hcju/isabelleps huggingface dataset:
\<tau>move2Sync6: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) (5 + length (compE2 o') + length (compE2 e)) None"
chain_subdiv_path_singleton: fixes \<gamma> :: "real \<Rightarrow> real \<times> real" shows "chain_subdiv_path \<gamma> {(1, \<gamma>)}"
add_diff_assoc2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "b - a + c = b + c - a"
temp1: assumes "e = Cvtop t1 Reinterpret t sx" shows "type_update tn' [TSome t] (Type [arity_1_result e]) = tm'"
interchange: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b"   and \<tau> :: "'b"   and \<sigma> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (V \<nu> \<mu>)"   and "partial_magma.arr V (V \<tau> \<sigma>)" shows "H (V \<nu> \<mu>) (V \<tau> \<sigma>) = V (H \<nu> \<tau>) (H \<mu> \<sigma>)"
cts_wp_DC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "bd_cts (wp a)"   and "bd_cts (wp b)"   and "healthy (wp a)"   and "healthy (wp b)" shows "bd_cts (wp (a \<Sqinter> b))"
i_def: shows "i \<equiv> closed_segment a b \<inter> {p. fst p = xl}"
slength_sappend: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "slength\<cdot>(xs :@ ys) = slength\<cdot>xs + slength\<cdot>ys"
quot_Neg: fixes A :: "fm" shows "\<guillemotleft>Neg A\<guillemotright> = Q_Neg \<guillemotleft>A\<guillemotright>"
SINK: shows "x \<in> SINK (g \<upharpoonleft> \<Gamma> / f)"
inj_img: fixes a :: "'s" assumes "ide a" shows "inj_on img (hom unity a)"
vifunion_upper: fixes i :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" assumes "i \<in>\<^sub>\<circ> I" shows "f i \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
wlp_if_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q s = (if bval b s then wlp \<pi> c\<^sub>1 Q s else wlp \<pi> c\<^sub>2 Q s)"
that_2: shows "(A ===> A ===> (=)) (\<le>) (\<le>)"
sorted_list_of_set_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
summand: fixes d :: "nat" assumes "d dvd n" shows "f d * dirichlet_prod_neutral (n div d) = (if n = d then f d else 0)"
ln_unique: fixes y :: "real"   and x :: "real" assumes "exp y = x" shows "real_ln x = y"
simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list" shows "filter_compliant_stateful_ACS_accu G M accu [] = accu"
index_append_vec_1: fixes i :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "i < dim_vec v + dim_vec w" shows "(v @\<^sub>v w) $ i = (if i < dim_vec v then v $ i else w $ (i - dim_vec v))"
act_dom_proj_eff_subset_act_dom_eff: fixes a :: "('d, 'f) fmap \<times> ('d, 'e) fmap"   and vs :: "'d set" shows "fmdom' (snd (action_proj a vs)) \<subseteq> fmdom' (snd a)"
hyps: shows "listset xs = {ys. length ys = length xs \<and> (\<forall>i<length xs. ys ! i \<in> xs ! i)}"
conj_fix_pt: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "bij f" shows "inv f ` Fix g = Fix (inv f \<circ> g \<circ> f)"
arithmetic_simps_62: fixes a :: "'a" shows "(0::'a) * a = (0::'a)"
zero_less_power2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times a 2) = (a \<noteq> zero)"
saturated_upto_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist"   and N :: "'a literal multiset set" shows "src.saturated_upto Sts N = (gd.inferences_from Sts (N - src.Rf N) \<subseteq> src.Ri Sts N)"
elims: fixes x :: "jvm_state"   and y :: "frame list" assumes "frames_of x = y" obtains xp :: "nat option" and h :: "heap" and frs :: "frame list" and sh :: "sheap" where "x = (xp, h, frs, sh)" and "y = frs"
simps_1: fixes y :: "real" shows "IEEE.round To_nearest y = (if y \<le> - threshold TYPE(('a, 'b) IEEE.float) then - \<infinity> else if threshold TYPE(('a, 'b) IEEE.float) \<le> y then \<infinity> else closest valof (\<lambda>a. even (fraction a)) (Collect is_finite) y)"
