> R}"
assms_3: shows "no_spoofing TYPE('pkt_ext) ipassmt rs"
mem_multiset_diff: fixes x :: "'b"   and A :: "'b multiset"   and y :: "'b" assumes "x \<in># A"   and "x \<noteq> y" shows "x \<in># A - {#y#}"
fib_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P 1"   and "\<And>n. \<lbrakk>P (n + 1); P n\<rbrakk> \<Longrightarrow> P (n + 2)" shows "P n"
homotopic_loops_def: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" shows "homotopic_loops s p q \<equiv> homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
sl2: shows "- 2 * a * e\<^sup>2 \<in> radical_sqrt"
a_consistent_edom_subsetD: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, e, S)" shows "edom ae \<subseteq> domA \<Gamma> \<union> upds S"
ipset_from_cidr_base_wellforemd: fixes l :: "nat"   and base :: "'b word" assumes "mask (LENGTH('b) - l) && base = 0" shows "ipset_from_cidr base l = {base..base || mask (LENGTH('b) - l)}"
less_imp_Suc_mult_le: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m < n" shows "Suc m * k \<le> n * k"
make_vert_block_map_blockmat: fixes n :: "nat" shows "blockmat (make_vert_block n) = mat1 (2 ^ n)"
witness_used: fixes used :: "'b set"   and i :: "'b"   and block :: "('c, 'b) fm list \<times> 'b" assumes "infinite UNIV"   and "finite used"   and "i \<notin> block_nominals block" shows "i \<notin> block_nominals (witness block ({i} \<union> used))"
simps_6: fixes x3 :: "'b" shows "vertex.SINK \<noteq> Inner x3"
inR0: shows "([spawn\<^bsub>\<iota>\<^esub> V], [spawn\<^bsub>\<iota>\<^esub> V]) \<in> R0"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "(nat \<times> nat) option" assumes "prev x xa = y" obtains "x = 0" and "xa = 0" and "y = None"   | "x = Suc 0" and "xa = 0" and "y = None"   | v :: "nat" where "x = Suc (Suc v)" and "xa = 0" and "y = Some (Suc v, Suc v - Suc 0)"   | j :: "nat" and i :: "nat" where "x = j" and "xa = Suc i" and "y = Some (j, i)"
G_valid_validn: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and K :: "nat" assumes "G_valid G" shows "G_validn K G"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "a1 >\<^sub>t a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = ta__" and "a2 = sa__" and "is_App ta__" and "fun ta__ \<ge>\<^sub>t sa__ \<or> arg ta__ \<ge>\<^sub>t sa__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tb__" and "a2 = sb__" and "head tb__ >\<^sub>h\<^sub>d head sb__" and "chkvar tb__ sb__" and "chksubs (>\<^sub>t) tb__ sb__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t = head s" and "chksubs (>\<^sub>t) t s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
assms_1: shows "is_map_to_sorted_list ordR Rk Rv Rm tsl"
dimensionI: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and E :: "'a set" assumes "ring R"   and "subfield K R"   and "ring.independent R K Us"   and "ring.Span R K Us = E" shows "ring.dimension R (length Us) K E"
A: fixes y :: "val" shows "insert_var (Suc x) (case_nat y \<sigma>) (cexpr_sem \<sigma> e) = case_nat y (insert_var x \<sigma> (cexpr_sem \<sigma> e))"
neutral: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "\<forall>x\<in>A. g x = \<^bold>1" shows "F.F g A = \<^bold>1"
balL_def: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "balL l k v r = (let lr = lsub r in if cl l = R then Node (Node (lsub l) B (key l) (val l) (rsub l)) R k v r else if r = Leaf then Node l R k v r else if cl r = B then balance l k v (Node (lsub r) R (key r) (val r) (rsub r)) else if lr = Leaf then Node l R k v r else if cl lr = B then Node (Node l B k v (lsub lr)) R (key lr) (val lr) (balance (rsub lr) (key r) (val r) (paint R (rsub r))) else Node l R k v r)"
uminus_set_def: fixes A :: "'c set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
\<sigma>_4: shows "\<forall>s\<in>subst_range \<sigma>. \<forall>u\<in>subst_range \<sigma>. (\<exists>v. v \<sqsubseteq> s \<and> v \<sqsubseteq> u) \<longrightarrow> s = u"
simps: fixes xs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs'))"
