hcju/isabelleps huggingface dataset:
MC: fixes n :: "nat"   and s :: "nat \<Rightarrow> 's"   and s' :: "'s" assumes "prob {\<omega> \<in> space M. \<forall>t\<le>n. X t \<omega> = s t} \<noteq> 0" shows "cond_prob M (\<lambda>\<omega>. X (Suc n) \<omega> = s') (\<lambda>\<omega>. \<forall>t\<le>n. X t \<omega> = s t) = cond_prob M (\<lambda>\<omega>. X (Suc n) \<omega> = s') (\<lambda>\<omega>. X n \<omega> = s n)"
IH: fixes env :: "v sem_env"   and s :: "'a state" assumes "\<forall>res. \<not> evaluate_prog False env s tops res" shows "\<exists>r. evaluate_prog True env s tops r \<and> snd r = Rerr (Rabort Rtimeout_error)"
Crypt_valid_pubSK: fixes C :: "agent"   and SKi :: "nat"   and evs :: "event list" assumes "Crypt (priSK RCA) \<lbrace>Agent C, Key SKi, onlySig\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_cr" shows "SKi = pubSK C"
pG: shows "mset_factors (image_mset poly_lift G) (poly_lift q)"
dropWhile_cong: fixes l :: "'g list"   and k :: "'g list"   and P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "dropWhile P l = dropWhile Q k"
vsubstAll_ident: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>z. z \<in> varsOf P \<longrightarrow> X #[z // z]_zs = X)) \<and> (goodAbs A \<longrightarrow> (\<forall>z. z \<in> varsOf P \<longrightarrow> A $[z // z]_zs = A))"
right_diff_distrib: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a *\<^sub>o\<^sub>w (b -\<^sub>o\<^sub>w c) = a *\<^sub>o\<^sub>w b -\<^sub>o\<^sub>w a *\<^sub>o\<^sub>w c"
pw_eq_iff: fixes S :: "'e nres"   and S' :: "'e nres" shows "(S = S') = (nofail S = nofail S' \<and> (\<forall>x. inres S x = inres S' x))"
Atm_Sbis: fixes atm :: "atom" assumes "Example_PL.compatAtm atm" shows "Atm atm \<approx>s Atm atm"
il_bot_unit: fixes x :: "'a" shows "x \<squnion> bot = x"
c_secure_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s\<^sub>0 :: "'a"   and I :: "('d \<times> 'd) set"   and D :: "'b \<Rightarrow> 'd" shows "c_secure step out s\<^sub>0 I D \<equiv> \<forall>x xs. out (foldl step s\<^sub>0 xs) x = out (foldl step s\<^sub>0 (c_ipurge I D (D x) xs)) x"
substAll_efficient_def: fixes g :: "'g" shows "i.substAll_efficient g \<equiv> let phiVals = Mapping.keys (uninst_code.ssa.phidefNodes (p_g g s1_phis) g); u = u_g g s1_uses g; p = p_g g s1_phis g; tp = uninst_code.ssa.trivial_phis (p_g g s1_phis) g; nou = uninst_code.ssa.useNodes_of (u_g g s1_uses) g; nop = uninst_code.ssa.phiNodes_of (p_g g s1_phis) g in while (\<lambda>((u, p), triv_phis, nodes_of_uses, nodes_of_phis). \<not> Set.is_empty triv_phis) (\<lambda>((u, p), triv_phis, nodes_of_uses, nodes_of_phis). let next = Max triv_phis; next' = uninst_code.substitution_code (\<lambda>g. p) g next; (u', p') = uninst_code.step_codem (\<lambda>_. u) (\<lambda>_. p) g next next' nodes_of_uses nodes_of_phis; tp' = uninst_code.triv_phis' (\<lambda>_. p') g next triv_phis nodes_of_phis; nou' = uninst_code.nodes_of_uses' g next next' phiVals nodes_of_uses; nop' = uninst_code.nodes_of_phis' g next next' nodes_of_phis in ((u', p'), tp', nou', nop')) ((u, p), tp, nou, nop)"
bit_minus_1_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (uminus one) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
list_all2_conv_all_nth: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
cprops_2: shows "\<forall>w\<in>space M. discount_factor r n w = c"
monoid_def: fixes G :: "('a, 'b) monoid_scheme" shows "Group.monoid G \<equiv> ((\<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G) \<and> (\<forall>x y z. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> z \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z))) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x) \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x)"
lift_Def_node: fixes V :: "'g"   and Def :: "'h \<Rightarrow> 'g set"   and n :: "'h"   and E :: "'h"   and X :: "'h"   and H :: "'g set"   and L :: "'g set" assumes "V \<in> Def n" shows "(Node n, V) \<in> lift_Def_set Def E X H L"