w: shows "w \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])"
U0_swap: shows "U0 ;; swap\<^sub>m = U1"
zorder_def: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "zorder f z = (THE n. \<exists>h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0))"
terminates_to_terminates_mark_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>mark_guards f c \<down> s"
prems: shows "sem\<^sub>e\<^sub>s\<^sub>t_d {} \<I> (to_est \<A>2)"
root_def: fixes tree :: "'b tree" shows "tree.root tree = (case tree of Node x1 x2 x3 \<Rightarrow> x2)"
distinct_27: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list"   and x61 :: "nat"   and x62 :: "recf" shows "Cn x41 x42 x43 \<noteq> Mn x61 x62"
H_outside: fixes x :: "'v"   and y :: "'v" assumes "x \<notin> A \<Gamma> \<or> y \<notin> B \<Gamma>" shows "H (x, y) = 0"
obs_def: shows "obs \<equiv> \<lambda>n S. {x. obsp n (\<lambda>x. x \<in> S) x}"
iMOD_gr0_add_neg: fixes m :: "nat"   and r :: "nat"   and k :: "nat" assumes "0 < m" shows "[ r, mod m ] \<oplus>- k = (if k \<le> r then [ r - k, mod m ] else [ (m + r mod m - k mod m) mod m, mod m ])"
submonoid_inverse_closed: fixes u :: "'a \<Rightarrow> 'a" assumes "symmetric.sub.invertible u"   and "u \<in> G" shows "symmetric.inverse u \<in> G"
subo: shows "Subobjs P C (Cs @ [D, D'] @ Cs')"
ArrLength: fixes AT :: "Arraytype" shows "treach (ArrT AT) IntgT"
c_suc_k: shows "c (n + Suc k) = (ts', m', \<S>')"
DFS_invar_axioms: shows "DFS_invar G tarjan_params s"
simps_2: fixes v :: "('a, 'b) Node"   and xs :: "('a, 'b) Node list" shows "FingerTreeStruc.app3 (Single v) xs Empty = FingerTreeStruc.rconsNlist (Single v) xs"
a: shows "f \<in> mor\<^bsub>C\<^esub>"
surj_iff_all: fixes f :: "'b \<Rightarrow> 'a" shows "surj f = (\<forall>x. f (inv f x) = x)"
mult_vec_def: shows "(**) = lift_poly_fun_2 (*)"
minus_in_Ints_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "(uminus x \<in> ring_1.Ints one plus zero minus) = (x \<in> ring_1.Ints one plus zero minus)"
sappend_scons: fixes x :: "'r"   and xs :: "[:'r:]"   and ys :: "[:'r:]" shows "(x :# xs) :@ ys = x :# xs :@ ys"
simps_2: fixes \<pi> :: "'b prm"   and x :: "'b" shows "\<pi> \<bullet> PVar x = PVar (\<pi> $ x)"
dghm_is_arr_HomCod_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
him_add: fixes p :: "quat"   and q :: "quat" shows "HIm (p + q) = HIm p + HIm q"
nqfree_aneg: fixes a :: "'a" shows "nqfree (aneg a)"
sams_chara: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "A' \<noteq> B"   and "Bet A B A'" shows "(SAMS A B C D E F) = (D E F LeA C B A')"
leftmost_witness: fixes x :: "symbol list"   and N :: "symbol"   and y :: "symbol list" shows "leftmost (length x) (x @ N # y) = (is_word x \<and> is_nonterminal N)"
invP: shows "A \<TTurnstile>\<^sub>A (PI \<rightarrow>) P"
ODNmap_mem: fixes D :: "'c Order"   and x :: "'c Order set"   and y :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D"   and "y \<in> ODnums"   and "y \<sqsubset> x" shows "ODNmap D y \<in> carrier D \<and> (\<forall>Y\<in>y. ord_equiv Y (Iod D (segment D (ODNmap D y))))"
star_slide: fixes x :: "'a" shows "top \<sqinter> x = x \<sqinter> top"
not_eSuc_ilei0: fixes n :: "enat" shows "\<not> eSuc n \<le> 0"
evalP_def: fixes M :: "model" shows "evalP M = snd (Rep_model M)"
normal_form_minus_normal_form: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "normal_form F (normal_form F p - normal_form F q) = normal_form F p - normal_form F q"
scale_cancel_left: fixes a :: "'b"   and x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" shows "(a *s x = a *s y) = (x = y \<or> a = (0::'b))"
us: shows "us = getFrN (map snd txs) (map fst txs) [fls] (length txs)"
