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If tree T maintains the invariant property, then the tree resulting from the application of the function 'tree_map_of' to T with 'pres' as an additional parameter, also maintains the invariant property.
: 1. The decomposition of a single-step rest message that sends a term sequence 'ts' is the same single-step rest message that sends 'ts'. 2. The decomposition of a single-step rest message that receives a term sequence 'ts' is the same single-step rest message that receives 'ts'. 3. The decomposition of a single-step ...
If the arity of a function f is greater than 0, then the application of function Γ to the function f with argument T is equal to the composition of function f with the map of function Γ over T.
If the lengths of two lists, ss and ts, are equal, then the result of the MATCH1 function, when applied to a singleton multiset containing a pair of functions (Fun f ss, Fun f ts) and a substitution σ, is equal to the result of the MATCH1 function when applied to a multiset created from zipping the lists ss and ts and ...
For any three elements A, B, and C, the star of the wand of A and B and C implies the wand of A and the star of B and C, given the context of delta.
If a semilattice exists for a binary operation (denoted by bold asterisk), then a semilattice also exists for the map2 function applied to the same binary operation. This is applicable for any function f, which is represented with the infix notation bold asterisk.
The lemma named "no_hyp_change_no_cons" assumes that "x" is greater than zero and there is no hypothesis change from "z" at indices "i" and "j" for "x". It shows that there is no consistent learning for the function "phi" with the set of adverse conditions at indices "i" and "j" for "s".
The lemma "Bex_inj_conv" assumes that the function "f" is injective. It shows that there exists an element "y" in the image of "f" over set "A" such that property "P" holds for the inverse of "f" applied to "y" if and only if there exists an element "x" in set "A" such that property "P" holds for "x".
The lemma named "equalmodstrick2" assumes that the modulus of "a" and "p" equals the modulus of "b" and "p". It shows that the modulus of the difference between "a" and "b" with respect to "p" equals zero.
If the following conditions are met in a well-formed program P: 1) the type of x in the derived class is a subtype of the type of x in the superclass, 2) the field f in the superclass has a certain type, and 3) the fields in the derived class are a certain set, then the field f is included in that set of fields in ...
The function "eexp(u n)" tends to "eexp l" as n tends to infinity if and only if the function "u n" tends to "l" as n tends to infinity. This is true for any filter "F".
The mapping of "x" in "xs" equals to "None" if and only if "x" is not in the set of the first elements of "xs".
The resampling process from set A to set B is lossless according to the single probability measure function.
This lemma, named "set_as_mapping_refined", is fixing two types of sets, "t" which is a set of red-black trees (RBT) and "xs" which is a set of doubly-linked lists (DList). It shows that the function "set_as_mapping" applied to the RBT set "t" is equal to a certain case expression "?C1" applied to the RBT set "t". Simi...
If the lists xs1 and xs2 are not empty, and for all elements in the combined list of xs1 and xs2, there does not exist a pair of elements r and s such that the element equals the sum of r and s, and if xs2 is sorted and contains distinct elements, then the result of applying the ACI_nPlus function to the sum of xs1 and...
If "a" derives "b" in one step at position "i" using rule "r", and "a" splits at position "i" into two parts, "α" and "Nβ", then "b" is equal to the concatenation of "α", the second part of rule "r", and "β".
If the following conditions are met: 1. For any two elements 'a' and 'b', if property 'p' holds for 'a', then it also holds for the sum of 'a' and 'b'. 2. Property 'p' does not hold for 'i'. 3. 'd' is a measured digit. 4. 'd' is a digit of level 'n'. 5. Property 'p' holds for the sum of 'i' and the measured digit 'd'....
If the function "ipurge_fail_aux_t_inv_1" with parameters I, D, U, xs, Y holds true, and the form of Y is a valid "ipurge_fail_aux_t_form", then the first element of the output of the function "ipurge_fail_aux_t_out" with Y as its parameter is equal to the output of the function "ipurge_tr_aux" with parameters I, D, U,...