preImage_insert: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and A :: "'d set" shows "preImage f (insert a A) = pre f a \<union> preImage f A"
eventually_Abs_filter: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
cring_simprules_6: shows "\<one> \<in> carrier R"
bet2_suma__suma: fixes A' :: "'p"   and B :: "'p"   and D' :: "'p"   and E :: "'p"   and A :: "'p"   and D :: "'p"   and C :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A' \<noteq> B"   and "D' \<noteq> E"   and "Bet A B A'"   and "Bet D E D'"   and "A B C D E F SumA G H I" shows "A' B C D' E F SumA G H I"
gec: shows "(b \<bullet> i - c) * sum ((\<lambda>K. content K / interv_diff K i) \<circ> (\<lambda>K. K \<inter> {x. c \<le> x \<bullet> i})) \<D> \<le> content (cbox a' b)"
orthogonal_imp_lin_indpt_list: fixes gs :: "'a vec list"   and n :: "nat" assumes "orthogonal gs"   and "set gs \<subseteq> carrier_vec n" shows "cof_vec_space.lin_indpt_list n gs"
rtranclp_eq_rtrancl_path: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)"
R_subset_Field: fixes R :: "('b \<times> 'b) set" shows "R \<subseteq> Field R \<times> Field R"
eval_neg_neg: fixes f :: "atom fm"   and L :: "real list" shows "eval (neg (neg f)) L = eval f L"
gram_schmidt_fs_int_axioms: shows "gram_schmidt_fs_int n fs"
to_nat_from_nat_id: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "x < CARD('b)" shows "mod_type.to_nat Rep (mod_type.from_nat Abs x) = x"
fin_dg_Obj_vfinite: shows "vfinite (\<AA>\<lparr>Obj\<rparr>)"
hom_mult: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "times_hom hom" shows "hom (x * y) = hom x * hom y"
a1: shows "fmdom' (snd a) \<subseteq> vs \<union> vs'"
ae_eq: shows "\<forall>w\<in>space M. (cls_val_process Mkt2 p 0 w = 0) = (cls_val_process Mkt p 0 w = 0)"
sphere_translation_subtract: fixes c :: "'b"   and a :: "'b"   and r :: "real" shows "sphere (c - a) r = (\<lambda>x. x - a) ` sphere c r"
divide_ln_mono: fixes x :: "real"   and y :: "real" assumes "3 \<le> x"   and "x \<le> y" shows "x / ln x \<le> y / ln y"
proj6_pair_snd: fixes dsn' :: "nat"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "\<pi>\<^sub>6 (dsn', snd r) = \<pi>\<^sub>6 r"
fold_true_invalid_fold_rev_false_invalid: fixes l :: "trilean list" assumes "fold (\<or>?) l true = invalid" shows "fold (\<or>?) (rev l) false = invalid"
valid_cfg_state_in_S: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "state cfg \<in> S"
continuous_map_componentwise_UNIV: fixes X :: "'a topology"   and Y :: "'b \<Rightarrow> 'c topology"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "continuous_map X (product_topology Y UNIV) f = (\<forall>k. continuous_map X (Y k) (\<lambda>x. f x k))"
a_absorb1: fixes x :: "'a"   and y :: "'a" shows "(ad x \<oplus> ad y) \<cdot> ad x = ad x"
simps_1: fixes f :: "expr" shows "f \<bullet> [] = f"
cases: fixes a1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and a2 :: "'e"   and a3 :: "'f \<Rightarrow> 'g"   and a4 :: "'h"   and a5 :: "'e"   and a6 :: "'f \<Rightarrow> 'g" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>\<^bsub>a4\<^esub> \<langle>a5, a6\<rangle>" obtains A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)" and l :: "'e" and g :: "('f, 'g) cconstraint" and a :: "'h" and r :: "'f list" and l' :: "'e" and u :: "'f \<Rightarrow> 'g" and u' :: "'f \<Rightarrow> 'g" where "a1 = A" and "a2 = l" and "a3 = u" and "a4 = a" and "a5 = l'" and "a6 = u'" and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" and "u \<turnstile> g" and "u' \<turnstile> inv_of A l'" and "u' = [r\<rightarrow>(0::'g)]u"
zero_neq_succ: fixes m :: "'a" shows "zero \<noteq> succ m"
f2r_ad_fun_hom_var: shows "\<R> \<circ> kad = ad_rel \<circ> \<R>"
assms_5: shows "pathfinish p = a - complex_of_real d"
rep_abs_rsp_left: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and x1 :: "'b"   and x2 :: "'b" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R (Rep (Abs x1)) x2"