that_2: fixes x' :: "'a"   and xs' :: "'a llist" assumes "xs = LCons x' xs'"   and "x \<in> lset xs'" shows "thesis"
inf_top_right: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf x top = x"
higher_deriv_compose_linear: fixes f :: "complex \<Rightarrow> complex"   and T :: "complex set"   and S :: "complex set"   and z :: "complex"   and u :: "complex"   and n :: "nat" assumes "f holomorphic_on T"   and "open S"   and "open T"   and "z \<in> S"   and "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T" shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u ^ n * (deriv ^^ n) f (u * z)"
wfp_on_iff_minimal: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on P A = (\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q))"
simps: fixes e :: "'c \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'b"   and p :: "('c \<times> 'c rose_tree list \<times> 'c rose_tree list) list"   and t :: "'c rose_tree" shows "zippers_rose_tree e h (p, t) = (blind_path e h p, embed_source_tree e t) # concat (map (zippers_rose_tree e h) (zipper_children (p, t)))"
right_adjoint_simps_5: fixes f :: "'a" assumes "is_left_adjoint f" shows "cod f\<^sup>* = f\<^sup>*"
assms_2: shows "suffix i w \<Turnstile>\<^sub>n \<psi>[\<F>\<G> \<psi> (suffix i w)]\<^sub>\<Pi>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>"
dist: fixes xc1 :: "'b"   and xc2 :: "'b" assumes "invar xc1"   and "invar xc2"   and "xc1 \<in> S0"   and "xc2 \<in> S0"   and "\<alpha> xc1 = \<alpha> xc2" shows "xc1 = xc2"
right_cancel: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "(b + a = c + a) = (b = c)"
UN: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "nat \<Rightarrow> 'c set" assumes "Dynkin_system \<Omega> M"   and "disjoint_family A"   and "range A \<subseteq> M" shows "\<Union> (range A) \<in> M"
RUV_call: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and Ml :: "'d set"   and Me :: "'d set"   and U :: "'a set"   and V :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 0"   and "(v, Ml, Me) \<in> RUV_cs fg U V"   and "mon_n fg u \<inter> Me = {}" shows "(u, mon fg p \<union> M \<union> Ml, Me) \<in> RUV_cs fg U V"
4: shows "\<forall>x y. d y \<sqinter> x \<squnion> ad y \<sqinter> x = top \<sqinter> x"
simps: fixes backt :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and c :: "'c"   and cs :: "'a"   and s :: "'d" shows "backtrack_cs backt c (cs, s) = (cs, backt c s)"
Suc: shows "eval r_leap [t, i, x] \<down>= encode_config (iterate t step ([(f, Partial_Recursive.list_decode x, [])], None))"
star_plus_one: fixes x :: "'a list set" shows "1 + x\<^sup>\<star> = x\<^sup>\<star>"
assms_1: shows "ts \<subseteq> existence_ivl0 x"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
a_1: shows "v\<^sub>a \<in> defs g a"
is_zero_row_upt_k_le: fixes i :: "'b"   and k :: "nat"   and A :: "(('c, 'd) vec, 'b) vec" assumes "is_zero_row_upt_k i (Suc k) A" shows "is_zero_row_upt_k i k A"
nonzero_mult_divide_mult_cancel_right2: 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 c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times c b) = divide a b"
adds_poly_mapping: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "(s adds t) = (lookup s \<le> lookup t)"
pr_id3_2': shows "pr_conv_3_to_1 (\<lambda>x y z. y) \<in> PrimRec1'"
knows2know_neg_s: fixes A :: "specID"   and m :: "Secrets" assumes "\<not> knows A [sE m]" shows "\<not> know A (sKS m)"
from_nat_def: fixes i :: "nat" shows "Bij_Nat.from_nat i = univ_list ! i"
up_dist_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "{ya. less_eq (inf x y) ya} = semilattice_inf.filter_sup inf less_eq {y. less_eq x y} {ya. less_eq y ya}"
spmf_lub_spmf: fixes Y :: "'a spmf set"   and x :: "'a" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "spmf (lub_spmf Y) x = (\<Squnion>p\<in>Y. spmf p x)"
