hcju/isabelleps huggingface dataset:
monad_state_alt_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_state_alt return bind get put alt \<equiv> monad_state return bind get put \<and> monad_alt return bind alt \<and> monad_state_alt_axioms get put alt"
eucl_of_list_mem_eucl_of_list_aform: fixes x :: "real list"   and a :: "(real \<times> real pdevs) list" assumes "x \<in> Joints a"   and "length a = DIM('a)" shows "eucl_of_list x \<in> Affine (eucl_of_list_aform a)"
union: fixes M :: "'b multiset"   and N :: "'b multiset" shows "\<Sum>\<^sub># (M + N) = \<Sum>\<^sub># M + \<Sum>\<^sub># N"
p_der: shows "pderiv p = smult (of_nat op) (xx ^ (op - 1)) * pp + xx ^ op * pderiv pp"
row_append_rows: fixes A :: "'b mat"   and B :: "'b mat"   and i :: "nat" assumes "dim_col A = dim_col B"   and "i < dim_row A + dim_row B" shows "row (A @\<^sub>r B) i = (if i < dim_row A then row A i else row B (i - dim_row A))"
ctxt_consn: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " \<Turnstile>\<^sub>n G  "   and " \<Turnstile>\<^sub>n Call : A " shows " \<Turnstile>\<^sub>n ({A} \<union> G)  "
cut_le_less_conv_if: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t = (if t \<in> I then insert t (I \<down>< t) else I \<down>< t)"
state_eq_holds: fixes s :: "nat option" shows "state_eq s = holds (\<lambda>x. statename x = s)"
not_alw: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw \<phi> xs) = ev (\<lambda>xs. \<not> \<phi> xs)"
principal_ideal_eq_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.comm_ring plus zero minus uminus times" shows "ring.principal_ideal plus zero uminus times S = (\<exists>a. ring.left_ideal_generated plus zero uminus times {a} = S)"
sel_1: fixes x1 :: "symbol \<times> symbol list"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat" shows "item_rule (Item x1 x2 x3 x4) = x1"
min_bot2: fixes x :: "'a" shows "min x (0::'a) = (0::'a)"
sFFalse: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and n :: "nat" assumes "\<not> pbval b ps"   and "vars b \<subseteq> dom ps" shows "wp\<^sub>3\<^sub>' (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
rtranclp_add_poly_p_polynomial_of_mset: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and p' :: "(char list multiset \<times> int) multiset"   and q' :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset" assumes "add_poly_p\<^sup>*\<^sup>* (p, q, r) (p', q', r')" shows "polynomial_of_mset r + (polynomial_of_mset p + polynomial_of_mset q) = polynomial_of_mset r' + (polynomial_of_mset p' + polynomial_of_mset q')"
bij_betw_vertices: shows "bij_betw f (\<Union> X) (\<Union> X)"
simps_9: fixes f1 :: "'a"   and f2 :: "('b, 'c) Node \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> ('b, 'c) Digit \<Rightarrow> ('b, 'c) FingerTreeStruc \<Rightarrow> ('b, 'c) Digit \<Rightarrow> 'a" shows "(case Empty of Empty \<Rightarrow> f1 | Single x \<Rightarrow> f2 x | Deep x xa xb xc \<Rightarrow> f3 x xa xb xc) = f1"
2: fixes q :: "'a poly"   and r :: "'a poly" assumes "0 < degree q"   and "degree q < degree p"   and "0 < degree r"   and "degree r < degree p"   and "p = q * r" shows "thesis"
simps_2: fixes yU :: "char list \<times> Unary_PCF.type" shows "fv \<langle>yU\<rangle> = {|yU|}"
mult_nonneg_nonneg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
residual_eq2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "f x = \<Sqinter>{y. x \<le> residual f y}"
antisym_conv2: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
decr_simps_eq: fixes js :: "'ord stream" assumes "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"   and "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js" shows "decr (shd js) js = decr (shd js) (stl js)"
size_remove1_mset_If: fixes M :: "'a multiset"   and x :: "'a" shows "size (remove1_mset x M) = size M - (if x \<in># M then 1 else 0)"