If for every element 'i' in set 'I', the function 'f' is differentiable at a point 'a', then the sum of the function 'f' over the set 'I' is also differentiable at the same point 'a'.
The lemma named "comp_char" shows that the composition of g and f equals: if g is sequentially followed by f, then if f equals Zero, it returns g, otherwise if g equals One, it returns f, else it returns null. If g is not sequentially followed by f, it returns null.
If a set 's' is independent in module 'm1' and function 'f' is injective on the span of 's' in 'm1', then the image of 's' under 'f' is independent in module 'm2'.
If X is a weak lens, then the quotient of X by X is equal to the identity lens.
The lemma named "fill_gholes_induct" is about a function P that takes a 'f gmctxt and a list of 'a and returns a boolean. It assumes that the number of gholes in C is equal to the length of xs. It also assumes that for any x, P of GMHole and a list containing x is true. Furthermore, it assumes that for any function f, ...
The sum of a set containing two elements, x and y, of a type 'a that is a join semilattice zero, is equal to the addition of x and y.
The function 'fpxs_of_fls' when applied to 0, returns 0.
The relation "crel_vs" holds if the function "dp" with the parameter "param" equals the value "v'" in the case that "v'" is not None, and "v" equals "dp param". This relation is established between "dp param" and the function "lookup" with the parameter "param".
The lemma named "rep_number_point_group_one" assumes that "x" is an element of "V". It shows that the cardinality (or size) of the set of all "g" in "G" such that "x" is an element of "g" is equal to 1.
If there exists an 'x' such that property 'P' holds for 'x', and for any 'x' for which property 'P' holds, property 'Q' also holds, then the least 'x' for which either property 'P' or property 'Q' holds is the same as the least 'x' for which property 'Q' holds. This is under the condition that 'a' is a well-ordered set...
If the execution move "exec_move" is valid for a given configuration consisting of the class instance "ci", program "P", thread "t", expression "e", heap "h", stack "s", thread action "ta", new heap "h'", and new stack "s'", then the same execution move is also valid when the expression "e" is replaced with a new array...
The stable segment of a list (xs) with respect to an element (y) remains the same after insertion sort with a comparison function (cmp) is applied to insert an element (x) into the list, given that the comparison of y and x does not result in equivalence.
The lemma named "atLeastLessThan_less_eq2" assumes that the function 'f' is an injection on the set of natural numbers from 0 to less than 'm', and the image of the set under 'f' is less than or equal to the set of natural numbers from 0 to less than 'n'. Under these assumptions, it shows that 'm' is less than or equal...
If delta is the inverse of epsilon under a partial bijective function, and epsilon extends gamma, and gamma is the inverse of beta, then delta extends beta.
If sequence 's' is less than sequence 't', then the sequence resulting from prepending element 'a' to 's' is less than the sequence resulting from prepending the same element 'a' to 't'.
The lemma named "map_project_simps" demonstrates two things: 1. The first part, "map_project_empty", shows that if you apply the function "List.map_project" with any function "f" to an empty set, the result will be an empty set. 2. The second part, "map_project_insert", shows that if you apply the function "List.map_...
The lemma named "bet_suma__sams" assumes that points A, B, C, D, E, F, SumA, G, H, I exist and that point H lies between points G and I. Under these assumptions, it shows that the Side-Angle-Side (SAS) congruence holds for triangles ABC and DEF.
A constant function "k" tends to "k" for any filter "F".
"coalg" holds for the parameters "car_final1", "car_final2", "str_final1", and "str_final2".
The probability equality 3 holds reflexively for any given set X, sigma-algebra α, and measure μ.
The first element of the transformed rule set is equal to the first element of the original rule set.
If "t2" is a well-formed directed acyclic graph, and a node "r1" with a subtree "(t1,e1)" is a subtree of "t2", and "t2" is equal to its normalized form, then the rank of the reverse of "r1" is less than or equal to the rank of the reverse of the root of "t1".