restrict_to_halfspace_def: fixes sctn :: "'c sctn"   and X :: "'c set" shows "restrict_to_halfspace sctn X = CHECK (\<lambda>_. ()) (normal sctn \<in> set Basis_list \<or> - normal sctn \<in> set Basis_list) \<bind> (\<lambda>_. let y = pstn sctn; b = normal sctn in ivl_rep X \<bind> (\<lambda>(i, s). let i' = if b \<le> (0::'c) then i + min (y - i \<bullet> b) 0 *\<^sub>R b else i; s' = if (0::'c) \<le> b then s + min (y - s \<bullet> b) 0 *\<^sub>R b else s in RETURN ({i'..s'} :::\<^sub>i \<langle>i_rnv\<rangle>\<^sub>ii_ivl)))"
H_contained_in_set_mult: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" assumes "second_isomorphism_grp H G S" shows "H \<subseteq> H <#>\<^bsub>G\<^esub> S"
unat_eq_1: fixes x :: "'b word" shows "(unat x = Suc 0) = (x = 1)"
arcs_ends_wf: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow> y" shows "x \<in> verts G \<and> y \<in> verts G"
oreturn_rule: fixes P :: "'b \<Rightarrow> bool"   and r :: "'b" assumes "P r" shows "ospec (Some r) P"
2: fixes f :: "nat \<Rightarrow> real" shows "is \<lhd> input_sizes (insert_weights s m f)"
sets_Collect_neg: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M" shows "{x \<in> space M. \<not> P x} \<in> sets M"
intros_2: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> analz H" shows "X \<in> analz H"
toList_inverse: fixes x :: "'c odlist" shows "odlist_Abs (toList x) = x"
WHILET_def: shows "WHILE\<^sub>T \<equiv> WHILE\<^sub>T\<^bsup>\<lambda>_. True\<^esup>"
AG_AG: fixes p :: "'b set" shows "\<^bold>A\<^bold>G \<^bold>A\<^bold>G p = \<^bold>A\<^bold>G p"
r_auxhyp: fixes e :: "nat" shows "\<phi> (the (eval r_auxhyp [e])) = prenum e"
sum_eq_sum_prepend: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list"   and ys :: "'b list"   and as :: "'b list" assumes "0 < i"   and "i \<le> j"   and "length xs < i"   and "length ys = length xs" shows "SWA.sum (xs @ as) i j = SWA.sum (ys @ as) i j"
connected_frontier_component_complement: fixes S :: "'a set"   and C :: "'a set" assumes "connected S"   and "C \<in> components (- S)" shows "connected (frontier C)"
nil_eqvt: fixes pi :: "('b \<times> 'b) list" shows "pi \<bullet> [] = []"
listgcd_greatest: fixes xs :: "'a list"   and y :: "'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> y dvd x" shows "y dvd some_gcd.listgcd xs"
chop_def: shows "i = i1 \<squnion> i2 \<and> (i1 = bot \<or> i2 = bot \<or> consec i1 i2)"
pr_SN: shows "SN {(x, y). pr_strict' x y}"
enc_atom_lang_Arbitrary_Except2: fixes r :: "nat"   and I :: "('b + 'b set) list"   and s :: "nat"   and p :: "'b"   and n :: "nat"   and a :: "'a" assumes "r < length I"   and "s < length I"   and "case I ! r of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "case I ! s of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "length I = n"   and "a \<in> set \<Sigma>" shows "[enc_atom I p a] \<in> lang n (Atom (Arbitrary_Except2 r s))"
Abs_dlist'_inverse: fixes y :: "'c list" assumes "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}" shows "list_of_dlist (Abs_dlist' y) = y"
Fubini': fixes M1 :: "'b measure"   and M2 :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> ennreal" assumes "pair_sigma_finite M1 M2"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f x y \<partial>M1 \<partial>M2 = \<integral>\<^sup>+ x. integral\<^sup>N M2 (f x) \<partial>M1"
complement_wf: shows "incidence_system \<V> \<B>\<^sup>C"
disj_simps_6: fixes P :: "fm"   and Q :: "fm" assumes "P \<noteq> T"   and "P \<noteq> F"   and "Q \<noteq> T"   and "Q \<noteq> F"   and "P \<noteq> Q" shows "Ferrack.disj P Q = Or P Q"
iT_Plus_empty: fixes k :: "nat" shows "{} \<oplus> k = {}"
hyps: shows "(\<Sum>i = 0..<l. if i \<in> R then f i else 0) = sum f (R \<inter> {..<l})"