word_uint_eqI: fixes a :: "'b word"   and b :: "'b word" assumes "uint a = uint b" shows "a = b"
y_def: shows "y = complex_of_real ((x - 1) / (x + 1))"
nat_power_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and m :: "nat" assumes "class.nat_power one times plus zero nat_power"   and "0 < n" shows "nat_power n (times a (semiring_1.of_nat one plus zero m)) = power.power one times (nat_power n a) m"
one_mult_mat_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "1\<^sub>m n *\<^sub>v v = v"
cMatch: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a"   and b :: "name" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "\<And>C. F C P a x P'" shows "F C ([b\<frown>b]P) a x P'"
sets: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> sets M"
asm: shows "falsifies\<^sub>c G (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x)))"
smcf_0_is_semifunctor: fixes \<AA> :: "V" assumes "semicategory \<alpha> \<AA>" shows "smcf_0 \<AA> : smc_0 \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<AA>"
less_eq_Sup: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and u :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq u v"   and "A \<noteq> {}" shows "less_eq u (Sup A)"
or_greater_eq: fixes l :: "int"   and k :: "int" assumes "0 \<le> l" shows "k \<le> k OR l"
skip_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b" shows "(\<sigma>, II) \<rightarrow>\<^sub>u (\<sigma>, II)"
add_to_counter_3: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r3 :: "(v, v) result"   and extra :: "nat" assumes "evaluate True env s e (s', r3)"   and "r3 \<noteq> Rerr (Rabort Rtimeout_error)" shows "evaluate True env (update_clock (\<lambda>_. clock s + extra) s) e (update_clock (\<lambda>_. clock s' + extra) s', r3)"
inverse_rel_def: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" shows "cinv\<^bsub>C\<^esub> f g \<equiv> f \<approx>>\<^bsub>C\<^esub> g \<and> f ;;\<^bsub>C\<^esub> g = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) \<and> g ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)"
unannotated: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "eval\<^sub>w_simplep c mem c' mem'" shows "eval\<^sub>wp \<langle>cxt_to_stmt E c, mds, mem\<rangle> \<langle>cxt_to_stmt E c', mds, mem'\<rangle>"
abs_le_D2: fixes a :: "'b"   and b :: "'b" assumes "\<bar>a\<bar> \<le> b" shows "- a \<le> b"
causal_Write: shows "causal_program_history is\<^sub>s\<^sub>b sb"
simps_1: fixes i :: "'a"   and j :: "'a"   and cs :: "'a list" shows "cycle_of_list (i # j # cs) = Fun.swap i j id \<circ> cycle_of_list (j # cs)"
comm_semigroup_axioms: shows "comm_semigroup f\<^sub>b"
mem_set_sum: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier M"   and "K \<subseteq> carrier M"   and "x \<in> H \<minusplus> K" shows "\<exists>h\<in>H. \<exists>k\<in>K. x = h \<plusminus> k"
lim_stream_def: fixes x :: "'a" shows "lim_stream x = distr (lim_sequence x) (stream_space M) to_stream"
chamber_distance_def: fixes C :: "'a set"   and D :: "'a set" shows "domain.chamber_distance C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. domain.gallery (C # Cs @ [D]))))"
axioms_2: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "ideal I R addition multiplication zero unit" shows "ideal_axioms I R multiplication"
basis_in_latticeI: fixes fs :: "'a vec list"   and n :: "nat"   and f :: "'a vec" assumes "set fs \<subseteq> carrier_vec n"   and "f \<in> set fs" shows "f \<in> vec_module.lattice_of n fs"
subset: fixes r :: "('a \<times> 'a) set"   and linear_cf :: "'a set \<Rightarrow> 'a set"   and Y :: "'a set"   and X :: "'a set" assumes "linear_cf r linear_cf"   and "linear_cf Y \<subseteq> X"   and "X \<subseteq> Y" shows "linear_cf Y = linear_cf X"
h0a: shows "dim_row (take_rows_from_matrix (alt_matrix_A signs subsets) S) = length (take_indices subsets S)"