ideD_1: fixes a :: "'a" assumes "Adj.C.ide a" shows "Adj.C.arr a"
mset_add: fixes a :: "'b"   and A :: "'b multiset" assumes "a \<in># A" obtains B :: "'b multiset" where "A = add_mset a B"
ND0aux2: fixes x :: "'a"   and y :: "'a"   and b :: "('a, 'b) Combinators list" assumes "DenyAllFromTo x y \<in> set b" shows "y \<in> set (net_list_aux b)"
last_upt: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "last [i..<j] = j - 1"
hs'_def: shows "hs' = map (\<lambda>h x. h (nat \<lfloor>x\<rfloor>)) hs"
lm098: fixes Y :: "'d set" shows "injections {} Y = {{}}"
acquired_reads_no_pending_write_in: fixes x :: "nat"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired_reads False xs A" shows "x \<in> acquired_reads False xs B"
vanishes_const: fixes c :: "rat" shows "Real.vanishes (\<lambda>n. c) = (c = 0)"
noninf_mem_Z: fixes x :: "ant" assumes "x \<in> Z\<^sub>\<infinity>"   and "x \<noteq> \<infinity>" shows "\<exists>z. x = ant z"
l8_8: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B A" shows "A = B"
infnorm_mul: fixes a :: "real"   and x :: "'a" shows "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
pp_of_lookup: fixes t :: "('a, 'b) pp" shows "pp_of_fun (lookup_pp t) = t"
Infinitesimal_star_of_mult: fixes x :: "'a star"   and r :: "'a" assumes "x \<in> Infinitesimal" shows "x * star_of r \<in> Infinitesimal"
subset_entailed: fixes N2 :: "'a literal multiset set"   and N1 :: "'a literal multiset set" assumes "N2 \<subseteq> N1" shows "N1 \<TTurnstile>\<G>e N2"
vars\<^sub>l\<^sub>s_def: fixes L :: "fterm literal set" shows "vars\<^sub>l\<^sub>s L \<equiv> \<Union> (vars\<^sub>l ` L)"
normalized_simps_2: fixes t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" shows "normalized (Some t # xs) = normalized xs"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list_R1"   and b :: "'c list_R1" assumes "rel_list_R1 R a b" obtains xa :: "unit + 'b \<times> 'b list_R1" and ya :: "unit + 'c \<times> 'c list_R1" where "a = list_R1 xa" and "b = list_R1 ya" and "rel_sum (=) (rel_prod R (rel_list_R1 R)) xa ya"
dsSk: fixes k :: "nat" assumes "k < Suc n" shows "matrix_sum d (\<lambda>k. f (Suc k)) n \<in> carrier_mat d d"
drop_none: fixes x :: "\<langle>'c\<rangle>\<^sub>\<bottom>" assumes "x \<noteq> None" shows "\<lfloor>\<lceil>x\<rceil>\<rfloor> = x"
intros_4: fixes x :: "real"   and y :: "real" assumes "x \<in> radical_sqrt"   and "y \<in> radical_sqrt" shows "x + y \<in> radical_sqrt"
get_child_nodes_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "h \<turnstile> ok get_child_nodes ptr" shows "ptr |\<in>| object_ptr_kinds h"
indoc_single: fixes a :: "'item"   and b :: "'item" shows "indoc a [b] = (a = b)"
act_sup_distl: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c"   and p :: "'d"   and q :: "'d" assumes "unital_quantale_module act" shows "act x (p \<squnion> q) = act x p \<squnion> act x q"
bdd_above_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set" assumes "class.preorder less_eq less"   and "preorder.bdd_above less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_above less_eq A"
exhaust: fixes y :: "('b, 'c) chain" obtains x1 :: "'b" where "y = \<lbrace> x1 \<rbrace>"   | x21 :: "'b" and x22 :: "'c" and x23 :: "('b, 'c) chain" where "y = \<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23"
aXb: shows "LeftDerives1 \<delta> i r (a @ [X] @ b)"
leg0_commutes: shows "trg_f.cod.leg0 \<cdot> trg_f.chine = trg_f.dom.leg0"
assms_2: shows "A B C Cong3 A' B' C'"
ceiling_add_le: fixes x :: "'b"   and y :: "'b" shows "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>"
inf_llist_neq_llist_of_1: fixes xs :: "'c list"   and f :: "nat \<Rightarrow> 'c" shows "llist_of xs \<noteq> inf_llist f"