abstract_primitive_in_doubt_allow_Allow2: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'c common_primitive match_expr"   and p :: "('c, 'd) tagged_packet_scheme"   and disc :: "'c common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "\<not> matches (\<beta>, in_doubt_allow) m Drop p" shows "\<not> matches (\<beta>, in_doubt_allow) (abstract_primitive disc m) Drop p"
closedin_closed_subset: fixes U :: "'c set"   and V :: "'c set"   and T :: "'c set"   and S :: "'c set" assumes "closedin (top_of_set U) V"   and "T \<subseteq> U"   and "S = V \<inter> T" shows "closedin (top_of_set T) S"
simps_3: fixes \<A> :: "'b ltl set"   and \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> and \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<and> \<A> \<Turnstile>\<^sub>P \<psi>)"
assms_3: shows "\<forall>w\<in>space M. 0 \<le> X w"
arc_to_block_to_ends: fixes e :: "'b" shows "{fst (arc_to_ends G e), snd (arc_to_ends G e)} = arc_to_block e"
Moebius_function_def: shows "Moebius_function \<equiv> \<lambda>t w z. exp (\<i> * complex_of_real t) * (z - w) / (1 - cnj w * z)"
mult_mat_vec_hom: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "vec\<^sub>h (A *\<^sub>v v) = mat\<^sub>h A *\<^sub>v vec\<^sub>h v"
yun_rel_mult: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and G :: "int poly"   and d :: "rat"   and g :: "rat poly" assumes "yun_rel F c f"   and "yun_rel G d g" shows "yun_rel (F * G) (c * d) (f * g)"
sync: shows "\<forall>a t t'. spr_jview a t = spr_jview a t' \<longrightarrow> tLength t = tLength t'"
of_nat_le_one_cancel_iff: fixes n :: "nat" shows "(real n \<le> 1) = (n \<le> 1)"
tendsto_norm_zero_iff: fixes f :: "'e \<Rightarrow> 'f"   and F :: "'e filter" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> (0::'f)) F"
and_mask_arith: fixes w :: "'b word"   and n :: "nat" shows "w AND mask n = w * 2 ^ (size w - n) div 2 ^ (size w - n)"
plus_vwb_lens: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e" assumes "vwb_lens x"   and "vwb_lens y"   and "x \<bowtie> y" shows "vwb_lens (x +\<^sub>L y)"
star_inductl_var_equiv: fixes x :: "'a"   and y :: "'a" shows "(qstar x \<cdot> y \<le> y) = (x \<cdot> y \<le> y)"
setdist_sym: fixes S :: "'b set"   and T :: "'b set" shows "setdist S T = setdist T S"
assoc_in_hom'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<guillemotleft>\<a>[\<mu>, \<nu>, \<tau>] : (local.dom \<mu> \<star> local.dom \<nu>) \<star> local.dom \<tau> \<Rightarrow> cod \<mu> \<star> cod \<nu> \<star> cod \<tau>\<guillemotright>"
All2_SUCC_E: fixes i :: "name"   and t :: "tm"   and A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "atom i \<sharp> t"   and "insert (All2 i t A) (insert (A(i::=t)) H) \<turnstile> B" shows "insert (All2 i (SUCC t) A) H \<turnstile> B"
hyps_2: obtains c :: "'a" where "poly p (x + y) = poly p x + c * y"
cases: fixes x :: "'a form" obtains i :: "nat" where "x = At i"   | f :: "'a" and fs :: "'a form list" where "x = Compound f fs"   | "x = ff"
comm_semiring_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.comm_semiring plus times \<equiv> class.ab_semigroup_add plus \<and> class.ab_semigroup_mult times \<and> class.comm_semiring_axioms plus times"
is_nonterminal_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and s :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_nonterminal \<NN> s = (s \<in> \<NN>)"
simps: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color" shows "local.leaves n p w = (\<forall>w'. local.possible n p w w' \<longrightarrow> w' p = w p)"
induct: fixes P :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and a0 :: "'b trm"   and a1 :: "'b trm" assumes "\<And>c M N. P (Const c) (M \<cdot> N)"   and "\<And>M N c. P (M \<cdot> N) (Const c)"   and "\<And>c v. P (Const c) (Var v)"   and "\<And>M N v. P (M \<cdot> N) (Var v)"   and "\<And>v M. P (Var v) M"   and "\<And>c d. P (Const c) (Const d)"   and "\<And>M N M' N'. \<lbrakk>P M M'; \<And>x2. unify M M' = Some x2 \<Longrightarrow> P (N \<lhd> x2) (N' \<lhd> x2)\<rbrakk> \<Longrightarrow> P (M \<cdot> N) (M' \<cdot> N')" shows "P a0 a1"