If a function 'f' is measurable in the product of two universal count spaces, then the function 'map_int_pair' with parameters 'f' and 'g' is measurable in the product of two integer spaces, given the assumption 'f'.
The lemma named "state_repcD" assumes that the configuration 'cfg' is in the set of configurations 'R_G.cfg_on' based on the state 's'. It shows that the state of the representative configuration 'repc' of 'cfg' is equal to the representative state 'reps' of 's'.
: 1. The product of a function 'f' and the shifted version of another function 'g' by 'r' is equal to the shifted version of the product of 'f' and 'g' by 'r'. 2. The product of the shifted version of a function 'f' by 'r' and another function 'g' is equal to the shifted version of the product of 'f' and 'g' by 'r'.
Applying the function "aezfun" to the sum of a list "as" is equal to the sum of applying "aezfun" to each element "a" in the list "as".
If "beta" is a bijective function, then "alpha" is a store renaming for "beta" and "sigma" in the context of "R-sub-S alpha beta sigma".
The well-founded relation "wcode_fourtimes_case_le" is well-founded.
If not not p is provable, then p is provable.
: For any index 'i', process 'p', instruction sequence 'is', shared memory 'θ', store buffer 'sb', delay set 'D', owned set 'O', and read set 'R', if 'i' is less than the length of the thread states 'ts' and the 'i'-th element of 'ts' is a tuple consisting of 'p', 'is', 'θ', 'sb', 'D', 'O', and 'R', and for all indice...
If 'x' and 'xa' are elements in the set resulting from the prefix match split of a word interval 'wi', and if the bitwise AND of 'pt' and the bitwise NOT of the mask of 'x' equals the prefix of 'x', and the bitwise AND of 'pt' and the bitwise NOT of the mask of 'xa' equals the prefix of 'xa', then 'x' must be equal to ...
If set A is a subset of set A' and set B is a subset of set B', then the unordered product of sets A and B is a subset of the unordered product of sets A' and B'.
If the first elements of all sublists in list 'l' are distinct, then the length of list 'l' is equal to the cardinality (or the number of unique elements) of the domain of the map created from list 'l'.
, given the precondition that twice the value of N is equal to 10, if we assign to N the value of twice N, then the postcondition is that N is equal to 10.
The lemma named "condition_1_part_4" is defined with the following conditions: - It fixes a matrix 'A' of any field type with modifiable types of columns and rows, and a natural number 'k'. - It defines 'ia' as 0 if for all 'm', 'm' is a zero row up to 'k' in 'A'. Otherwise, it is the natural number equivalent of the ...
If a parallel composition "A" is secure, and the typing condition holds for the unlabeled version of "A", and "A'" is an element in the set resulting from the transition relation applied to "A" and an empty list, then the typing condition also holds for the unlabeled version of "A'".
The lemma named "conga_diff56" assumes that triangle ABC is congruent to triangle A'B'C'. It shows that point C' is not equal to point B'.
The lemma named "alpha_equiv_refl" assumes that a function "fmpred" equals to "\<Gamma>". It shows that "alpha_equiv" of "\<Gamma>", "t", and "t" is true. This lemma is suggesting that alpha equivalence is reflexive under these conditions.
If "xss" is a flip alternate sublist chain, then the reverse of "xss" is also a flip alternate sublist chain.
If the lexicographic power-product order of 's' and 't' holds, then the lexicographic power-product order of 's + u' and 't + u' also holds, where 's', 't', and 'u' are elements of an ordered commutative monoid with addition and an ordered additive semigroup with an implication from less than to less than or equal to.
For any two words 'x' and 'y' of any length, the XOR (exclusive OR) operation on 'x' and 'y' is equivalent to the OR operation on the AND operation of 'x' and the NOT operation of 'y', and the AND operation of the NOT operation of 'x' and 'y'.
In the context of a vector space, the lemma named "min_gen_is_basis" is defined. It fixes a set A and assumes that A is minimal with respect to the property that it is a subset of the carrier of the vector space and is a generating set. The lemma then shows that A is a basis.