b'B: shows "emb b B \<in> B \<rightarrow> carrier K"
mult_left_sub_dist_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> y \<oplus> x \<odot> z \<preceq> x \<odot> (y \<oplus> z)"
Ctxt_lowerm: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and m :: "nat" assumes " \<Turnstile>\<^sub>n G  "   and "m < n" shows " \<Turnstile>\<^sub>m G  "
real_root_lt_1_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x < 1) = (x < 1)"
set_bset_inject: fixes x :: "'c set['d]"   and y :: "'c set['d]" shows "(set_bset x = set_bset y) = (x = y)"
mask_eq_take_bit_minus_one: 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 "mask n = take_bit n (uminus one)"
Trie_inject: fixes x :: "('c, 'd) Trie.trie"   and y :: "('c, 'd) Trie.trie" assumes "x \<in> {t. invar_trie t}"   and "y \<in> {t. invar_trie t}" shows "(Trie2.trie.Trie x = Trie2.trie.Trie y) = (x = y)"
GC_subsumption_step: fixes Dl :: "'a literal multiset \<times> label"   and N :: "('a literal multiset \<times> label) set"   and Cl :: "'a literal multiset \<times> label" assumes "Dl \<in> N"   and "strictly_subsumes (fst Dl) (fst Cl) \<or> subsumes (fst Dl) (fst Cl) \<and> snd Dl \<sqsubset>l snd Cl" shows "N \<union> {Cl} \<leadsto>GC N"
assms: shows "0 \<in> interior (convex hull {a, b, c})"
simps_2: fixes i :: "'b memref"   and "is" :: "'b memref list" shows "all_unshared (i # is) = (case i of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then L \<union> all_unshared is else all_unshared is | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> L \<union> all_unshared is | _ \<Rightarrow> all_unshared is)"
assms_2: shows "is_induced_by_cell w' r\<^sub>0 \<omega>"
that: fixes cs' :: "'edge list"   and rs' :: "'edge list" assumes "valid_node m'"   and "valid_call_list cs' m'"   and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)"   and "valid_return_list rs' m'"   and "length rs' = length cs'"   and "ms' = targetnodes rs'"   and "upd_cs cs [a] = cs'" shows "thesis"
no_small_separators: fixes S :: "'a set" assumes "S \<subseteq> V"   and "v0 \<notin> S"   and "v1 \<notin> S"   and "\<And>xs. v0 \<leadsto>xs\<leadsto> v1 \<Longrightarrow> set xs \<inter> S \<noteq> {}" shows "n \<le> card S"
mem_simps_11: fixes A :: "'c"   and C :: "'c set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
mutex_not_gval: fixes x :: "'a gexp"   and y :: "'a gexp"   and s :: "'a \<Rightarrow> value option" assumes "mutex x y" shows "gval (gAnd y x) s \<noteq> true"
inv_solve_left': fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and c :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P"   and "c \<in> carrier P" shows "(\<ominus>\<^bsub>P\<^esub> b \<oplus>\<^bsub>P\<^esub> c = a) = (c = b \<oplus>\<^bsub>P\<^esub> a)"
it_to_it_correct: fixes it :: "('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and S :: "'b set" assumes "set_iterator it S" shows "set_iterator (it_to_it it) S"
rep_eq: fixes x :: "'e['g, 'f]" shows "fromQ (x\<^sup>-\<^sup>\<one>) = inverse (fromQ x)"
vec_times_mat_eq_def: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" shows "[y\<^sub>v*A]=c = ((\<forall>i<dim_vec c. (A\<^sup>T *\<^sub>v y) $ i = c $ i) \<and> dim_col A\<^sup>T = dim_vec y \<and> dim_row A\<^sup>T = dim_vec c)"
pOp_assocTr41: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "d \<in> carrier M" shows "a \<plusminus> b \<plusminus> c \<plusminus> d = a \<plusminus> b \<plusminus> (c \<plusminus> d)"
plays_from_strategy_chain_Suc: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list"   and n :: "nat" shows "prefix (strategy_progression \<sigma> p $ n) (strategy_progression \<sigma> p $ Suc n)"
H: fixes c :: "real"   and x :: "'a" shows "eNorm N1 (c *\<^sub>R x) + eNorm N2 (c *\<^sub>R x) \<le> ennreal \<bar>c\<bar> * (eNorm N1 x + eNorm N2 x)"