Cg12: fixes h :: "face" assumes "h \<in> \<F> g" shows "\<E> h \<noteq> (\<E> f\<^sub>1)\<inverse> \<and> \<E> h \<noteq> (\<E> f\<^sub>2)\<inverse>"
rep_eq: fixes x :: "'c"   and xa :: "'c fset" shows "fset (finsert x xa) = insert x (fset xa)"
viable_if: assumes "1 + r < u"   and "d < 1 + r" shows "viable_market Mkt"
eq_iff: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "ordering less_eq less" shows "(a = b) = (less_eq a b \<and> less_eq b a)"
raw_match'_Type_conds_2: fixes a :: "String.literal"   and Ts :: "typ list"   and b :: "String.literal"   and Us :: "typ list"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' (Ty a Ts) (Ty b Us) subs = Some subs'" shows "length Ts = length Us"
uint64_of_nat_def: shows "uint64_of_nat = uint64_of_int \<circ> int"
set_cases_2: fixes e :: "'d"   and a :: "('e, 'd) Term.term" assumes "e \<in> vars_term a" obtains "a = Var e"   | z1 :: "'e" and z2 :: "('e, 'd) Term.term list" and x :: "('e, 'd) Term.term" where "a = Fun z1 z2" and "x \<in> set z2" and "e \<in> vars_term x"
w_defs_1: shows "w1 = fst (bv_chop w (Suc l))"
mod_dist_1: fixes b :: "bool"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> \<up> b) = (snd h = {} \<and> b)"
fmran'_fmfilter_eq: fixes p :: "'b \<Rightarrow> bool"   and fm :: "('b, 'a) fmap" shows "fmran' (fmfilter p fm) = {y |y. \<exists>x\<in>fmdom' fm. p x \<and> fmlookup fm x = Some y}"
steps_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and q :: "nat"   and w :: "bool list list" assumes "aut_dfa A n"   and "dfa_is_node A q"   and "list_all (is_alph n) w" shows "dfa_is_node A (dfa_steps A q w)"
ext_inject: fixes initial :: "char list \<Rightarrow> nat option"   and final :: "char list \<Rightarrow> nat option"   and more :: "'a"   and initial' :: "char list \<Rightarrow> nat option"   and final' :: "char list \<Rightarrow> nat option"   and more' :: "'a" shows "(\<lparr>initial = initial, final = final, \<dots> = more\<rparr> = \<lparr>initial = initial', final = final', \<dots> = more'\<rparr>) = (initial = initial' \<and> final = final' \<and> more = more')"
clean_no_outstanding_volatile_Write\<^sub>s\<^sub>b: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<not> \<D>" shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"
append_cols_mult_right_id: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and m :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat n 1"   and "B \<in> carrier_mat n (m - 1)"   and "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"   and "D \<in> carrier_mat (m - 1) (m - 1)" shows "(A @\<^sub>c B) * C = A @\<^sub>c B * D"
CatExpId: fixes X :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" assumes "X \<in> obj\<^bsub>CatExp A B\<^esub>" shows "id\<^bsub>CatExp A B\<^esub> X = IdNatTrans X"
inf_left_commute: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "inf_prod x (inf_prod y z) = inf_prod y (inf_prod x z)"
expandFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'e"   and C' :: "'f" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* C'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* C'" obtains p :: "(name \<times> name) list" and \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P' :: "'b" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" and "(p \<bullet> \<Psi>\<^sub>P) \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'" and "distinctPerm p" and "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>" and "A\<^sub>P' \<sharp>* P'" and "A\<^sub>P' \<sharp>* \<alpha>" and "A\<^sub>P' \<sharp>* (p \<bullet> \<alpha>)" and "A\<^sub>P' \<sharp>* C" and "bn (p \<bullet> \<alpha>) \<sharp>* C'" and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>" and "bn (p \<bullet> \<alpha>) \<sharp>* P'" and "distinct A\<^sub>P'"