If the sum of a given index 'i' and the length of list 'xs' is less than or equal to the length of list 'ys', and for all indices 'jj' less than the length of 'xs', the 'jj'th element of 'xs' is equal to the element at index 'i+jj' in 'ys', then 'xs' is a sublist of 'ys' at index 'i'.
Walking through an immutable array (IArray) that is the result of appending an element 'x' to an array 'xs', starting from a base 'b', using a function 'f', is equivalent to applying the function 'f' to the result of walking through the original array 'xs' from the same base 'b' and then applying 'f' to 'x'.
The lemma named "f_proj" assumes that "x" and "y" form a direct product and "g" is a map. It shows that the composition of "F f x g y" and "x" equals "f".
The intersection of the intervals from some unspecified point to 'n' and from the same unspecified point to 'n prime' is equal to the interval from the unspecified point to the minimum of 'n' and 'n prime'.
If the action "HStartBallot" transitions the system from state "s" to state "s'" for a given process "p", and if the conditions "Inv2a_innermost" for the current state "s", process "p", and the disk block of process "p" in state "s", "HInv4b" for the current state "s" and process "p", and "HInv4a" for the current state...
If we assume that set A is finite and is a subset of G, and the cardinality (size) of the sumset of A with itself is less than or equal to K times the real number representation of the cardinality of A, and both r and s are greater than 0, then it shows that the cardinality of the difference set between the iterated su...
The action of the abstract configuration is equal to the abstraction of the action of the configuration.
The lemma named "between_normalize_if_near_0" assumes that the norm of a point 'p' is less than or equal to 1. It shows that 'p' is between 0 and the normalized value of 'p'.
The lemma named "expandHnf" is defined with respect to a process P of type pi and a set S of processes of type pi. It assumes that the pair (P, S) is in the sumComposeSet and that for every process P in set S, P is in unique head normal form (uhnf) and P is valid. It shows that there exists a process P' such that P' is...
A monomial with coefficient 1 and degree 0 is equal to 1.
If a monad writer is defined with a return function, a bind function of type 'a and 'm, and a tell function of type 'w and 'm, then a monad writer can also be defined with the same return function, a bind function of type 'a and a pair of 'r and 'm (represented as envT), and a tell function of type 'w and a pair of 'r ...
: Assuming that "t0" is a list decision tree, and for every "x" in the vertices of "t1", there exists a "v" in the set of the root of "t0" or in the path vertices of "t1" leading to the head of "x", such that "v" leads to the head of "x" in the tree "T". Also, assuming that "t1" is in the first element of the set of s...
The lemma named "card_of_Un_ordLess_infinite_Field" assumes that the field of the relation 'r' is not finite, 'r' is a cardinal order, and the cardinality of sets 'A' and 'B' is less than the order of 'r'. It shows that the cardinality of the union of sets 'A' and 'B' is also less than the order of 'r'.
For a given matrix 'A' of type 'euclidean_ring_gcd', the determinant of 'A' is equal to the following expression: First, we convert the matrix 'A' to an iarray and compute its echelon form using the determinant function 'echelon_form_of_det_iarrays' and the Euclidean algorithm 'euclid_ext2'. Let's call this new iarra...
If a weighted transition from state 's' with context 'c' to state 's'' with context 'c'' exists in the transitive closure of the transition relation of the flowgraph, then the intersection of the set of monitors for the weight 'w' in the flowgraph and the set of monitors for the context 'c' in the flowgraph is empty.
If X is a part of H, then there exists a Y in H such that X is a part of Y.
The lemma named "inv_CONJ" assumes that the function 'f' is bijective. It shows that the inverse of the conjunction of 'f' is equal to the conjunction of the inverse of 'f'.
The nth bit of the unsigned cast of a word 'w' of any length 'a' to a word of any length 'b' is equal to the nth bit of 'w' if and only if 'n' is less than the minimum of the lengths of 'a' and 'b'.