split_between: fixes vs :: "'a list"   and r :: "'a"   and v :: "'a"   and u :: "'a" assumes "distinct vs"   and "r \<in> set vs"   and "v \<in> set vs"   and "u \<in> set (between vs r v)" shows "between vs r v = (if r = u then [] else between vs r u @ [u]) @ between vs u v"
set_intros_2: fixes x2 :: "'b" shows "x2 \<in> set_literal (x2\<inverse>)"
set_remdups: fixes xs :: "'k list" shows "set (remdups xs) = set xs"
d_IN_add: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" shows "d_IN (\<lambda>e. f e + g e) x = d_IN f x + d_IN g x"
fmsubset_alt_def: fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap" shows "(m \<subseteq>\<^sub>f n) = fmpred (\<lambda>k v. fmlookup n k = Some v) m"
AA28: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<diamond>\<langle>A\<rangle>_v = \<diamond>\<langle>A\<rangle>_v"
generate_check_filter: fixes a :: "nat list"   and b :: "nat list" shows "generate_check a b = filter (\<lambda>(xs, ys). suffs (C\<^sub>1 b (fst ys)) a xs \<and> suffs C\<^sub>2 b ys) (alls2 (B\<^sub>1 b) B\<^sub>2 a b)"
mem_extend_set_iff: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and z :: "'f"   and A :: "'d set" assumes "Extend h" shows "(z \<in> extend_set h A) = (fst (inv h z) \<in> A)"
atoms_of_cnf_alt: fixes c :: "'b literal set set" shows "atoms_of_cnf c = \<Union> ((`) atoms_of_lit ` c)"
range_from_nat_into: fixes A :: "'a set" assumes "A \<noteq> {}"   and "countable A" shows "range (from_nat_into A) = A"
arc_top_edge: fixes x :: "'a" assumes "arc x" shows "x\<^sup>T * top * x = x\<^sup>T * x"
simps: fixes a :: "('test, 'atom) com" shows "siso a = (\<exists>c. a = c \<and> (\<forall>x xa xb. (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> siso xa) \<and> (\<forall>x xa xb xc. x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>c (xb, xc) \<longrightarrow> (\<exists>t'. xc \<approx> t' \<and> (c, xa) \<rightarrow>c (xb, t'))) \<and> (\<forall>x xa xb. x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>t xb \<longrightarrow> (\<exists>t'. xb \<approx> t' \<and> (c, xa) \<rightarrow>t t')))"
ide_prod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.ide C (elementary_category_with_binary_products.prod C pr0 pr1 a b)"
adopt_node_node_in_disconnected_nodes: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"   and "known_ptrs h"   and "type_wf h" shows "node_ptr \<in> set disc_nodes"
simps: fixes a1 :: "'state"   and a2 :: "'state" shows "scut a1 a2 = (\<exists>p a. a1 = p \<and> a2 = ex a p \<and> p \<in> nodes \<and> en a p \<and> a \<notin> cuts)"
env_restr_env_restr_subset: fixes d1 :: "'a set"   and d2 :: "'a set"   and x :: "'a \<Rightarrow> 'b" assumes "d1 \<subseteq> d2" shows "x f|` d2 f|` d1 = x f|` d1"
proj_mono_prefix: fixes t :: "cnf llist"   and t' :: "cnf llist"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" assumes "lprefix t t'" shows "lprefix (dynamic_component.proj tCMP active c t) (dynamic_component.proj tCMP active c t')"
rel_nondetTE: fixes M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "('i, 'g) nondetT"   and m' :: "('j, 'h) nondetT" assumes "rel_nondetT M m m'" obtains x :: "'g" and y :: "'h" where "m = NondetT x" and "m' = NondetT y" and "M x y"
img_fst: fixes a :: "'b"   and b :: "'c"   and S :: "('b \<times> 'c) set" assumes "(a, b) \<in> S" shows "a \<in> fst ` S"
red_singleton_zero: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "\<not> red {0} p q"
ge_right_comm: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Ge Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Ge Bet Cong A B D C"
restrict0_const_diff_fun_space: fixes c :: "real" shows "restrict0 sub.carrier (\<lambda>x. c) \<in> sub.diff_fun_space"