a_consistent_app\<^sub>1: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, App e x, S)" shows "a_consistent (ae, inc\<cdot>a, as) (\<Gamma>, e, Arg x # S)"
baseE_1: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "p \<in> sparsegrid' dm" shows "base ds p \<in> grid (start dm) ({0..<dm} - ds)"
simps_3: fixes S1 :: "com"   and S2 :: "com" shows "ps_P.well_com (S1;; S2) = (ps_P.well_com S1 \<and> ps_P.well_com S2)"
bind_game_def: fixes \<A> :: "'ck \<Rightarrow> ('commit \<times> 'plain \<times> 'opening \<times> 'plain \<times> 'opening) spmf" shows "bind_game \<A> = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A> ck \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. let b = verify vk m c d; b' = verify vk m' c d' in return_spmf (b \<and> b')))) ELSE return_spmf False"
fps_binomial_deriv: fixes c :: "'a" shows "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)"
2: fixes n :: "nat" shows "\<forall>v\<in>fv (t \<cdot> var_rename n). n < snd v"
subI: fixes f :: "'a \<Rightarrow> ('a, 'b, 'c) sp\<^sub>\<mu>"   and a :: "'a" shows "(f a, Get f) \<in> sub"
disc_2: fixes x2 :: "'b" shows "\<not> is_dyn_operand (OpUbx1 x2)"
finite_ReZ_segments_constI: fixes g :: "real \<Rightarrow> complex"   and c :: "complex"   and z :: "complex" assumes "\<forall>t. 0 < t \<and> t < 1 \<longrightarrow> g t = c" shows "finite_ReZ_segments g z"
bind_cong: fixes m :: "'a nres"   and m' :: "'a nres"   and f :: "'a \<Rightarrow> 'b nres"   and f' :: "'a \<Rightarrow> 'b nres" assumes "m = m'"   and "\<And>x. RETURN x \<le> m' \<Longrightarrow> f x = f' x" shows "m \<bind> f = m' \<bind> f'"
Lambda_def: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "Lambda A f \<equiv> Repl A (\<lambda>x. Opair x (f x))"
l11_aux: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and A' :: "'a"   and E :: "'a"   and D :: "'a"   and D' :: "'a"   and A0 :: "'a"   and D0 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet B A A'"   and "Tarski_neutral_dimensionless.Out Bet E D D'"   and "Cong B A' E D'"   and "Bet B A A0"   and "Bet E D D0"   and "Cong A A0 E D"   and "Cong D D0 B A" shows "Cong B A0 E D0 \<and> Cong A' A0 D' D0"
obj_src: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.obj (Span.src \<mu>)"
sel_2: fixes x2 :: "'g" shows "projr (Inr x2) = x2"
gt_half_sum: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b" shows "less (divide (plus a b) (plus one one)) b"
strict_trans1: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
subseq_of_lengthI: fixes ys :: "'c list"   and xs :: "'c list"   and n :: "nat" assumes "ys \<in> set (subseqs xs)"   and "length ys = n" shows "subseq_of_length n xs ys"
out: shows "outs_gpv \<I> gpv \<subseteq> A"
Neg: fixes p :: "('a, 'b) fm" shows "\<turnstile> \<^bold>\<not> \<^bold>\<not> p \<^bold>\<longrightarrow> p"
disc_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "is_Fun (Fun x21 x22)"
not_le_imp_less: fixes x :: "'a"   and y :: "'a" assumes "\<not> x \<le> y" shows "y < x"
assms_1: shows "\<Z>_vfsequence \<alpha> \<SS>"
insert_var_rel_set_rel: fixes \<V> :: "char list set"   and \<V>' :: "nat set"   and yb :: "char list"   and x2 :: "nat" assumes "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel"   and "(yb, x2) \<in> var_rel" shows "(insert yb \<V>, insert x2 \<V>') \<in> \<langle>var_rel\<rangle>set_rel"
distinct_5: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "SKIP \<noteq> IF x41 THEN x42 ELSE x43"
preserves_reflects_arr: fixes f :: "'a1" shows "A2_B.arr (local.curry F G \<tau> f) = A1.arr f"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "pred x xa" obtains f :: "('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" where "x = \<Up>f__" and "xa = cf__ # cfs__"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "Q__ (fst cfa__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "Qa__ (fst cfb__, r)"   | 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" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "Q cf \<and> cfs \<noteq> []"