The lemma named "clauseTrueAndClauseFalseImpliesInconsistent" establishes that for a given clause and valuation, if the clause is true under the valuation and the clause is also false under the same valuation, then the valuation is inconsistent.
If function 'f' is a bijection, then mapping the variables of a monomial 'm' with coefficient 'c' using function 'f' is equivalent to creating a monomial with the permutation of 'm' using the inverse of 'f' (considering the entire universe of possible inputs) and the same coefficient 'c'.
"ERETURN x" is less than or equal to "ESPEC Φ Ψ" if and only if "Ψ x" is true.
The modulus of a hypercomplex number, which is created from a hyperreal number that is in turn created from a hypernatural number, is equal to the original hypernatural number.
The lemma "taut_split_subst" assumes that "x" is a very well-behaved lens. It shows that a predicate "P" is true if and only if for all values "v", the predicate "P" remains true when "v" is substituted for "x".
If we assume "b" is an element of the value range of "r", then we can obtain an "a" such that the ordered pair of "a" and "b" is an element of "r".
For all elements in the list produced by the function "zeros" with parameters n and v, the function "is_alph" with parameter v is true.
The function "normalized_sort" applied to a set of elements "cs" and a sort "s" is equivalent to the function "exenormalized_sort" applied to the same elements "cs" and sort "s". This is a code-level equivalence in the Isabelle proof assistant.
The lemma named "rule_states_finite" asserts that the set of states defined by a rule "r" is finite.
The set of unique vertices in the edge undirected graph is equal to the set of vertices 'S_verts'.
If "a" is not equal to "b", then the inverse of the matrix "P a b" is equal to the reciprocal of the difference between "a" and "b" multiplied by the matrix with the first row as [1, -b] and the second row as [-1, a].
If a proposition is true, then it is necessarily true.
The lemma named "maxdeg_homogenize" assumes that "x" is not an element of the union of the set of individual elements (indets) of "F". It shows that the maximum degree (maxdeg) of the homogenization of "F" with respect to "x" is equal to the maximum degree of "F".
If a point-set 'ps' is a subset of the set of points 'V' and the cardinality of 'ps' is 2, then the block index of 'ps' in the block set 'B' is equal to 'Lambda' if and only if the size of the multiset of blocks 'G' in which 'ps' is a subset is zero.
In the context of a semicategory, the composition operation of the semicategory is a subset of the set Vset alpha.
The lemma named "path_split_general" assumes that for all lists "zs", the list "xs" is not equal to the concatenation of the list "ys" and "zs". It then concludes that there exist a number "j" and a list "zs" such that "xs" equals the concatenation of the first "j" elements of "ys" and "zs", "j" is less than the length...
If a program 'p' is well-formed and a state 'σ' is in the states of the program 'p', then it is not the case that executing the body of program 'p' from state 'σ' leads to no state (or an error).
The lemma named "landau_symbol_if_at_top_eq" assumes that "L" is a Landau symbol with respect to "L'" and "Lr". It shows that if "L" at the top is equal to a function "f" at point "a" and function "g" at all other points, then "L" at the top is equal to function "g". This is applicable for any "x" in a linearly ordered...
The lemma mkM_simps consists of five statements: 1. The set of worlds in the model created from T is equal to T. 2. If a pair (t, t') is in the relations of the model created from T for a certain agent a, then the judgement view of agent a on t is equal to the judgement view of agent a on t'. 3. If a pair (t, t') is i...
"G is below the square of S" if and only if "the co-call field of G is a subset of S".
The lemma "Con_initial_left" shows that if a transition "t" followed by a set of transitions "T" is residuated with a set "U", then the residuation of the singleton set containing only "t" with "U" also holds.
For a given semantic context Γ, the execution of the sequence of commands "c1" followed by the sequence of "c2" and "c3" from state "s" to state "t" is equivalent to the execution of the sequence of the sequence of "c1" and "c2" followed by "c3" from state "s" to state "t".