x: shows "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x"
r_inv_of_ZX: shows "ZX * ZX\<^sup>\<dagger> = 1\<^sub>m 2"
Entry_path: fixes n :: "'node" assumes "valid_node n" shows "\<exists>as. (_Entry_) -as\<rightarrow>* n"
inv_implications_nonneg_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'e, 'f) configuration_scheme" assumes "dataflow_topology summary results_in" shows "inv_implications_nonneg c = (\<forall>loc t. 0 \<le> zcount (c_imp c loc) t)"
degen_invar: fixes ecnv :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" shows "b_graph (degeneralize_ext ecnv)"
ket_one_is_state: shows "state 1 |Deutsch.one\<rangle>"
parent_child_rel_child_in_heap: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptr parent"   and "(parent, child_ptr) \<in> parent_child_rel h" shows "child_ptr |\<in>| object_ptr_kinds h"
eventually_at_filter: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<forall>\<^sub>F x in topological_space.nhds open a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
lin_independent_Cons_conditions: fixes v :: "'v"   and vs :: "'v list" assumes "v \<in> V"   and "set vs \<subseteq> V"   and "v \<notin> RSpan vs"   and "R_lin_independent vs" shows "R_lin_independent (v # vs)"
not_one_le_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
diff_delta_lnexp_cf3_def: shows "diff_delta_lnexp_cf3 \<equiv> \<lambda>x. x ^ 6 / (numer_cf3 (- x) * numer_cf3 x)"
sop_distr2_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and i :: "'g \<Rightarrow> 'g"   and A :: "'g set"   and z :: "'g" shows "sop_distr2 R s f i A z = (\<forall>a\<in>carrier R. \<forall>x\<in>addition_set f (aug_pm_set z i A). \<forall>y\<in>addition_set f (aug_pm_set z i A). a \<^sub>s\<cdot> (x \<^sub>f+ y) = a \<^sub>s\<cdot> x \<^sub>f+ a \<^sub>s\<cdot> y)"
transaction_fresh_def: fixes prot_transaction :: "('e, 'f, 'g, 'h) prot_transaction" shows "transaction_fresh prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x1)"
reachNT_reach: fixes s :: "'state" assumes "reachNT s" shows "reach s"
ag_inv_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<plusminus> y = \<zero>" shows "y = -\<^sub>a x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a"   and f6 :: "'a"   and f7 :: "'a"   and f8 :: "'a"   and f9 :: "'a"   and f10 :: "'a"   and f11 :: "'a"   and f12 :: "'a"   and f13 :: "'a"   and f14 :: "'a"   and f15 :: "'a"   and f16 :: "'a"   and f17 :: "'a"   and bop :: "bop" shows "P (case bop of Eq \<Rightarrow> f1 | NotEq \<Rightarrow> f2 | LessThan \<Rightarrow> f3 | LessOrEqual \<Rightarrow> f4 | GreaterThan \<Rightarrow> f5 | GreaterOrEqual \<Rightarrow> f6 | Add \<Rightarrow> f7 | Subtract \<Rightarrow> f8 | Mult \<Rightarrow> f9 | Div \<Rightarrow> f10 | Mod \<Rightarrow> f11 | BinAnd \<Rightarrow> f12 | BinOr \<Rightarrow> f13 | BinXor \<Rightarrow> f14 | ShiftLeft \<Rightarrow> f15 | ShiftRightZeros \<Rightarrow> f16 | ShiftRightSigned \<Rightarrow> f17) = ((bop = Eq \<longrightarrow> P f1) \<and> (bop = NotEq \<longrightarrow> P f2) \<and> (bop = LessThan \<longrightarrow> P f3) \<and> (bop = LessOrEqual \<longrightarrow> P f4) \<and> (bop = GreaterThan \<longrightarrow> P f5) \<and> (bop = GreaterOrEqual \<longrightarrow> P f6) \<and> (bop = Add \<longrightarrow> P f7) \<and> (bop = Subtract \<longrightarrow> P f8) \<and> (bop = Mult \<longrightarrow> P f9) \<and> (bop = Div \<longrightarrow> P f10) \<and> (bop = Mod \<longrightarrow> P f11) \<and> (bop = BinAnd \<longrightarrow> P f12) \<and> (bop = BinOr \<longrightarrow> P f13) \<and> (bop = BinXor \<longrightarrow> P f14) \<and> (bop = ShiftLeft \<longrightarrow> P f15) \<and> (bop = ShiftRightZeros \<longrightarrow> P f16) \<and> (bop = ShiftRightSigned \<longrightarrow> P f17))"