path_take_to: fixes l1 :: "'loc"   and l2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and m :: "nat"   and l2' :: "'loc" assumes "path l1 l2 xs"   and "m < length xs"   and "(case xs ! m of (s, l, t) \<Rightarrow> t) = l2'" shows "path l1 l2' (take (m + 1) xs)"
intros_2: fixes r :: "Vertex \<Rightarrow> Vertex"   and s :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations"   and "s \<in> complex_rotations" shows "r \<circ> s \<in> complex_rotations"
2: shows "(w @ w\<^sub>1) @ v\<^sub>1 =\<^sub>F w @ w\<^sub>2"
abst_action_repcs: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "abst (action (repcs s cfg)) = action cfg"
ipresIGSwap_def: fixes h :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'h, 'p, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 't, 'u, 'o, 'i, 'v, 'w) model_scheme" shows "ipresIGSwap h MOD MOD' \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> h (igSwap MOD zs z1 z2 X) = igSwap MOD' zs z1 z2 (h X)"
src_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>C (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
HFail_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HInv1 s"   and "HFail s s' p" shows "HInv1 s'"
reach_init: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set" assumes "reach T \<subseteq> I" shows "init T \<subseteq> I"
ih_2: assumes "getCn state' \<noteq> 1"   and "InvariantUniq (getM (applyExplain (getCl state') state'))"   and "InvariantCFalse (getConflictFlag (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state'))"   and "InvariantCEntailed (getConflictFlag (applyExplain (getCl state') state')) F0 (getC (applyExplain (getCl state') state'))"   and "InvariantClCharacterization (getCl (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantCnCharacterization (getCn (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantClCurrentLevel (getCl (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantUniqC (getC (applyExplain (getCl state') state'))"   and "InvariantGetReasonIsReason (getReason (applyExplain (getCl state') state')) (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (set (getQ (applyExplain (getCl state') state')))"   and "InvariantEquivalentZL (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) F0"   and "getConflictFlag (applyExplain (getCl state') state')"   and "0 < currentLevel (getM (applyExplain (getCl state') state'))" shows "let state' = applyExplainUIP (applyExplain (getCl state') state') in InvariantCFalse (getConflictFlag state') (getM state') (getC state') \<and> InvariantCEntailed (getConflictFlag state') F0 (getC state') \<and> InvariantClCharacterization (getCl state') (getC state') (getM state') \<and> InvariantCnCharacterization (getCn state') (getC state') (getM state') \<and> InvariantClCurrentLevel (getCl state') (getM state') \<and> InvariantUniqC (getC state')"
positively_invariant_def: fixes M :: "'a set" shows "rev.positively_invariant M = (\<forall>x\<in>M. rev.trapped_forward x M)"
hyps_2: fixes thesis :: "bool" assumes "\<And>P'''. \<lbrakk>\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''; \<Psi> \<rhd> P' \<sim> P'''\<rbrakk> \<Longrightarrow> thesis"   and "\<Psi> \<rhd> P \<sim> P''" shows "thesis"
bind_map_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and g :: "'b \<Rightarrow> 'a spmf" shows "bind_spmf (map_spmf f p) g = bind_spmf p (g \<circ> f)"
subst_pp_single: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'e"   and e :: "nat" shows "subst_pp f (monomial e x) = f x ^ e"
deg_const: fixes a :: "'a" assumes "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
norm_vec_def: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec" shows "norm x = L2_set (\<lambda>i. norm (x $h i)) UNIV"