assms_1: fixes x :: "'a" assumes "x \<in> set l'" shows "f x = f' x"
gamma2_cont: shows "continuous_on ({0..1} - s' \<inter> {0..1}) \<gamma>2"
distinct_3: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "Skip \<noteq> Assign x31 x32"
csol_3: shows "csol t0 x0 t t0 = x0"
assms: fixes a :: "'a" assumes "a \<in> set al" shows "f a = g a"
simps_4: fixes r :: "'b rexp"   and s :: "'b rexp" shows "ACI_norm_alt (Plus r s) = ACI_nPlus (ACI_norm_alt r) (ACI_norm_alt s)"
R_Sup: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A} is R healthy"
select_convs_2: fixes carrier :: "'d set"   and mult :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and more :: "'e" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
openin_Inter: fixes \<F> :: "'b set set"   and T :: "'b topology" assumes "finite \<F>"   and "\<F> \<noteq> {}"   and "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter> \<F>)"
cases: fixes a1 :: "'b"   and a2 :: "'b \<Rightarrow> real"   and a3 :: "intv" assumes "intv_elem a1 a2 a3" obtains u :: "'b \<Rightarrow> real" and x :: "'b" and d :: "nat" where "a1 = xa__" and "a2 = u__" and "a3 = Const d__" and "u__ xa__ = real d__"   | d :: "nat" and u :: "'b \<Rightarrow> real" and x :: "'b" where "a1 = xb__" and "a2 = ua__" and "a3 = Intv d" and "real d < ua__ xb__" and "ua__ xb__ < real (d + 1)"   | c :: "nat" and u :: "'b \<Rightarrow> real" and x :: "'b" where "a1 = x" and "a2 = u" and "a3 = Greater c" and "real c < u x"
exhaust_disc: fixes step :: "('c, 'd) step" obtains "step = Done"   | "is_Skip step"   | "is_Yield step"
induces_extended_monoidal_category: shows "extended_monoidal_category (\<cdot>\<^sub>C) T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C"
Mp_ident_iff: fixes f :: "int poly" shows "(r.Mp f = f) = (\<forall>n. poly.coeff f n \<in> {0..<p * q})"
x_within_bounds: shows "x \<in> {a..b} - s"
b0: shows "inj_map (Some \<circ> \<pi>)"
f0: shows "\<langle>|v\<rangle>| \<Otimes> \<langle>|s\<rangle>| = |v\<rangle> \<Otimes> |s\<rangle>\<^sup>\<dagger>"
emeasure_pmf_single: fixes M :: "'a pmf"   and x :: "'a" shows "emeasure (measure_pmf M) {x} = ennreal (pmf M x)"
rfb_Id_pres: shows "fb\<^sub>\<R> Id = id"
funion_absorb1: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B" shows "A |\<union>| B = B"
basis_indep: fixes X :: "'a set" assumes "X_insert.basis X" shows "indep_in (insert x X) X"
plus_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less a b" shows "less (plus a c) (plus b c)"
ladj_adj: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "f \<stileturn> g" shows "f = ladj g"
2: shows "map ((!) (as @ bs) \<circ> (+) (length as)) ys = map ((!) bs) ys"
ismaxcoeff_mono: fixes t :: "MIR.num"   and c :: "int"   and c' :: "int" assumes "ismaxcoeff t c"   and "c \<le> c'" shows "ismaxcoeff t c'"
nc_vec_par_closed: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<cdot> nc \<parallel> ((y \<sqinter> nc) \<cdot> nc) \<cdot> nc = (x \<sqinter> nc) \<cdot> nc \<parallel> ((y \<sqinter> nc) \<cdot> nc)"
Exh_Up: fixes z :: "'b\<^sub>\<bottom>" shows "z = \<bottom> \<or> (\<exists>x. z = up\<cdot>x)"
lowlink_set_discovered: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c" assumes "DFS_invar G param s" shows "param_DFS_defs.lowlink_set G s v \<subseteq> dom (discovered s)"
that: fixes x :: "'a" assumes "x \<in> A"   and "Max (f ` A) = f x" shows "thesis"
L3: shows "c_graph (chf A) = {c_pair x y |x y. x \<in> A \<and> y = 0 \<or> x \<notin> A \<and> y = 1}"
elims_2: fixes x :: "'a"   and xa :: "'b"   and xb :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "qFreshAbs x xa xb" obtains xs :: "'a" and xc :: "'b" and ys :: "'a" and y :: "'b" and X :: "('c, 'd, 'a, 'b, 'e) qTerm" where "x = xs" and "xa = xc" and "xb = qAbs ys y X" and "xs = ys \<and> xc = y \<or> qFresh xs xc X"