assms: shows "\<forall>(d, e, f)\<in>set neq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
cod_simp: fixes f :: "'a" assumes "S.arr f" shows "S.cod f = cod f"
simps_9: fixes f1 :: "'a"   and f2 :: "num \<Rightarrow> 'a"   and f3 :: "num \<Rightarrow> 'a" shows "(case num.One of num.One \<Rightarrow> f1 | num.Bit0 x \<Rightarrow> f2 x | num.Bit1 x \<Rightarrow> f3 x) = f1"
that_2: shows "1 - min 1 (max 0 s) = (1 - min 1 (max 0 p)) * (1 - min 1 (max 0 q))"
semiring_norm_77: fixes n :: "num" shows "(num.One < num.Bit1 n) = True"
assms: shows "mopup_aft_erase_c (2 * n + 4, l, Bk # xs) lm n ires"
x_dif: shows "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> g y = h y"
that: shows "A \<turnstile>\<^sub>R res' \<sim> res"
llength_LCons: fixes x :: "'b"   and xs :: "'b llist" shows "llength (LCons x xs) = eSuc (llength xs)"
open_core_eq: fixes a :: "nat set" assumes "a open" shows "core a = a"
length_removeAll_less: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "length (removeAll x xs) < length xs"
Re_Hv: fixes v :: "(real, 3) vec" shows "quat.Re (Hv v) = 0"
empty: fixes A :: "'a set set" shows "{} \<in> tiling A"
dual_bot: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual bot = top"
uint_of_uint32_def: shows "uint_of_uint32 \<equiv> map_fun Rep_uint32 Abs_uint ucast"
inv_into_into: fixes x :: "'f"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
cases_scheme: fixes r :: "('a, 'b) monoid_scheme" obtains carrier :: "'a set" and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and one :: "'a" and more :: "'b" where "r = \<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>"
assms_2: fixes k :: "nat" assumes "k < n" shows "prime (p k)"
uniformity_discrete_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
inv_check_left_moving_in_middle_Oc_Bk: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and list :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and s' :: "nat" assumes "inv_on_left_moving_in_middle_B (as, lm) (s, Oc # list, Bk # r) ires" shows "inv_check_left_moving_in_middle (as, lm) (s', list, Oc # Bk # r) ires"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" assumes "Problem.monot wtFsym wtPsym arOf resOf parOf \<Phi>" shows "MonotProblem_axioms wtFsym wtPsym arOf resOf parOf \<Phi>"
is_lubI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" assumes "class.po below"   and "po.is_ub below S x"   and "\<And>u. po.is_ub below S u \<Longrightarrow> below x u" shows "po.is_lub below S x"
card_subset: fixes Y :: "'b set"   and n :: "nat"   and X :: "'b set" assumes "card Y = n"   and "Y \<subseteq> X" shows "n \<le> card X \<or> infinite X"
GatewaySystem_L2: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "ServiceCenter i a"   and "GatewayReq req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "ack t = [init_state]"   and "req (Suc t) = [init]"   and "\<forall>t1\<le>t. req t1 = []"   and "\<forall>m\<le>k + 2. req (t + m) \<noteq> [send]"   and "req (t + 3 + k) = [send]"   and "inf_last_ti dt t \<noteq> []"   and "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" shows "vc (2 * d + (t + (4 + k))) = [vc_com]"
induct: fixes x1 :: "'a"   and x2 :: "[:'a:]"   and P :: "'a \<Rightarrow> [:'a:] \<Rightarrow> bool" assumes "slistmem x1 x2"   and "\<And>x xs. \<lbrakk>x \<noteq> \<bottom>; xs \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P x (x :# xs)"   and "\<And>x xs y. \<lbrakk>slistmem x xs; P x xs; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P x (y :# xs)" shows "P x1 x2"
surjective: fixes r :: "('a, 'b) simple_packet_scheme" shows "(r::('a, 'b) simple_packet_scheme) = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, \<dots> = simple_packet.more r\<rparr>"